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[
"A Targeted Survey for H I Clouds in Galaxy Groups",
"A Targeted Survey for H I Clouds in Galaxy Groups"
] | [
"Martin A Zwaan [email protected] \nKapteyn Astronomical Institute\nP.O. Box 8009700 AVGroningenThe Netherlands\n"
] | [
"Kapteyn Astronomical Institute\nP.O. Box 8009700 AVGroningenThe Netherlands"
] | [
"Mon. Not. R. Astron. Soc"
] | Five galaxy groups with properties similar to those of the Local Group have been surveyed for H i clouds with the Arecibo Telescope. In total 300 pointings have been observed on grids of approximately 2.5 × 1.5 Mpc centred on the groups. The 4.5σ detection limit on the minimal detectable H i masses is approximately 7 × 10 6 M ⊙ (H 0 = 65 km s −1 Mpc −1 ). All detections could be attributed to optical galaxies; no significant detections of H i clouds have been made. This null result leads to the conclusion that the total H i mass of intragroup clouds must be less than 10 per cent of the total H i mass of galaxy groups and less than 0.05 per cent of the dynamical mass. The recent hypothesis that Galactic high velocity clouds are Local Group satellite galaxies is highly inconsistent with these observations. | 10.1046/j.1365-8711.2001.04514.x | [
"https://arxiv.org/pdf/astro-ph/0103328v1.pdf"
] | 16,190,381 | astro-ph/0103328 | 7e787eaef9728c3d1853d2cc1fbb975c10502557 |
A Targeted Survey for H I Clouds in Galaxy Groups
24 October 2018
Martin A Zwaan [email protected]
Kapteyn Astronomical Institute
P.O. Box 8009700 AVGroningenThe Netherlands
A Targeted Survey for H I Clouds in Galaxy Groups
Mon. Not. R. Astron. Soc
000000024 October 2018(MN L A T E X style file v1.4) Accepted ... Received ...ISM: clouds -intergalactic medium -Local Group -galaxies: luminosity function, mass function -radio lines: ISM
Five galaxy groups with properties similar to those of the Local Group have been surveyed for H i clouds with the Arecibo Telescope. In total 300 pointings have been observed on grids of approximately 2.5 × 1.5 Mpc centred on the groups. The 4.5σ detection limit on the minimal detectable H i masses is approximately 7 × 10 6 M ⊙ (H 0 = 65 km s −1 Mpc −1 ). All detections could be attributed to optical galaxies; no significant detections of H i clouds have been made. This null result leads to the conclusion that the total H i mass of intragroup clouds must be less than 10 per cent of the total H i mass of galaxy groups and less than 0.05 per cent of the dynamical mass. The recent hypothesis that Galactic high velocity clouds are Local Group satellite galaxies is highly inconsistent with these observations.
INTRODUCTION
Groups of galaxies have been the subject of many H i studies. Famous H i maps of, for example, the M81 group (van der Hulst 1979, Appleton, Davies & Stephenson 1981, Yun, Ho & Lo 1994 show that these groups often host tidal H i features (see also e.g., van Driel et al. 1992, Li & Seaquist 1994, Rand 1994, and Haynes, Giovanelli & Chincarini 1984 for a review). The incidence of H i features was quantified by Haynes (1981) who found that six galaxy groups from a sample of 15 show H i appendages from at least one member galaxy. These data might suggest that H i filaments and intragroup clouds are commonplace throughout galaxy groups, but such a conclusion could be misleading since H i surveys are normally concentrated on the inner parts of galaxy groups and often on interacting pairs within the groups. Systematic searches throughout the volumes occupied by the groups are rare because the projected sizes of groups on the sky are too large to be covered by aperture synthesis instruments. A notable exception is the survey described by Lo & Sargent (1979) who systematically searched for H i clouds throughout the volumes around the M81, CVnI and NGC 1023 groups. No clouds were detected to the H i mass limit of 4 × 10 7 M⊙, although several new dwarf galaxies were discovered. Lo & Sargent (1979) discussed their null-result in the context of Galactic High Velocity Clouds (HVCs). They concluded that HVCs are unlikely to be intergalactic gas clouds in the Local Group (LG) because they should have detected ⋆ Present address: Astrophysics Group, School of Physics, University of Melbourne, Victoria 3010, Australia an equivalent population in the external groups. Similar conclusions were reached by Giovanelli (1981) and Wakker & van Woerden (1997). However, the idea that the HVCs instead of being small clouds close to the Milky Way galaxy, are clouds of primordial composition at LG distances has recently seen a revival. Originally proposed by Oort (1966), subsequently discussed by Verschuur (1969) and Hulsbosch (1975), the idea has been refined by Blitz et al. (1999, hereafter BSTHB), who added a dark matter component to the clouds. The problem with earlier considerations of the extragalactic origin of HVCs was their derived distance. Hulsbosch (1975) concluded on the basis of the virial theorem that typical distances would be approximately 10 Mpc, which places the clouds outside the LG. BSTHB show that if the HVCs are built from the same material that galaxies are made of (typical baryon content fB of 10 per cent), their distances would reduce to ∼ 1 Mpc. Moreover, they show that the clouds' distribution on the sky resembles that of LG dwarfs, and their kinematics as an ensemble can be well modelled if they are distributed throughout the LG. Braun & Burton (1999, hereafter BB) define a subsample of 65 compact HVCs and come to essentially the same conclusions about their subsample. Note that the BSTHB and BB models do not apply to the HVCs associated with the Magellanic Stream, which are likely the result of tidal interactions between the Milky Way and the Magellanic Clouds (Putman et al. 1998) Placed at LG distances, the HVCs have H i masses of ∼3×10 7 M⊙ and typical diameters of 30 kpc. In this picture, approximately 500 to 1000 clouds are distributed throughout the LG, together adding approximately 4 × 10 10 M⊙ to the
LG H i mass. Interestingly, this number of HVCs is in reasonable agreement with the number of mini halos that are predicted by numerical simulations of the hierarchical LG formation (Klypin et al. 1999;Moore et al. 1999). Corrected for incompleteness, the BB sample comprises ∼ 100 clouds. Using the updated average distance of 650 kpc (Braun & Burton 2000), the total H i mass in their clouds would be approximately 10 9 M⊙. Charlton, Churchill & Rigby (2000) show that the statistics of Mgii and Lyman limit absorbers in the spectra of quasars is in conflict with the hypothesis that the HVCs are intragroup material (see also . For analogous HVC populations to exist around intervening galaxies as well as in the LG, the typical distances would need to be less than 200 kpc from the LG barycenter. A very similar conclusion has been reached by Zwaan & Briggs (2000) who show that existing H i surveys impose strong constraints on the existence of extragalactic HVCs. If H i clouds exist in other groups or around galaxies, with properties similar to those proposed for the LG, several instances should already have been detected in large H i surveys such as the Arecibo survey discussed by Zwaan et al. (1997).
In this paper we present additional evidence by means of a targeted survey of five galaxy groups with the Arecibo telescope. The selection of the targets is discussed in § 2 and the data acquisition and analysis is summarised in § 3. In § 4 the detections are presented, and in § 5 we discuss the implications on the existence of intragroup H i clouds. To enable direct comparison between the surveyed groups and the LG, we adopt distances to the groups based on their redshift velocities and a Hubble constant of H0 = 65 h65 km s −1 Mpc −1 .
SAMPLE SELECTION
In order to make the targeted survey for H i clouds most efficient, we have selected the galaxy groups according to the following criteria: 1) The distance to the groups must be such that the expected H i cloud diameters match the projected extent of the Arecibo beam. BSTHB estimate that the cloud diameters are approximately 30 kpc. The 3 ′ beam of the Arecibo Telescope subtends d beam = 0.87D kpc at a distance D Mpc. The optimal group distance at which the clouds fill the beam is therefore 30 Mpc. We have selected groups with radial velocities in the range ∼ 1800 km s −1 to 3000 km s −1 . 2) The declination must be in the range 10 • to 30 • so that the groups are accessible to the Arecibo Telescope and can be tracked for at least one hour. 3) To enable comparison with the LG, the global properties of the groups, such as the integral H i mass, luminosity, and dynamical mass, must be comparable to those of the LG.
The list of galaxy groups compiled by Garcia (1993) was found to be the most successful in meeting the above listed criteria. This catalogue was compiled from the LEDA † galaxy sample, and is basically a cross section of groups found via a percolation method (Huchra & Geller 1982) † We have made use of the Lyon-Meudon Extragalactic Database (LEDA) supplied by the LEDA team at the CRAL-Observatoire de Lyon (France). and groups identified via the hierarchical clustering method (Materne 1978). Table 1 gives the measured and derived properties of the selected groups. The groups are named after their brightest member.
We have made crude estimates of the groups' dynamical masses by applying the 'projected mass estimator' which is discussed by Heisler, Tremaine & Bahcall (1985) and is defined as
MPM = 32/π G(Nm − 1.5)ΣiV 2 i Ri ,(1)
where Nm is the number of members, Vi is the radial velocity of member i with respect to the group mean velocity, and Ri is the projected distance of member i from the centre of the group. Since the groups have only 3 to 6 identified members, the errors on the mass estimates are large (approximately MPM/ √ Nm). The radius of the zero-velocity surface, R0, beyond which galaxies participate in the Hubble expansion, can be calculated via (Sandage 1986)
R0 = 8GT 2 π 2 M dyn 1/3 ,(2)
where T is the age of the group. We take the ages of all groups to be 14 Gyr.
For comparison, we also give in Table 1 the most recent published properties of the LG, taken from Mateo (1998), Courteau & van den Bergh (1999), and van den Bergh (2000). For the LG, the dynamical mass is estimated using the virial theorem, but it has been shown by Heisler et al. (1985) that the virial theorem and the 'projected mass estimator' give very similar results. It is evident from the table that the selected groups have properties not very dissimilar from those of the LG: the selected groups are on average equally gas rich, luminous, and massive and have a similar radial extent as the LG. Note that the centres of the NGC 5962 and NGC 5970 groups are only 1.6h −1 65 Mpc apart and that their formal zero velocity radii are overlapping. Their difference in v hel is only 15 km s −1 . It is suggestive that these two groups actually form one gravitationally bound system. If we assume that they form one group, its dynamical mass would become log M dyn = 13.2 − log h65, and its zero velocity radius would be 2.3 h −1 65 Mpc. In the remainder of this paper we regard the NGC 5962/5970 group as one group.
OBSERVATIONAL STRATEGY
Observations were carried out with the refurbished Arecibo ‡ Telescope during five nights in the period from April 18 until 24, 1999 and on June 8 and 9, 1999. The data were taken with the L-narrow Gregorian receiver which has a measured system temperature of 32 K, and a gain of 10K Jy −1 . Spectra of 2048 channels were recorded for two polarisations over a bandwidth of 12.5 MHz, centred on the frequency of the 21cm line redshifted to the mean velocity of each galaxy ‡ The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under a cooperative agreement with the National Science Foundation. (1) Most luminous member, (2) and (3) Unweighted average RA and declination, (4) Number of known member with measured redshifts, (5) Unweighted average radial velocity, (6) Radial velocity dispersion. The uncertainty on this is approximately σr/ √ Nm, (7) Integral B-band luminosity, (8) Integral H i mass, (9) Rough estimate of dynamical projected mass (see text), (10) Estimate of the 'zero-velocity radius' (see text). a Number of LG members included in the Garcia (1993) catalogue if the LG were to be at a distance of 30 Mpc, b Taken from Courteau & van den Bergh (1999), c Taken from van den Bergh (2000), d Dwarfs from Mateo (1998) and giants from van den Bergh (2000).
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Name α 2000 δ 2000 Nm v σr log L B log M HI log M dyn R 0 (h m) ( • ′ ) (km s −1 ) (km s −1 ) (h −2 65 L ⊙ ) (h −2 65 M ⊙ ) (h −1 65 M ⊙ ) (h −
group. This setting results in a total velocity coverage of ∼ 2600 km s −1 and a velocity resolution of 1.3 km s −1 before Hanning smoothing. Spectra were dumped every 60 seconds.
The pointings were arranged in rectangular grids centred on the galaxy groups. The grids were built from several series of five pointings on lines of constant declination, where each pointing was separated by 5 minutes in hour angle. The integration time per pointing was 4 minutes and integrations were separated exactly 5 minutes in time. The remaining minute was used to slew the telescope back. This strategy insures that the same path on the telescope dish was traced during each of the five pointing in a series. No separate OFF scans were taken, but for each scan a synthetic OFF scan was computed by averaging the other four scans in the same series. This synthetic OFF scan yj can be written as
yj = Nx − xj N − 1 ,(3)
where N is the number of pointings in an array, xj is an individual scan andx is the average of all scans in one series. For this type of observation, this observing technique is superior in its efficiency compared to a traditional observing scheme where separate ON/OFF pairs are taken for each pointing. For this ON/OFF scheme the total noise depends on integration time as σ ∝ 1/TON + 1/TOFF. Normally, the integration times for the ON and the OFF scans are taken to be equal so that σ ∝ 2/ √ Ttot, where Ttot = TON + TOFF is the total integration time needed for each pointing. For our strategy, where we make composite OFF scans from other scans in the array, the noise
σ ′ ∝ 1/T ′ tot + 1/([N − 1]T ′ tot ),
where N is the number of pointings in one series and T ′ tot is the time spend on each pointing. From this we can derive
Ttot T ′ tot = 4(N − 1) N σ ′2 σ 2 .(4)
This shows that in our case, where N = 5, the technique is a factor of 3.2 faster than traditional ON/OFF procedures. In other words, in the same amount of time a factor 1.8 higher signal to noise can be achieved. Fig. 1 shows a schematic view of the pointings that were observed, together with the positions of the group galaxies.
The circles indicate the sizes of the groups characterised by the zero-gravity radius R0. The grids of pointings sparsely sample rectangular areas of approximately 2.5 × 1.5 Mpc.
Calibration of the data was performed by observing continuum sources with known flux densities. Separate scans and polarisations were averaged and Hanning smoothed. Polynomials were fitted to regions free from obvious signals, and subtracted from the spectra in order to obtain flat baselines. A first order polynomial (linear baseline) was generally found to be sufficient. Each spectrum was then Gaussian smoothed to resolutions 5, 10, 25 and 50 km s −1 . The resulting noise level was on average 0.75 mJy at 10 km s −1 resolution, which corresponds to a minimal detectable H i column density level of 1.2 × 10 18 cm −2 (4.5σ). The lowest detectable H i mass at 30 Mpc was 7.1 × 10 6 M⊙ for a 10 km s −1 broad signal. The H i mass limit increases to 1.6 × 10 7 M⊙ for signals 50 km s −1 wide.
All spectra were checked for 4.5σ peaks in the full resolution and the smoothed spectra. The list of 4.5σ peaks was first checked for false positives due to RFI by re-analysing the separate 60 second scans for both polarisations. All peaks that could not unambiguously be attributed to RFI were re-observed for confirmation on 8 and 9 June 1999. The new observations were conducted following an ON/OFF fashion, spending 8 minutes ON and 8 minutes OFF on each position. The resulting noise level for these observations was therefore a factor 0.89 lower than for the original observations carried out in April 1999.
THE DETECTIONS
Three pointings were deliberately aimed at the positions of known galaxies. This was done to check the positional accuracy of the pointing method, and to obtain a confirmation of the flux calibration. The left three panels of Fig. 2 turns out to be part of the H i filament in the galaxy pair NGC 6500/6501, that has been mapped with the WSRT by van Driel, Davies & Appleton (1988). The authors describe the H i structure as a 'classical bridge and tail configuration of a double galaxy interaction'. This is obviously of tidal origin, and not a primordial H i cloud.
The right panels of Fig. 2 show the three other confirmed detections. The first one is due to UGC 11037. This galaxy was not intentionally pointed at, but one of the pointings happened to fall very close (∼ 1.3 ′ ) to this galaxy. There is a similar explanation for the detection in the spectrum in the central right panel. This pointing was only 2.6 ′ separated from the centre of NGC 6574. The redshift of the signal agrees very well with the measured redshift of NGC 6574.
Finally, the lower right spectrum shows a detection at 15 h 39 m 00 s , 12 • 30 ′ 00 ′′ (α, δ) in the direction of the NGC 5962/5970 group. There is no catalogued galaxy at this position, but there is an obvious optical galaxy visible on the DSS, only 3 ′ to the north. The observed H i redshift is consistent with membership in the NGC 5962/5970 group. From the DSS we estimate the brightness of the galaxy to be mB = 17 (assuming B − V = 0.5). The number density of galaxies brighter that mB = 17 is approximately 4.0 per square degree (Metcalfe et al. 1995). The probability of encountering a galaxy of such brightness within a 3 ′ radius is therefore approximately 3 per cent. Furthermore, the velocity width of the signal is 150 ± 15 km s −1 , a factor of ∼ 7 larger than the typical width of the HVCs in the Wakker & van Woerden (1991) catalogue, and broader than any known HVC. We therefore conclude that this H i signal is very likely associated with the optical galaxy. Additional 21cm observations on this field are required to confirm this.
In summary, no H i clouds of the type predicted by the BSTHB scenario have been found. Two detections were made that could not unambiguously be identified as known optical galaxies. One is a known tidal H i filament in the NGC 6500/6501 pair, similar to the Magellanic Stream seen in the Local Group (Putman et al. 1998). The other detection is very likely the result of a uncataloged member of the NGC 5970 group.
SPACE DENSITY OF H I CLOUDS
We now use the null-result of the Arecibo group survey to derive upper limits to the space density of H i clouds in galaxy groups and discuss the cosmological significance of intragroup H i clouds.
HVCs as intragroup clouds
An explanation for Galactic HVCs that is of widespread current interest is provided by BSTHB who suggest that most of the HVCs are actually distributed throughout the LG and each cloud contains a few ×10 7 M⊙ of H i. We perform Monte Carlo simulations to test this scenario by filling the five observed galaxy groups with synthetic populations of clouds following a recipe outlined by BSTHB. For the cloud properties we use the measured solid angles Ω, velocity widths ∆V , and average brightness temperatures TB for Galactic HVCs from Wakker & van Woerden (1991). Virial distances rg are calculated for each cloud individually. The values of rg are directly proportional to the assumed ratio of baryon to total mass fB. If fB is 0.1, the virial distances rg are found to be approximately 1 Mpc. At those distances, the distribution of HVCs is in agreement with the kinematics of the LG, which was one of the main motivations of BSTHB to propose the extragalactic HVC scenario.
Within the groups, the clouds are placed at rg from the group's barycenter, in a random direction. This situation would resemble that in the LG, except that the substructure that BSTHB attribute to LG dynamics is not simulated in the models of the external groups. The radial column density distribution for each cloud is first assumed to be flat. The average column density is calculated by taking NHI = MHI/(πr 2 cloud ), where MHI is the cloud H i mass based on its value of rg and its observed flux, and r cloud is the cloud radius, calculated from its measured solid angle and rg. For each model, a number N clouds per group is drawn randomly from the Galactic HVC parent population.
The synthetic cloud ensembles are 'observed' with patterns of beams following the observational strategy of our survey. A detection is counted if the fraction of the flux of a cloud within the beam exceeds the detection threshold. The simulations are run with velocity resolutions of 5, 10, 25 and 50 km s −1 , similar to the searching of the real data. The same simulation is run 100 times for each group in order to obtain reliable error estimates on the expected number of detections.
First, we assume that the number of clouds per group, N , is invariant over the different groups. If N = 100, which is substantially lower than the number of HVCs observed around the Milky Way Galaxy, the expected total number of detections is 6 ± 3, where the error indicates the 1σ variation around the mean. Note that N ≈ 1000 in the BSTHB scenario and N ≈ 100 in the BB scenario. The expected number of detections increases linearly with increasing N . Next, we drop the restriction that all groups contain an equal number of clouds and instead scale N with the dynamical mass of the group. This seems like a more logical thing to do since the H i mass and luminosity are also observed to scale in direct proportion to the dynamical mass. However, N could of course be dependent on the dynamical state of the groups. In groups that have formed more recently, the primordial clouds are likely to have been less efficient in merging than in older groups. We have no detailed information on the dynamical state of the groups, and therefore simply assume that N scales proportional to M dyn . We find that the expected total number of detections rises under this assumption: 23 ± 8 clouds are expected if the number of clouds in a group with LG mass is N = 100. The reason for this increase is that the average M dyn for the external groups is slightly higher than that for the LG. The conclusion from this exercise is that the hypothesis set forward by BSTHB that HVCs are in-falling gas clouds in the LG is highly inconsistent with the observations, unless the LG is an unusual group. If the LG is a representative group and the five surveyed external groups are similar to the LG, our survey should have detected at least 30 clouds (assuming N = 500).
Constraints on intragroup H I cloud properties
A graphical representation of the constraints on intragroup H i clouds is presented in Fig. 3. This figure shows the combined constraints on the mean H i mass of clouds, and the number of clouds in each group. The lines show 68, 90, 95, and 99 per cent confidence levels at which the existence of a cloud population can be excluded. Again we have made use of the observed parameters of Galactic HVCs to model cloud populations in the external groups and the number of clouds N is again scaled with M dyn . For reference, the cloud populations proposed by BSTHB and BB are indicated by hatched boxes, the size of which reflects the uncertainty in the number and average H i mass. The horizontal arrow indicates the effect of changing the mean distance of the BB clouds from 1 Mpc to 650 kpc from the Local Group barycenter. This latter value is preferred by Braun & Burton (2000) after they have estimated the distance to one HVC by comparing the measured H i column density and the angular size of the cool core. Both the BSTHB and the BB populations are inconsistent with the observations at the > 99 per cent confidence level.
The distribution of H i column densities in HVCs often show a core-halo structure (Wakker & van Woerden 1997). Braun & Burton (2000) present high resolution WSRT imaging of six compact HVCs and show that the morphology can be described by a diffuse halo that encompasses one or more compact cores. We test the influence of this morphology on the detection efficiency in our survey by designing clouds with cores which account for 50 per cent of the total flux and have a radius Rcore = R cloud /5. The remaining 50 per per cent confidence levels on the hypothesis that the existence of a group population can be rejected. The dashed line is the 95 per cent confidence level assuming that the clouds can be described by a core-halo model in which 50 per cent of the flux is in a core with radius R cloud /5. cent of the flux is distributed over a halo with a flat H i column density distribution. The 95 per cent confidence level on this population is indicated by a dashed line. It is clear that the detection efficiency is not significantly changed by this modification of the cloud structure. Fig. 4 is similar to Fig. 3, but here the number of clouds per group is non-variant. Again in this case, neither proposed population of clouds can be reconciled with our observations.
Significance of intragroup clouds
How do these upper limits compare to the hierarchical formation scenarios of galaxy groups? Klypin et al. (1999) and Moore et al. (1999) show that in numerical simulations of a hierarchical universe the relative amount of dark matter substructure halos is scale-invariant. The predicted relative number of dark matter halos is similar in clusters, groups and galaxies. However, only in clusters does the predicted number of clumps agree with observed luminosity functions; on galaxy and group scale the simulations predict an excess over the observed number of satellites by a factor of 10, especially for halos with circular velocities < 50 km s −1 .
One of the proposed solutions for this problem of missing satellites is provided by the BSTHB hypothesis. However, the evidence presented in this paper, by Charlton et al. (2000) and by Zwaan & Briggs (2000) seem to rule out this solution. Only a very limited number of clouds with MHI ∼ 10 7 M⊙ could exist in galaxy groups. A similar conclusion has been reached by Verheijen et al. (2000), who systematically survey a region of the Ursa Major cluster of galaxies and find no H i clouds to a limit of 10 7 M⊙. From Fig. 3 we conclude that the intragroup H i clouds contribute a maximum of 1.0 × 10 9 M⊙ of H i to the total group mass. This implies that no more than 10 per cent of all the H i in groups can reside in clouds with masses greater than MHI = 7 × 10 6 M⊙. The H i mass in clouds must be less than 0.05 per cent of the total dynamical mass of the groups. In the non-variant N case, these numbers rise to 20 and 0.1 per cent. The dynamical mass of a cloud population that is still permitted by the observations is more difficult to constrain. If we assume that the cold gas (H i and He i) is the only baryonic component in the clouds, and the baryon fraction fB = 0.10 (a factor normally observed in galaxies and clusters, Fukugita, Hogan & Peebles 1998), then the total contribution of the clouds to the dynamical mass of the groups must be less than 1 per cent. Note that Klypin et al. (1999) estimate that the total mass in the predicted dark matter satellites amounts to approximately 5 per cent of the total group mass. Such a high prediction can only be brought into agreement with our survey results if the clouds' fB is lowered to 0.02. However, the median distance of the clouds from the groups' barycentres would then reduce to ∼ 200 kpc. It is not clear whether this is still consistent with the hierarchical model predictions in which the dark matter satellites are distributed throughout the groups.
A solution to the problem of missing satellites might be that the cold neutral gas is only a minor contributor to the total baryonic content of the clouds making the H i so insignificant that it can not be detected in 21cm surveys. This situation could occur if a large fraction of the gas reservoirs in the satellites are ionised by the intergalactic background. Klypin et al. (1999) and Moore et al. (1999) discuss gas ejection by early generation supernova-driven winds and inhibit-ing gas cooling and star formation by photo-evaporation as possible explanation of the absence of cold gas and stars in the satellites.
Solutions of a different kind can be found by changing the predicted number of clouds instead of modifying the baryons within the clouds. This can be achieved by suppressing of the primordial density fluctuation spectrum on small scales (Kamionkowski & Liddle 1999). Self-interacting dark matter (e.g., Spergel & Steinhardt 1999) and other darkmatter flavours (fluid dark matter, repulsive dark matter) have been suggested as possible explanations for a less efficient formation of small mass halos.
SUMMARY
The conclusion reached by Lo & Sargent (1979) that Galactic HVCs are unlikely to be intergalactic gas in the Local Group (LG) remains sound and intact under scrutiny of a new 21cm survey with the refurbished Arecibo Telescope. This new survey consists of 300 pointings in five nearby galaxy groups and is sensitive to H i masses of approximately 7 × 10 6 M⊙, depending on the velocity spread and distance of the signals. Two detections have been made that are not unambiguously caused by known optically selected galaxies. One is a known tidal H i filament in the NGC 6500/6501 pair, comparable to the Magellanic Stream (Putman et al. 1998). The other detection very likely originates in an uncataloged member of the NGC 5970 group. We therefore conclude that we have made no significant detections of H i clouds in galaxy groups.
We use this null-result to estimate constraints on the proposed population of H i clouds in groups, suggested by Blitz et al. (1999) and Braun & Burton (1999). These authors present a scenario in which the Galactic high velocity clouds (HVCs) are actually distributed throughout the LG at typical distances of a few hundred kpc to 1.5 Mpc. Fig. 3 shows the combined upper limits on the number of clouds per galaxy group, and the average H i mass on such clouds. The Blitz et al. (1999) scenario can be ruled out with > 99 per cent confidence levels, assuming that the LG is typical of the five groups studied here. The integral amount of H i in intragroup clouds is typically less than 10 per cent of the groups' total H i mass, and less than 0.05 per cent of the total dynamical mass of the groups.
The absence of clouds in groups seems to present a problem for hierarchical structure formation scenarios that predict many satellites within groups. At present it remains unclear whether the solution to this problem lies in modifying the descriptions of hierarchical formation so that the predicted number of satellites drops, or that the baryons in the clouds are simply hiding from detection.
Figure 1 .
1Positions of groups and pointings on the sky. The grey dots represent the individual galaxies in the groups, the big circles show the surfaces of zero-gravity. These are typically 1-2 Mpc. For the NGC 5962 and NGC 5970 groups their joint surface of zero-gravity is also indicated. The groups are named after their brightest members.
Figure 2 .
2'Detections' in the Arecibo group survey. The left three spectra are the result of pointings aimed at known galaxies. The secondary peak in the spectrum of NGC 6500 is caused by a known filamentary H i structure in the NGC 6500/6501 pair. The upper right and middle right are serendipitous detections of known galaxies. The lower right spectrum is probably the result of a nearby optical galaxy. All spectra are smoothed to an effective resolution of 15 km s −1 .
Figure 3 .
3Combined constraint on the baryon fraction f B and the number of clouds N per group. The number of clouds in each group is normalised using the dynamical mass estimates, the number on the vertical axis is the assumed number in the Local Group. The average H i masses of the cloud populations are indicated on the top axis. The contours represent 68, 90, 95 (fat line), and 99
Figure 4 .
4Same asFig. 3, but here the number of clouds per group, N , is equal for all galaxy groups.
Table 1 .
1Properties of surveyed groups
show the spectra of NGC 5789, UGC 11168 and NGC 6500. The latter one shows a secondary peak at ∼ 3500 km s −1 . ThisNGC 5798
NGC 5962
NGC 5970
NGC 6278
NGC 6500
NGC 6574
c 0000 RAS, MNRAS 000, 1-7
ACKNOWLEDGEMENTS I thank F. Briggs and the referee, V. Kilborn, for useful comments and discussion and K. O'Neil for doing part of the observing for this project.
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| [] |
[
"On the use of chaotic dynamics for mobile network design and analysis: towards a trace data generator",
"On the use of chaotic dynamics for mobile network design and analysis: towards a trace data generator"
] | [
"Martin Rosalie \nLaboratoire Génome et Développement des Plantes\nUMR\nUniv. Perpignan Via Domitia\n5096, F-66860PerpignanFrance\n\nLaboratoire Génome et Développement des Plantes\nUMR\nCentre National pour la Recherche Scientifique\n5096, F-66860PerpignanFrance\n",
"Serge Chaumette \nUniv. Bordeaux\nLaBRI\n5800, F-33400TalenceUMRFrance\n"
] | [
"Laboratoire Génome et Développement des Plantes\nUMR\nUniv. Perpignan Via Domitia\n5096, F-66860PerpignanFrance",
"Laboratoire Génome et Développement des Plantes\nUMR\nCentre National pour la Recherche Scientifique\n5096, F-66860PerpignanFrance",
"Univ. Bordeaux\nLaBRI\n5800, F-33400TalenceUMRFrance"
] | [] | With the constant increase of the number of autonomous vehicles and connected objects, tools to understand and reproduce their mobility models are required. We focus on chaotic dynamics and review their applications in the design of mobility models. We also provide a review of the nonlinear tools used to characterize mobility models, as it can be found in the literature. Finally, we propose a method to generate traces for a given scenario involving moving people, using tools from the nonlinear analysis domain usually dedicated to topological analysis of chaotic attractors. | 10.1080/10236198.2023.2194452 | [
"https://export.arxiv.org/pdf/2303.16583v1.pdf"
] | 257,804,686 | 2303.16583 | ab9966091134b8b35b889210b2a314b5ce859b15 |
On the use of chaotic dynamics for mobile network design and analysis: towards a trace data generator
29 Mar 2023 March 30, 2023
Martin Rosalie
Laboratoire Génome et Développement des Plantes
UMR
Univ. Perpignan Via Domitia
5096, F-66860PerpignanFrance
Laboratoire Génome et Développement des Plantes
UMR
Centre National pour la Recherche Scientifique
5096, F-66860PerpignanFrance
Serge Chaumette
Univ. Bordeaux
LaBRI
5800, F-33400TalenceUMRFrance
On the use of chaotic dynamics for mobile network design and analysis: towards a trace data generator
29 Mar 2023 March 30, 2023Mobile networksMobility modelsChaotic dynamicsPoincaré sec- tionFirst return map
With the constant increase of the number of autonomous vehicles and connected objects, tools to understand and reproduce their mobility models are required. We focus on chaotic dynamics and review their applications in the design of mobility models. We also provide a review of the nonlinear tools used to characterize mobility models, as it can be found in the literature. Finally, we propose a method to generate traces for a given scenario involving moving people, using tools from the nonlinear analysis domain usually dedicated to topological analysis of chaotic attractors.
Introduction
The number of applications that use autonomous devices, for instance robots or Unmanned Aerial Vehicles (UAVs), increases nowadays. In this context, defining and analysing their mobility is particularly important. A mobility model describes the behaviour of an entity considering its capacities, possible moves and speed. The mobility models are described either analytically at the individual level, or by the interactions between the parts of the system (between UAVs, UAVs and planes, UAVs and points to survey, etc.). The resulting behaviours described with these simple rules can induce the emergence of a global intelligent behaviour. Inversely, from the resulting behaviour of such a swarm, these initial simple rules are hard to discover. A similar phenomenon occurs for chaotic dynamics where a chaotic process appears to be random while it arises from a deterministic process. Therefore, the idea is to study the connections between the two concepts.
Literature on chaotic dynamics and nonlinear dynamics has been developed at the end of the century while the main concepts of this theory come from the early nineties (see Fig. 1 of [1]). Chaos is observed, described and analysed in numerous domains, for instance: electronic circuits [29], chemical reactions [47], laser behaviours [30] or biological models [62]. The first example of the use of chaos (from the Chua system [9]) to design a mobility model has been proposed in 1997 [49]. The authors proposed to use the properties of chaotic dynamics to ensure a good coverage of an area. Chaotic dynamics is defined as follows: the solution of a deterministic process is chaotic if it is sensitive to initial conditions, aperiodic and globally time invariant. These properties induce that this chaotic solution will be unpredictable when considering a long term behaviour. Nowadays, several tools improve the understanding of a chaotic behaviour, chaotic mechanism and bifurcation diagrams and some authors use them to provide mobility models. Because "little is known of the potential relationship between swarm and chaotic systems" [21], the main goal of this paper is to explore the uses of chaotic dynamics in the domain of mobility models. This can be done either by designing chaotic deterministic models or by using tools coming from nonlinear analysis.
In this paper we first present a review of mobile networks including chaotic dynamics using chaotic maps or ordinary differential equations. We then review the tools used to analyse these systems. In section 3, we propose a method to generate traces using nonlinear analysis tools. This method is supported by numerical simulations using the Lorenz system and permits to reproduce congestion and distribution patterns. We finally present the conclusion along with our future work in section 4.
Mobility models and chaotic dynamics
In this section, we present models from the literature including chaotic dynamics and the tools used to analyse them. In these models, chaos is obtained from well-known discrete or continuous systems. The chaotic variables of these systems are used to design mobility models. Sections 2.1 and 2.2 detail the two main approaches used to generate chaotic dynamics for mobility models using: a chaotic attractor from a discrete system or a chaotic attractor solution of a continuous system. Section 2.3 is dedicated to the tools from nonlinear analysis and their applications. The last section (Sec. 2.4) details our previous contributions in this domain in which we proposed to use periodic orbits of chaotic dynamics to define mobility models of UAVs in order to enhance the coverage of an area.
Models using chaotic maps
Chaotic maps are nonlinear recurrent relations. The most used in the literature is the logistic map defined by x n+1 = αx n (1 − x n ). Introduced by Verhulst [58] it represents the growth of population x at each step where α is the growth rate. This map contains only one nonlinear term and can exhibit both periodic and chaotic dynamic. When varying α from 2 to 2.4, its bifurcation diagram results in a period doubling cascade: a classical route to chaos found in several systems. For details on the logistic map, the reader is referred to [5]. Charrier et al. [7] propose to use the logistic map to reproduce the behaviour of flocks. In the original paper [39], Reynolds introduced boids to reproduce the flocking behaviour with three rules (collision avoidance, velocity matching and flock centring) that generates a force vector for each agent in the swarm. In the Charrier et al. model the synchronization between agents is performed by the environment: the control parameter of the logistic map is updated depending on the neighbourhood of each agent. This model does not use the standard rules of flocking but reproduces their behaviour. The authors use a bifurcation diagram to emphasize the convergence of the system and to show the global dynamics of their agents and the transition to chaotic dynamics.
A chaotic map (the standard map [24]) can also be used to produce chaotic motion for a robot [28]. From such a two-dimensional map, a planning is assigned to the robot using two coordinate points obtained from the chaotic map. Then a robot visits these points in their order of appearance. As the purpose of the robot is to cover a square surface, the authors evaluate their system using coverage rate. Curia et al. [11] proposed to use another map, the Hénon map, to move a mobile robot with unpredictable trajectories. Their mobility model combines a guiding line that the robot follows with a chaotic motion obtained from the map. While the robot follows the guideline, the chaotic motion controls the evolution of the robot around this line. Here again, the authors use a bifurcation diagram to underline the chaotic properties of their system. Even if their system is made of six equations (including the Hénon map), they prove that it has a chaotic solution when a = 1.4 as it is the case for the Hénon map. The additional equations dedicated to the movement around the line do not influence the chaotic dynamics.
Models using ordinary differential equations systems
In this section, we present models that use a set of ordinary differential equations as a source of chaotic dynamics. We are now considering chaotic continuous solutions instead of discrete solution from a map. However, there is a way to discretely represent these continuous chaotic dynamics. These solutions are embedded in the phase space, for instance the Lorenz attractor [26] is a famous attractor in a three-dimensional phase space. Introduced by H. Poincaré [37]; the Poincaré section is a transversal surface of the flow that provides a discrete description of the chaotic dynamics obtained from continuous systems. The original idea is to consider only the discrete points when the flow crosses a Poincaré section instead of the whole trajectory in the phase space. The discrete sequence of points contains a synthesis of the dynamical properties of the attractors. Most of the references presented below use the chaotic dynamics from the chaotic attractor using a Poincaré section.
There are many examples of such systems where nonlinearity induces chaotic behaviour. Among the most studied one: Lorenz, Rössler, Chua systems; there are several articles about robots, the mobility models of which use a set of ordinary differential equations. For instance, Nakamura & Sekiguchi [34] use the Arnold equations to model the behaviour of a mobile robot with chaotic motion. They prove that the coverage of their chaotic robot is better than the coverage of a robot using a random walk. Further to this work, Bae et al. [3] proposed a "chaotic UAV" with the Chua system, Arnold equations and Van Der Pol equations. They also introduce an obstacle avoidance method without decreasing the performance of the model in terms of coverage.
Fallahi & Leung [16] proposed a cooperative set of four mobile robots synchronized using Chen [8] and Lorenz [26] systems. They used one of the variables of these systems to define the movements of their robots. In their system, one of the robots is the master, and the others are synchronized with it. This system is efficient compared to unsynchronized robots or random walks in terms of coverage rate and travelled distances. Similarly, Mukhopadhyay & Leung [33] present synchronized robots with chaotic path planners. They add a symbolic dynamic description of their robots. The symbolic dynamic is a nonlinear analysis tool used to describe chaotic dynamics. Its purpose is to label a discrete trajectory obtained from a Poincaré section. Thus, it gives a symbol according to the dynamical aspect of the solution depending on the topological period of the system. At the end, the solution is no longer a variable but a sequence of symbols indicating dynamical aspects of the system studied. The authors use this sequence of symbols to evaluate the synchronization rate between their robots. We also would like to mention the work of Bezzo et al. [4] where synchronization of chaos is used to detect changes in the topology of a mobile robotic network. The authors study the motion of mobile agents through an unknown environment with obstacles (see also [54,55] for details on this method).
Volos and co-workers [59] proposed another use of chaos which consists in designing a path planning generator for autonomous mobile robots. From the double scroll chaotic circuit (Chua system [9]), they obtained a chaotic true random bit generator. They use it to define the path planning of the robots i.e., the list and the order of the points the robots have to visit. The efficient coverage rate and the unpredictability of the robot trajectories are the main characteristics of this system. This system is similar to the system using chaotic map [28]. Comparing their coverage rate in terms of the number of planned points, the system designed by Volos and co-workers [59] is ten times better. From the same authors, similar results are obtained when the Arnold map is used to generate waypoints for path planning [12]. Finally, we would like to point out a recent work done by Pimentel-Romero and co-workers [36] using Poincaré sections of chaotic attractors as threshold to generate random numbers. They conclude that these particular Random Number Generators (RNGs) using chaotic dynamics are efficient for generating random paths for autonomous mobile robots.
Another way to include chaos to support mobile robot mobility is presented by Rosyid et al. [48]. Their method does not use any well-known Ordinary Differential Equations (ODE) system to drive the robots. Robots communicate using sound that all can hear. Each robot moves in a direction depending on the "total amount" of sound received. This is modelled by an ODE that details how this system is synchronized because their relative positions influence the sound emitted and received. The authors use the coverage rate to compare their system to the previously presented methods for robots driven with Lorenz or Arnold equations to prove that their model has better performance. They also compute the Largest Lyapunov Exponent (LLE) to ensure that chaos occurs in their system. This value is a measure of the separation rate of two infinitely initially closed trajectories [60,35]. The LLE refers to the predictability of a system and is commonly used as an indicator of a chaotic behaviour. This is a metric approach that does not permit to distinguish chaotic solutions that have distinct structures in the state space. The reader is referred to [31,17,6] for details about chaotic mechanisms (e.g. folding mechanism or tearing mechanism). The 0-1 test can also be used as an indicator to distinguish chaotic dynamics from periodic one (the reader is referred to [19] for details). We also mention that diffusion coefficient can be computed for first return maps to measure the difference between the dynamics when a parameter is varied [20,23].
We recently proposed mobility models [46,45,44] using chaotic behaviour based on the Rössler system using Poincaré section and periodic orbits. In the next section we will present nonlinear tools used to analyse mobility models. Then, in section 2.4, we present our mobility models and the nonlinear tools used to build and analyse them.
Nonlinear analysis tools
We gave above examples of mobility models built from chaotic dynamics. Some of them have been analysed with tools coming from the domain of nonlinear analysis. In this section we present studies carried out on mobility models using these nonlinear tools to understand and describe their behaviour.
In 2014, Timme & Casadiego wrote an article entitled "Revealing networks from dynamics" [56]. This paper gives an overview of approaches considering collective nonlinear network dynamics, but few details are given about tools from the domain of nonlinear analysis that could be used when chaotic systems are identified. These tools are well introduced by Qu et al. [38] in their paper about emergence in swarming systems. The authors detail the emergence phenomenon and the following tools: Lyapunov Exponents, Attractor, Recurrence plot, Poincaré section. They also extract periodic orbits from a Poincaré section. The "periodic" term of periodic orbits refers to the state space and not to the time space (topological period). Periodic orbits are time invariant while the system evolves in a chaotic state (from initial condition, the solution evolves and successively visits the unstable periodic orbits). From a chaotic time series and with a Poincaré section, orbits can be extracted, and this acquisition is a preliminary step of the topological characterization [17]. For dissipative systems, the purpose of this method is to obtain the structure of the chaotic mechanism from a topological invariant (the linking number) computed between periodic orbits (the reader is referred to [18] for details).
Hazan et al. [22] opt for this approach because the aim of their work is to classify the behaviour of robots using periodic orbits. The authors explain that their method is not based on a metric (for instance the LLE) because it is "highly sensitive to perturbations such as noise contamination" [22]. They use the topological characterization tool after building an embedding (they reconstruct a phase space from one variable) of the behaviour from the x-axis motion of the robot. They build a Poincaré section in order to extract periodic orbits and then describe the behaviour of the robots using the linking numbers between the orbits.
Das and co-authors [14,13] propose to use Lyapunov exponents to distinguish transitions in a multi-agent swarm system. The system is designed to solve an optimization problem where the agents have to reach a particular point. Computing Lyapunov exponents enables the authors to find the range of parameters where chaos occurs and where the system is no longer periodic. The authors propose an application of their system: each robot is an automatic fire extinguisher, and they have to reach a burning place. "In any swarming dynamics, emergence of chaos is a very important situation to be dealt with" [13] and this is illustrated by the work of Wu and co-workers [61]. In the latter article, the authors use Lyapunov exponents to analyse their swarming system and concluded that the chaos in swarm model becomes weaker while the emergence becomes stronger. The Lyapunov exponents are also used to analyse a swarm model of Self Propelled Particles (SPP) by Shiraishi and Aizawa [53,52]. This tool permits to understand the relations between the behaviour of the system and the number of agents: "the Lyapunov exponents reflect the biological sensitivity hidden behind the motion of swarm" [53].
Chaos in neuronal network is also studied with Lyapunov exponents [15] or more recently, using reconstructed attractors with time-delay coordinates and with a Poincaré section [25]. These latest tools are robust to well define the chaotic mechanism because the first return map to the Poincaré section with unimodal structure indicates that there is a stretching and folding mechanism: it is a signature of the classical "horseshoe" mechanism (also known as folding mechanism). This type of chaotic mechanism is also present in the work of Sato and co-workers [51,50] as illustrated by Fig. 6 of [50]. They work on a multiagent system using reinforcement learning that is modelled with coupled differential equations. This is applied to game theory: Matching Pennies and Rock-Scissors-Paper games. The stretching and folding mechanisms describe the effect of mutual adaptation and memory loss with non-transitive structure for their system. This leads to Hamiltonian chaos if there is no memory loss and to a dissipative system where there is memory loss. The dissipative system exhibits limit cycles, intermittency and deterministic chaos. To study their system, they employ Lyapunov exponents, Poincaré section, bifurcation diagram and extract periodic orbits. These tools are also used to study languages and learning mechanisms where chaotic dynamics appears [32].
Mobility models based on periodic orbits
We recently proposed mobility models using chaotic behaviour based on the Rössler system [46,45,44]. These mobility models permit to enhance the coverage of an area compared to random mobility models. We used the first return map from a Poincaré section of a chaotic attractor solution of the Rössler system and considered the periodic orbits to build efficient mobility models in terms of coverage rate. The Rössler system [47] is given by the equations
ẋ = −y − ż y = x + aẏ z = b + z(x − c)(1)
and its Poincaré section is defined as follows: These tools are used to obtain the topological structure of the Rössler attractor [43] where ρ n ∈ [0; 1] is the normalized value of y n in the Poincaré section (Fig. 2). We introduce a new concept to provide trajectories for UAVs: the dynamics of the first return map enable us to obtain a local direction. The first return map is a step-by-step process used to update the direction of the UAVs based on a three symbols dynamic (L for left, R for right and A for ahead). The periodic orbits of an attractor are considered as its skeleton because they structure the dynamics of the system. From the first return map, we extract the periodic orbits of attractors to obtain these recurrent points often visited with the same order. Thus, our UAVs can follow these specific patterns that allow them to explore a wide area. We obtained straight lines and wide turns with respectively period one orbit (AAAA . . . for ahead, ahead, ahead, . . . ) and period two orbit (ARARAR . . . for ahead, right, ahead, . . . ) (Fig. 2). The period one orbit leads to a straight forward line to enable exploration while the period two orbit enables the UAV to make large right turns to change its direction. The reader is referred to [44] for details about the periodic orbits of attractors for mobility models of UAVs. The increase in performance provided by these mobility models using first return maps indicates that they deserve further investigation. In the next section, we propose a method to generate traces for mobile agents using this specific tool from nonlinear analysis. We would like to answer the following question: from a global point of view, can first return maps be useful to produce and/or analyse traces of UAVs?
P = {(y n , z n )|x n = 0,ẋ n > 0}(2)
Traces generation from multi-components Poincaré section
In this section we first present the concept of partial first return map as a tool to describe data with an unknown part. Conversely to the classical first return map, this partial first return map allows input and output which is a mandatory property to provide data traces. The main purpose of this article is to analyse and generate traces of agents in an open environment which means that the agents can be added or removed in the model. Then, section 3.2 provides a scenario where a partial first return map can describe the behaviour of agents. In section 3.3 we present a proof of concept supported by numerical experiments using a theoretical dynamical system: the Lorenz system [26].
Concept
The purpose of this section is to propose a methodology to generate traces of mobile entities named agents (robots, persons, UAVs, . . . ) using the most accurate tools related to the chaotic behaviour. We consider that the agents move in a well-defined area (the environment). The agents can enter, move and exit this area after a while. We suppose that the behaviour of the agents is mainly induced by constraints of the environment that force them to follow certain paths . As a consequence, we can make an analogy between the environment and the phase space of a deterministic dynamical system. We do consider that these paths are similar to unstable periodic orbits of an attractor in a phase space that agents might follow. The classical method applied in this case is to reconstruct the whole phase space from one measured variable (for instance one coordinate of the position). Contrary to Hazan et al. [22], we do not consider that the traces contain the whole dynamical properties, but only a portion of it because of the capability of the considered robot to enter and exit the area. They use the standard way to reconstruct the phase space. For the same reason, we are not able to use even better global modelling methodology for time series data [27].
Based on our experiment in mobility models design [46,45,44], we can say that an approach using the orbits of an attractor as guidelines is very efficient in terms of coverage of an area. This efficiency is due to the patterns followed by the UAV during the exploration process. Such an approach including patterns repetition can be applied to the generation of traces of mobile agents. Fig. 2 is the first return map of a Poincaré section of a Rössler attractor used in several research projects [46,45,44]. This first return map is an unimodal map made of an increasing branch and a decreasing branch: this illustrates the folding mechanism ("horseshoe" mechanism). However, this first return map details the whole chaotic dynamics of a bounded and globally time invariant process of a given chaotic attractor. Thus, the periodic points of Fig. 2 describe the entire periodic orbits and considering the analogy with the mobility model, this prevents entrance or exit of agents. As we are aiming to obtain a trace data generator, this global perspective is a drawback to overcome, because the agents cannot be considered as permanently evolving in a dedicated well-defined environment. Therefore, we propose a new method using partial data from a chaotic attractor.
For attractors bounded by high genus torus, the Poincaré section is made of several components that can be used to properly describe chaotic dynamics using symbolic dynamics [43]. For instance, the multispiral chaotic attractor (Fig. 3) introduced by Aziz-Alaoui [2] is bounded by a genus-5 torus. To describe the dynamics in a discrete way, the Poincaré section has to detail the transitions between the spirals. Consequently, the Poincaré section will be made of several components. These components are chosen accordingly to the bounding torus theory [57] using the fixed points. For the multispiral attractor (Fig. 3) this theory indicates that four components are required to build the Poincaré section. The Poincaré section is no longer one plane but a set of planes. The flow crosses these planes and from a continuous flow we obtain discrete values. We used a concatenation of these values to build one variable representing the whole Poincaré section: ρ n . Each component is represented by a range of values in ρ n : component c i for ρ n ∈ [i − 1, i] (see Fig. 3). Thus, ρ n is a variable between 0 and 4 that describes the Poincaré section. The four components synthesized in ρ n describe the entire dynamic of the system.
In a first return map to this Poincaré section, orbits can be extracted, and a symbolic dynamic can be assigned as it has been done for the Rössler attractor (See Fig. 2 that details periodic points associated to periodic orbits). Fig. 4 shows a first return map of a Poincaré section made of four components for the First return map of a Poincaré section describing a multispiral attractor (Fig. 3) using a Poincaré section made of four components [43]. multispiral attractor (Fig. 3) by plotting ρ n+1 versus ρ n . This first return map contains both horseshoe mechanisms and tearing mechanisms. This map is a discrete description of the flow and permits to obtain information concerning possible transitions between components of the Poincaré section. The reader is referred to [43] for details on the methodology used to build a Poincaré section with several components using the properties of fixed points and the bounding torus theory. One of the way to build a bounding torus is to place the hole where the fixed point of the attractor are. For the multispiral attractor there are details in [43] and for the Rössler system, there are details in [40] including parameters variation ensuring the robustness of the method.
In this multispiral attractor there are transitions from a component to the same component (e.g. c 2 to c 2 ) and transitions to another component (e.g. c 2 to c 3 ). The first return map (Fig. 4) details the chaotic mechanism occurring to perform these transitions and describes the whole dynamics of the system. The novelty of our approach lies on the use of only a part of the data to introduce chaotic mechanisms as a model for traces generation. We develop a new tool to handle this unknown part of the data: the partial first return map. A partial return map is an incomplete map with at least one component with incoming flow, the initial components, and at least one component with outgoing flow, the final components. The flow is split in such a way that the partial first return map describes the transitions between the initial components and the final components using transitional components. With this repartition of components, we propose to follow a particular ordering to build the partial first return map: the initial components, the transitional components and the final components.
Our new concept is to consider only a part of the trajectory in the environment. Thus, a partial first return map is only an uncompleted first return map with transitions between a subset of components of a Poincaré section. This permits to represent experimental data without taking into account the rest of the trajectories. Consequently, this missing or unknown data can be considered as input and output for traces model indicating where agents enter or leave the environment. In the next section we first present a scenario for our methodology and in section 3.3 we detail the method to build partial first return map.
Scenario
We consider the following scenario. The environment is an exhibition centre where we want to reproduce the behaviour of the visitors. We consider one entry door and one exit door. The exposition is composed of two rooms (Fig. 5). The room 1 is accessible from the entry and the room 2 is before the exit. Visitors can stay in room 1 as long as they want before leaving the exhibition. As there are two rooms to visit in the exhibition centre, we consider the transition between these rooms. Thus, we have visitors coming from the entry door and going to the first room. There is a transition from the entry to room 1 and also from room 1 to room 2. Figure 5: Scenario using an exhibition centre with two rooms. The shape is intentionally similar to the considered phase space (Fig. 6) to underline the asset of our method but this is not mandatory: only transitions between components are significant.
These transitions refer to the way chaotic multispiral attractors are analysed. As there are multiple spirals, and since the trajectory evolves from one spiral to another, it is required to consider these transitions. The Poincaré section is composed of several components to handle these multiples transitions. The analogy of transition between spirals is with transitions from one room to another room to generate traces. Moreover, this approach does not prevent to include more components to highlight patterns as it has been done to provide templates of attractors (See Fig. 15 of [43]). For instance, for a Malasoma attractor bounded by a genus one torus, a Poincaré section made of four components has been used to detail torsions or permutations movements [42]. This kind of mechanism can be used to reproduce the movements of the agents.
In this particular scenario, we consider some parts of our model as components of a Poincaré section: the entry and the exit as two components of a Poincaré section and the transition between room 1 and 2 is another compo-nent. To build the first return map, we set up the following order: the entry is an initial component, the transition between room 1 and 2 is a transitional component and the exit is a final component.
Numerical experimentation using the Lorenz system
We perform experimentations using the Lorenz system [26]
ẋ = σ(y − x) y = Rx − y − xż z = −βz + xy(3)
in order to illustrate our methodology on a chaotic attractor bounded by a torus with a genus higher than one. This system is solved using a 4 th order Runge-Kutta method. We obtain an attractor solution to this system for the parameters values R = 70, β = 8 3 and σ = 10. Fig. 6 represents the projection of the attractor in the plane (x, y). The hatch part is not used to perform the experimentation of our method because we do not consider the whole solution. As a consequence, a partial first return map is obtained from the data (Fig. 6) with a list of three components:
• initial component A: Entry
• transitional component B: Transition where an agent decides to stay in room 1 or to proceed to room 2
• final component C: Exit
These components are given by the following equations: P A = {(y n , z n )|x n = 0,ẋ n > 0} P B = {(y n , z n )|x n = 10,ẋ n < 0} P C = {(y n , z n )|x n = 0,ẋ n < 0}
The three components are represented Fig. 6 with arrows showing the flow of the attractor between them. Here we choose arbitrarily 10 for our partition based on bounding torus theory, but we remind that fixed point of the differential equations system can also be used to ensure good partition according to this theory.
As we have done for attractors bounded by high genus torus [43], we build one variable ρ n with different values to represent the position of the component in the partial first return map depending on its value:
• initial component A: ρ n < 1;
• transitional component B: 1 < ρ n < 2 and
• final component C: 2 < ρ n < 3. As introduced in one of our previous work [41], we choose to follow the orientation convention that gives the values of each component from the inside to the outside of the attractor. This convention is mandatory to compare chaotic mechanisms of attractors [41]. Thus, the ρ n values close to and lower than 1, 2 and 3 are respectively associated to positions in the components close to the letters A, B and C (Fig. 6).
We build the partial first return map (Fig. 7) based on ρ n to highlight the possible transitions between all components. It describes the transition between components X → Y with X the abscissa an Y the ordinate:
• The absence of points with ordinate value for the component A underlines the fact that it is an initial component. In this map, it is not possible to reach component A. From the dynamical point of view, even though we do not have the whole dynamical system, such kind of maps gives details on the possible chaotic mechanisms of the system. For instance, if we only consider the (A, B) area, it is an unimodal map with an increasing and a decreasing branch describing the classical "horseshoe" mechanism (stretching and folding mechanisms). This is illustrated by the positions of the points β and γ in the partial first return map (Fig. 7). These two points have the same ordinate but not the same abscissa. Using the map, we obtain only one image from these points: the point δ. In the range [β; γ] there exist pairs of points in that interval for which this happens because there is only one image for two fibres. In terms of mobility model analysis, it means that even if two agents do not come from the same position in the entry (component A) they can reach the same position in the next component B. Such a mechanism illustrates congestion, i.e. the convergence of agents to the same point. The topological description of a chaotic attractor can be viewed as a series of mechanisms: stretching and folding are enough to generate chaos. Considering the flow of the attractor, a folding mechanism is responsible for gathering trajectories before a stretching mechanism. The congestion is the result of a folding mechanism (continuous map with several branches) because trajectory will collapse to the same area. Now considering points with abscissa in component B, their ordinate are in components B and C. For an agent coming from component B, there is a split into two different places. The chaotic mechanism associated to such behaviour is the tearing mechanism, which is well known for Lorenz attractors (for details about mechanisms in the Lorenz system, see [6]). The coloured areas (Fig. 7) underline such a split directly from component A where two points (agents) close to α will move to B but one will stay in component B (green area of Fig. 7) while the other will proceed to component C (cyan area of Fig. 7). This mechanism is similar in chaotic In the mobility model, it highlights the fact that agents can stay in the same room or proceed to the last room before leaving the exhibition.
Finally, we can use one dynamical system (e.g. Lorenz system) to generate the global pattern of the traces with respects to some conditions based on the transition between areas. This is a proof of concept validated by numerical simulations, a preliminary step before considering experimental data instead of a solution to a dynamical system. From data, we are expecting to find some patterns similar to tearing mechanism or folding mechanism by identify their dynamical signature in partial first return map generated from the data. Between the mechanisms of transitions in experimental real world data we will then look for a dynamical system with similar dynamical properties to use it as a trace generator. The dynamical system will not be directly extracted from the data with an algorithm. One differential equations system, with a set of parameters, has to be found in a database of research articles detailing topology of chaotic attractors (including the list of chaotic mechanisms). The additional search of an appropriate dynamical system including equations and parameters can be achieved using a method we developed providing templates of attractors directly from a bifurcation diagram [40]. This article illustrates the richness of non-equivalent chaotic dynamics that could be found in one dynamical system, and it also provides various chaotic mechanisms with their parameters. To complete, we have to mention that multiple attractors can provide same mechanisms. For instance, the multispiral attractor can be used instead of the Lorenz system because they share common chaotic mechanism. Once a system has been found, the traces can be obtained by solving the dynamical system with initial points (the mobile agent) in the initial components to let them evolve in the environment by visiting the other components and escape via the final components. We thus propose to find a model fitting the data by giving a similar partial first return map. This model could provide a good mobility model, even if there is a suppressed or hidden part that is not used. We consider agents as particles in a flow where the chaotic dynamics provide enough variability despite the deterministic process to be used as a mobility model with our methodology.
Conclusion
In systems containing mobile entities, we emphasize the fact that chaos can be built, as well as it can emerge, by synchronization of the entities or by interaction with the environment. In both cases, if a chaotic state is observed, then dedicated tools can be employed to analyse it. For instance, bifurcation diagrams can illustrate transition from limit cycles (periodic solution) to various type of chaotic dynamics. We have seen that, Lyapunov exponents are indicators to distinguish chaos from hyper-chaos. However, for chaotic dynamics, this measure is not well adapted to detail the chaotic mechanism. Thus, others tools (Poincaré section or periodic orbits for instance) leading to topological characterizations can separate non-equivalent chaos, and provide more accurate analysis of these chaotic dynamics. The recent improvements concerning comparison of chaotic attractors and topological characterization method can be used to identify or distinguish chaotic mechanisms and consequently identify and distinguish particular behaviours of dynamical mobile networks.
Consequently, we have focused on the dynamical structure exhibited by the transitions between components of a Poincaré section. These transitions are significant and reliable to design a mobility model with congestion or distribution of agents in a given area. The procedure is applied on a Lorenz system to highlight the advantages of our methodology. The exhibited structure can be used to both generate traces and analyse them. For instance, with the constant increase in the number of connected devices carried by users, there are several ways to collect real traces with their approval. It has been done for the students of the University Politehnica of Bucharest [10] where their social interactions have been studied. In our future work we will apply our methodology to analyse agents traces obtained by such measurements to reproduce and generate traces by finding the most appropriate dynamical system.
Figure 1 :
1Chaotic attractor solution of the Rössler system (1) (values of parameters a = 0.1775, b = 0.215 and c = 5.995) with the Poincaré section (2) represented by an arrow.
Figure 2 :Figure 3 :
23First return map to the Poincaré section of the Rössler attractor(Fig. 1). This map is partitioned in three parts that give the UAV directions: L (left), A (ahead) and R (right). Orbits of period 1 and 2 illustrate patterns (AAAAA. . . ) and (ARARA. . . ), respectively straight lines and large turns. These patterns are efficient to cover an area with UAVs[44]. The multispiral attractor defined in[2] with the fixed points and the four components of the Poincaré section: c 1 , c 2 , c 3 and c 4[43]. This attractor is bounded by a genus-5 torus (the five aligned holes of the bounding torus are the fixed points and indicated with a dot in a circle ).
Figure 4 :
4Figure 4: First return map of a Poincaré section describing a multispiral attractor (Fig. 3) using a Poincaré section made of four components [43].
Figure 6 :
6Phase portrait of an attractor solution to the Lorenz system (3) for the parameters values R = 70, β = 8 3 and σ = 10. Arrows highlight the possible transitions from a component to another component (A to B, B to B or C).
•
A → B in the case (A, B) and B → B in the case (B, B) indicate that the next choice for the agent depends on the value of component B: stay in room 1 or proceed to room 2. • B → C in the case (B, C) indicates that the agent will leave the room 1 to the room 2 and then leave the exhibition • The absence of points with abscissa value for the component C underlines the fact that it is a final component. From component C there is no successor point.
Figure 7 :
7Partial first return map based on ρ n illustrating the possible transitions between the components. Even if some parts of this figure are empty, we chose to show them to make the partial first return map mechanism understandable. Transitions from A to B are possible and generate congestion; for instance, β and γ have the same image (δ). Conversely, after the component B, there is a tearing mechanism that results in by the separation of closed points (the green and cyan areas underline this split).
AcknowledgementsThe authors also would like to thank the reviewers for their useful comments and remarks that have made it possible to improve this article.
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| [] |
[
"A NEW CATALOG OF ORBITS OF 152 GLOBULAR CLUSTERS FROM Gaia EDR3",
"A NEW CATALOG OF ORBITS OF 152 GLOBULAR CLUSTERS FROM Gaia EDR3"
] | [
"A T Bajkova \nRussian Academy of Sciences\n\n",
"V V Bobylev \nRussian Academy of Sciences\n\n"
] | [
"Russian Academy of Sciences\n",
"Russian Academy of Sciences\n"
] | [] | This paper provides a new catalog of orbits and their parameters for a practically complete list of currently known galactic globular clusters (GCs), compiled by Vasiliev (2019) based on the most accurate modern measurements of their velocities and positions. The integration of the orbits of 152 globular clusters for 5 Gyr backward was performed using the new average proper motions obtained from the Gaia EDR3 catalog (Vasiliev and Baumgardt, 2021) and new average distances (Baumgardt and Vasiliev, 2021) in the axisymmetric three-component potential with spherical bulge, disk component, and spherical dark Navarro-Frank-White halo (Bajkova and Bobylev, 2016). The new orbital parameters are compared with the orbital parameters constructed by us earlier (Bajkova and Bobylev, 2021) in the same gravitational potential using proper motions obtained from the Gaia DR2 catalog (Vasiliev, 2019) and with the distances from the Harris catalog (2010). | 10.31725/0367-7966-2022-227-2 | [
"https://export.arxiv.org/pdf/2212.00739v2.pdf"
] | 254,125,162 | 2212.00739 | bb45fb06b2bdf14266d2f2b34c4c474ca47459d3 |
A NEW CATALOG OF ORBITS OF 152 GLOBULAR CLUSTERS FROM Gaia EDR3
24 May 2023
A T Bajkova
Russian Academy of Sciences
V V Bobylev
Russian Academy of Sciences
A NEW CATALOG OF ORBITS OF 152 GLOBULAR CLUSTERS FROM Gaia EDR3
24 May 2023Globular clusters: Galaxy (Milky Way) 1
This paper provides a new catalog of orbits and their parameters for a practically complete list of currently known galactic globular clusters (GCs), compiled by Vasiliev (2019) based on the most accurate modern measurements of their velocities and positions. The integration of the orbits of 152 globular clusters for 5 Gyr backward was performed using the new average proper motions obtained from the Gaia EDR3 catalog (Vasiliev and Baumgardt, 2021) and new average distances (Baumgardt and Vasiliev, 2021) in the axisymmetric three-component potential with spherical bulge, disk component, and spherical dark Navarro-Frank-White halo (Bajkova and Bobylev, 2016). The new orbital parameters are compared with the orbital parameters constructed by us earlier (Bajkova and Bobylev, 2021) in the same gravitational potential using proper motions obtained from the Gaia DR2 catalog (Vasiliev, 2019) and with the distances from the Harris catalog (2010).
Introduction
Globular clusters (GCs) are among the most interesting objects in our Galaxy. Their study allows us to understand the birth and evolution of the Galaxy, since they are the oldest stellar formations. Their age is almost equal to the age of the Universe. Currently, about 170 GCs of the Milky Way are known. According to theoretical estimates, the number of GCs in the Milky Way can be on the order of 200.
One of the methods for investigation of GCs is to study their orbital motion, which has become possible thanks to high-precision measurements of their spatial velocities and positions from the Gaia spacecraft. The appearance of catalogs of mean proper motions already according to the data of the second release of DR2 in combination with other astrometric data on radial velocities and positions of GCs made it possible to study the orbital motion of almost all currently known GCs (for example, Helmi Among the catalogs of astrometric data with proper motions from Gaia DR2 (Brown et al., 2018), we especially note the catalog by Vasiliev (2019) for about 150 GCs, which allows one to build the 6D phase space needed to calculate the orbits. This catalog was used by us to study the orbital properties of GCs, and on this basis, we developed a new method for dividing GCs into subsystems of the Galaxy: bulge, thick disk, and halo (Bajkova et al., 2020), based on the bimodal nature of the distribution of the L Z /ecc parameter , where L Z is the Z component of the angular momentum, ecc is the eccentricity of the orbit.
In the work of , we present a catalog of the orbits of 152 GCs and their orbital parameters, and also propose a modified classification by Massari et al. (2019) according to subsystems of the Galaxy, based on the obtained orbital properties of GCs.
With the advent of a new, more accurate version of the GCs proper motion catalog ) based on Gaia EDR3 measurement data (Brown et al., 2021), as well as new average GCs distances , a natural problem arises of constructing a new catalog of GCs orbits and refining their parameters. This work is dedicated to this aim.
The work is structured as follows. In the first section, a brief description and substantiation of the adopted model of the gravitational potential is given, in which the GCs orbits are integrated, the equations of motion are given, and formulas are given for calculating the orbital parameters. The second section describes the data, compares the average proper motions and their uncertainties obtained from the data of the Gaia DR2 and EDR3 catalogs, and compares the new GCs distances with the previously used distances from the Harris (2010) catalog. The third section is devoted to a presentation of the results of the work. A catalog of GCs orbits and their parameters calculated from new data is given, a comparison of the main orbital parameters with the parameters published in the work of is made. The Conclusion gives a short summary of the main results of the work.
1 Method
Model of axisymmetric galactic potential
The axisymmetric gravitational potential of the Galaxy is presented as the sum of three components: the central spherical bulge Φ b (r(R, Z)), the disk Φ d (r(R, Z)), and the massive spherical dark matter halo Φ h (r(R, Z)) (see and references therein):
Φ(R, Z) = Φ b (r(R, Z)) + Φ d (r(R, Z)) + Φ h (r(R, Z)).(1)
Here we use a cylindrical coordinate system (R, ψ, Z) with origin at the center of the Galaxy. In a rectangular Cartesian coordinate system (X, Y, Z) with origin at the center of the Galaxy, the distance to the star (spherical radius) is
r 2 = X 2 + Y 2 + Z 2 = R 2 + Z 2 .
The gravitational potential is expressed in units of 100 km 2 s −2 , distances -in kpc, masses -in units of the mass of the Galaxy, M 0 = 2.325 × 10 7 M ⊙ , the gravitational constant is G = 1. The potentials of the bulge Φ b (r(R, Z)) and the disk Φ d (r(R, Z)) are expressed in the form proposed by Miyamoto and Nagai (1975):
Φ b (r) = − M b (r 2 + b 2 b ) 1/2 ,(2)Φ d (R, Z) = − M d R 2 + a d + Z 2 + b 2 d 2 1/2 ,(3)
where M b , M d are the masses of these components, and b b , a d , b d are the scale lengths of the components in kpc.
To describe the halo component, we used the Navarro-Frank-White (NFW) expression presented in Navarro et al. (1997):
Φ h (r) = − M h r ln 1 + r a h ,(4)
where M h is the mass, a h is the scale length. The model of the galactic potential adopted by us, which for brevity we denote as NFWBB, has parameters obtained as a result of their fitting to the data on the circular velocities of clouds of ionized hydrogen HI, maser sources, and various halo objects with large galactocentric distances R up to ∼ 200 kpc from Bhattacharjee et al. (2014) (see Fig. 1). In addition, when adjusting the parameters, restrictions were used on the local dynamic density of matter ρ ⊙ = 0.1M ⊙ pc −3 and the force acting perpendicular to the plane of the Galaxy |K z=1.1 |/2πG = 77M ⊙ pc −2 (Irrgang et al., 2013).
NFWBB model parameters are given in Table 1. The corresponding model rotation curve up to distances R = 200 kpc is shown in Fig. 1. When constructing the rotation curve, we used the values R ⊙ = 8.3 kpc for the galactocentric distance of the Sun and V ⊙ = 244 km s −1 for the linear velocity of rotation of the Local Standard of Rest around the center of the Galaxy, as adopted by Bhattacharjee et al ( 2014). The mass of the Galaxy according to this model ) is equal to M G (R≤200 kpc) = 0.75±0.19×10 12 M ⊙ . This value is in good agreement with some modern independent estimates. So, for example, the lower estimate of the mass of the NFW halo, obtained quite recently by Koppelman, Helmi (2021)
Parameter
Value
M b [M 0 ] 443±27 M d [M 0 ] 2798±84 M h [M 0 ] 12474±3289 b b [kpc] 0.2672±0.0090 a d [kpc] 4.40±0.73 b d [kpc] 0.3084±0.0050 a h [kpc] 7.7±2.1 R ⊙ [kpc] 8.30 V ⊙ [km s −1 ] 243.9 M G (R≤200kpc) 0.75±0.19 [10 12 M ⊙ ]
from the data on the velocities of runaway halo stars, is M G (R≤200 kpc) = 0.67 +0.30 −0.15 × 10 12 M ⊙ . In Fig. 1, in addition to the available data, we also plot (blue dots) the circular velocities of the thick disk GCs identified by us with orbital eccentricities < 0.2, which show good agreement with data on maser sources in the interval of galactocentric distances 2 < R <20 kpc.
The NFWBB model of the gravitational potential of the Milky Way seems to us the most realistic compared to other known models, since it is supported by data at large galactocentric distances, which is very important when integrating the orbits of distant globular clusters and clusters with large apocentric distances, and also gives good agreement with modern estimates local parameters and a number of independent estimates of the mass of the Galaxy (Bajkova and Bobylev, 2017), a thorough review of which is also given in the recent paper by Wang et al. (2020).
Orbit integration
The equation of motion of a test particle in an axisymmetric gravitational potential can be obtained from the Lagrangian of the £ system (see Appendix A in Irrgang et al. (2013)):
£(R, Z,Ṙ,ψ,Ż) = 0.5(Ṙ 2 + (Rψ) 2 +Ż 2 ) − Φ(R, Z).(5)
Introducing the canonical moments "NFWBB model" u 1:2 "Bhattacharjee data" u 1:2:3 "masers data" u 3:2:9:7 "HI data" u 4:5 "Disk GCs data" u 5:7 we obtain the Lagrange equations in the form of a system of six first-order differential equations:
p R = ∂£/∂Ṙ =Ṙ, p ψ = ∂£/∂φ = R 2ψ , p Z = ∂£/∂Ż =Ż,(6)Ṙ = p R , ψ = p ψ /R 2 , Z = p Z , p R = −∂Φ(R, Z)/∂R + p 2 ψ /R 3 , p ψ = 0, p Z = −∂Φ(R, Z)/∂Z.(7)
To integrate the equations (7), we used the fourth-order Runge-Kutta algorithm. Integration was carried out 5 Gyr backward. As shown by , the galactic potential can be considered stationary in this time interval.
The peculiar velocity of the Sun relative to the Local Standard of Rest was taken equal to (u ⊙ , v ⊙ , w ⊙ ) = (11.1, 12.2, 7.3) ± (0.7, 0.5, 0.4) km s −1 (Schönrich et al., 2010). Here we use heliocentric velocities in a moving Cartesian coordinate system with the velocity u directed towards the Galactic center, v in the direction of the Galaxy's rotation and w perpendicular to the plane of the Galaxy and directed towards the north pole of the Galaxy.
Let the initial positions and spatial velocities of the test particle in the heliocentric coordinate system be equal to (x o , y o , z o , u o , v o , w o ). Then the initial positions (X, Y, Z) and velocities (U, V, W ) of the test particle in Cartesian galactic coordinates are given by the formulas:
X = R ⊙ − x o , Y = y o , Z = z o + h ⊙ , R = √ X 2 + Y 2 , U = u o + u ⊙ , V = v o + v ⊙ + V 0 , W = w o + w ⊙ ,(8)
where R ⊙ and V ⊙ are the Galactocentric distance and linear velocity of rotation of the Local Standard of Rest around the center of the Galaxy, h ⊙ = 16 pc is the height of the Sun above the Galactic plane, Π and Θ are the radial and circular velocities, respectively.
In this work, we calculate the following parameters of the orbits of globular clusters using well-known formulas (the units of measurement of the parameters are given in Table 2):
(1)initial distance of the GC from the center of the Galaxy d GC :
d GC = √ X 2 + Y 2 + Z 2 ;(9)
(2)radial velocity Π:
Π = −U X R + V Y R ;(10)
(3) circular velocity Θ:
Θ = U Y R + V X R ;(11)
(4) total 3D velocity V tot :
V tot = √ Π 2 + Θ 2 + W 2 ;(12)
(5)apocentric distance (apo) of the orbit;
(6)pericentric distance (peri) of the orbit;
(7)eccentricity (ecc) of the orbit:
ecc = apo − peri apo + peri ;(13)
(8)angular momentum components:
L X = Y × W − Z × V ;(14)L Y = Z × U − X × W ;(15)L Z = X × V − Y × U ;(16)
(9) orbital inclination θ:
θ = arccos( L Z L ),(17)
where L = L 2 X + L 2 Y + L 2 Z is total orbital momentum; (10)orbit period T r ;
(11)total energy E:
E = Φ(R, Z) + V 2 tot 2 .(18)
The uncertainties of the orbital parameters were calculated by the Monte Carlo method using 100 iterations, taking into account the uncertainties in the initial coordinates and GC velocities, as well as the errors in the peculiar velocity of the Sun.
Data
For the 152 globular clusters we studied earlier with data mainly from the Vasiliev (2019) catalog, we took new average values of proper motions and their uncertainties from the new Vasiliev and Baumgardt (2021) catalog obtained from the Gaia EDR3 catalog data , as well as new average distances from . All other astrometric data (radial velocities, coordinates) remained the same. In Fig. 2 we give a comparison of the mean proper motions from these two catalogs obtained from measurements of Gaia DR2 and Gaia EDR3. As follows from the figure, the new values of proper motions for a number of GCs noticeably differ from the old ones. At the same time, the accuracy of measuring new proper motions increased on average by a factor of two. differ quite significantly, which, together with changes in proper motions, as will be shown below, significantly affected the orbital motion (and, accordingly, the orbital parameters) of many GCs. Table 2 gives the orbital parameters d GC , Π, Θ, V tot , apo, peri, ecc, θ, T r , L Z , E (see Section 1.2) of 152 GCs calculated for the new EDR3 average proper motions (Vasiliev and Baumgardt 2021) as well as new average distances . A comparison of the main orbital parameters of the GC calculated using the average proper motions of Gaia EDR3 with similar parameters calculated from the data of the Gaia DR2 catalog is shown in Fig. 4. As can be seen from the figures, a significant difference is observed for a number of GCs.
Results
As the analysis shows, the most distant objects and objects with highly elongated orbits in the radial direction have undergone the greatest change in their orbital properties. We also note that the classification of Massari (2019) of GCs by subgroups, modified by , has been preserved, since the parameter L Z /ecc, as can be seen from the figure, has not undergone major changes.
The new catalog of orbits in two projections (X, Y ) and (R, Z) of 152 globular clusters is shown in Fig. 5, which can be compared with the previous one published by .
Conclusion
The emergence of more and more accurate astrometric data on the coordinates and spatial velocities of globular clusters makes it possible to study their motion in three-dimensional space by integrating orbits in the gravitational potential of the Galaxy.
Already thanks to the Gaia DR2 data (Helmi et al., 2018;Baumgardt et al., 2019;Vasiliev, 2019) on the proper motions of almost all currently known globular clusters, it became possible to study their kinematics and dynamics, and to classify GCs by subsystems of the Milky Way in order to determine objects that were formed directly in the Galaxy, or introduced from outside as a result of accretion from other (dwarf) galaxies surrounding the Milky Way (Massari et al., 2019). The creation of a catalog of orbits and their parameters for almost all GCs with known data on the 6D phase space required for orbit integration provides highly informative material for subsequent research. Figure 4: Comparison of GC orbit parameters (apo, peri, ecc, L Z , L Z /ecc, E, V tot , θ, T r ) obtained from the DR2 catalog (horizontal axis) and the EDR3 catalog (vertical axis). Each panel has a matching line. 15 Figure 5: Orbits of the GC of the Milky Way galaxy in two projections (X, Y ) and (R, Z). The blue circle marks the beginning of the orbit. 16
et al., 2018; Baumgardt et al., 2019; Vasiliev, 2019; Bajkova et al., 2020; Bajkova and Bobylev, 2021).
Figure 1 :
1Rotation curve corresponding to the NFWBB potential model. The blue dots show the circular velocities of the disk GCs with orbital eccentricities < 0.2.
Figure 2 :
2Comparison of GC proper motions (in α (µ α ) and δ(µ δ ), indicated on the graphs as mua and mud, respectively) and their uncertainties (Emua) and (Emud) , respectively) from the Vasiliev (2019) catalog (Gaia DR2, horizontal axis)) and the Vasiliev and Baumgardt (2021) catalog (Gaia EDR3, vertical axis). Each panel has a matching line.
Figure 3
3compares the GC distances to the Sun, taken from the Harris (2010) catalog when compiling the previous catalog of orbits, and the average distances from the Baumgardt and Vasiliev (2021) catalog, which we used when compiling the new catalog. The left panel reflects distances up to 50 kpc, while the right panel reflects distances up to 15 kpc. From the above figures (especially for small d Sun ), it can be seen that the distances for a number of
Figure 3 :
3Comparison of GC heliocentric distances (d Sun ) from the Harris (2010) catalog (horizontal axis) and the Baumgardt and Vasiliev (2021) catalog (vertical axis). The left panel shows distances up to 50 kpc, the right panel shows distances up to 15 kpc. Each panel has a matching line.
Table 1 :
1Potential model parameters, M 0 = 2.325 × 10 7 M ⊙
Table 2 :
2Orbital properties of the GCs. For each GC the parameter values are obtained as a result of integration of the orbit for 5 Gyr backward.Name
d
GC
Π
Θ
Vtot
apo
peri
ecc
incl. θ
Tr
L
Z
E
[kpc]
[km s −1
]
[km s −1
] [km s −1
]
[kpc]
[kpc]
[degr.]
[Myr] [kpc km s −1
]
[km 2
s −2
]
NGC 104
7.6
6 +8
−4
191 +5
−5
196 +5
−5
7.7 +0.1
−0.1
5.51 +0.26
−0.21
0.16 +0.02
−0.02
28 +1
−0
116 +4
−3
1323 +34
−33
−126364 +1301
−933
NGC 288
12.3
4 +1
−1
−45 +19
−16
68 +11
−10
12.4 +0.4
−0.5
1.44 +0.55
−0.58
0.79 +0.08
−0.07
121 +5
−11
142 +7
−5
−374 +160
−136
−115903 +1953
−2171
NGC 362
9.7
133 +9
−17
0 +13
−9
149 +8
−14
11.9 +0.5
−0.2
0.08 +0.35
−0.00
0.99 +0.00
−0.06
90 +11
−20
130 +7
−4
3 +95
−65
−119870 +2494
−1943
Whiting 1
35.2 −236 +14
−10
82 +8
−15
250 +9
−15
79.0 +12.5
−11.5
22.25 +1.15
−2.03
0.56 +0.05
−0.04
75 +2
−2
1406 +243
−231
1893 +224
−323
−38160 +3070
−3664
NGC 1261
18.3
−98 +8
−6
−22 +6
−4
121 +6
−8
21.3 +0.8
−0.7
0.84 +0.22
−0.23
0.92 +0.02
−0.01
122 +6
−8
248 +9
−11
−289 +85
−61
−90868 +1747
−1625
Pal 1
17.5
43 +7
−3
215 +1
−3
221 +1
−2
19.4 +0.8
−0.3
14.89 +0.37
−0.38
0.13 +0.02
−0.01
15 +0
−1
358 +13
−5
3691 +90
−48
−77319 +1282
−600
E 1
120.3
−21 +35
−49
−6 +52
−47
60 +56
−0
129.4 +26.7
−7.0
5.20 +31.55
−0.00
0.92 +0.00
−0.35
100 +21
−34
2140 +923
−0
−485 +4161
−3811
−29985 +5096
−0
Eridanus
89.8
−80 +17
−9
22 +11
−21
163 +8
−12
159.2 +12.1
−15.7
13.85 +5.37
−4.11
0.84 +0.04
−0.06
74 +14
−6
2900 +321
−366
1567 +793
−1494
−25451 +1362
−1878
Pal 2
34.3
−107 +4
−3
22 +17
−10
110 +4
−2
39.7 +0.6
−2.2
1.35 +1.13
−0.60
0.93 +0.03
−0.05
14 +15
−8
492 +11
−29
754 +548
−344
−64910 +689
−2083
NGC 1851
16.8
104 +1
−3
−4 +5
−3
132 +4
−4
19.9 +0.5
−0.5
0.17 +0.12
−0.06
0.98 +0.01
−0.01
97 +5
−7
228 +5
−8
−66 +72
−52
−94603 +985
−1354
NGC 1904
19.2
45 +5
−4
9 +4
−9
46 +5
−3
19.9 +0.4
−0.3
0.24 +0.21
−0.17
0.98 +0.01
−0.02
68 +24
−9
220 +6
−4
158 +68
−172
−95806 +1326
−563
NGC 2298
15.2
−88 +7
−8
−14 +6
−9
112 +9
−5
16.9 +0.4
−0.4
0.51 +0.25
−0.24
0.94 +0.03
−0.03
105 +8
−7
188 +6
−3
−212 +91
−129
−103115 +1523
−729
NGC 2419
96.0
−5 +7
−2
46 +8
−20
73 +5
−16
97.8 +4.3
−2.9
18.02 +2.06
−4.65
0.69 +0.07
−0.03
52 +16
−10
1690 +86
−98
4018 +778
−1757
−34413 +953
−1094
Pyxis
38.6
−243 +4
−4
−22 +8
−5
292 +6
−7
173.6 +18.1
−19.5
18.60 +0.85
−1.28
0.81 +0.01
−0.01
97 +1
−2
3310 +442
−466
−861 +284
−193
−23684 +1548
−2006
NGC 2808
11.6
−159 +1
−1
36 +3
−3
166 +1
−1
14.9 +0.3
−0.3
0.90 +0.08
−0.07
0.89 +0.01
−0.02
11 +2
−1
162 +4
−2
410 +31
−29
−110129 +1092
−1052
E 3
9.2
42 +17
−7
248 +12
−9
271 +13
−8
12.4 +1.6
−0.8
9.05 +0.30
−0.45
0.15 +0.07
−0.03
28 +2
−1
214 +17
−9
2184 +135
−131
−99850 +3920
−2651
Pal 3
98.2 −146 +26
−16
56 +23
−32
168 +17
−26
148.5 +49.3
−29.3
68.08 +16.19
−20.09
0.37 +0.13
−0.05
74 +9
−7
3742 +1066
−925
4168 +1807
−2405
−22356 +3501
−3779
NGC 3201
9.0
−113 +6
−9
−297 +6
−6
351 +6
−5
24.9 +2.0
−1.6
8.29 +0.20
−0.23
0.50 +0.03
−0.02
152 +1
−1
354 +27
−22
−2671 +92
−92
−77493 +2730
−2444
Pal 4
104.1
−2 +23
−19
2 +30
−12
49 +17
−0
108.7 +5.7
−6.6
4.10 +8.74
−1.74
0.93 +0.03
−0.15
88 +10
−14
1712 +139
−65
91 +1166
−462
−33868 +1358
−807
Crater
147.0
−88 +45
−50
−26 +88
−63
106 +87
−6
149.9 +340.5
−45.9
71.70 +74.52
−14.77
0.35 +0.30
−0.08
99 +21
−26
3854 +1158
−3208
−2527 +8541
−6201
−21992 +13587
−1114
NGC 4147
20.8
42 +3
−6
7 +10
−4
137 +2
−3
25.5 +1.0
−0.6
0.79 +0.47
−0.15
0.94 +0.01
−0.03
83 +3
−7
304 +13
−7
73 +104
−36
−82528 +1719
−888
NGC 4372
7.3
16 +6
−8
132 +4
−5
148 +5
−5
7.3 +0.2
−0.1
2.96 +0.16
−0.16
0.42 +0.03
−0.02
27 +1
−1
98 +2
−4
959 +38
−40
−139945 +1706
−1384
Rup 106
18.0
−243 +4
−4
90 +8
−6
261 +4
−4
36.8 +2.6
−3.4
4.48 +0.45
−0.40
0.78 +0.02
−0.02
46 +3
−3
480 +41
−50
1585 +162
−165
−66205 +2557
−3720
NGC 4590
10.4
−171 +4
−8
293 +3
−8
340 +2
−4
30.5 +1.4
−1.2
8.94 +0.29
−0.22
0.55 +0.01
−0.01
41 +2
−1
438 +22
−19
2464 +71
−75
−69766 +1722
−1527
NGC 4833
7.2
101 +11
−13
37 +8
−7
116 +11
−12
8.0 +0.3
−0.3
0.64 +0.16
−0.09
0.85 +0.02
−0.03
39 +7
−6
84 +4
−3
266 +57
−55
−145170 +2075
−2171
NGC 5024
19.1
−98 +4
−4
138 +7
−6
184 +6
−4
23.0 +1.5
−0.7
9.15 +0.98
−0.51
0.43 +0.02
−0.02
75 +1
−1
344 +30
−14
772 +52
−34
−78916 +3067
−1551
NGC 5053
18.1
−91 +4
−3
136 +3
−6
168 +3
−6
18.1 +0.2
−0.8
10.87 +0.28
−1.01
0.25 +0.03
−0.02
76 +0
−1
304 +5
−19
738 +25
−38
−84327 +460
−2712
NGC 5139
6.6
−63 +5
−5
−76 +5
−4
131 +6
−7
7.1 +0.2
−0.1
1.28 +0.13
−0.14
0.70 +0.03
−0.03
135 +4
−3
82 +3
−2
−489 +38
−35
−147265 +1628
−1497
NGC 5272
12.2
−38 +4
−4
142 +5
−6
199 +3
−4
15.9 +0.7
−0.4
5.14 +0.28
−0.23
0.51 +0.02
−0.01
57 +1
−1
212 +10
−6
988 +33
−39
−98433 +2126
−1193
NGC 5286
8.5
−220 +2
−1
−31 +8
−11
223 +1
−1
13.0 +0.7
−0.5
0.54 +0.20
−0.07
0.92 +0.01
−0.03
116 +8
−6
144 +7
−5
−258 +66
−87
−116387 +2525
−2131
NGC 5466
16.5
169 +15
−14
−141 +18
−11
315 +9
−11
52.9 +4.6
−6.9
5.93 +0.34
−0.81
0.80 +0.02
−0.01
108 +2
−1
736 +76
−114
−816 +96
−61
−53178 +2505
−4606
NGC 5634
21.8
−45 +6
−15
41 +10
−19
66 +7
−2
22.3 +0.8
−0.7
2.29 +0.43
−0.24
0.81 +0.02
−0.03
70 +9
−4
268 +12
−9
383 +99
−171
−87537 +1759
−1473
NGC 5694
29.1 −182 +6
−11
−46 +10
−12
254 +12
−6
71.0 +8.8
−4.9
2.81 +0.99
−0.75
0.92 +0.02
−0.02
134 +5
−5
1004 +159
−87
−1062 +251
−318
−45021 +3362
−2142
IC 4499
15.7
−243 +4
−2
−74 +7
−9
262 +2
−2
29.9 +1.7
−1.5
6.44 +0.47
−0.37
0.65 +0.02
−0.03
113 +3
−2
406 +24
−21
−1056 +87
−130
−72415 +2140
−1899
NGC 5824
25.3
−42 +13
−9
105 +17
−12
213 +11
−9
36.4 +2.9
−2.7
13.54 +2.28
−1.67
0.46 +0.04
−0.05
58 +2
−3
576 +67
−56
2343 +412
−300
−60920 +3013
−2792
Pal 5
17.2
−52 +2
−3
138 +31
−16
148 +29
−15
17.6 +1.0
−0.8
7.93 +3.14
−1.42
0.38 +0.07
−0.14
67 +3
−2
260 +44
−20
962 +249
−180
−89799 +6001
−3264
NGC 5897
7.4
87 +15
−26
98 +12
−23
159 +9
−19
8.8 +0.4
−0.4
1.94 +0.35
−0.51
0.64 +0.07
−0.05
60 +5
−3
106 +7
−6
369 +76
−110
−131407 +2594
−3428
NGC 5904
6.3 −290 +12
−11
126 +8
−8
364 +9
−9
23.2 +1.9
−2.0
2.26 +0.23
−0.19
0.82 +0.02
−0.02
72 +2
−2
284 +24
−25
404 +28
−35
−85760 +3511
−4080
Table 2 :
2Orbital properties of the GC. Continued from previous page.Name
d
GC
Π
Θ
Vtot
apo
peri
ecc
incl. θ
Tr
L
Z
E
[kpc]
[km s −1
]
[km s −1
] [km s −1
]
[kpc]
[kpc]
[degr.]
[Myr]
[kpc km s −1
]
[km 2
s −2
]
NGC 5927
4.8
−51 +9
−11
231 +9
−4
237 +8
−3
5.5 +0.3
−0.2
4.13 +0.25
−0.21
0.15 +0.03
−0.03
9 +1
−1
84 +6
−2
1098 +66
−41
−146878 +3102
−1748
NGC 5946
5.2
19 +12
−11
7 +3
−10
100 +4
−4
5.8 +0.3
−0.3
0.06 +0.08
−0.01
0.98 +0.01
−0.03
86 +6
−1
56 +3
−1
37 +14
−52
−165455 +2817
−1113
ESO 224-8
12.6
−44 +18
−24
257 +17
−16
261 +17
−13
16.8 +3.4
−1.2
11.84 +0.84
−0.98
0.17 +0.08
−0.02
7 +0
−1
290 +50
−23
3226 +323
−206
−85573 +6123
−3392
NGC 5986
4.8
64 +7
−17
27 +6
−7
70 +8
−15
5.6 +0.1
−0.6
0.20 +0.10
−0.05
0.93 +0.02
−0.04
63 +5
−7
58 +2
−3
112 +22
−29
−167349 +1141
−2618
FSR 1716
4.2
87 +9
−19
162 +8
−7
217 +6
−7
5.2 +0.0
−0.6
2.20 +0.31
−0.08
0.41 +0.00
−0.10
35 +2
−2
68 +3
−4
676 +42
−38
−160568 +2517
−2838
Pal 14
68.5
126 +9
−15
−13 +11
−12
177 +9
−11
127.1 +15.6
−10.9
1.49 +2.44
−0.47
0.98 +0.00
−0.04
130 +4
−37
2058 +344
−222
−597 +537
−587
−30562 +2348
−1823
BH 184
4.4
40 +7
−24
119 +10
−6
154 +7
−5
4.7 +0.2
−0.3
1.65 +0.21
−0.16
0.48 +0.03
−0.05
36 +2
−2
58 +2
−4
522 +49
−42
−169132 +2552
−3715
NGC 6093
3.9
37 +12
−15
27 +6
−16
78 +9
−11
4.2 +0.4
−0.2
0.45 +0.17
−0.26
0.80 +0.12
−0.06
79 +6
−2
48 +1
−3
50 +10
−29
−173732 +1571
−2111
NGC 6121
6.6
−58 +1
−2
47 +9
−8
75 +7
−5
6.8 +0.1
−0.1
0.62 +0.17
−0.13
0.83 +0.03
−0.04
5 +1
−0
74 +1
−2
306 +58
−52
−155030 +474
−998
NGC 6101
10.3
−24 +23
−19
−308 +4
−1
363 +2
−4
36.3 +1.6
−1.9
10.14 +0.33
−0.43
0.56 +0.01
−0.01
143 +1
−2
532 +27
−31
−2947 +140
−95
−63205 +1597
−2044
NGC 6144
2.5
−180 +60
−25
−119 +36
−54
220 +3
−2
3.4 +0.2
−0.2
1.56 +0.25
−0.19
0.37 +0.07
−0.08
106 +6
−4
50 +3
−5
−146 +35
−51
−174592 +1679
−1361
NGC 6139
3.5
−1 +14
−19
73 +4
−11
150 +3
−7
3.6 +0.1
−0.2
0.97 +0.04
−0.16
0.57 +0.05
−0.01
61 +4
−2
50 +0
−4
237 +8
−37
−178098 +1677
−3389
Terzan 3
2.5
−20 +28
−31
215 +6
−9
234 +7
−6
3.1 +0.3
−0.1
2.33 +0.10
−0.24
0.14 +0.05
−0.00
36 +3
−3
42 +6
−2
472 +43
−42
−176378 +3635
−2851
NGC 6171
3.9
−1 +2
−2
98 +9
−4
114 +8
−3
4.0 +0.2
−0.2
1.07 +0.20
−0.10
0.57 +0.03
−0.05
42 +2
−2
50 +2
−3
308 +42
−25
−174517 +2798
−2825
ESO 452-11
2.2
−47 +7
−9
22 +8
−13
108 +7
−4
3.0 +0.2
−0.3
0.06 +0.05
−0.02
0.96 +0.01
−0.03
67 +11
−6
28 +2
−3
34 +17
−22
−201978 +3413
−3281
NGC 6205
8.7
15 +2
−5
−28 +4
−5
84 +6
−3
8.8 +0.2
−0.2
0.97 +0.10
−0.11
0.80 +0.02
−0.02
108 +3
−2
102 +3
−2
−205 +30
−35
−133640 +953
−1236
NGC 6229
29.5
32 +6
−5
10 +3
−4
58 +5
−5
30.6 +1.3
−1.3
0.57 +0.22
−0.23
0.96 +0.02
−0.01
64 +11
−12
368 +18
−18
227 +58
−96
−75072 +1590
−1875
NGC 6218
4.7
−11 +4
−4
128 +6
−5
154 +6
−5
4.9 +0.2
−0.1
2.08 +0.16
−0.11
0.40 +0.03
−0.03
40 +1
−2
64 +2
−2
524 +42
−21
−159771 +2293
−1319
FSR 1735
3.2
−96 +9
−6
35 +15
−14
166 +8
−6
4.2 +0.3
−0.2
0.21 +0.16
−0.08
0.90 +0.04
−0.06
74 +6
−6
42 +5
−0
114 +56
−42
−183293 +5174
−920
NGC 6235
4.3
159 +7
−4
213 +20
−25
269 +18
−19
7.2 +0.9
−1.1
3.13 +0.34
−0.60
0.39 +0.04
−0.01
50 +10
−5
98 +12
−15
705 +138
−234
−137634 +5829
−8991
NGC 6254
4.5
−91 +4
−4
118 +7
−8
157 +5
−5
4.8 +0.2
−0.2
1.78 +0.15
−0.13
0.46 +0.03
−0.04
43 +2
−1
74 +1
−3
470 +31
−29
−162398 +2162
−1569
NGC 6256
2.0
−49 +31
−34
191 +9
−16
208 +5
−6
2.4 +0.1
−0.1
1.53 +0.48
−0.48
0.22 +0.16
−0.12
25 +4
−3
36 +6
Table 2 :
2Orbital properties of the GC. Continued from previous page.Name
d
GC
Π
Θ
Vtot
apo
peri
ecc
incl. θ
Tr
L
Z
E
[kpc]
[km s −1
]
[km s −1
] [km s −1
]
[kpc]
[kpc]
[degr.]
[Myr]
[kpc km s −1
]
[km 2
s −2
]
Terzan 2
0.8
−104 +63
−25
−74 +26
−36
137 +3
−1
1.0 +0.3
−0.3
0.13 +0.04
−0.04
0.76 +0.10
−0.16
156 +3
−9
10 +4
−3
−56 +11
−10
−252725 +13021
−13814
NGC 6366
5.3
94 +2
−2
135 +2
−3
176 +2
−3
5.9 +0.1
−0.1
2.24 +0.06
−0.08
0.45 +0.01
−0.01
32 +1
−1
76 +2
−2
711 +20
−26
−152836 +1168
−1273
Terzan 4
0.9
13 +11
−25
61 +9
−5
118 +8
−3
0.9 +0.3
−0.2
0.18 +0.02
−0.03
0.68 +0.05
−0.05
58 +4
−4
10 +4
−2
55 +21
−13
−250150 +13896
−14886
BH 229
1.4
−54 +11
−8
7 +12
−19
250 +16
−10
2.4 +0.5
−0.7
0.24 +0.42
−0.30
0.82 +0.18
−0.24
88 +4
−3
26 +7
−9
10 +14
−17
−205120 +14227
−22643
FSR 1758
3.4
45 +27
−41
−343 +5
−3
396 +4
−5
12.0 +1.2
−1.3
3.31 +0.37
−0.39
0.57 +0.01
−0.02
148 +1
−1
152 +16
−17
−1133 +128
−121
−115174 +5171
−6059
NGC 6362
5.2
18 +16
−19
124 +8
−9
160 +6
−4
5.4 +0.2
−0.1
2.48 +0.16
−0.20
0.37 +0.03
−0.02
45 +3
−3
72 +3
−3
585 +41
−36
−153270 +2099
−1855
Liller 1
0.8
80 +27
−45
−70 +37
−32
109 +21
−18
0.8 +0.3
−0.0
0.12 +0.11
−0.06
0.75 +0.12
−0.11
160 +15
−33
10 +3
−2
−54 +30
−47
−261115 +16395
−4011
NGC 6380
2.1
−82 +8
−8
−27 +18
−9
87 +8
−9
2.4 +0.2
−0.3
0.10 +0.05
−0.06
0.92 +0.04
−0.03
153 +12
−38
28 +3
−4
−54 +37
−24
−212458 +4778
−5472
Terzan 1
2.7
−72 +2
−2
98 +9
−9
121 +8
−7
2.8 +0.2
−0.2
0.67 +0.13
−0.11
0.62 +0.04
−0.05
3 +0
−0
38 +2
−3
259 +34
−33
−200171 +3293
−3450
Pismis 26
1.9
44 +18
−27
238 +6
−6
296 +6
−3
3.3 +0.5
−0.4
1.76 +0.21
−0.18
0.30 +0.02
−0.01
38 +1
−1
52 +11
−2
429 +65
−52
−178784 +6979
−5703
NGC 6388
3.9
−29 +8
−15
−92 +7
−3
98 +5
−5
4.2 +0.2
−0.3
1.00 +0.12
−0.24
0.61 +0.07
−0.02
155 +2
−5
52 +3
−4
−338 +57
−27
−178537 +3036
−5190
NGC 6402
4.0
−18 +13
−15
46 +5
−5
53 +8
−4
4.7 +0.1
−0.3
0.27 +0.12
−0.02
0.89 +0.01
−0.05
46 +5
−5
52 +1
−4
149 +17
−19
−177531 +1637
−2128
NGC 6401
0.8
257 +8
−76
34 +114
−132
285 +2
−2
2.0 +0.6
−0.4
0.06 +0.36
−0.05
0.94 +0.02
−0.25
77 +33
−38
18 +12
−3
19 +119
−88
−221130 +14747
−3396
NGC 6397
6.1
41 +3
−6
118 +4
−8
181 +5
−7
6.5 +0.1
−0.2
2.57 +0.14
−0.26
0.43 +0.04
−0.02
47 +3
−1
84 +2
−4
721 +16
−59
−145511 +1125
−2578
Pal 6
1.3
−199 +4
−4
−10 +25
−21
270 +10
−3
2.8 +0.8
−0.8
0.08 +0.08
−0.07
0.95 +0.07
−0.11
93 +6
−6
28 +8
−7
−13 +28
−36
−202849 +14610
−15486
NGC 6426
14.4
−111 +12
−12
94 +6
−9
149 +12
−14
16.7 +0.8
−0.7
3.28 +0.26
−0.41
0.67 +0.03
−0.01
26 +2
−2
204 +11
−12
1240 +95
−136
−100200 +2396
−2590
Djorg 1
1.7
−291 +48
−41
294 +39
−48
414 +8
−10
8.6 +2.2
−1.6
1.06 +0.18
−0.15
0.78 +0.02
−0.02
19 +3
−4
94 +24
−16
487 +97
−82
−140346 +13303
−12077
Terzan 5
1.8
75 +4
−2
62 +7
−5
102 +5
−2
1.9 +0.3
−0.2
0.22 +0.06
−0.05
0.80 +0.02
−0.03
32 +5
−3
24 +3
−4
108 +25
−20
−220380 +6113
−5974
NGC 6440
1.3
94 +4
−19
−22 +29
−28
104 +3
−9
1.5 +0.1
−0.2
0.05 +0.12
−0.02
0.93 +0.04
−0.15
104 +17
−19
14 +2
−1
−24 +34
−34
−233545 +4082
−591
NGC 6441
4.7
17 +4
−7
99 +10
−18
103 +10
−17
4.7 +0.7
−0.6
1.43 +0.26
−0.35
0.53 +0.08
−0.03
18 +3
−2
58 +8
−7
447 +90
−107
−169955 +7831
−7536
Terzan 6
1.1
−129 +48
−24
−79 +11
−28
153 +4
−4
1.3 +0.4
−0.7
0.17 +0.04
−0.04
0.77 +0.05
−0.16
164 +6
−19
16 +5
−9
−83 +34
−20
−236580 +13730
−37032
NGC 6453
2.0
−109 +16
−7
20 +32
−18
178 +9
−4
2.6 +0.6
−0.9
0.20 +0.43
−0.03
0.86 +0.04
−0.38
84 +6
−9
32 +8
−8
38 +36
−36
−201623 +10681
−14973
NGC 6496
2.8
28 +23
−31
264 +9
−11
269 +11
−11
4.6 +0.4
−0.3
2.35 +0.22
−0.15
0.32 +0.05
−0.05
37 +1
−2
64 +4
−5
581 +46
−31
−161433 +3819
−2581
Terzan 9
2.6
−45 +4
−2
52 +14
−4
84 +9
−3
2.7 +0.2
−0.2
0.28 +0.09
−0.05
0.81 +0.02
−0.05
45 +2
−7
32 +2
−2
132 +39
−17
−205042 +3295
−4485
Djorg 2
0.7
8 +99
−77
−206 +62
−13
218 +4
−4
0.8 +0.6
−0.2
0.50 +0.30
−0.21
0.21 +0.31
−0.10
146 +6
−10
14 +8
−5
−126 +51
−81
−246126 +22397
−10440
NGC 6517
3.2
58 +4
−16
41 +10
−4
78 +4
−9
3.7 +0.2
−0.2
0.23 +0.09
−0.03
0.88 +0.02
−0.04
53 +2
−8
42 +2
−2
125 +31
−12
−190163 +2504
−2700
Terzan 10
2.1
228 +25
−46
94 +54
−30
335 +9
−10
5.3 +1.0
−1.3
0.57 +0.12
−0.09
0.81 +0.06
−0.11
70 +5
−9
70 +10
−16
191 +31
−32
−162093 +10123
−15831
NGC 6522
1.1
26 +5
−5
102 +7
−8
206 +11
−6
1.4 +0.5
−0.2
0.42 +0.35
−0.11
0.54 +0.06
−0.12
59 +3
−1
24 +8
−7
105 +54
−27
−222763 +17892
−10688
NGC 6535
4.1
109 +5
−7
−69 +5
−8
135 +4
−3
4.8 +0.1
−0.2
0.80 +0.11
−0.11
0.72 +0.03
−0.04
163 +0
−2
56 +2
−2
−273 +27
−31
−172105 +804
−2255
NGC 6528
0.8 −201 +217
−66
103 +62
−38
230 +4
−2
1.1 +0.8
−0.4
0.23 +0.15
−0.11
0.66 +0.23
−0.33
70 +5
−3
16 +7
−6
53 +15
−17
−238646 +17943
−12004
NGC 6539
3.1
1 +16
−16
118 +4
−7
214 +8
−7
3.5 +0.1
−0.2
1.85 +0.27
−0.14
0.31 +0.04
−0.07
57 +2
−2
62 +7
−11
349 +16
−30
−172417 +2008
−2527
NGC 6540
2.5
16 +3
−3
136 +5
−6
150 +5
−5
2.6 +0.3
−0.2
1.14 +0.19
−0.12
0.38 +0.04
−0.03
26 +1
−1
40 +4
−3
331 +50
−28
−198910 +6319
−3631
NGC 6544
5.7
10 +2
−1
40 +5
−9
87 +3
−6
6.0 +0.1
−0.2
0.45 +0.08
−0.13
0.86 +0.04
−0.02
63 +5
−3
64 +2
−2
229 +27
−52
−162598 +1056
−1837
NGC 6541
2.2
117 +33
−47
197 +22
−21
255 +7
−3
3.7 +0.4
−0.2
1.29 +0.23
−0.15
0.49 +0.07
−0.08
41 +3
−4
50 +4
−3
333 +34
−15
−175101 +4019
−1532
ESO 280-06
13.3
34 +7
−3
21 +16
−11
83 +9
−1
13.7 +0.6
−0.9
0.73 +0.45
−0.22
0.90 +0.03
−0.06
71 +10
−14
156 +8
−9
257 +191
−135
−112187 +2270
−3282
NGC 6553
3.0
29 +5
−3
249 +1
−1
251 +1
−1
3.9 +0.2
−0.3
2.96 +0.18
−0.30
0.13 +0.02
−0.01
5 +1
−0
62 +3
−3
756 +46
−74
−168744 +2830
−4839
NGC 6558
1.2
187 +3
−1
91 +6
−13
209 +3
−4
1.6 +0.4
−0.5
0.26 +0.17
−0.08
0.72 +0.13
−0.25
65 +11
−6
18 +5
−4
79 +25
−38
−219183 +7780
−11838
Pal 7
4.5
−102 +6
−5
268 +6
−4
287 +5
−3
7.2 +0.5
−0.2
3.78 +0.24
−0.11
0.31 +0.01
−0.01
10 +0
−1
100 +6
−4
1184 +85
−39
−138577 +4025
−1922
Table 2 :
2Orbital properties of the GC. Continued from previous page. ] [km s −1 ] [km s −1 1.87 +0.15 −0.13 0.36 +0.02 0.49 +0.11 −0.14 0.73 +0.08 0.12 +0.10 −0.13 0.91 +0.10 0.87 +0.11 −0.17 0.66 +0.05 14.41 +0.76 −1.06 0.56 +0.04Name
d
GC
Π
Θ
Vtot
apo
peri
ecc
incl. θ
Tr
L
Z
E
[kpc]
[km s −1
]
[kpc]
[kpc]
[degr.]
[Myr]
[kpc km s −1
]
[km 2
s −2
]
Terzan 12
3.3
−94 +5
−2
160 +8
−6
211 +6
−5
4.0 +0.3
−0.3
−0.03
31 +2
−2
54 +5
−2
524 +45
−40
−174529 +3905
−4020
NGC 6569
2.5
−40 +4
−4
164 +14
−25
170 +14
−24
2.6 +0.6
−0.4
1.46 +0.25
−0.43
0.28 +0.15
−0.05
29 +10
−7
46 +7
−8
354 +75
−95
−189983 +6899
−7687
ESO 456-78
2.3
62 +4
−4
203 +2
−3
249 +4
−3
3.2 +0.3
−0.3
1.81 +0.28
−0.26
0.27 +0.04
−0.02
32 +2
−1
58 +4
−6
458 +64
−59
−180481 +5218
−5189
NGC 6584
6.9
198 +12
−16
105 +29
−12
327 +12
−9
18.6 +2.5
−1.2
1.79 +0.77
−0.27
0.82 +0.03
−0.05
50 +3
−4
220 +34
−15
603 +231
−83
−96417 +6158
−3082
NGC 6624
1.2
−23 +44
−26
60 +4
−18
138 +6
−5
1.7 +0.3
−0.1
0.08 +0.10
−0.05
0.91 +0.05
−0.09
74 +2
−4
20 +4
−4
32 +16
−11
−227013 +7470
−2603
NGC 6626
3.1
−27 +4
−2
61 +12
−6
113 +7
−4
3.2 +0.3
−0.3
−0.06
58 +3
−5
44 +2
−9
190 +32
−28
−191218 +2777
−6026
NGC 6638
2.2
64 +9
−12
10 +13
−16
70 +6
−5
2.7 +0.3
−0.2
0.04 +0.06
−0.01
0.97 +0.01
−0.04
81 +13
−10
24 +7
−3
20 +18
−30
−206839 +8133
−4599
NGC 6637
1.7
43 +20
−113
89 +9
−58
128 +7
−6
2.4 +0.2
−0.2
−0.09
75 +11
−2
22 +2
−0
47 +12
−37
−211371 +4131
−1668
NGC 6642
1.7
112 +6
−9
29 +24
−36
126 +8
−5
2.2 +0.2
−0.1
0.09 +0.11
−0.05
0.92 +0.04
−0.05
43 +58
−18
24 +4
−5
42 +42
−51
−215290 +4209
−2746
NGC 6652
2.1
−55 +12
−7
10 +16
−23
175 +9
−7
3.6 +0.5
−0.4
0.03 +0.08
−0.01
0.98 +0.01
−0.02
81 +11
−6
32 +5
−3
10 +21
−15
−192830 +7171
−3378
NGC 6656
5.1
177 +2
−1
199 +1
−1
305 +8
−4
9.8 +0.4
−0.3
2.99 +0.09
−0.11
0.53 +0.01
−0.00
34 +3
−2
124 +6
−3
1013 +26
−33
−125755 +2366
−1764
Pal 8
4.0
−17 +13
−18
78 +11
−11
85 +15
−11
4.2 +0.2
−0.2
−0.03
29 +3
−2
52 +1
−4
295 +39
−47
−178566 +2806
−3432
NGC 6681
2.2
223 +16
−45
9 +87
−45
287 +4
−6
5.0 +0.6
−0.5
0.48 +0.27
−0.18
0.82 +0.07
−0.10
88 +13
−8
56 +4
−3
8 +45
−43
−163960 +3687
−3159
NGC 6712
3.6
141 +3
−4
5 +10
−8
213 +5
−6
5.6 +0.3
−0.4
0.07 +0.05
−0.03
0.98 +0.01
−0.02
88 +3
−4
58 +4
−4
18 +35
−27
−168037 +3989
−3245
NGC 6715
18.4
231 +3
−3
48 +13
−16
311 +5
−12
51.8 +8.7
−8.2
−0.03
80 +3
−3
826 +153
−143
836 +231
−266
−50574 +3996
−4779
NGC 6717
2.4
−1 +27
−18
111 +5
−7
114 +7
−5
2.7 +0.5
−0.2
0.64 +0.15
−0.07
The recent appearance of a new, more accurate version of the catalog of proper motions , as well as new, more accurate average distances , made it possible to create a new version of the catalog of orbits of 152 GCs and their parameters, which was the result of this work.The authors are grateful to the referee for useful remarks that contributed to the improvement of the article.
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| [] |
[
"On minimal representation-infinite algebras",
"On minimal representation-infinite algebras",
"On minimal representation-infinite algebras",
"On minimal representation-infinite algebras"
] | [
"Klaus Bongartz \nUniversität Wuppertal\n\n",
"Klaus Bongartz \nUniversität Wuppertal\n\n"
] | [
"Universität Wuppertal\n",
"Universität Wuppertal\n"
] | [] | Irrtum verlässt uns nie, doch ziehet ein höher Bedürfnis immer den strebenden Geist leise zur Wahrheit hinan XenienAbstractWe consider finite dimensional basic associative algebras over an algebraically closed field and we classify those that are not distributive and minimal representation-infinite. As a consequence the number of isomorphism classes of all minimal representation-infinite algebras of any fixed dimension is finite and there are Z-forms for these. We show that tame concealed algebras are minimal representation-infinite and that the classification of all minimal representation-infinite algebras would lead to a useless unreadable list. * [email protected] | null | [
"https://export.arxiv.org/pdf/1705.10858v5.pdf"
] | 119,318,551 | 1705.10858 | 258e9e0baaf6aa7c6cad70963c94b993736496fa |
On minimal representation-infinite algebras
19 May 2023
Klaus Bongartz
Universität Wuppertal
On minimal representation-infinite algebras
19 May 2023
Irrtum verlässt uns nie, doch ziehet ein höher Bedürfnis immer den strebenden Geist leise zur Wahrheit hinan XenienAbstractWe consider finite dimensional basic associative algebras over an algebraically closed field and we classify those that are not distributive and minimal representation-infinite. As a consequence the number of isomorphism classes of all minimal representation-infinite algebras of any fixed dimension is finite and there are Z-forms for these. We show that tame concealed algebras are minimal representation-infinite and that the classification of all minimal representation-infinite algebras would lead to a useless unreadable list. * [email protected]
Introduction
Our algebras A are basic, associative and of finite dimension over an algebraically closed field k. Such an A is given by its quiver Q and an admissible ideal I. The A-modules are left-modules of finite dimension and we think of these often as representations of Q satisfying the relations imposed by I. The category of these modules is denoted by mod A. An algebra A is representationfinite if it has only finitely many isomorphism classes of indecomposable modules and minimal representation-infinite if it is not representation-finite, but any proper quotient is. Finally A is distributive if its lattice of two-sided ideals is distributive.
In 1957 Jans showed in [24] that a non-distributive algebra is strongly unbounded i.e. that there exist infinitely many d such that there are infinitely many isomorphism classes of indecomposables of dimension d. Furthermore he mentions two conjectures of Brauer and Thrall: The first says that A is representation-finite if there is a bound on the dimensions of indecomposables and the second says that otherwise A is strongly unbounded.
The first conjecture was 1968 positively answered by Roiter in [32] using brilliant elementary arguments and for the generalization to artinian rings 1974 in [1] Auslander considered almost split sequences in disguise. The proof [3] of the second conjecture by Bautista in 1985 required some of the new concepts of representation theory introduced after 1968 and also an intensive study of representation-finite and distributive minimal representation-infinite algebras. This was done between 1970 and 1985 by several people who turned their attention afterwards to other directions.
However, some natural questions remained unanswered e.g.: Can there be gaps in the lengths of the indecomposables? Is there a finite dimensional representation-infinite algebra which is smallest with respect to representation embeddings? Are there only finitely many isomorphism classes of minimal representation-infinite algebras in each dimension?
I answered the first two questions in two former publications [11,13] and here I answer the third. To this end we define five families of algebras depending on parameters by a picture of their quivers and by giving afterwards the relations and the possible values of the parameters.
A(p,q) q q q q q q ✠ ❅ ❅ | ❅ ❅ | ✠ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ a 1 b 1 a p b q B(p,q) q q q q q q q ✠ ❆ ❆ ❆ ❆ ❯ ❄ ❅ ❅ |❄ ✁ ✁ ✁ ✁ ☛ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ c a 1 b 1 a p b q C(p) q q ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ρ p ρ 1 α β q q q q q q q ❅ ❅ ■ ❆ ❆ ❆ ❆ ❯ ✁ ✁ ✁ ✁ ☛ ✠ ❄ ❄ ❅ ❅ | ✒ D(p,q) q q ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ q q q q q q q q q ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ρ p ρ 1 α β α 1 α q+1 ❅ ❅ ■ ❅ ❅ ❅ ❅ | ✠ ✠ ❅ ❅ | ✠ ❄ ❄ ❅ ❅ | ❅ ❅ | ✒ E(p,q,r) q q q q q q q q q q q q q q q q ✒❅ ❅ | ❄ ❄ ✠ ❅ ❅ ■ γ 1 γ p α 1 α q β 1 β r ✲ ✲ ✲ ✒❅ ❅ | ✠ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✻ ✻ ❅ ❅ ■ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
For the family A(p, q) there is no relation and one has p, q ≥ 0. In the family B(p, q) only the possibilities p ≥ q = 1 and 4 ≥ p ≥ q = 2 are allowed and the sum of all three paths between the source and the sink is a relation. Thus so far we have just the tame canonical algebras. In the remaining cases all parameters p, q, r ≥ 1 are allowed. C(p) has one zero-relation ρ 1 ρ p which also holds for D(p, q) where in addition the two paths between the source and the sink give a commutativity relation. Finally the relations α 1 α q ,γ 1 γ p and γ 1 β r . . . β 1 α q define E(p, q, r).
Recall that for each algebra A with a source a and a sink z one obtains another 'glued' algebra by identifying a and z to one point x and by adding in the new quiver all paths of length 2 with x as an interior point to the relations.
Our first result says:
Theorem 1. An algebra over an algebraically closed field is basic minimal representation-infinite and not distributive if and only if it is isomorphic to an algebra listed above or to its glued version.
This has an interesting consequence whose analogue for representation-finite algebras is not true because non-standard algebras exist.
Theorem 2. Let d be a natural number. There is a finite list of Z-algebras which are free of rank d as Z-modules such that for each algebraically closed field k the algebras A ⊗ Z k form a list of basic minimal representation-infinite algebras of dimension d.
In fact both theorems and their proofs remain valid for k-split algebras over any field.
The proof of theorem 1 given on 15 pages is the heart of the article. There are similarities to the proof of the structure theorems for non-deep contours in the central article [2] of Bautista, Gabriel, Roiter and Salmerón about multiplicative bases. In section 2 we subdivide the problem into three different types of algebras, where the first two are related by the glueing procedure described above. The algebras of type 2 are analyzed in section three and the slightly more delicate algebras of type 3 in section 4.
In section 5 theorem 2 is derived from theorem 1 and my former results on coverings in [11].
In the next section it is shown that tame concealed algebras are minimal representation-infinite. Somewhat surprisingly this is nowhere mentioned in the recent literature. The mathematical and historical relations between the characterization of tame concealed algebras by Happel and Vossieck and my results on critical simply connected algebras are clarified.
At the end we prove that all basic distributive minimal representationinfinite algebras can be obtained by a glueing process from a critical line or a critical simply connected algebra. However, in the second case a complete classification remains out of reach. Throughout this article A is a basic associative algebra of finite dimension over a field k, N denotes the radical of A and S the socle of A as a bimodule. We assume that A/N is a product of copies of k which holds always if A is basic and k is algebraically closed. By a fundamental observation of Gabriel there is then a quiver Q and a surjective algebra homomorphism π from the path-algebra kQ to A whose kernel is contained in the ideal generated by all paths in Q of length 2. We fix such a presentation and we write often v instead of π(v). Thus we get in A a decomposition of 1 as a sum of pairwise orthogonal primitive idempotents 1 = x∈Q0 e x where e x is the image of the path of length 0 through the point x.
We denote by I the lattice of two-sided ideals of A, by I ′ the sublattice of the ideals contained in N and by B(x, y) the lattice of e x Ae x − e y Ae ysubbimodules of e x Ae y . For such a subbimodule J we denote by rad J the radical as a bimodule and the higher radicals rad i J are defined by induction. We have rad e x Ae x = e x N e x for any x and rad J = e x (N J + JN )e y for any subbimodule. The algebra is called distributive provided I is a distributive lattice.
We refine a little bit the important observations of Jans [24] and Kupisch [25] on distributive algebras. Proposition 1. Keeping the above assumptions and notations we have: i) If J is a subbimodule of e x Ae y and J the two-sided ideal generated by J then we have J = N J + JN + J and e x J e y = J ∩ e x Ae y = J.
ii) The map I → e x Ie y is a surjective lattice homomorphism from I to B(x, y) for all x, y.
iii) For two points x, y the following are equivalent:
(a) B(x, y) is distributive. (b) dim (rad i e x Ae y /rad i+1 e x Ae y ) ≤ 1 for all i.
(c) e x Ae y is a uniserial bimodule i.e. it has a unique chain of subbimodules.
iv) Equivalent are:
(a) I is distributive.
(b) I ′ is distributive.
(c) All the lattices B(x, y) are distributive.
v) The ring e x Ae x is uniserial if and only if its radical is 0 or generated by one element α x .
vi) Let x, y be two points such that e x Ae x and e y Ae y are both uniserial. Then e x Ae y is uniserial as a bimodule if and only if for i = 0 and i = 1 we have dim (rad i e x Ae y /rad i+1 e x Ae y ) ≤ 1. In that case e x Ae y is uniserial as a left e x Ae x -or a right e y Ae y -module.
Proof. Statement i) is immediately clear and also that the map I → e x Ie y preserves intersections, sums and inclusions. The surjectivity follows from the last equation in i). Suppose now that the vector space V = (rad i e x Ae y /rad i+1 e x Ae y ) has dimension ≥ 2 for some i. Then rad i e x Ae y lies in N . Namely for x = y we have e x Ae y ⊆ N and for x = y we have i ≥ 1. In V there is a plane containing three different lines violating the law of distributivity. Their preimages L 1 , L 2 , L 3 under the canonical projection are subbimodules also violating distributivity and so B(x, y) is not distributive. Similarily one gets that I ′ is not distributive by looking at the two-sided ideals generated by the L i and using part i).
We have just seen that the distributivity of B(x, y) implies for all i that dim(rad i e x Ae y /rad i+1 e x Ae y ) ≤ 1. It follows easily that B(x, y) is uniserial whence distributive. So part iii) is true.
If I is distributive so is its sublattice I ′ . From this we obtain by the argument from above that dim(rad i e x Ae y /rad i+1 e x Ae y ) ≤ 1 for all i and all x, y. Thus all B(x, y) are distributive by part iii). Using the relation I = ⊕ x,y e x Ie y valid for any two-sided ideal one gets that I is distributive.
Part v) is trivial. If one of the spaces e x Ae x , e y Ae y or e x Ae y has dimension ≤ 1 then part vi) is obvious. In the other case let a be a generator of e x Ae y . Then α x a and aα y are not linearly independent modulo rad 2 e x Ae y . Up to symmetry we can assume that we have α x a = ξaα y + r for some scalar ξ and some r ∈ rad 2 e x Ae y . Then we obtain α p x aα q y = ξ p aα p+q y + r(p, q) with some r(p, q) ∈ rad p+q+1 e x Ae y for all p and q by induction on p. Now the elements α i x aα n−i y with i ≥ n generate rad n e x Aey for any n and this space is zero for large n. By descending induction it follows that all rad i e x Ae y are generated by the aα j y with j ≥ i. Thus e x Ae y is cyclic as a module over e y Ae y whence uniserial.
The subdivision
We show that the minimal non-distributive algebras fall into three disjoint classes. A pair (a, z) of points is called critical if the bimodule e z Ae a is not uniserial. The critical index i(a, z) of such a pair pair is then the smallest natural number such that rad i e z Ae a /rad i+1 e z Ae a has dimension ≥ 2. Furthermore, given a point x in Q, we denote by I x the two-sided ideal generated by all paths of lengths 2 with x as the interior point. Recall that x is called a node if I x ⊆ I holds.
Proposition 2. Let A = kQ/I be an algebra which is not distributive but any proper quotient is. Then the following holds: i) For any critical pair (a, z) with critical index i we have rad i+1 e z Ae a = 0 and S(a, z) := rad i e z Ae a is a bimodule of dimension 2 which is contained in S.
ii) There is only one critical pair (a, z) and we have S = S(a, z). Moreover we are in one of the following three situations:
(a) ( type 1 ) a = z, i(a, z) = 1, e a Ae a ≃ k[X, Y ]/(X, Y ) 2 and I a ⊆ I. (b) (type 2) a = z, i(a, z) = 0, a is a source, z a sink in Q and for e = e a + e z the algebra eAe is isomorphic to the path-algebra of the Kronecker quiver K 2 consisting of two parallel arrows. (c) (type 3) a = z, i(a, z) = 1 and for e = e a + e z the algebra eAe is isomorphic to the path algebra of the quiver with one loop α in a, one arrow β from a to z and one loop γ in z divided by the relations α 2 = γ 2 = γβα = 0.
Proof. We consider the two-sided ideal J generated by rad i+1 e z Ae a . Then we have e z Ae a ∩J = rad i+1 e z Ae a whence the quotient A/J is still not distributive. By minimality we have J = 0 and a fortiori rad i+1 e z Ae a = 0. Similarly, if V := (N rad i e z Ae a + rad i e z Ae a N ) = 0 we look at the non-zero two-sided ideal J it generates. Because of J ∩ e z Ae a = 0 the proper quotient A/J is again not distributive and so J = 0 and a fortiori V = 0. This means that rad i e z Ae a is contained in S. If the dimension of rad i e z Ae a is strictly greater than 2 we choose a non-zero subbimodule J of codimension 2 in rad i e z Ae a . Then J is even a two-sided ideal and A/J is still not distributive. This contradiction shows that dim S(a, z) = 2.
There is at least one critical pair (a, z) and we have S = S(a, z) ⊕ S ′ for some two-sided ideal S ′ . This ideal is zero because A/S ′ is still rerepresentationinfinite. Thus we have S = S(a, z) and there is only one critical pair. We discuss the different possibilities.
For a = z we have i = i(a, z) = 1 and e a Ae a ≃ k[X, Y ]/(X, Y ) 2 . For any path p = βα of length 2 with interior point a we consider the two-sided ideal J generated by p. For any paths v, w we have that e a vβ and αwte a are in rad e a Ae a whence their product vanishes and J ∩ e a Ae a = 0. Thus A/J is still not distributive and we have J = 0 by minimality. Thus we have I a ⊆ I.
For a = z all e x Ae x are uniserial rings and we can apply the last part of proposition 1 to see that only i = i(a, z) = 0 and i = 1 are possible. In the case i = 0 we have S(a, z) = e z Ae a . Take an element f in some e a N e y . Then the two-sided ideal J generated by f is spanned by products vf w and the intersection with e z Ae a by products e z vf we a . This product vanishes because f annihilates the element e z v from S(a, z). Thus A/J is still not distributive and we conclude J = 0 whence f = 0. It follows that x is a source. Dually z is a sink and so eAe has the wanted form.
Finally we look at the case a = z, i = 1 and S = rade z Ae x . Let f be in rad 2 e a Ae a and let J be the two-sided ideal generated by f . Then the intersection of J with e z Ae a is spanned by products e z vf we a which are all 0. We get that J = 0 and f = 0, i.e. dim e a Ae a = 2 and dually dim e z Ae z = 2. Let f be an element in e a Ae z such that the intersection of the ideal J generated by f with e z Ae a is not 0. Then there is a product e z vf we a = 0. The non-zero products f we a and e z vf show that the quiver of eAe has no loops and so it is an oriented cycle. But then eAe is uniserial. Thus the intersection J ∩ e z Ae a is zero, J = 0 and f = 0. It follows that eAe has the wanted shape.
A non-distributive algebra satisfies the second Brauer-thrall conjecture as Jans has shown by direct calculations already in [24]. A proof of his results via representation embeddings is given in [13, section 3.1].
Glueing and separating: the relation between the first two types
We recall and refine a little bit the well-known constructions of glueing a source and a sink or separating a node into a source and a sink [26]. So let Q be a quiver with a proper source a and a proper sink z. Denote by Q ′ the quotient obtained by identifying a and z to one point x. The other points of Q ′ and the arrows are just dashed versions of those of Q. It follows from the universality of path-algebras that there is an algebra-homomorphism φ : kQ ′ −→ kQ with φ(α ′ ) = α for each arrow, φ(e y ′ ) = e y for all y different from a and z and φ(e x ) = e a + e z . The image of φ is the subalgebra B of kQ generated by f = e a + e z , by the other idempotents and by all arrows. We denote by I(x) the ideal of kQ ′ generated by all path β ′ α ′ where x is the endpoint of α ′ . We have φ(β ′ α ′ ) = φ(β ′ e x α ′ ) = β(e a + e z )α = 0 because there is no arrow ending in a or starting in z. Thus I(x) lies in the kernel of φ. On the other hand the paths in Q ′ of length at least 1 and not having x as an interior point are in bijection under φ with all proper paths in Q. Thus φ induces an isomorphism kQ ′ /I(x) ≃ B.
Reversely one can start with a quiver Q ′ containing a point of transition x and separate this point into an emitter a and a receiver z to obtain a quiver Q with a proper source a and a proper sink z. Clearly these operations on quivers are inverse to each other.
There is an exact functor F from mod kQ to mod kQ ′ /I(x) defined in the language of representations by F M (x) = M (a)⊕M (z), F M (y ′ ) = M (y) for y ′ = x and by the obvious action on the arrows. This functor maps indecomposables to indecomposables and it hits all indecomposables up to isomorphism. The two simples corresponding to the points a and z are the only two non-isomorphic indecomposables that become isomorphic. In fact F is just the restriction to B if one identifies B with kQ ′ /I(x).
Proposition 3.
We keep all the assumptions and notations introduced above. Let J be a two-sided admissible ideal in kQ such that A = kQ/J is finitedimensional and let J ′ be the inverse image of J under φ and define A ′ = kQ ′ /J ′ . Then we have:
i) A is distributive iff A ′ is distributive. ii) A is minimal representation-infinite iff A ′ is so.
iii) A is a non-distributive minimal representation-infinite algebra of type 2 with respect to a and z iff A ′ is one of type 1 with respect to x.
Proof. For an algebra C we denote by I ′ (C) the lattice of two-sided ideals of C contained in the radical. Recall that C is distributive iff I ′ (C) is distributive. Now I ′ (A ′ ) and I ′ (B/J ) are isomorphic. Any two-sided ideal of B/J contained in the radical is automatically a two-sided ideal in A. Thus part i) is proven. The functor F applied to the full subcategories of representations annihilated by J resp. by J ′ shows that A is representation-infinite iff A ′ is so. Now a representation-infinite algebra C is minimal representation-infinite iff all quotients C/I with I contained in the radical are representation-finite. For A and for A ′ these ideals correspond each other and the representation types of the quotients coincide. Part ii) follows.
We know already that
A is non-distributive minimal representation-infinite iff A ′ is so. We have e x A ′ e x ≃ f (B/J)f = f (rad A) f ⊕ kf and f Af = f (rad A) f ⊕ ke a ⊕ ke z .
Part iii) follows from proposition 2 by comparing the dimensions.
General remarks on the proof
The basic minimal representation-infinite algebras that are not distributive of types 2 or 3 will be studied in the next two sections. We call such an algebra suspicious. The only critical pair is denoted by (a, z) and the two-dimensional two-sided socle by S.
We consider the algebra often as a k-category with the points of Q as objects and with the A(x, y) = e y Ae x as morphism spaces. Any non-zero morphism f ∈ A(x, y) can be prolongated to a non-zero morphism gf h ∈ S and so we have A(a, y) = 0 = A(y, z) for all y. A path p in Q is called a zero-path resp. a nonzero-path if p = 0 resp. p = 0. Recall that we work with a fixed presentation. Any non-zero path p from x to y can be prolongated to a non-zero-path p 2 pp 1 with p 2 pp 1 ∈ S. Such a path is called long.
Observe that all A(x, x) are uniserial and all A(x, y) are uniserial for (x, y) = (a, z). A point x is called thin if dim A(x, x) = 1 and thick otherwise. Given two morphisms f, g ∈ A(x, y) we write f ∼ g if both elements generate the same subspace of A(x, y). For three thin points x 1 , x 2 , x 3 with (x 1 , x 3 ) = (a, z) and morphisms f, g ∈ A(x 1 , x 2 ), h ∈ A(x 2 , x 3 ) one has a nice cancellation property: hf ∼ hg ∼ 0 implies f ∼ g. We often use that the situation is self-dual. In particular there is a dual cancellation result.
Our main method to derive all the wanted results is to look at a full subcategory A ′ of A supported by 5 points at most and at its quiver Q ′ . Then any proper quotient of A ′ has to be representation-finite. To exclude certain possibilities we will construct a quotient of A ′ which is defined by zero-relations. Then there is a Galois-coveringà ′ given by an infinite tree with relations. The group is free and it acts freely so that it is sufficient to find a representation-infinite tree-algebra as a full convex subcategory ofà ′ and this is always easy.
Our method is based on the elementary part of Galois-coverings as defined by Gabriel ( see [12, theorem 16, part a)] ). It was applied again and again in Gabriels proof for the structure and disjointness theorems of non-deep contours ( see [2, remark 3.8 ] ). But there the situation is more complicated and one cannot always reduce to a quotient A ′ given by zero-relations. In fact the only representation-infinite algebras we need to know are quiver algebras of types A n ,D n andẼ 6 .
3 Algebras of type 2
Thick points
Throughout this section A is a suspicious algebra of type 2 with quiver Q. Thus we have a source a and a sink z. Let b be a thick point which is of course different from a and z. We choose a generator r of rad A(b, b) as well as generators s resp. t of A(a, b) resp. A(b, z) as modules over A(b, b). Lemma 1. The full subcategory A ′ supported by a, b, z is given by the quiver Q ′ with arrows α : a → b, β : b → z and ρ : b → b and the relation ρ 2 = 0.
Proof. Because a is a source and z is a sink the quiver Q ′ contains the three arrows mentioned above. We denote by n be the greatest integer with r n = 0. Then we have tr n s = 0 by the prolongation property. If the elements tr i s with 0 ≤ i ≤ n do not generate S = A(a, z) there is an arrow from a to z which implies the contradiction that the separated quiver is a quiver of typeà 3 . Thus Q ′ has only three arrows.
The full subcategory A ′′ supported by b and z is representation-finite with the quiver containing ρ and β and defined by the relation ρ n+1 = 0. The universal cover of A ′′ shows that n ≤ 2 holds.
Suppose ρ 2 = 0. Because of dim S = 2 there is a non-trivial linear relation x 0 βα + x 1 βρα + x 2 βρ 2 α. For x 0 = 0 we can replace the presentation π by a new presentation π ′ : kQ ′ −→ A ′ by defining π ′ (α) = π(x 0 α + x 1 ρα + x 2 ρ 2 α) and so A ′ is defined by the relations ρ 3 and βα. Then the two other paths produce a basis of S. Similarly for x 1 = 0 = x 0 we can reduce to the relations ρ 3 and βρα. So in all cases A ′ is defined by zero-relations. One finds in the corresponding Galois-coveringà ′ as convex subcategories for βα = 0 a quiver of typeD 4 or for βα = 0 a tame concealed algebra of typeẼ 6 which both are annihilated by all liftings of the path βρ 2 α. This is a contradiction. ii) b is the only thick point.
iii) One has ξρ m = 0 for all arrows ξ = β and therefore dim
A(b, x) ≤ 1 for all x with b = x = z.
Proof. We choose a path β with β = t. If β is not an arrow we choose a decomposition β = β 1 β 2 where β 1 : y → z is an arrow. Then a, b, y, z are four different points in Q ( y = b follows from r 2 = 0 ) and we look at the full subcategory A ′ supported by these four points and its quiver Q ′ which contains the arrows α : a → b, γ : b → y and β 1 : y → z. The two elements γα and γρα are linearly independent in A(a, y) because their products with β 1 are so in A(a, z). Thus A(a, y) is cyclic over A(y, y) and so we get dim A(a, y) = 2 = dim A(y, y) from the last lemma and similarly dim A(a, b)
= 2 = dim A(b, b) = dim A(b, y)
. It follows that A(b, y) is uniserial from both sides. Thus there are only two possibilities for the quiver Q ′ of A ′ : Either one adds an arrow ǫ : y → b and the relations (γǫ) 2 = (ǫγ) 2 hold or one adds a loop ρ in b and a loop σ in y and the relation γρ = σγ holds. In the second case one can even divide by σγ and one gets also a zero-relation algebra. Inà ′ one finds in both cases easily a convex subcategory with quiver an extended Dynkin diagram of typeD 5 that is annihilated by the liftings of the path βρα. This contradiction shows that β is an arrow in Q and so is α by duality. Let b ′ be another thick point. Then the quiver Q ′ of the full subcategory A ′ supported at a, b, b ′ , z contains the arrows α :
a → b, β : b → z, α ′ : a → b ′ and β ′ : b ′ → z. If ρ or ρ ′ factorize it contains also arrows b → b ′ and b ′ → b
and then the separated quiver to the proper quotient A ′ /rad 2 A ′ contains a quiver of typeà 5 . The same holds if the loops survive.
If part iii) is not true we find a long path p 2 ξρ m p 1 . Because α is an arrow we have ρ m p 1 ∼ ρα and because β is an arrow we get ii) Any long path q not starting with α has no interior point in common with βρα.
p 2 ξ ∼ βρ i.e. p 2 ξρ m p 1 ∼ βρ 2 α ∼ 0. This is a contradiction. The second statement follows because A(b, x) is cyclic over A(b, b).
Uniqueness and disjointness for long paths
Proof. Let p = ζ n ζ n−1 . . . ζ 1 α be a third long path. We will derive a contradiction. First assume m = 1. For ζ := ζ 1 = ρ we would get ζ 2 = β from part iii) of lemma 2 and so p = βρα.
Thus ζ : b → d is different from ρ.
We consider the full subcategory A ′ supported by the four points a, b, d, z and its quiver Q ′ in which the arrows α, β, ρ, ζ still exist. An arrow ζ ′ : d → b would occur in a long path p 2 ζ ′ p 1 . But a and d are thin and so ζα generates A(a, d). Thus ζ ′ ζ is not a zero-path and ρ not an arrow. Thus there is an arrow ζ ′ : d → z because there is a long path containing ζ. We have ζ ′ ζ ∼ βρ and we divide A ′ by βρ to get a zero-relation algebra having the obviousD 4 -quiver as a convex subcategory in its universal cover. The situation is shown in figure 1.
figure 1 q q q q ✛ ❄ ❄ ❄ ❅ ❅ ❅ ❅ ❅ ■ ✖✕ ✗✔ α β ζ ζ ′ ρ figure 2 q q q q q ✛ ✛ ✲ ❄ ❄ ✻ α β ζ ζ ′ ρ ′ ρ ′′ figure 3 q q q q q ✛ ✛ ✲ ❄ ❄ ❅ ❅ ❅ ❅ ❅ ■ α β ζ ζ ′ ρ ′ ρ ′′
Thus we have m ≥ 2. We consider ρ ′ := ρ 1 : b → c and ζ := ζ 1 : b → d and we assume ζ 1 = ρ 1 . This time we study the full subcategory A ′ supported by a, b, c, d, z and its quiver Q ′ . Here d = b because r is not irreducible, d = z because ζ = β and finally d = c because ζ = ρ ′ . Thus Q ′ has 5 points and the arrows α, β, ζ, ρ ′ survive. Now a, c, d are all thin so that all A(x, y) between any of these points have dimension ≤ 1. From part iii) of lemma 2 we also have dim A(b, c) = dim A(b, d) = 1. We want to show that A(c, d) = 0. If not there is a non-zero path δ : c → d and so a long path p 2 δp 1 . We have p 1 ∼ ρ ′ α and p 2 δρ ′ α = 0 contradicting that ζ is an arrow. An analogous reasoning shows A(d, c) = 0. From A(c, b) = 0 we obtain an arrow ρ ′′ : c → b and there is no loop at b.
For A(d, b) = 0 one gets an arrow ζ ′ : d → b and then there is no arrow d → z because ζ ′ occurs in a long path p 2 ζ ′ ζα. The quiver Q ′ is shown in figure 2. We have ζ ′ ζ ∼ ρ ′′ ρ ′ ∼ ρ. We divide A ′ by ρ and obtain a zero-relation algebra with aD 4 -quiver in its universal cover.
For A(d, b) = 0 there is an arrow ζ ′ : d → z because ζ belongs to a long path. The situation is illustrated by figure 3. We have ζ ′ ζ ∼ βρ. Dividing by this we end up with another zero-relation algebra with aD 4 -quiver in its universal cover.
We have shown that ζ 1 = ρ 1 and we will show by induction on i for 1 ≤ i ≤ m that ζ i exists and coincides with ρ i . The start for the induction was just shown and we explain the step from i − 1 to i. Consider ζ j : d j−1 → d j and ρ j :
c j−1 → c j ( c 0 = d 0 = b ) for all j ≤ i − 1. Since c i−1 = d i−1 = z the arrow ζ i exists. Set c := c i and d := d i . Any non-zero path δ : d → c lies on a long path p 2 δp 1 ∼ p 2 δζ i ρ i−1 . . . ρ 1 α ∼ p 2 ρ i . . . ρ 1 α whence δζ i ∼ ρ i which is a contradiction. Thus A(d, c) = 0. For i = m i.e. c i = b one
has also d i = b and one shows similarly A(c, d) = 0. We look as before at the full subcategory A ′ supported by a, b, c, d, z and again we end up with the two cases shown in the figures 2 and 3. Argueing as above we always get a contradiction. Finally for i = m we consider the full subquiver supported by a, b, d, z and we are in the situation of figure 1 and get the same contradiction. Thus we obtain ζ m . . . ζ 1 α = ρ m . . . ρ 1 α and this path can only be prolongated by β.
Finally we consider a long path p from a to z which does not start with α.
Suppose we have a proper decomposition p = p 2 p 1 such that the end-point d of p 1 lies on ρ. For d = b we have dim A(d, z) = 1 and we find a subpath ρ ′ of ρ such that p 2 ∼ βρ ′ . Then βρ ′ p 1 is a long path ending with β but not starting with α. This contradicts the dual of part i). For d = b we have p 1 ∼ ρα because α is an arrow and then p 2 ρα is a long path starting with α and therefore ending with β. Thus p ends with β and we obtain the contradiction that p starts with α again by the dual of part i).
Lemma 4. Let α ′ : a → b ′ be an arrow such that the interior points of all long paths starting with α ′ are thin. Then there is only one such path.
Proof. Let p = ζ n . . . ζ 2 ζ 1 and q = ξ m . . . ξ 2 ξ 1 be two different long paths with
ζ 1 = ξ 1 = α ′ . Because of dim A(b ′ , z) = 1 we have p ∼ q.
By symmetry we can assume that n ≥ m. Then ξ j = ζ j for all 1 ≤ j ≤ m implies n = m and p = q because z is a sink and so ζ m+1 cannot exist. So let j > 1 be the smallest index with ζ j : c → d = ξ j : c → e. Then we have A(d, e) = 0. Namely a non-zero path δ : d → e would occur in a long path p 2 δζ j ζ j−1 . . . ζ 1 and so by cancellation δζ j ∼ ξ j which is a contradiction. Symmetrically we have A(e, d) = 0. Now we consider the full subcategory A ′ supported by a, z, d, e and its quiver Q ′ . It has arrows a → d, a → e, d → z and e → z. Because of dim A(a, z) = 2 and p ∼ q we have also an arrow a → z. Then the separated quiver of A ′ contains a quiver of typeD 5 .
Suspicious algebras of type 2
Proposition 4. The suspicious algebras of type 2 are exactly the algebras listed in the first four families.
Proof. Of course all the algebras in the four families are not distributive. The algebras in the first two families are tame concealed, whence in particular minimal representation-infinite by proposition 6 in section 6.2.
For an algebra in one of the families 3 or 4 one has to look at quotients by a one-dimensional ideal generated by x 0 βα + x 1 βρα. In fact, by changing the presentation slightly only the values 1 or 0 have to be considered for x 0 and x 1 . One obtains two non-isomorphic quotients for C(p) and three for D(p, q) which are all representation-finite by the finiteness-criterion ( see [12, theorem 27] or section 8 ).
Reversely, let A be a suspicious algebra of type 2. Observe that all points occur in a long path. Assume first that there is a thick point b. If there is only one arrow α : a → b starting at a we obtain an algebra of the family C(p) by lemma 3. Thus let α ′ : a → b ′ be a second arrow where b ′ = z by lemma 1. For the uniquely determined long path p starting wih α ′ we have p = x 0 βα + x 1 βρα. For x 0 = 0 we change the presentation to obtain an algebra of type D(p, q). For x 0 = 0 we look at the full subcategory A ′ supported by a, b, b ′ , z and we get an algebra defined by the relation p = βρα. Dividing out by p one obtains a zerorelation algebra containing a quiver-algebra of typeẼ 6 in its universal cover.
Thus A is not minimal representation-infinite. Finally there cannot be a third arrow α ′′ : a → b ′′ . Namely the full subcategory A ′ supported by a, b, b ′ , b ′′ is then already representation-infinite becauseà ′ contains a quiver of typeD 7 .
So we can assume that there is no thick point. By lemma 4 each arrow α i : a → b i starting at a can be prolongated to a uniquely determined long path p i and these paths have no interior points in common by the dual of lemma 4 and because all points are thin. If only two arrows start at a then p 1 and p 2 are a basis of S and a is isomorphic to an A(p, q). So assume there are three arrows starting at a. If p 1 ∼ p 2 we look at the full subcategory A ′ supported by a, b 1 , b 2 , z. Dividing out by p 1 we obtain a zero-relation-algebra containing a quiver of typeD 5 in its universal cover. By symmetry we can assume that for i = j the vectors p i and p j are linearly independent. Because of dim S = 2 we have a relation x 1 p 1 + x 2 p 2 + x 3 p 3 = 0 with x i = 0 for all i. Changing the presentation slightly we find that A belongs to the family B(p, q). The conditions on p and q follow from the fact that the full subcategory supported by all points except a is representation-finite.
The case where more than three arrows start at a is excluded because A is minimal representation-infinite.
Algebras of type 3 4.1 Each point divides exactly one of the morphisms r, s or t
Now we study suspicious algebras of type 3. We fix morphisms s, r, t generating rad A(a, a), rad A(z, z) and A(a, z) as bimodules. A point x divides a non-zero morphism f if x is an interior pont of a path p with p ∼ f . We consider full subcategories A ′ containing a, z and a third point b that varies. The quiver of A ′ is then denoted by Q ′ and the possible arrows by α 1 : a → b,α 2 : b → a, γ 1 : b → z, γ 2 : z → b, σ : a → a, µ : b → b, ρ : z → z and β : a → z. The situation is illustrated in figure 4.
figure 4 q q q ✲ ✒ ✠ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ | ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ■ ✒✑ ✓✏ ✒✑ ✓✏ ■ ✠ ✲ ✒✑ ✓✏ α 1 α 2 γ 2 γ 1 β σ τ ρ a b z
The first lemma restricts the shapes of the possible quivers Q ′ .
Lemma 5. Using the above notation we have:
i) Q ′ cannot contain the quiver Q 1 consisting of the four arrows α 1 , α 2 , γ 1 , γ 2 .
ii) Q ′ cannot contain the quiver Q 2 consisting of α 1 ,α 2 ,γ 1 and ρ.
iii) Q ′ cannot contain Q 3 given by the arrows σ,β ρ,α 1 and γ 1 .
Proof. Suppose Q ′ contains Q 1 . The arrow α 2 is part of a long path qα 2 p and so α 2 p generates the one-dimensional radical of A(a, a) and there is no loop at a. Dually there is no loop at z. There is also no loop at b because otherwise the separated quiver to A ′ contains a quiver of typeD 5 . Thus either Q ′ coincides with Q 1 or one has to add β. In both cases we have s ∼ α 2 α 1 and r ∼ γ 2 γ 1 .
First we treat the case without β. From 0 = ts we see that α 1 α 2 α 1 is not a zero-path so that A(a, b) is cyclic over A(b, b). Dually we get that A(b, z) is cyclic over A(b, b) which is a uniserial ring whose radical is generated by X := α 1 α 2 or by Y = γ 2 γ 1 . Up to duality we can assume that X is a generator. From Xα 1 ∼ α 1 s we get 0 = α 1 s 2 = X 2 α 1 and it follows that A(a, z) is generated as a vector space by γ 1 α 1 and by γ 1 Xα 1 in contradiction to dim A(a, z) = 3.
Thus let β belong to Q ′ . Then γ 1 α 1 lies in S and so it is annihilated on both sides by all elements in N . Furthermore 0 = rt shows that γ 1 γ 2 β is a non-zero path. But γ 2 β belongs to rad A(b, a) and so it is α 1 f or gα 1 for some elements f, g in N . In the first case the contradiction 0 = rt is immediate and also for g ∼ α 1 α 2 . The only remaining case is g ∼ (γ 2 γ 1 ) i α 1 for some i ≥ 0 and again 0 = rt follows.
Next assume that Q ′ contains Q 2 but not Q 1 . Then γ 2 and σ do not exist. If β belongs to Q ′ then τ does not as the separated quiver shows. Then γ 1 α 1 lies in S and βα 2 in the radical of A(b, z) and so it is proportional to ργ 1 or to γ 1 (α 1 α 2 ) i for some i ≥ 1. In both cases the contradiction 0 = ts ∼ βα 2 α 1 ∼ 0 follows. So we have Q ′ = Q 2 or one has to add τ .
We treat the case witout τ first. We have t ∼ γ 1 (α 1 α 2 ) i α 1 for some i ≥ 0. From ts = 0 = s 2 we get i = 0 and ts ∼ γ 1 α 1 α 2 α 1 . On the other hand rt ∼ ργ 1 α 1 ∼ γ 1 (α 1 α 2 ) i α 1 for some i ≥ 1 implies rt ∼ ts or rt = 0. Both cases are a contradiction.
So assume finally that τ exists in Q ′ . Let n be the largest natural number such that τ n is not a zero-path. Then also γ 1 (τ ) n α 1 is not a zero path. For n ≥ 2 the space A(a, b) is only transit and A(b, z) only cotransit. We look at the full subcategory A ′′ supported by b and z. If n ≥ 3 we divide it by the relations γ 1 τ 2 and by ργ 1 . The remaining zero-relation algebra has an obvious Galois-covering containing a quiver of typeẼ 6 . Thus we get τ 3 = 0. From ts = 0 we obtain that α 1 α 2 is a non-zero path and therefore proportional to τ 2 . It follows the contradiction ts ∼ rt.
The last part is trivially excluded because the separated quiver contains a quiver of typeD 5 . Lemma 6. Any point b different from a and z divides exactly one of the morphisms r, s, t.
Proof. We always look at the full subcategory A ′ supported by a, b, z and its quiver Q ′ and we show first that b divides at most one of the morphisms.
If b divides s there is a non-zero path p in Q ′ from a to a with b as an interior point and z is not an interior point because of A(z, a) = 0. Thus α 1 and α 2 belong to Q ′ . Dually, if b divides r, γ 1 and γ 2 belong to Q ′ . Thus part i) of the last lemma implies that b cannot divide s and r.
If b divides s and t but not r then α 1 , α 2 and ρ exist, but not γ 2 which would produce a non-zero-path from z to z saying that b divides r. If γ 1 does not exist then β does and t ∼ β is true. Then any path p from a to z with interior point b satisfies p ∈ S. Thus γ 1 exists and the contradiction Q 2 ⊆ Q ′ follows. The case that b divides r and t is excluded by duality.
Finally, if b divides none of s, r, t then Q 3 is contained in Q ′ . Namely, α 2 does not exist because b does not divide s and so σ exists. Dually γ 2 does not exist but ρ does. Since t does not factor through b the arrow β exists. Finally, there is a non-zero path from a to z with b as an interior point because the identity at b is a non-zero path and this enforces the two arrows α 1 and γ 1 . iv) Any path q in Q with q = s coincides with p.
The uniqueness of the two cycles and the bridge
Proof. The arrows α 1 ,α 2 ,ρ and β exist because b divides s, but neither r nor t. An additional arrow γ 1 is excluded as shown in the proof of part ii) of lemma 5 and an arrow γ 2 implies that b divides r. Suppose now that Q ′ contains also τ and let n be the largest natural number such that τ n is not a zero-path. Then also βα 2 τ n α 1 is not a zero path. We look at the full subcategory A ′′ supported by b and z and its quiver Q ′′ that contains the two loops τ and ρ and one arrow ǫ from b to z induced from βα 2 . Because ρβ lies in S we have ρǫ = 0. For n ≥ 2 we find in the Galois-covering of A ′′ a quiver of typeẼ 6 . Now A ′′ is already a full subcategory of the proper quotient of A ′ by βα 2 τ n α 1 and therefore A ′′ is representation-finite. Thus τ 2 is a zero-path and so is α 1 α 2 . We can arrange by a slight change of the presentation that in addition α 2 α 1 is a zero path. Then A ′ / βα 2 τ is a special biserial algebra containing the cyclic word τ −1 α 1 (ρβ) −1 α 2 β. Thus A ′ is not minimal representation-infinite. This contradiction shows that Q ′ has only four arrows.
If b is not thin α 1 α 2 is not a zero-path and so it can be prolongated to a non-zero path βα 2 α 1 α 2 α 1 contradicting s 2 = 0. If δ 1 δ m is not a zero-path we can prolongate it to a non zero-path δ 1 δ m p 1 and so the end-point of δ 1 is not thin.
To see that dimA(a, b), dimA(b, a) and dim A(b, z) are 1 we can assume that b is an interior point of p. For dim A(a, b) ≥ 2 the space A(a, b) is cyclic over A(a, a) and so f s = 0 where f ∼ q for the subpath q of p leading from a to b and where s ∼ p. This contradicts the fact that α 1 α m is a zero-path. The proof for A(b, a) is similar. Finally A(b, z) is generated by βα 2 .
Part iv) is clear for m = 1. So suppose q = δ ′ n δ ′ n−1 . . . δ ′ 1 is another path with q ∼ s. Then we have also n > 1 and because a is not an interior point of p or q there is a smallest index i with δ i = δ ′ i . Let b = b ′ be the ending points of δ i and δ ′ i and let c be the starting point. We decompose p and q as p = p 2 δ i p 1 and q
= q 2 δ ′ i p 1 . For b = a we obtain δ i ∼ q 2 δ ′ i which is impossible. The same is true for b ′ = a. We claim that A(b, b ′ ) = 0. If not then A(b, b ′ ) = k because A(b, b ′ ) is uniserial. Thus there is a path q ′ : b → b ′ such that q ′ is a basis of A(b, b ′ )
and this path can be prolongated to a path q ′ p ′′ from a to b ′ which is is not a zero-path. Since A(a, b ′ ) and A(a, b) have dimension we obtain that
q ′ p ′′ ∼ q ′ δ i p 1 ∼ δ ′ i p 1 .
Thus q ′ δ i and δ ′ i are two non-zero path between c and b ′ where A(c, b ′ ) has dimension one for c = a by part iii) and also for c dividing s because A(c, b) is uniserial. This implies that δ ′ i is not an arrow, a contradiction. Of course also β : a → z and ρ : z → z belong to Q ′ . Now we divide A ′ by the non-trivial ideal I a to obtain a zero-relation algebra where the path ρβ is not killed. In the universal cover we find a quiver of typeD 7 . ii) Any path q with q ∼ t coincides with p.
Proof. As usual we look at the full subcategory A ′ supported by a, b, z and its quiver Q ′ and we use the notations from figure 4. There is no arrow α 2 because b does not divide s and dually there is no arrow γ 2 . Furthermore β does not belong to Q ′ since b divides t. Thus σ, ρ, α 1 and γ 1 exist. Suppose that there is a loop τ in addition. If τ 3 is not a zero-path also γ 1 τ 3 is none and we have ργ 1 = γ 1 (x 2 τ 2 + x 3 τ 3 + . . .). In the fullsubcategory A ′′ supported by b, z and its quiver Q ′′ we introduce the relations γ 1 τ 2 and ργ 1 . Then τ 3 is not annihilated and we find in the universal cover of the resulting zero-relation algebra a quiver of typeẼ 6 . Thus τ 3 is a zero-path. We obtain in A ′ the contradiction γ 1 α 1 σ ∼ γ 1 τ i α 1 ∼ ργ 1 α 1 where i = 2 if τ 2 is not a zero path and i = 1 in the other case. The proof of part i) is complete.
Let q = β ′ n β ′ n−1 . . . β ′ 1 be another path with q ∼ t. Let i be the smallest index with β i = β ′ i and write p = p 2 β i p 1 and q = q 2 β ′ i p 1 . Let c be the starting point of β i and β ′ i and let b, b ′ be the two different end points. We consider the full subcategory A ′ supported by a, b, b ′ , z and its quiver Q ′ and we claim that it contains arrows ζ : a → b and ζ ′ : a → b ′ . This is clear if p 1 has length 0. Thus assume c = a. If the morphism β i p 1 does not induce an arrow in Q ′ then there is a path
ξ : b → b ′ in Q with ξβ ′ i p 1 ∼ β i p 1 because A(b, b ′ )
has dimension one at most as a uniserial bimodule over k. This is also true for A(c, b ′ ) and so we get the contradiction β i ∼ ξβ ′ i . Thus Q ′ contains the arrows σ, ζ and ζ ′ . We claim that A(b, b ′ ) = 0. If not there is an arrow ξ : b → b ′ in Q ′ that can be prolongated to a path q with non-zero q ∈ S. Then ξζ is not a zero-path and we find ξζ ∼ ζ ′ σ because ζ ′ is an arrow. From q 2 ξ = xρp 2 + yp 2 and q 2 ξζσ = 0 we obtain the contradiction βσ ∼ q 2 ζ ′ σ ∼ q 2 ξζ ∼ ρp 2 ζ ∼ ρβ. Symmetrically we get A(b ′ , b) = 0.
Finally the full subcategoryA ′′ supported by a, b, b ′ is a zero-relation algebra having a quiver of typeD 5 in its universal cover.
Suspicious algebras of type 3
Proposition 5. The suspicious algebras of type 3 are exactly the algebras E(p, q, r).
Proof. For an algebra A in the fifth family define p = β r . . . β 1 α q . . . α 1 and q = γ p . . . γ 1 β r . . . β 1 . Then A is a special biserial algebra with q −1 p as the only primitive cyclic word up to inversion and cyclic permutation. Any proper quotient is still special biserial but without any cyclic word and so it is representation-finite.
Reversely, let A be an algebra of type 3 with quiver Q and let α = α q . . . α 1 , β = β r . . . β 1 , γ = γ p . . . γ 1 be the three uniquely determined paths giving s, t, r. Since all interior points of α are thin by lemma 7 the interior points are pairwise different. The same holds for β by lemma 8 and for γ by the dual of lemma 7. Furthermore the union of the interior points is disjoint by lemma 6 and Q 0 consists in these interior points and a and z. We show that any arrow in Q occurs already in one of the three paths. So let φ : x → y be an arrow. First take x = a. For y = a resp. y = z we get q = 1 and φ = α 1 resp. r = 1 and φ = β 1 . If y is thin, there is always a non zero-path from a to y and we always have dim A(a, y) = 1. Thus φ is α 1 or β 1 . Next we look at x = z and a thin point y. Then we have dim A(z, y) = 0 if y divides s or t and dim A(z, y) = 1 if y divides r. Thus only φ = γ 1 is possible. Thus there is no additional arrow starting in a thick point. By duality we can assume now that x and y are thin.
We consider always a prolongation qφp of φ such that qφp ∈ S. First assume that x divides s and also y. Let x be the endpoint of α i and y the endpoint of α j . For i > j we can assume that p = α i α i−1 . . . α 1 because of dim A(a, x) = 1 and then the non-zero path φp runs twice through y contradicting the fact that y is thin. For i < j we have φ ∼ α i . . . α j+1 whence there is an arrow only for i = j + 1. Next suppose y divides t. Then there is a non-zero path of length ≥ 2 from x to y in Q ′ and because of dim A(x, y) = 1 there can be no arrow φ. Finally assume that y divides r. Then we get φp ∼ r ′ t and qφ ∼ ts ′ for some non-zero morphisms r ′ ∈ A(z, y) and s ′ ∈ A(a, x). The contradiction ts ∼ rt follows.
Next assume that x is the ending point of β i for some i. If y divides s or if it is the ending point of β j with j < i then there is a long path qφp running twice through the thin point x which is a contradiction. For any other y we have a non-zero-path from x to y and so there is an arrow only if β i+1 ends in y. Up to duality the only remaining case is when x edivides r and y divides s. This would give a long path running first through z and then through a which is excluded by A(z, a) = 0.
We have determined the quiver of A. We know already that the two zerorelations α 1 α m and γ 1 γ p hold in A. If γ 1 β r . . . β 1 α m is not a zero-path it can be prolongated to a long path contradicting rts = 0.
5 Two consequences of theorem 1
Accessible modules for non-distributive algebras
Ringel defined in [29] the notion of an accessible module of finite length: To start with all modules of length 1 are accessible and a module of length n ≥ 2 is accessible if it is indecomposable and if it has an accessible submodule or quotient of length n − 1. It is known since a long time [9,10] that any indecomposable is accessible provided that the field is algebraically closed and A is representation-finite or tame concealed. Ringel has shown in [29] the next result which follows also from theorem 1.. Theorem 3. Let A be a finite-dimensional algebra over an algebraically closed field. If there is an indecomposable of length n there is also an accessible of length n.
Proof. We can assume that A is minimal representation-infinite and basic and we have to show that accessible modules exist in all dimensions. The case when A is distributive is the difficult one and this case is treated in [11] without mentioning the new terminus 'accessible'.
For a non-distributive algebra Ringel has given in [29] a nice direct argument. Of course one can now alternatively inspect the list in theorem 1. It suffices to look at the 'unglued' algebras. The first two families consist of tame concealed algebras and then all indecomposables are accessible. The same is true by [30] for the last family containing only special biserial algebras. In the remaining two cases there is an obvious Galois-cover with fundamental group Z that contains a tame-concealed algebra B of typeD n as a convex subcategory. The push-downs of the indecomposable B-modules provide accessibles in each dimension.
The proof of theorem 2
Fix a natural number d. We want to find a finite list L of Z-algebras A such that for any algebraically closed field k the extended algebras A ⊗ k are a representative system of isomorphism classes of basic minimal representationinfinite algebras of dimension d. We consider first the non-distributive algebras.
The relations imposed on any of the quivers Q occurring in theorem 1 and also their glued versions make sense already in ZQ and the quotient algebra A is always a free Z-module. Define L 1 as the set of Z-algebras obtained that way which are free of rank d. By scalar extension one obtains for all fields a list of non-distributive minimal representation-infinite algebras.
To treat the distributive algebras we need some concepts and highly nontrivial results all described in [12]. We choose any algebraically closed field k. There is only a finite list L ′ 2 of equivalence classes of ray categories P such that the linearization kP is minimal representation-infinite of dimension d. By the finiteness-criterion this property is independent of the chosen field. Furthermore any basic distributive minimal representation-infinite algebra is isomorphic to the linearization of a ray category. Finally kP and kP ′ are isomorphic if and only if P and P ′ are equivalent categories. We take L 2 as the finite set of algebras ZP with P in L ′ 2 and define L as the union of L 1 and L 2 .
6 Tame concealed and critical simply connected algebras 6
.1 Three notions of minimality
For an algebra A of infinite representation type we can ask whether all proper quotients A/I are representation-finite resp. only quotients A/ e x for an arbitrary point x in Q resp. only quotients A/ e x for a source or a sink x in Q. In this way we obtain the set M of ( isomorphism classes of ) minimal representation-infinite algebras resp. the set A of representation-infinite algebras such that almost all indecomposables ( up to isomorphism ) are sincere resp. the set B of representation-infinite algebras such that almost all indecomposables are extremal or -equivalently -such that all proper convex subcategories are representation-finite. We call algebras in B critical. Sometimes e.g. in [23,28,33] algebras in A are already called minimal representation-infinite.
The inclusions M ⊆ A ⊆ B are proper but restricted to algebras having a simply connected component all three sets coincide. First we note the following: Lemma 9. Suppose a critical algebra A has a preprojective component Z. Then Z contains all indecomposable projectives but no injective. Almost all indecomposables in Z are sincere and so A is tilted from a path-algebra kK. If K is Euclidean then A is tame concealed.
Proof. If Z contains an indecomposable injective De x A it contains such a module for a sink x. Only the finitely many predecessors in Z of this module can be extremal and so A is not critical. Thus Z contains no injective and so it is infinite with almost all modules extremal whence sincere by [6]. Thus Z contains a section K and A is tilted from kK by a tilting module T by [28]. For tame K only a preprojective ( or preinjective ) tilting module leads to a critical algebra A as explained in section 4.2 (8) in Ringels classical book [28].
Tame concealed algebras are minimal representationinfinite
I observed this already in [8] but it went unnoticed. Here is the proof.
Proposition 6. Any tame concealed algebra A is minimal representation-infinite.
Proof. Let I be a non-zero ideal in A and take a non-zero element a in the intersection of I and the radical. Suppose a lies in e y Ae x . The multiplication with a from the right induces a minimal projective resolution Here we plug in a sincere indecomposable U which we identify with the corresponding representation. Then we obtain an exact sequence
0 → Hom(C, U ) → U (x) → U (y) → DHom(U, DT rC) → 0,
where the middle arrow is just the multiplication map U (a) : U (x) → U (y).
Thus we see that U (a) = 0 impies Hom(C, U ) ≃ U (x) = 0 and 0 = U (y) ≃ Hom(U, DT rC) and so U and C are regular in the same tube T . Let h be the number of regular simples in T . Their dimension vectors add up to a vector where all components are different from zero. Since A is directed we have dim C(x) = 1 and so the regular length of C is less than 2h. This implies dim Hom(C, U ) ≤ 2 in the uniserial category T . Thus we have dim U (x) ≤ 2 which shows that the regular length of U is less than 3h.
We have shown that a annihilates only finitely many sincere indecomposables ( up to isomorphism ). But by the easy implication of theorem 2 in [23] almost all indecomposables are sincere.
The relations between tame concealed and critical simply connected algebras
In the criterion of [7] to decide whether an algebra A is representation-finite I consider a directed categoryà as a Galois-covering of A such that each finite subcategory C has a finite convex hullĈ with a simply connected component in its Auslander-Reiten quiver. Such a component is given by a graded tree (T, g) as in [4, section 6]. Hereà is locally representation-finite iff A is representationfinite by [21]. If there is a finite subcategory ofà of infinite representation type one gets just by throwing away successively extremal points a critical convex subcategorŷ C which has then exactly one simply connected component in its Auslander-Reiten quiver. Typical examples for such a critical category are the algebras D(p, q, r) of figure 5. Here the left hand side is a commutative diagram that disappears for p = 0 and the analogous statement holds for r = 0 on the other side.
c 2 c p b z 1 z 2 z q d f 1 f 2 f r e ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ✠ ✠ ❅ ❅ | ❅ ❅ | ❄ ❄ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
For the critical subcategories I proved in [8]:
Theorem 4. A critical subcategoryĈ contained inà has a simply connected component given by a Euclidean tree T . For T =D n only the algebrasD(p, q, r) and B(n − 3, 1) are possible.
Thus the critical algebras are all tame concealed by lemma 9. Happel and Vossieck have determined in [23] the tame concealed algebras by looking at the possible preprojective tilting modules up to T rD-translation and by calculating their endomorphism algebras. The 'frames' of these are depicted in the HV-list: Theorem 5. An algebra A is tame concealed iff it occurs in the HV-list.
They proved also in [23]: Theorem 6. Let A be a basic connected algebra of finite dimension. Then A is tame concealed or a generalized Kronecker algebra iff A has an infinite preprojective component and A/AeA is representation-finite for each idempotent e = 0.
Thus the tame concealed algebras are exactly the basic minimal representationinfinite algebras with a preprojective component.
In the literature there are many false or misleading statements concerning the last three theorems and the finiteness criterion. Here are some comments on this.
• Theorem 5 was presented by Happel at a conference in september 1982 at Luminy. The caseà n is easy,D n affords some combinatorial considerations and the remaining three cases a computer and some drawing. Theorem 6 was not mentioned in Happels talk.
• At the same conference theorem 4 was presented in my talk as a part of a general finiteness criterion. For the exceptional casesẼ n I had only the gradings of the representation-infinite algebras as unreadable computerlists. So my condition was that all convex subcategories with at most 9 points have to be representaton-finite. Up to the action of the Galoisgroup only finitely many convex subcategories have to be considered.
My lists contained also some algebras which are not critical. In [8] I removed these and then my lists are in accordance with the HV-list. The obvious lemma 9 was overlooked by me at that time and it is still not mentioned in [15] that contains several incorrect statements.
• Theorem 6 is published in the article [23] submitted in november 1982 .
To prove the difficult direction a result of Ovsienko on quadratic forms [27] and tilting theory are used.
The proof of theorem 6 is much more elegant and shorter than my barehanded proof of theorem 4 using only the inductive method from [4]. Both proofs are completely different contradicting [33, chapter XIV].
However, for the algebras occurring in the finiteness criterion my result is better than theorem 6 because convex subcategories are easy to detect. Also the caseD n is much easier to analyze with my inductive method and this could have been used in [33,chapter XIV] where on 14 pages only the cases n ≤ 6 are treated.
• In July 1982 at a meeting in Bielefeld I had mentioned in a private conversation with Ringel theorem 4 that I just found before. The result was not known to him but he conjectured that there is a connection to Ovsienkos result that was unknown to me.
• What Ringel writes in [31] about the importance of the HV-list is not true. Neither the proof of BT2 nor the article on multiplicative bases depend on the HV-list.
• It is aggravating that the finiteness criterion [7] and its further developments in [12] are not mentioned in the recent literature. I will say more about this in a forthcoming survey on representation-finite selfinjective algebras, coverings and so on.
Towards the classification
A line L of length e inP is a convex subcategory living on a linear subquiver x 1 → x 2 . . . x e−1 ← x e without any relation. The line is called critical if π(x 1 ) = π(x e ) are both sources or both sinks in L and if π(x 2 ) = π(x e−1 ) holds.
Proposition 8.
We keep all the assumptions and notations and we assume that d := dim kP < ∞. Then we have:
i) Any line L inP of length 2d + 1 contains a critical line as a subline.
ii) For any critical line L of length e the push-down N of the indecomposablẽ P -module M with support L has infinitely many pairwise non-isomorphic quotients of dimension e − 1.
Proof. Up to duality we can assume that x 1 is a sink in L which we write down thereby marking all sinks s 1 , s 2 , . . . s r . We obtain
x 1 = s 1 ← . . . → s 2 → . . . → s i ← . . . → s r ← . . . x 2d+1 ,
where s r = x 2d+1 is possible. Thus we get 2d + 1 ≤ r i=1 dim DkP ( , s i ) and so three sinks are mapped onto the same point under π. We can assume that these points are s 1 , s i and s r . There is a critical subline with two of these as extremal points. The first assertion is proven and the second is shown in lemma 3.2 of [7]. Now we can show that any minimal representation-infinite algebra A is isomorphic to a glued algebra C R where C is a critical line or a critical algebra and R an equivalence relation. For the case of triangular algebras and for the notion of minimality defining the algebras in A a slightgly weaker statement was obtained in [16]. .
We distinguish three cases depending on the structure of the universal cover P which is not locally representation-finite by [21].
Case I: Each finite full subcategory is representation-finite. Then there are indecomposableÃ-modules with arbitrarily large support B which is always a convex full subcategory. Therefore B belongs to the list LSS and we find a critical line C with part i) of proposition 8. Part ii) and proposition 7 show that A is glued from C by an appropriate R.
Case II:Ã contains a critical algebra C of typeD n , but none of typeẼ n . Then we take for M ′ a progenerator of C. The powers of M ′ have infinitely many pairwise non-isomorphic indecomposable modules and the same holds for the extension M of M ′ by zero and its push-down N by basic properties of the push-down functor. Thus A is a glued algebra by proposition 7. The same argument applies in the last case.
Case III: There is a critical convex full subcategory C of typeẼ n .
Unfortunately for a given C there are many equivalence relations R such that C R is not minimal representation-finite even if we restrict to those R such that the induced morphism q : Q C → Q C /R is injective on arrows with a common source or sink as p : Q C → Q A is.
For example let C be the quiver-algebra of the quiver shown in figure 6. There are 53 isomorphism classes of proper glueings but only 9 of them are minimal representation-infinite. The smallest of these algebras has two points x, y and two arrows α : x → y, β : x → x subject to the relations β 4 = αβ 3 = 0. This algebra is minimal represenation-infinite but wild since its universal cover contains a hyper-critical quiver algebra. Proposition 9. Any basic distributive minimal representation-infinite algebra A is defined by zero-relations and by at most three commutativity relations. A is a zero-relation algebra if it is obtained by glueing a zero-relation algebra.
At the end we discuss shortly the different cases. Of course we can always choose a critical line or a critical algebra C of minimal cardinality.
Case I is solved completely by Ringel in [30]. Only special biserial algebras occur and so all glueings are tame and Ringel also studies the module categories.
Case II is more complicated and there is in general no chance to describe the module categories as the above example shows. Nevertheless this case seems to allow a classification into finitely many families. I started this project by finding necessary conditions on R ensuring that the algebra is minimal representationinfinite, but I finally flinched from producing another list.
Case III means to classify the minimal representation-infinite ray categories with at most 9 points and this is a finite problem. But already the case of 3 points treated by Fischbacher in his diploma thesis published in [19] leads to very many algebras and this shows that a general classification makes no sense.
Lemma 2 .
2We keep the above assumptions and notations. i) There are arrows α : a → b, β : b → z in Q and an oriented cycle ρ := ρ m ρ m−1 . . . ρ 1 in b of length m ≥ 1 such that βα and βρα give a basis of S.
Lemma 3 .
3Let b be the thick point with an oriented cycle ρ := ρ m ρ m−1 . . . ρ 1 in b of length m ≥ 1 such that ρ = r. Then the following holds: i) Any long path p starting with α coincides with βα or βρα.
Lemma 7 .
7Suppose b divides s. Choose a path p = δ m δ m−1 . . . δ 1 with p = s. We consider the full subcategory A ′ supported by a, b, z and its quiver Q ′ . Then the following holds: i) Q ′ contains only the arrows α 1 , α 2 , β, ρ defined in figure 4. ii) b is a thin point and δ 1 δ m = 0.iii) A(a, b), A(b, a) and A(b, z) all have dimension one.
Thus we have A(b, b ′ ) = 0 and symmetrically A(b ′ , b) = 0. Now consider the full subcategory A ′ supported by a, b, b ′ , z and its quiver Q ′ . Because A(b, a) = 0 but A(b, b ′ ) = 0 and A(z, b) = 0 there is an arrow b → a. Symmetrically there is an arrow b ′ → a. Similarly we have A(a, b) = 0 but A(b ′ , b) = A(z, b) = 0 leading to an arrow a → b. Again by symmetry we also have an arrow a → b ′ .
Lemma 8 .
8Suppose b divides t. Choose a path p = β m β m−1 . . . β 1 with p ∼ t. Then we have: i) b is a thin point and A(b, a) = A(z, b) = 0.
Ae y →
yAe x → C → 0 with an indecomposable module C. This induces by general facts used in the existence proof for almost split sequences ( see [20, section 1.3] [34] ) after dualizing an exact sequence of functors 0 → Hom(C, ) → Hom(Ae x , ) → Hom(Ae y , ) → DHom( , DT rC) → 0.
figure 5
at the HV-list shows:
] ) and their properties as surveyed in. freely use ray categories. freely use ray categories ( [2] ) and their properties as surveyed in
P is interval-finite and any finite subset ofP lies in a finite full convex subcategory C ofà = kP which has a simply connected preprojective component in its module category. Here C,à and A are standard and so they admit a presentation induced by zero-paths and contours. Now we take an embedding i : C →à of a finite full convex subcatgory C and the composition p : C → A with kπ :à → A. We denote by i and p also the induced morphisms at the level of the quivers Q C , Qà and Q A and their path-categories. We obtain an equivalence relation R p on the point set (Q C ) 0 of Q C having the non-empty fibres of p as the equivalence classes. More general for any equivalence relation R on (Q C ) 0 we have the quotient quiver Q C /R with the equivalence classes as points and the natural surjective quiver-morphism q : Q C → Q C /R extending to a functor beteen the pathcategories again denoted by q. Thus let A be a basic distributive minimal representation-infinite algebra. By the important theorem 2 of [11] A is isomorphic to the linearization kP of its associated ray category P , the universal cover π :P → P has a free fundamental group. We define the glued algebra C R as the quotient of k(Q C /R) by the ideal generated by the path qv where v is a zero-path in C, by the differences qu − qw where (u, wThus let A be a basic distributive minimal representation-infinite algebra. By the important theorem 2 of [11] A is isomorphic to the linearization kP of its associated ray category P , the universal cover π :P → P has a free fundamental group,P is interval-finite and any finite subset ofP lies in a finite full convex subcategory C ofà = kP which has a simply connected preprojective component in its module category. Here C,à and A are standard and so they admit a presentation induced by zero-paths and contours. Now we take an embedding i : C →à of a finite full convex subcatgory C and the composition p : C → A with kπ :à → A. We denote by i and p also the induced morphisms at the level of the quivers Q C , Qà and Q A and their path-categories. We obtain an equivalence relation R p on the point set (Q C ) 0 of Q C having the non-empty fibres of p as the equivalence classes. More general for any equivalence relation R on (Q C ) 0 we have the quotient quiver Q C /R with the equivalence classes as points and the natural surjective quiver-morphism q : Q C → Q C /R extending to a functor beteen the path- categories again denoted by q. We define the glued algebra C R as the quotient of k(Q C /R) by the ideal generated by the path qv where v is a zero-path in C, by the differences qu − qw where (u, w)
N is faithful because A is minimal representation-infinite and the powers of N have infinitely many pairwise non-isomorphic indecomposable quotients. Let v be a path in Q A from x to y and let v ′ be a lifting in QÃ from x ′ to y ′. Proof. N is faithful because A is minimal representation-infinite and the powers of N have infinitely many pairwise non-isomorphic indecomposable quotients. Let v be a path in Q A from x to y and let v ′ be a lifting in QÃ from x ′ to y ′ .
Since C is convex v ′ is a path in C if and only if x ′ and y ′ both belong to C. By the definition of the push-down N (v)N (x) → N (y) acts 'diagonally' through the various liftings M (v ′ ) : M (x ′ ) → M (y ′ ). Since C is convex v ′ is a path in C if and only if x ′ and y ′ both belong to C. By the definition of the push-down N (v)N (x) → N (y) acts 'diagonally' through the various liftings M (v ′ ) : M (x ′ ) → M (y ′ ).
In particular paths of length 0 or 1 have a lifting in C and so the morphism from the quiver of C to the quiver of A is surjective and it identifies Q A with Q C /R. If v has no lifting in C it annihilates N and so it is a zero-path. If v has a lifting v ′ in C. Then v is a zero-path in A if and only if v ′ is a zero-path inà if and only if v ′ is a zero-path in C. N Thus, v) = 0 unless there is a lifting v ′ in C. Similarly for a contour (u, w) in A the path u is not a zero-path and so N (u) = 0 implies that there is a lifting u ′ in C with starting point x ′ . Then the lifting w ′ of w starting in x ′ also lies in C and (u ′ , w ′ ) is a contour in C mapping to the given contourThus N (v) = 0 unless there is a lifting v ′ in C. In particular paths of length 0 or 1 have a lifting in C and so the morphism from the quiver of C to the quiver of A is surjective and it identifies Q A with Q C /R. If v has no lifting in C it annihilates N and so it is a zero-path. If v has a lifting v ′ in C. Then v is a zero-path in A if and only if v ′ is a zero-path inà if and only if v ′ is a zero-path in C. Similarly for a contour (u, w) in A the path u is not a zero-path and so N (u) = 0 implies that there is a lifting u ′ in C with starting point x ′ . Then the lifting w ′ of w starting in x ′ also lies in C and (u ′ , w ′ ) is a contour in C mapping to the given contour.
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On a theorem of Nazarova and Roiter. O Bretscher, G Todorov, Proc. ICRA IV. ICRA IV1177O.Bretscher, G.Todorov: On a theorem of Nazarova and Roiter, Proc. ICRA IV, Lecture Notes 1177, ( 1986), 50-54.
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Une nouvelle preuve d'un théorème de Nazarova et Roiter. (French) [A new proof of a theorem of Nazarova and Roiter. U Fischbacher, C. R. Acad. Sci. Paris Sér. I Math. 3009U.Fischbacher: Une nouvelle preuve d'un théorème de Nazarova et Roiter. (French) [A new proof of a theorem of Nazarova and Roiter] C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 9, 259-262.
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| [] |
[
"Global Atmospheric Models for Cosmic Ray Detectors",
"Global Atmospheric Models for Cosmic Ray Detectors"
] | [
"Martin Will [email protected] \nInstitut für Technologie\nInstitut für Kernphysik\nInstitut de Fisica d'Altes Energies\nKarlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain\n",
"Pierre Auger Collaboration \nInstitut für Technologie\nInstitut für Kernphysik\nInstitut de Fisica d'Altes Energies\nKarlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain\n"
] | [
"Institut für Technologie\nInstitut für Kernphysik\nInstitut de Fisica d'Altes Energies\nKarlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain",
"Institut für Technologie\nInstitut für Kernphysik\nInstitut de Fisica d'Altes Energies\nKarlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain"
] | [] | The knowledge of atmospheric parameters -such as temperature, pressure, and humidity -is very important for a proper reconstruction of air showers, especially with the fluorescence technique. The Global Data Assimilation System (GDAS) provides altitude-dependent profiles of these state variables of the atmosphere and several more. Every three hours, a new data set on 23 constant pressure level plus an additional surface values is available for the entire globe. These GDAS data are now used in the standard air shower reconstruction of the Pierre Auger Observatory. The validity of the data was verified by comparisons with monthly models that were averaged from on-site meteorological radio soundings and weather station measurements obtained at the Observatory in Malargüe. Comparisons of reconstructions using the GDAS data and the monthly models are also presented. Since GDAS is a global model, the data can potentially be used for other cosmic and gamma ray detectors. Several studies were already performed or are underway for several locations worldwide. As an example, a study performed in Colorado as part of an Atmospheric R&D for a possible future cosmic ray observatory is presented. | null | [
"https://export.arxiv.org/pdf/1402.4782v1.pdf"
] | 118,834,260 | 1402.4782 | f1d765c4d783141e0dd1d2848642fc8def35bcb4 |
Global Atmospheric Models for Cosmic Ray Detectors
19 Feb 2014 ATMOHEAD WORKSHOP, 2013
Martin Will [email protected]
Institut für Technologie
Institut für Kernphysik
Institut de Fisica d'Altes Energies
Karlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain
Pierre Auger Collaboration
Institut für Technologie
Institut für Kernphysik
Institut de Fisica d'Altes Energies
Karlsruher, Karlsruhe, Bellaterra, BarcelonaGermany, Spain
Global Atmospheric Models for Cosmic Ray Detectors
19 Feb 2014 ATMOHEAD WORKSHOP, 20132 Full author list:cosmic raysextensive air showersatmospheric monitoringatmospheric models
The knowledge of atmospheric parameters -such as temperature, pressure, and humidity -is very important for a proper reconstruction of air showers, especially with the fluorescence technique. The Global Data Assimilation System (GDAS) provides altitude-dependent profiles of these state variables of the atmosphere and several more. Every three hours, a new data set on 23 constant pressure level plus an additional surface values is available for the entire globe. These GDAS data are now used in the standard air shower reconstruction of the Pierre Auger Observatory. The validity of the data was verified by comparisons with monthly models that were averaged from on-site meteorological radio soundings and weather station measurements obtained at the Observatory in Malargüe. Comparisons of reconstructions using the GDAS data and the monthly models are also presented. Since GDAS is a global model, the data can potentially be used for other cosmic and gamma ray detectors. Several studies were already performed or are underway for several locations worldwide. As an example, a study performed in Colorado as part of an Atmospheric R&D for a possible future cosmic ray observatory is presented.
Introduction
A cosmic ray particle entering the atmosphere can initiate an extensive air shower. The secondary shower particles excite nitrogen molecules in the air which emit a characteristic, isotropic emission in the UV range as part of their deexcitation process. The light can then be observed by an optical telescope, typically consisting of a collecting mirror and a camera. To properly reconstruct the properties of such air showers, the atmospheric conditions at the site have to be known in order to correct for Rayleigh scattering effects and to estimate the fluorescence yield of the air shower [1]. Height-dependent profiles of temperature, pressure and humidity as well as weather conditions near the ground are relevant.
The Pierre Auger Observatory [2] is a cosmic ray detector located near Malargüe in the Mendoza province in Argentina. It consists of a Surface Detector (SD) array and five Fluorescence Detector (FD) buildings [3]. Between 2002 and 2010, atmospheric conditions over the Observatory were measured by intermittent meteorological radio soundings. Additionally, ground-based weather stations measure surface data continuously in order to provide the atmospheric parameters to properly reconstruct the measured air showers.
In south-east Colorado, several balloon soundings were performed as part of an atmospheric R&D project. The aim of this effort was to study possible enhancements and performance improvements for the Pierre Auger Observatory, as well as explore technological advancements for a possible future ground-based observatory. The ground station used for the soundings was a mobile and slightly advanced version of the equipment used in Argentina. The launches were performed at two sites, the Atmospheric Monitoring Telescope (AMT) and the Distant Raman Laser Facility (DRLF) [4]. The sites are about 40 km apart and are both equipped with identical weather stations.
Performing radio soundings imposes a large burden, both in terms of funds and manpower. We investigated the possibility of using data from the Global Data Assimilation System (GDAS) [5], a global atmospheric model, for the site of the Pierre Auger Observatory [6,7]. GDAS data are publicly available free of charge via READY (Real-time Environmental Applications and Display sYstem). Each data set contains all the main state variables as a function of altitude. The data gathered in Colorado were also compared to GDAS data in order to evaluate the possibility to use GDAS also in different locations and for a possible future ground-based cosmic ray detector.
Global Data Assimilation System
Data assimilation is a process in numerical weather prediction in which the development of a model incorporates the real behavior of the atmosphere as found in meteorological observations [8]. The atmospheric models describe the atmospheric state at a given time and position. The first step in performing a full data assimilation is to collect data from meteorological instruments placed all over the world. Using the current atmospheric conditions, a future state -e. g. 3 hours ahead -is forecast using numerical weather prediction. Finally, data assimilation is used to adjust the model output to the measured atmospheric state, resulting in a 3dimensional image of the atmosphere. At a given time, the value of a state variable is known from observations. For the same time, a model forecast for this variable from a previous iteration a few hours earlier exists. The data assimilation step combines observation and forecast. This analysis is the initial point for the weather prediction model to create the forecast for a later time, when this process is repeated.
The Global Data Assimilation System is an atmospheric model developed at the National Centers for Environmen- For the site of the Pierre Auger Observatory, applicable GDAS data are available starting June 2005. Because of the lateral homogeneity of the atmospheric variables across the Auger array [1], one location is sufficient to describe the atmospheric conditions. The grid point at 35 • S and 69 • W was chosen, at the north-eastern edge of the Observatory. The grid point for the Colorado R&D site is 38 • N and 102 • W, about 40 km to the east of the DRLF and 60 km to the north-east of the AMT. Since the terrain is very similar to the Argentinian high desert, horizontal uniformity can be assumed. This assumption was verified by radiosonde launches at different starting positions.
For the air shower analyses of the Pierre Auger Observatory, the main state variables of the atmosphere -temperature, pressure and relative humidity -are needed at several altitudes. They are provided directly by the GDAS surface and upper air data. From those, air density and atmospheric depth profiles are calculated.
GDAS vs. Measurements
To validate the quality of GDAS data and to verify their applicability to air shower reconstructions for the Pierre Auger Observatory, we compare the GDAS data with local soundings from weather balloons and ground-based weather stations. Comparisons using the data from the Colorado site are also shown.
GDAS vs. Weather Balloon Soundings
Local radio soundings are performed above the array of the Pierre Auger Observatory since 2002, but not on a regular basis. To provide a set of atmospheric data for every measured event, the profiles from the ascents were averaged to obtain local models, called Malargüe Monthly Models (MM) [9]. The MM have been compiled using data until the end of 2008. The uncertainties for each variable are given by the standard deviation of the differences within each month together with the absolute uncertainties of the sensors measuring the corresponding quantity.
Comparing the monthly models with ascent data until the end of 2008 shows, by construction, only small deviations [1]. In the comparison displayed in the top panels of Fig. 1, radiosonde data from 2009 and 2010 are used to illustrate the strength of the GDAS data, the data set of local soundings being independent of the MM. The error bars denote the RMS of the differences at each height. These uncertainties are larger for the MM than for GDAS data. In contrast, the GDAS data represent the local conditions in 2009 and 2010 much better and the intrinsic uncertainty is consistently small. For earlier years, the GDAS data fit the measured data equally well or better than the MM which were developed using the data from these years.
In the bottom panels of Fig. 1, the same comparison is shown between the radiosonde data measured in Colorado and the corresponding GDAS data. The differences are of the same order and the error bars are similar to the results at the Pierre Auger Observatory site.
The GDAS data fit the radiosonde data in the upper part of the atmosphere, especially in the field of view of the fluorescence detectors. Possible inconsistencies between local measurements and GDAS data close to the ground are investigated using weather station data.
GDAS vs. Ground Weather Stations
Five ground weather stations continuously monitor atmospheric values at the Pierre Auger Observatory. They are mounted between about 2 to 5 m above ground level at four FD stations, and one was set up near the center of the array at the Central Laser Facility (CLF). For the Colorado R&D site, two identical weather stations were set up, one at the DRLF laser facility and one at the AMT telescope site. To make sure that the GDAS data describe the conditions at the ground reasonably well, the values provided by the GDAS data set are compared to all available weather station data. The profiles built using GDAS data are interpolated at the height of the station.
In Fig. 2, the differences between measured weather station data and GDAS data are shown. For the stations at the CLF and the FD site Loma Amarilla (LA) all data measured in 2009 were used (top panel). For the Colorado R&D site, data taken between January 2010 and June 2011 were used for both the AMT and DRLF sites. Temperature, pressure (not shown), and vapor pressure are in similar agreement as GDAS data with local sounding data close to ground (cf. Fig. 1). The mean difference in temperature is 1.3 K for the CLF, −0.3 K for the LA, 0.5 K for the DRLF and 0.7 for the AMT station. For vapor pressure, the means are −0.2 hPa (CLF), −0.7 hPa (LA), 0.2 hPa (DRLF) and 0.4 hPa (AMT). The differences between the GDAS and the weather station data are of the same order as the difference in data of two different stations [6].
The GDAS data fit the measured data at the Observatory and the R&D site very well and are better suited for use in air shower reconstructions and simulation than monthly mean models. This reduces the need for laborious and costly radiosonde launches to sporadic checks of the consistency of the GDAS data.
Air Shower Reconstruction
To study the effects caused by using GDAS data in the air shower reconstruction of the Pierre Auger Observatory, all air shower data between June 1, 2005 and the end of 2010 were used. The change of the description of the atmosphere will mainly affect the reconstruction of the fluorescence data. Varying atmospheric conditions alter the fluorescence light production and transmission [1]. The fluorescence model we use determines the fluorescence light as a function of atmospheric conditions [10], parameterized using results from the AIRFLY experiment [11,12].
Data Reconstruction
The following analysis is based on three sets of reconstructions. The first set, FY, is the reconstruction applying an atmosphere-dependent fluorescence yield calculation without temperature-dependent collisional cross sections and humidity quenching [13]. The MM are used in the calculations. For the second set, FY mod , all atmospheric effects in the fluorescence calculation are taken into account. Again, the MM are used. For the third set, FY GDAS mod , the MM are exchanged with the new GDAS data in combination with the modified fluorescence calculation. Comparing the reconstruction sets with each other, the variation of the reconstructed primary energy E and the position of shower maximum X max can be determined, see Fig. 3.
Using GDAS data in the reconstruction instead of MM affects E only slightly. The mean of the difference FY GDAS mod minus FY mod is 0.4% with an RMS of 1.4%. For the reconstructed X max , only a small shift of −1.1 g cm −2 is found with an RMS of 6.0 g cm −2 . Comparing the full atmosphere-dependent reconstruction FY GDAS mod with FY, a clear shift in E can be seen: an increase in E by 5.2% (RMS 1.5%) and a decrease of X max by −1.9 g cm −2 (RMS 6.3 g cm −2 ). These modified fluorescence settings are now used in the Auger reconstruction, in conjunction with other improvements to the procedure, see [14].
The description of atmospheric conditions close to ground is very difficult in monthly mean profiles since the fluctuations in temperature and humidity are larger below 4 km than in the upper layers of the atmosphere. Consequently, a more precise description of actual atmospheric conditions with GDAS than with MM will alter the reconstruction for those air showers which penetrate deeply into the atmosphere. The full atmosphere-dependent fluorescence calculation alters the light yield for conditions with very low temperatures, corresponding to higher altitudes. Showers reaching their maximum in the altitude range between 3 and 7 km show a difference in E around 5%, see very deep X max are reconstructed with a 7-8% higher energy than using the atmosphere-independent fluorescence calculation. The X max sensitivity to the different parameterizations of the atmosphere and fluorescence yield (Fig. 3, lower right) is consistent to what has been reported in [15].
Impact on Reconstruction Uncertainties
To study the effect that GDAS data have on the uncertainties of air shower reconstructions, air showers induced by protons and iron nuclei are simulated with energies between 10 17.5 eV and 10 20 eV. The fluorescence light is generated using temperature-dependent cross sections and water vapor quenching. The times of the simulated events correspond to 109 radio soundings between August 2002 and December 2008 so that realistic atmospheric profiles can be used in the simulation. All launches were performed at night during cloud-free conditions. After the atmospheric transmission, the detector optics and electronics are simulated. The resulting data are reconstructed using the radiosonde data, as well as the GDAS data. Some basic quality cuts are applied to the simulated showers. The same study has been performed to determine the uncertainties of the MM [16]. The systematic error due to different atmospheres was found to be less than 1% in E and less than 2 g cm −2 in X max . Between 10 17.5 eV and 10 20 eV, energy-dependent reconstruction uncertainties of ±1% and ±5 g cm −2 for low energies and up to ±2% and ±7 g cm −2 for high energies were found.
In Fig. 4, the influence on the reconstruction due to GDAS data is shown. A deviation from zero indicates a systematic error, the gray error bands denote the true RMS spread of all simulated events and are a measure of the reconstruction uncertainty due to the atmospheric parameterization using GDAS. The red bands indicate the same RMS spread for the reconstructions using the MM. The systematic shifts in E are below 1%, and the shifts in X max are less than 0.5 g cm −2 . The RMS spread for GDAS is considerably smaller than for the MM, ±0.9% and ±2.0 g cm −2 for low energies, ±1.3% and ±3.5 g cm −2 for high energies. The E uncertainty at low energies is comparable to that introduced by the MM. At high energies, the uncertainty is almost half. For X max , the uncertainties at all energies are halved.
This study of the reconstruction uncertainties using different atmospheric parameterizations further demonstrates the advantages of GDAS data over the MM.
Conclusion
The comparison of GDAS data for the site of the Pierre Auger Observatory in Argentina with local atmospheric measurements validated the adequate accuracy of the 3hourly GDAS data. An air shower reconstruction analysis confirmed the applicability of GDAS for Auger reconstructions and simulations, giving improved accuracy when incorporating GDAS data instead of MM. Also, the value of using an atmosphere-dependent fluorescence description has been demonstrated. For the Colorado R&D site, the differences between the measured radiosonde data and GDAS are of the same order as in Argentina, further supporting the general validity of GDAS data as an atmospheric description to be used in current and future cosmic ray observatories.
Figure 1 :
1Top: Difference between measured individual radiosonde data and the corresponding GDAS data (black dots) and MM (red squares) versus height for all ascents performed at the Pierre Auger Observatory in Argentina in 2009 and 2010. Bottom: Difference between radiosonde data and the GDAS data (blue dots) versus height for all ascents performed at the Colorado R&D site in 2009 and 2010. tal Prediction of the National Oceanic and Atmospheric Administration. The numerical weather prediction model used is the Global Forecast System. Data are available for every three hours at 23 constant pressure levels -from 1000 hPa (≈ sea level) to 20 hPa (≈ 26 km) -on a global 1 • -spaced latitude-longitude grid (180 • by 360 • ). Each data set is complemented by data for the surface. The data are made available online [5].
Figure 2 :
2Difference between data measured at weather stations and from GDAS ('GDAS' minus 'weather station') in temperature and water vapor pressure are shown. Top: Data from 2009 of the CLF (dashed line) and Loma Amarilla (solid line) stations at the Pierre Auger Observatory. Bottom: Data taken between January 2010 and June 2011 at the AMT (dashed line) and DRLF (solid line) stations in Colorado.
Figure 3 :
3Difference of reconstructed E (top) and X max (bottom), plotted versus geometrical height of X max in the right panels. Dashed black line or black dots for FY GDAS mod minus FY mod , and solid red line or open red squares for FY GDAS mod minus FY.
Figure 4 :
4Energy difference (left) and X max difference (right) vs. reconstructed FD energy for simulated showers. Gray bands denote the true RMS spread for the GDAS reconstructions, the red band indicates the RMS for the reconstructions using monthly models.
Acknowledgment:We would like to thank the organizers of the workshop AtmoHEAD: Atmospheric Monitoring for High-Energy Astroparticle Detectors in Saclay, France, 2013 for the inspiring meeting. Part of these investigations are supported by the Bundesministerium für Bildung und Forschung (BMBF) under contracts 05A08VK1 and 05A11VK1. Furthermore, these studies would not have been possible without the entire Pierre Auger Collaboration and the local staff of the Pierre Auger Observatory.
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Will for the Pierre Auger Collaboration. B Keilhauer, M , Eur. Phys. J. Plus. 12796B. Keilhauer, M. Will for the Pierre Auger Collaboration, Eur. Phys. J. Plus, 2012, 127: 96.
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B Keilhauer, M Unger, arXiv:0906.5487Proc. 31 st ICRC. 31 st ICRCŁódź, Polandastro-phB. Keilhauer, M. Unger, Proc. 31 st ICRC, Łódź, Poland, 2009, arXiv:0906.5487 [astro-ph].
| [] |
[
"Automatic quantification of the microvascular density on whole slide images, applied to paediatric brain tumours",
"Automatic quantification of the microvascular density on whole slide images, applied to paediatric brain tumours"
] | [
"Christophe Deroulers christophederoulers*[email protected] \nLaboratoire IMNC\nUMR 8165\nUniv Paris Diderot\nCNRS\nUniv Paris-Sud\nF-91405OrsayFrance\n",
"Volodia Dangouloff-Ros [email protected] \nDepartment of Paediatric Radiology\nHôpital Necker EnfantsMalades\nAP-HP\n75105ParisFrance\n\nINSERM U1000\nParisFrance\n",
"Mathilde Badoual [email protected] \nLaboratoire IMNC\nUMR 8165\nUniv Paris Diderot\nCNRS\nUniv Paris-Sud\nF-91405OrsayFrance\n",
"Pascale Varlet [email protected] \nINSERM U1000\nParisFrance\n\nUniv Paris Descartes\nParisFrance\n\nDepartment of Neuropathology\nCentre Hospitalier Sainte-Anne\nParisFrance\n",
"Nathalie Boddaert [email protected] \nDepartment of Paediatric Radiology\nHôpital Necker EnfantsMalades\nAP-HP\n75105ParisFrance\n\nINSERM U1000\nParisFrance\n\nUniv Paris Descartes\nParisFrance\n\nInstitut Imagine\nUMR 1163\nParisFrance\n"
] | [
"Laboratoire IMNC\nUMR 8165\nUniv Paris Diderot\nCNRS\nUniv Paris-Sud\nF-91405OrsayFrance",
"Department of Paediatric Radiology\nHôpital Necker EnfantsMalades\nAP-HP\n75105ParisFrance",
"INSERM U1000\nParisFrance",
"Laboratoire IMNC\nUMR 8165\nUniv Paris Diderot\nCNRS\nUniv Paris-Sud\nF-91405OrsayFrance",
"INSERM U1000\nParisFrance",
"Univ Paris Descartes\nParisFrance",
"Department of Neuropathology\nCentre Hospitalier Sainte-Anne\nParisFrance",
"Department of Paediatric Radiology\nHôpital Necker EnfantsMalades\nAP-HP\n75105ParisFrance",
"INSERM U1000\nParisFrance",
"Univ Paris Descartes\nParisFrance",
"Institut Imagine\nUMR 1163\nParisFrance"
] | [] | Background: Angiogenesis is a key phenomenon for tumour progression, diagnosis and treatment in brain tumours and more generally in oncology. Presently, its precise, direct quantitative assessment can only be done on whole tissue sections immunostained to reveal vascular endothelial cells. But this is a tremendous task for the pathologist and a challenge for the computer since digitised whole tissue sections, whole slide images (WSI), contain typically around ten gigapixels. Methods: We define and implement an algorithm that determines automatically, on a WSI at objective magnification 40×, the regions of tissue, the regions without blur and the regions of large puddles of red blood cells, and constructs the mask of blur-free, significant tissue on the WSI.Then it calibrates automatically the optical density ratios of the immunostaining of the vessel walls and of the counterstaining, performs a colour deconvolution inside the regions of blur-free tissue, and finds the vessel walls inside these regions by selecting, on the image resulting from the colour deconvolution, zones which satisfy a double-threshold criterion. The two thresholds involved are automatically computed from the WSI so as to cope with variations in staining and digitisation parameters. A mask of vessel wall regions on the WSI is produced.The density of microvessels is finally computed as the fraction of the area of significant tissue which is occupied by vessel walls.We apply this algorithm to a set of 186 WSI of paediatric brain tumours from World Health Organisation grades I to IV. Results: The algorithm and its implementation are able to distinguish on the WSI the significant tissue and the vessel walls. The segmentations are of very good quality although the set of slides is very heterogeneous (in tumour type, in staining and digitisation parameters, and inside WSI themselves, where the tissue was often very fragmented). The computation time is of the order of a fraction of an hour for each WSI even though a modest desktop computer is used (a 2012 Mac mini) and the average size of WSI is 7 gigapixels. The computed microvascular density is found to be robust. We find that it strongly correlates with the tumour grade. 1 arXiv:1709.02309v1 [q-bio.TO] 7 Sep 2017Conclusions: We have introduced a method of automatic segmentation of significant, blur-free tissue and of vessel walls, and of quantification of the density of microvessels, in WSI. We successfully tested it on a large variety of brain tumour tissue samples. This method requires no training and estimates automatically several important parameters of the segmentation. It is robust and can easily be applied to other tumour types and other stainings. It should improve the reproducibility of quantitative estimates in pathology while sparing the pathologist time and effort. | 10.17629/www.diagnosticpathology.eu-2016-2:209 | [
"https://arxiv.org/pdf/1709.02309v1.pdf"
] | 44,191,348 | 1709.02309 | 3a24a16d9b1234714c0ae380a00ef6b7d2d2c9fb |
Automatic quantification of the microvascular density on whole slide images, applied to paediatric brain tumours
Christophe Deroulers christophederoulers*[email protected]
Laboratoire IMNC
UMR 8165
Univ Paris Diderot
CNRS
Univ Paris-Sud
F-91405OrsayFrance
Volodia Dangouloff-Ros [email protected]
Department of Paediatric Radiology
Hôpital Necker EnfantsMalades
AP-HP
75105ParisFrance
INSERM U1000
ParisFrance
Mathilde Badoual [email protected]
Laboratoire IMNC
UMR 8165
Univ Paris Diderot
CNRS
Univ Paris-Sud
F-91405OrsayFrance
Pascale Varlet [email protected]
INSERM U1000
ParisFrance
Univ Paris Descartes
ParisFrance
Department of Neuropathology
Centre Hospitalier Sainte-Anne
ParisFrance
Nathalie Boddaert [email protected]
Department of Paediatric Radiology
Hôpital Necker EnfantsMalades
AP-HP
75105ParisFrance
INSERM U1000
ParisFrance
Univ Paris Descartes
ParisFrance
Institut Imagine
UMR 1163
ParisFrance
Automatic quantification of the microvascular density on whole slide images, applied to paediatric brain tumours
Background: Angiogenesis is a key phenomenon for tumour progression, diagnosis and treatment in brain tumours and more generally in oncology. Presently, its precise, direct quantitative assessment can only be done on whole tissue sections immunostained to reveal vascular endothelial cells. But this is a tremendous task for the pathologist and a challenge for the computer since digitised whole tissue sections, whole slide images (WSI), contain typically around ten gigapixels. Methods: We define and implement an algorithm that determines automatically, on a WSI at objective magnification 40×, the regions of tissue, the regions without blur and the regions of large puddles of red blood cells, and constructs the mask of blur-free, significant tissue on the WSI.Then it calibrates automatically the optical density ratios of the immunostaining of the vessel walls and of the counterstaining, performs a colour deconvolution inside the regions of blur-free tissue, and finds the vessel walls inside these regions by selecting, on the image resulting from the colour deconvolution, zones which satisfy a double-threshold criterion. The two thresholds involved are automatically computed from the WSI so as to cope with variations in staining and digitisation parameters. A mask of vessel wall regions on the WSI is produced.The density of microvessels is finally computed as the fraction of the area of significant tissue which is occupied by vessel walls.We apply this algorithm to a set of 186 WSI of paediatric brain tumours from World Health Organisation grades I to IV. Results: The algorithm and its implementation are able to distinguish on the WSI the significant tissue and the vessel walls. The segmentations are of very good quality although the set of slides is very heterogeneous (in tumour type, in staining and digitisation parameters, and inside WSI themselves, where the tissue was often very fragmented). The computation time is of the order of a fraction of an hour for each WSI even though a modest desktop computer is used (a 2012 Mac mini) and the average size of WSI is 7 gigapixels. The computed microvascular density is found to be robust. We find that it strongly correlates with the tumour grade. 1 arXiv:1709.02309v1 [q-bio.TO] 7 Sep 2017Conclusions: We have introduced a method of automatic segmentation of significant, blur-free tissue and of vessel walls, and of quantification of the density of microvessels, in WSI. We successfully tested it on a large variety of brain tumour tissue samples. This method requires no training and estimates automatically several important parameters of the segmentation. It is robust and can easily be applied to other tumour types and other stainings. It should improve the reproducibility of quantitative estimates in pathology while sparing the pathologist time and effort.
Introduction
Angiogenesis is one of the key features of tumour progression, sustaining growth and sometimes enabling a change of aggressiveness when it starts [1]. In brain tumours [2,3], it is a crucial histology criterion used in diagnosis and to classify the disease into the proper World Health Organisation (WHO) grade [4]. Therefore, it is of great importance to be able to quantify in a reliable and robust way the status of the tumour vascular system. Although there exist several noninvasive, macroscopic imaging techniques [5][6][7], not all of them are innocuous (they may use ionising radiations or contrast agents), and they don't yield a direct access to the geometric parameters of the vasculature. In contrast, after proper immunohistochemical staining [8,9], biopsy samples reveal directly the tumour microvessels. It has been shown that the microvascular density, as measured on histology sections, is of prognostic significance in several brain tumours [10,11].
However, assessing manually the density of microvessels on whole histology sections is a tremendous task, very hard to perform for a human, and prone to much inter-(and even intra-) individual variability and lack of reproducibility. Quantifying the vascularity only on a few randomly chosen regions, or on a few "representative" regions on the section [12] will make the task easier (shorter) for the pathologist, but will increase the measurement variability and might reinforce the subjectivity of the task.
Luckily, virtual microscopy and the digitisation of pathology slides have become quite common over the last few years, allowing the use of the computer to perform various quantitative and reproducible measurements on histology sections [13]. In the beginning of this digital era, for cost and material reasons, it was not possible to measure the parameters of microvessels at full resolution (20x or 40x), and it was suggested to use images at resolution around 1x [14,15]. However, slide scanners produce now high resolution microscopy images of whole slides in a short time (at most a few minutes) and for a reasonable price [16], and using them for quantification should improve the precision of the results. The drawback of this high resolution is the very large size of the resulting files, which require specific software tools to be managed, like the ones some of us have already developed [17][18][19].
Such a quantification of microvessels on highresolution images has already been undertaken by several groups [20][21][22]. However, most of them were limited to small excerpts of the whole slide images (WSI), and possibly to relatively homogeneous sets of slides. Here, we report on a set of techniques we have developed to assess the density of microvessels on WSI of sizes a few tens of gigapixels, immunostained with CD34 (to reveal vascular endothelial cells), within a few minutes minutes, in a robust way, with a careful determination of zones of tissue without blur, and with as little intervention of the pathologist as possible. In particular, no training of an algorithm is necessary, hence the time-consuming task of manual segmentation of a number of vessels to feed to the computer is spared.
Although immunofluorescence is able to provide valuable additional quantitative information about the vasculature, such as 3D aspects [23], it requires the use of more elaborate microscopy and it not yet compatible with clinical routine. Therefore, we stick to classical bright field microscopy of immunostained pathology thin sections.
To demonstrate the versatility and scalability of our method, we applied it to a series of 129 human patients (186 WSI in total) suffering brain tumours of 19 different combinations of histological type and location, ranging from WHO grade I to grade IV. Since WSI are very large (our largest image had 162688 × 98816 pixels), they can't be opened in full in a standard computer's memory (we would need up to 60 GiB of RAM), and we had to develop strategies to treat them entirely without restricting ourselves to small excerpts. We used only open source software, or software we developed based on existing open source libraries, to avoid black-box algorithms, to promote interoperability and reproducibility, to reduce costs, and to avoid conflicts of interest [24].
Material and Methods
Our goal is to quantify the density of microvessels on a WSI as the ratio of the area occupied by vascular endothelial cells to the area occupied by the tissue. We assume that the WSI was obtained as a (possibly pyramidal) tiled TIFF or BigTIFF [25] file by digitisation at objective magnification 20× or 40× of a 5µm-thick tissue section of formalin-fixed, paraffin-embedded tissue, and that immunostaining with a CD34 antibody was performed so that microvessels appear brown whereas cell nuclei appear blue. The total area of tissue is typically of a few square centimetres.
If some deviation from this protocol is in order, it should be easy to adapt our method. E.g., if the WSI is stored in another format, it can be converted to TIFF using the free software NDPITools [17,19] that we developed or OpenSlide [26,27]. If the colours after immunostaining are different, two parameters can be changed (see below).
Method overview
The aim is first to select the zone of tissue on the WSI, then the zone of vessel walls inside the zone of tissue, and finally to measure the areas of the two and compute their ratio. Because of unavoidable slide-to-slide variations in staining and digitisation parameters (e.g. light intensity or temperature colour), both zone selections will require prior calibration steps. And, due to technical details, the work flow will be slightly more complex. We must:
• exclude from the WSI regions where the image is not sharp enough to recognise vessel walls accurately
• exclude from the WSI regions which look like tissue but should not be counted as such: es-sentially large puddles of red blood cells, coverslip boundaries and dust
• not count as vessel walls extra-vascular CD34positive tumour cells.
A scheme of the whole process is shown in Figure 1.
Preparatory steps
From the full-resolution 40× image, a 20× image was generated by bilinear interpolation and stored into a JPEG-compressed [28] tiled TIFF [29] file. Indeed, such an image proved of sufficient quality for several of the steps below while saving computation time.
Then, a mosaic of the 20× image was made and stored into JPEG files. This is a decomposition of the original image in rectangular pieces of equal sizes, stored into independent files for easy independent treatment, such that the original image is recovered if the pieces are reassembled together. We requested that each piece need at most 128 MiB to be stored (uncompressed) in RAM and that the dimensions of each piece be multiples of 8 pixels. This can be easily achieved using the -m and -M options of the tiffmakemosaic software [17,19].
Selection of sufficiently sharp zones
We used a variation of the method of blur quantification of [30,31]. On each piece of the 20× mosaic, after decompression into RGB colour space, we applied a colour space transformation into HSV colour space as defined by the vips program [32], extracted the V channel and convoluted the resulting image with the Laplacian kernel
-1 -1 -1 -1 8 -1 -1 -1 -1 .
Then, considering the result of the convolution as a mosaic of blocks of 8 × 8 pixels, we computed the S 2 score [31], namely the ratio of the sum of pixel intensities which are at least 10% of the maximum intensity in the block to the sum of all 64 pixel intensities (or 0 if the denominator is vanishing). Such a division into blocks of 8 × 8 pixels is natural since it is at the basis of the JPEG compression used by most slide scanners and used by our method to store the 20× image while saving disk space; artefacts due to this blocking are already present in the original WSI.
We generated a graylevel image where the S 2 value of each block of the 20× image was encoded as the intensity of one pixel (between 0 and 255 included). The resulting image, which can be deemed a sharpness map, is 16 times smaller (in linear dimension) than the original 40× image, thus has 256 times less pixels, and could easily be stored in a single file and opened at once in the computer's RAM.
Finally, this 2.5× sharpness map was transformed (using the ImageJ software [33,34]) into a mask of sharp regions in the following way: pixel intensities were averaged over regions of radius 2 pixels; a mask of regions where the resulting intensity is 43 and above was created (these are the sharp regions); sharp regions of less than 20 pixels of area (at resolution 2.5×) were turned into blurred regions, then blurred regions of less than 20 pixels of area were turned into sharp regions. The final mask of sharp regions was written on the disk.
Selection of tissue zones
We used as a first criterion to distinguish tissue from background the value of brightness of pixels (B in HSB colour space as defined by ImageJ [33]). Therefore, we needed to calibrate the brightness of pixels in the background.
This calibration may be influenced by the sides of the coverslip (and zones beyond) which are visible on 10 of our WSI. For each of these few images, we manually contoured the side of the coverslip on a downscaled 0.625× image and stored the resulting contour (union of polygons) as ImageJ's roi files which will be subsequently read at the proper stage.
We generated a 2.5× image from the full resolution image. With ImageJ, we selected on this image the non-excluded zones (sides of the coverslip), then we extracted the brightness of pixels (between 0 and 255) and selected pixels the brightness of which differed by less than ±1.2 from the Gaussian-weighted average (with standard deviation σ = 0.005) over their vicinity. We selected connected regions of at least 160 pixels among these "uniform" pixels and took the intersection with regions where the brightness was 127 or above. This defined the reference regions for the background.
We measured the histogram of the brightness value of the pixels in the reference regions. It always exhibits a peak of occurrence numbers of levels be-tween 200 and 255. We measured the right end b bg,r of this peak, as the largest brightness level which occurs at least once. Then we measured the left end b bg,l of this peak using the following algorithm: starting from the right end, we scanned occurrence numbers of decreasing brightness levels. We stopped when the current occurrence number was below half of the largest observed occurrence number so far and either was zero or was larger than the last seen occurrence number.
These left-and right-end define rather accurately the brightness of pixels belonging to the background and we stored them in a text file for later reuse. They had to be determined for each WSI because of marked variability: the left-end ranged from 212 to 234 while the right-end ranged from 232 to 249.
In addition, we selected pixels within the reference regions which had a brightness between the leftend of the peak and the left-end plus three (included) and computed their average values of R, G and B (hereafter called R bg , G bg , B bg ). We stored these values in a text file for later use.
The actual selection of tissue zones was performed on pieces of a mosaic of the image at resolution 20×. We produced with tiffmakemosaic a mosaic such that no piece required more than 100 MiB of RAM to be opened, with overlap of 256 pixels between adjacent pieces. Pieces were stored as TIFF files with zip compression (rather than JPEG compression to avoid another information loss and to facilitate opening by ImageJ). On each piece, ImageJ was used to produce a mask of the tissue zones in the following way. We selected the pixels the brightness of which was outside the interval measured previously as "background intensity peak", [b bg,l , b bg,r ]call them A-pixels. Then, among the connected regions formed by these pixels, we selected those which contained at least one pixel the saturation of which (in HSB space) was larger than the average of the saturation of all A-pixels -thus constructing the B-regions. This was to prevent selecting uniform regions with unusual high or low brightness, which could be a large piece of dust or a pen stroke (uniformly black region).
The resulting regions could contain "holes", some of which were regions of background inside a region of tissue. We noticed that almost all holes of less than ≈ 10000 pixels (at 20× magnification) had to be considered as tissue. Therefore, we included in the B-regions the holes of area less than 10000 pixels and which did not touch the boundaries of the mosaic piece. Indeed, a hole touching a boundary either was a small part of a bigger hole (more than 10000 pixels) extending on another mosaic piece, which should not be restored into the B-regions, or a small part of a small hole which would quite probably be entirely included into an adjacent mosaic piece, because mosaic pieces overlapped by 256 pixels. The exception would concern only small holes (less than 10000 pixels), a dimension of which exceeded 256 pixels, that is rather elongated holes, which was rather uncommon.
Similarly, we found that holes having a fractallike shape should also be restored as tissue in the B-regions. They were characterised in the following way: after we performed an erode then a dilate operation on them, their area was less than 10000 pixels, and they did not touch the boundary of the mosaic piece.
Then we made the boundaries of the B-regions (tissue regions) more regular applying an erode then a dilate on their mask. This eliminated the small overhangs or invaginations of a few pixels.
We eliminated from the B-regions the connected regions of less than 20000 pixels which were at a distance 30 pixels or larger from a region or 20000 pixels or more. Indeed, most of those were found to be non significant (dust, isolated cells, small pieces of tissue torn apart). And in the end we are interested only in the fraction of tissue area covered by vessels, which we neither overestimate nor underestimate by mistakenly removing a small significant area of tissue. This last operation was performed in ImageJ by a combination of morphological operations, including thresholding at 15 the distance map from the B-regions.
Finally, we found that B-regions with too low saturation (greyish regions) could exist and should be disregarded as tissue (tissue contains at least blue cell nuclei or brown vessel walls). Therefore, after convoluting by a Gaussian kernel of standard deviation σ = 3 pixels the image of the saturation in HSB colour space, we applied the isodata algorithm [35] to find an automatic threshold on this image and we defined as high-saturation pixels the pixels the saturation of which was above the isodata threshold and above 30. Only B-regions containing at least one high-saturation pixel were kept as tissue regions.
The mask (binary image) of the tissue regions of each mosaic piece was saved as PNG files, then all PNG files were combined into a large, 20× resolution, mask of tissue thanks to an in-house developed C program for the sake of speed and RAM economy. The whole mask was stored on the disk as a 1-bitper-pixel Deflate-compressed [29] TIFF file, which achieves a very high level of compression. Then the pieces of the mosaic with overlap were erased from the disk.
If there existed, for this WSI, a .roi file defining the side(s) of the coverslip on the image, the mask of the region to exclude was formed as a binary image of resolution 0.625× and the mask of tissue was replaced by the result of a logical and of the former mask of the tissue and the inverse of the mask of the region to exclude blown up to resolution 20×. This logical operation was performed thanks to another in-house developed C program, again for the sake of speed and RAM economy.
Finally, the logical and of the mask of sharp regions and of the mask of tissue was computed through the same C program and stored on the disk as a bilevel, Deflate-compressed TIFF file. Again, although the resulting mask is a very large image (1,76 gigapixels in average), this form of compression is very efficient: the average disk size of the mask was 4,36 MiB, ranging from 0,26 MiB to 16,38 MiB.
A detail from an example of such a mask of sharp tissue is shown, superimposed onto the corresponding WSI, in Figure 2.
Removal of puddles of red blood cells
This part of the method is rather cumbersome and empirical, but we found it useful since, on several WSI of brain tumours, the puddles of red blood cells occupied more than 30% of the regions of blur-free tissue. It was developed on two example WSI but proved efficient on all.
We constructed a mask of large puddles of red blood cells on the WSI at objective magnification 40× in the following way. We changed the colour space to HSB (as defined in ImageJ) and operated on the B channel.
First (step 1), on the result of the contour detection (application of a 3×3 Sobel filter, ImageJ's Find Edges command), we selected connected regions of at least 20 pixels at levels 250 and above, and pixels at the distance at most 10 pixels from these regions, then retained, from these enlarged connected regions, only those which had at least 10,000 pixels.
Second (step 2), we selected pixels of brightness exactly 255 and performed a morphological close operation on the resulting mask. This is to select the sides of the red blood cells, which appear very bright in bright field microscopy (because of the refringence property of these cells) and elongated (because of the circular shape of these cells). We skeletonised the result and eliminated the connected regions of at most 9 pixels. Then we extended the remaining skeletons to pixels within the distance of 20 pixels and eliminated the connected regions of less than 20,000 pixels. Defining the threshold b := 255 − 0.35(255 − b bg,r ), were b bg,r was defined during background's brightness calibration (see above), we retained, among the remaining regions, only those which contained at least one connected regions of 40 pixels or more with brightness above b and of circularity at most 0.2. We also added the connected regions of 2800 pixels and above which were constructed by added to these latter regions pixels at distance at most 2800.
We took the union of the regions defined in the two preceding steps (1 and 2). We added the pixels at distance at most 20 pixels of these regions and inside the convex hull of at least one of these regions. This formed a mask of puddles of red blood cells. Finally, we added to the mask the holes of less than 100,000 pixels if contained, and stored the mask on the disk as a 1-bit-per-pixel Deflate-compressed tiled TIFF image as before.
A detail from an example of such a mask of large puddles of red blood cells is shown, superimposed onto the corresponding WSI, in Figure 3.
Removing from the mask of sharp tissue pixels considered, according to this mask of puddles of red blood cells, we built the mask of significant, blur-free tissue.
Calibration of optical density ratios of stains Since the vascular endothelial cells were marked with a brown staining over a blue counterstaining, information relevant to the microvessels are be entirely contained in the brown channel after we perform a colour deconvolution [36].
To perform this linear change in colour space, we needed to know the optical densities (o.d.) in R, G and B channels (or, more precisely, only the ratios of these three o.d. to form their vector in the RGB space [36]) of the two stains, brown and blue. We couldn't use standard values from the literature nor common values for all WSI since, as for the brightness of the background above, there was a strong slide-to-slide variability, as can be seen on Figure 4.
We used the following procedure on each piece of the 20× mosaic without overlap formed earlier. Using ImageJ, we performed a change of colour space to HSB. On one hand, we selected pixels with brightness at most 198, saturation at least 70 and hue outside the interval [80,199]. Among them, we kept only connected regions of at least 25 pixels. Finally, we measured the average optical densities in the R, G, and B channels of these pixels and the number of remaining pixels and wrote these numbers to a text file. This gave the contribution of markedly brown areas. On the other hand, we repeated the procedure with pixels, the saturation of which was at least 70 and with hue inside the interval [80,199] to get the contribution of markedly blue areas.
Then we aggregated the results from all mosaic pieces to compute the average over the whole slide of the optical densities in R, G, and B channels of the brown and blue stains and wrote them to a text file.
The precise values of the thresholds above are irrelevant: the important thing is that the loose limits on the hue 80 and 199 clearly separate brown from blue (they can of course be adapted to other colours) and that the loose limits on the saturation and brightness select as representative areas for calibration regions markedly brown resp. blue. In average, the calibration of the optical densities of the brown staining rested on 15,4 megapixels (min: 56,3 kilopixels, max: 288 megapixels) and the calibration of the optical densities of the blue rested on 104 megapixels (min: 1,09 megapixels, max: 842 megapixels).
Selection of vessel walls
The whole process of actual selection of vessel walls operated only on the areas of significant, blur-free (sharp) tissue of the WSI, which were indicated by the mask constructed earlier.
First, the histogram H B of the brown optical density (o.d.) of each pixel after colour deconvolution was constructed. This was achieved through an in-house developed C program which took as input the WSI at resolution 20×, the mask of sharp tissue and the parameters of the colour deconvolution (average o.d. determined earlier) and operated independently on each tile of the WSI to save RAM and allow the use of parallel processing on a computer with multicore CPUs. Here, the restriction to areas of sharp tissue also avoided to bias the histogram with values from irrelevant pixels (e.g. dust).
Then, a global (but WSI-specific) threshold on the brown o.d. for vessel walls on the WSI, hereafter called b A , was automatically determined from the histogram in the following way. First, we computed b bg , the brown o.d. of a pixel with colour (R bg , G bg , B bg ) stored earlier (average colour of the darkest pixels of the background reference regions). Then, we applied the isodata automatic method of threshold computation [35] on the part of the histogram H B concerning o.d. above b bg .
We could not rely on the full histogram to determine b A since, on some of the WSI, cell nuclei were so dense in the tissue that they would manifest themselves as a peak in the low o.d. region of the histogram H B , so that the value b 1 computed by the isodata algorithm would be influenced by them instead on yielding information on the vessel walls only. Disregarding the low o.d. values of brown, that is, o.d. values below those of the background, was a simple and efficient way to solve this problem.
Then, a second global (but also WSI-dependent) threshold, called b B , was computed by a new application of the isodata algorithm on the part of the histogram H B concerning o.d. above b A .
The actual segmentation of the vessel walls consisted essentially in looking for connected sets of pixels, inside the sharp tissue, which had brown o.d. above (or equal to) b A and which contained at least a small regions where brown o.d. was above (or equal to) b B . The second condition avoided that pale brown regions be inaccurately recognised as vessel walls (this concerned for instance a few isolated CD34-positive tumour cells).
This was performed by treating in turn each rectangular zone of size roughly 3840 × 3840 of the WSI at resolution 20×. The zone was extracted using the tifffastcrop program from the LargeTIFFTools [17,18] (which can extract very quickly a rectangular zone from a (possibly huge) tiled TIFF image, twice as fast as the extract area command of vips). The corresponding zone from the mask of sharp tissue was extracted. Then an ImageJ macro selected pixels from the extract of the WSI inside the sharp tissue zones according to the mask, performed colour deconvolution and created a mask of the brown o.d. of pixels which satisfy one of the three conditions: the brown o.d. is above (or equal to) b A , the brightness (in HSB colour space) is at most 30, or the brightness is at most 40 and the saturation at most 127. The two latter conditions were necessary because almost black pixels composing some of the vessel walls may have low values of brown o.d. (as already noticed for brown staining [36]) and may be missed by the first condition.
The mask was post-processed in the following way. Let us call A-regions the connected regions of pixels selected so far on the basis of b A (discarding regions of less than 75 pixels). We convoluted the image of brown o.d. with a Gaussian kernel of standard deviation 0.5 pixel, then kept pixels with intensity b B and above, then connected regions of at least 25 pixels of these pixels. Finally, we kept as vessel walls in the current rectangular zone the Aregions containing at least a B-region. We recorded their mask on the disk as a PNG file.
As in an earlier step, we used our C program to merge all masks of vessel walls in rectangular zones into a single mask at 20× resolution stored in a bilevel, Deflate-compressed tiled TIFF file, where vessel walls were black and the background was white.
Review of the segmentation results
For each slide, we produced a set of files in Deep-Zoom format from the WSI at resolution 40× using vips. We also produced sets of files in DeepZoom format for the mask of sharp tissue and the mask of vessel walls. And we produced in DeepZoom format the image of the contours of vessel walls, where all pixels are transparent, except the pixels which belong to a boundary of a vessel wall (black pixels surrounded by at least one white pixel).
These sets of files were uploaded to a secured web server, along with a simple HTML file (automatically generated by a simple Perl script) calling the JavaScript OpenSeadragon [37] library to display the superimposition of the slide and, according to what the user selects, of the different masks or contours.
This allowed a convenient quality control of the segmentation of tissue and vessels by the pathologist from his office or a meeting room at the Hospital, even though the whole image processing was performed on a computer in a physics laboratory. Indeed, all that was needed for this visualisation was a standard desktop computer with a JavaScript capable web browser. A typical session of quality check is displayed on Figure 6.
Material of our cohort
We applied our method to a cohort of 129 human patients suffering from brain tumours ranging from WHO grade I to grade IV and in various locations: posterior fossa, thalamus and hemispheres. The detailed numbers are given in Table 1. Such a variety of tumour types served as a challenge to our method (see if it is really robust even without human intervention) and was also meant to check how much the microvascular density was correlated with tumour grade.
Each sample was prepared the same way: a 5µm-thick tissue section of formalin-fixed, paraffinembedded tissue was immunostained with a monoclonal mouse anti-human CD34 antibody (QBend-10, Dako R , Agilent Technologies, Santa Clara, California, USA). The reaction was carried out in an automated immunohistochemistry instrument (Benchmark, Ventana Medical Systems R , Hoffman La Roche, Basel, Switzerland). Patients were excluded if the pathological sample was insufficient to perform CD34 immunohistochemical analysis.
The resulting 186 tissue sections mounted on glass slides were digitised by a Hamamatsu NanoZoomer at objective resolution 40×, which produced a NDPI file per slide. The average size of the files was 748.5 MiB (max: 2.54 GiB), representing in total ≈100 GiB of compressed data. The average size of the images was 7.03 gigapixels (max: 16.08 gigapixels).
Results
Quality of the segmentation
The quality of the segmentation was reviewed during a collective meeting in the hospital (involving pathologist, radiologists and physicists). Each slide was displayed thanks to an overhead projector connected to a computer, itself using a web browser to retrieve images from our web server. Out of the 186 slides, 30 had unfortunately to be excluded because they displayed extensive areas of extra-vascular CD34positive cells.
On a few other slides, the pathologist was able to select zones free from extra-vascular CD34 signal. These zones were defined using ImageJ as unions of polygons on the 2.5× resolution versions of the WSI and stored on the disk as ImageJ's .roi files. The corresponding slides were reprocessed by disregarding all pixels outside the selected zones for tissue and vessel wall selection.
On two other slides, it proved sufficient to manually raise slightly the threshold b A and to reprocess the slide (vessel selection step) to prevent selecting most extra-vascular CD34-positive cells.
Otherwise, the segmentation was judged of very good quality by the pathologist. An example of the complete segmentation is shown in Figure 5.
Density of microvessels
On each WSI, we counted the number of selected pixels on the mask of sharp tissue and on the mask of vessel walls, and we took the ratio to get the density of microvessels. The measurements were done by an in-house C program for the sake of speed and RAM economy, again treating images tile-wise to enable multicore parallel processing.
The distribution (histogram) of the densities is displayed in Figure 7. We find that it is remarkably correlated with the tumour grade.
Computation time performance
All processing of the WSI was done on a Mac mini computer with 16 GiB of RAM and a quad-core i7 CPU at 2.3 GHz bought in October 2012. Although this computer was rather modest in regard of today's standards, we found the overall treatment time to be quite acceptable. Figure 8 shows the distribution of the computing time of the vessel wall segmentation (including colour deconvolution), which is one of the most timeconsuming operations. One can see that most WSI could be treated in less than 15 minutes. This has to be compared to the time necessary to transfer to the web server the OpenSeadragon files of the WSI and of the masks: the latter was larger even using a large bandwidth network connection.
Let us also notice that selecting the vessel walls only in the areas of sharp tissue of the WSI saved a substantial amount of computation time, since the average fraction of the WSI occupied by the sharp tissue was only 26.8% (ranging from 1.9% to 63.6%). Most of the rest of the WSI was usually background, empty space.
Discussion
Inter-slide variability and robustness As we already discussed, several of the physical parameters of the WSI (colour temperature of the background, optical densities of the stains,...) vary largely from one slide to the next. This is why our method includes several steps of calibration. Some previous studies [22] used values of the parameters of the segmentation common to all slides, but this was for smaller sets of slides. In our heterogeneous set of 186 slides, this would yield wrong results. For instance, the thresholds b A resp. b B on brown o.d. for vessel segmentation have the following statistics: average 93.2, min. 44, max. 160 resp. average 178.5, min. 159, max. 250.
We believe that other parameters (parameters which were not chosen after a calibration) fall in two categories: parameters, the value of which is not very relevant, and parameters which might influence the final result of the WSI. In the first category fall e.g. the precise limits of the interval of hue used to define the reference areas for the calibration of brown and blue optical densities. Changing these limits will alter only very marginally (if at all) the measured o.d., hence the vessel segmentation. It is therefore not worth to give too much attention to them.
Parameters in the second category include e.g. the threshold used to fix the limit between sharp and blurred tissue, and most of the geometrical parameters of the segmentation of tissue (the minimal / maximal number of pixels of connected regions to be kept / disregarded, the maximal distance of a small piece of tissue to a large piece to be considered as significant,...). We did only a manual calibration of them, based on a representative set of a few slides or a few excerpts of slides. But, even if a different choice for their value could change the final measured microvessel density, we think that this change is less important than in the case of calibrated parameters (like stains' o.d.) and that it would be systematic, affecting the measure roughly in the same way on all WSI. Hence fixing these values doesn't preclude the use of our automatic measurement for assistance to tumour grading.
Finally, let us remark that, even with these possible limitations in mind, our method should be much more reproducible than manual measurements performed on a few chosen excerpts in each slide even by a trained pathologist [15], if the latter is at all doable on such a large set of slides. We prefer to save the pathologist's time for quality control and make our overall study cheaper by the extensive use of the computer. Notice also that, among our 186 slides, many have very fragmented tissue (all the more that we are dealing with fragile brain tissue), making it very difficult to estimate accurately the area of tissue. And, of course, a systematic manual vessel segmentation is not in order: there is an average of 16571 vessel fragments in the sharp tissue area of our slides (min. 58, max. 72614).
Impact of red blood cells
The fraction of blur-free tissue occupied by puddles of red blood cells, as determined by the method above, ranged from 0.04% to 37.2%. In average, such spreads of red blood cells occupied 4.87% of the blur-free tissue, so they were not a crucial issue for the determination of the microvascular density for most slides. But for 8 of our 186 slides they occupied more than 25% of the apparent tissue, and for 21 of them they occupied more than 10% of the apparent tissue. For these slides, had we counted the puddles of red blood cells as tissue, we would have underestimated the density of microvessels by 10 to 25%, which is huge in comparison, e.g., to the difference of microvascular density between low-grade and high-grade tumours (Figure 7). Therefore, we believe that the step of automatic detection of puddles of red blood cells, although cumbersome and time-consuming, is necessary. We couldn't save time by performing it on the 10× magnification WSI instead of the 40× WSI because the specifically high intensity pixels of the red blood cells' sides (due to their refringence) was lost during the resolution reduction.
Uncertainties
We tried to assess the uncertainties on the measurement of the microvessel density against the most important parameters of the method in the following way.
• Changing by one (over 255) the brown o.d. threshold b A changed by 0.5% the total area of the regions recognised as vessel walls. Therefore, the uncertainty on b A and b B , as determined automatically by a thresholding algorithm [35] up to a few units over 255, has little influence on the final result. But remem-ber, once again, that this automatic threshold varies strongly from one slide to the next one: the standard deviation on b A is 21.1.
• Eroding or dilating either the regions recognised as vessel walls or the regions recognised as tissue by one pixel at resolution 20× changed by ≈7% the area. This is more serious and means that one has little latitude on morphological post-processing of the segmented regions. However, a change by 7% is still acceptable if one wants to use the microvessel density to distinguish between low-grade and high-grade tumours, owing to the large spreading of the density (see Figure 7). It should be interesting to redo the measurement using the WSI at magnification 40× and see if the uncertainty against eroding or dilating is smaller as one could expect, at the expense of a computation time four times larger.
In a context of strongly heterogeneous tissue, measuring the microvessel density on whole tissue sections also contributes to reduce the uncertainties by allowing the measurement to rest on a large zone of tumour tissue, hence reducing the risk to measure accidentally a zone with especially low or high vessel density.
Additionally, if one wants to draw a link between measurements at the microscopic scale (on WSI) where individual cells are resolved, and at macroscopic scale (using e.g. MRI) where the resolution is routinely of the order of 1 mm, one has to perform the microvessel density measurement on the largest (but still relevant) piece of tissue. In this spirit, and to further confirm the quality and relevance of our measurement on brain tumour tissues, it has been shown that the microvascular density we measured is in good correlation with the result of a noninvasive, macroscopic measurement using the ASL modality of MRI [11].
Further development
We plan to extend this work in several directions. First of all, the overall process can still be optimised to reduce the treatment time of each slide.
Then, much information is still left unexploited: on one hand, we plan to perform morphometry analyses on the segmented vessel walls [20,22,38]. This could serve as a basis for a system of computer aided diagnosis of some of the tumours. And this would yield precious data to develop a theoretical model of angiogenesis in brain tumours, which hopefully could guide treatments in the long term, in the spirit of what is being done e.g. for adult low-grade gliomas [39].
On the other hand, no information about cell nuclei has been exploited yet. It should be relatively easy to perform segmentation on e.g. the blue channel after colour deconvolution and get quantitative parameters in the same way as for vessels: density of nuclei, morphometry...
And, beyond morphometry on the black-andwhite masks resulting from the mere segmentation (thresholding) of biological objects, it could also be possible to extract more information from the virtual slides by continuously varying the threshold defining A-regions. In this way, a series of segmentations is built and can be analysed as would be a time series [40], revealing more aspects of the disease than a static picture taken at a single time point.
Finally, angiogenesis has been shown to be of significant value for diagnosis and prognostic more generally in oncology [41][42][43], so that our method can readily be applied to other tumours with the same of similar immunostainings (CD31, CD34).
Conclusions
We have introduced an automatic and training-free method of quantification of the density of microvessels on whole tissue sections immunostained with the CD34 antibody and digitised by a slide scanner. This method is, to our knowledge, the first one to include a careful determination of areas of tissue without blur and puddles of red blood cells before the proper segmentation of vessel walls.
We tested in on a large set of WSI (186) of a very large variety of brain tumours. Using a very reasonable amount of computation time on a quite affordable computer system (an Intel Core i7 CPU with 16 MiB of RAM), this method produced results of very good quality, even though an overall check of the segmented WSI by the pathologist was necessary, in particular because of extra-vascular CD34-positive tumour cells. It should be helpful in computer-aided diagnosis systems and easily reused for other stainings/tumours, especially because it uses only open source software (like ImageJ or vips) or well-described algorithms, and because its architecture is simple and modular and its parameters easy to understand and modify (e.g. to adapt it to other colours than brown and blue). For each WSI, after calibrating the optical densities in the red, green and blue channels of the brown (CD34) staining as defined in [36] according to the procedure described in the main text, we subtracted the blue optical density to the red optical density -let's call this difference ∆. This histogram shows the distribution of the values of ∆ over the 186 WSI. One can see that there is a strong variability, which prevents one to use a single set of optical densities to perform colour deconvolution on all WSI. The user is viewing, in the window of his JavaScript-capable web browser (here, Firefox), a detail of one of the whole slide images at full resolution (40×), on which he has superimposed the contours of the vessel walls (displayed in cyan on top of the slide). He can interactively add/remove contours and masks (displayed by shading the slide) to/from the list of displayed information, zoom in and out and drag the slide to explore it with his mouse. The red (resp. blue) histogram shows the distribution of the microvascular density (fraction of sharp tissue area covered with CD34-stained vessel walls) measured by our method on the WSI of samples from highgrade (resp. low-grade) tumours. Although the two histograms (here shown in log-lin scale) are relatively broad, there is a clear distinction between the typical microvascular densities of low-and high-grade tumours. We argue that the measurement uncertainties of the microvascular density (see Discussion in the main text) are much lower than the difference between these typical values.
Figures
Figure 1 -
1Overview of the whole method of selection of sharp tissue (top) and of vessel walls inside sharp tissue (bottom) The blue boxes represent real, full colour images; the checkerboard boxes represent masks (black-on-white images resulting from the selection of objects); the red boxes represent automatically determined quantitative parameters; the brown box represents the image of the brown optical densities of pixels of the original 20× image; the green boxes represent the final measured quantities.
Figure 2 -
2Example of a determination of tissue without blur. Top left: excerpt of the original image (one of our WSI). Top right: the mask (logical and combination of the mask of tissue and of the mask of sharp regions) is superimposed onto the WSI. Only regions considered, after our method, as sharp tissue are shaded. One can see in particular that tears inside the tissue are properly not counted as tissue (some of them are marked 'T'). Blurred regions (some of them are marked 'B') tend to form three vertical bands because of the way the image was acquired by the scanner. They will be excluded from the quantification process. Orange scale bar = 1 mm. Bottom left: detail of the top right image at the boundary of a blurred region. Bottom right: the mask of sharp regions is superimposed onto the detail of the WSI shown on the bottom left. One sees the transition between "fully blurred", on the left-hand side of the image, to "fully sharp", on the right-hand side of the image. Green scale bar = 20 microns.
Figure 3 -Figure 4 -
34Example of a determination of large puddles of red blood cells. Top: original image (excerpt of one of our WSI). Bottom: the same image, over which the mask of the large puddles of red blood cells was superimposed. Scale bar = 20 microns. Slide-to-slide variability of the optical densities of the brown staining.
Figure 5 -
5Example of a segmentation of tissue and vessel walls (detail). On this excerpt of a WSI at resolution 40×, the final result of the segmentation by our method is shown. The areas considered as sharp tissue are shaded, and the areas inside sharp tissue considered as vessel walls on the basis of the CD34 immunostaining are contoured in cyan. Scale bar = 20 microns.
Figure 6 -
6A typical session of quality control of the segmentation in a standard web browser
Figure 7 -
7Distribution of the density of microvessels in the tissue for patients suffering a high-grade resp. low-grade tumour
AcknowledgementsWe thank the CNRS/IN2P3 computing centre for hosting the website we used to view WSI and control the quality of segmentation.CD and MB belong to the CNRS consortium CellTiss and to the Labex P2IO.Competing interestsThe authors declare that they have no competing interests.Tables
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| [] |
[
"A formal moment map on Diff 0 (M )",
"A formal moment map on Diff 0 (M )"
] | [
"La Laurent [email protected] \nUniversité libre de Bruxelles and Haute-École Bruxelles-Brabant\nBelgium\n",
"Fuente-Gravy \nUniversité libre de Bruxelles and Haute-École Bruxelles-Brabant\nBelgium\n"
] | [
"Université libre de Bruxelles and Haute-École Bruxelles-Brabant\nBelgium",
"Université libre de Bruxelles and Haute-École Bruxelles-Brabant\nBelgium"
] | [] | In the framework of deformation quantization, we obtain a deformation of Donaldson moment map on Diff 0 (M ), the connected component of the group of diffeomorphisms of a symplectic manifold (M, ω) admitting another symplectic structure χ. | null | [
"https://arxiv.org/pdf/2203.12287v1.pdf"
] | 247,618,947 | 2203.12287 | 57b5ea420cd7611693b9cae95b6cfefd31353c6c |
A formal moment map on Diff 0 (M )
23 Mar 2022 March 24, 2022
La Laurent [email protected]
Université libre de Bruxelles and Haute-École Bruxelles-Brabant
Belgium
Fuente-Gravy
Université libre de Bruxelles and Haute-École Bruxelles-Brabant
Belgium
A formal moment map on Diff 0 (M )
23 Mar 2022 March 24, 2022Symplectic connectionsMoment mapDeformation quantizationHamiltonian diffeomor- phismsdiffeomorphisms group Mathematics Subject Classification (2010): 53D5553D2058D05
In the framework of deformation quantization, we obtain a deformation of Donaldson moment map on Diff 0 (M ), the connected component of the group of diffeomorphisms of a symplectic manifold (M, ω) admitting another symplectic structure χ.
Introduction
We exhibit another formal moment map on an infinite dimensional space using the technique from [12]. This formal moment map appears to be a deformation of Donaldson moment map [4] on Diff 0 (M) the connected component of the group of diffeomorphism of a closed symplectic manifold (M, ω) in the presence of an extra symplectic form χ.
When the manifold M is Kähler and both ω and χ are Kähler forms, Donaldson [4] deduced from his moment map a flow of Kähler metrics named the J-flow. As pointed out by Chen [3], existence of solutions of the J-flow leads to informations on the Mabuchi K-energy which plays a key role in the cscK metric problem.
Our deformation framework is deformation quantization [2]. We consider a Fedosov star product algebra bundle V over Diff 0 (M). On the fiber above f ∈ Diff 0 (M), the star product is obtained from Fedosov construction parametrised with the formal 2-form νf * χ and symplectic connection ∇ f depending on f , see Subsection 2.2.
We use the construction of Andersen-Masulli-Schätz [1] to obtain a formal connection on V. We show its curvature acts by inner derivations, that is the * -commutator with a formal function.
Acting as in the prequantisation picture, we consider the trace of the curvature of the formal connection as a first Chern form of a line bundle. We show this form is a deformation of the symplectic form on Diff 0 (M) that is preserved by the action of Ham(M, ω), the group of Hamiltonian diffeomorphisms of (M, ω), on Diff 0 (M).
Finally, we show that a formal moment map for the action of Ham(M, ω) on Diff 0 (M) is given by the trace of star product. At first order, our result recovers Donaldson moment map.
The result of this paper explain what was presented as a coïncidence in [11] linking first terms of traces for star products and moment maps on infinite dimensional space. It justifies the closed Fedosov star product problem as a problem of finding zeroes of a formal moment map as suggested from the works [11,10] and [7,8].
2 A bundle of Fedosov * -product algebras on Diff 0 (M ) Throughout this short paper, we consider a closed symplectic manifold (M, ω) of dimension 2m admitting another symplectic form χ. We also deal with infinite dimensional manifolds and Lie groups, we will follow the theory from [14].
Y : f → Y f := Y • f on Diff 0 (M). The vector field Y as a flow Fl Y t obtained from the flow of φ Y t of Y on M and defined by Fl Y t (f ) := φ Y t • f, for all f ∈ Diff 0 (M).(1)
This flow satisfies
d dt Fl Y t (f ) := Y Fl Y t (f )
, for all f ∈ Diff 0 (M). The Lie bracket of two right invariant vector fields Y and Z, build out of Y, Z ∈ X(M) is given by
[Y, Z] Diff 0 f := [Y, Z] • f, for all f ∈ Diff 0 (M),(2)
where the bracket on the RHS is the usual bracket of vector fields on M.
The symplectic form on Diff 0 (M) we will study is obtained from χ and ω, it is defined as
Ω Diff 0 f (Y f , Z f ) := M χ f (·) (Y f (·) , Z f (·) ) ω n n! .
To check d Diff 0 Ω Diff 0 vanishes, one uses the fact that dχ = 0 and remember that the formula of the differential for forms on Diff 0 (M) writes for a p-form Θ on Diff 0 (M),
(d Diff 0 Θ) f (Y 0 • f, . . . , Y p • f ) := p i=1 (−1) i (Y i ) f Θ(Y 0 , . . .Ŷ i . . . , Y p ) + 0≤i<j≤p (−1) i+j+1 Θ f ([Y i , Y j ] Diff 0 (M ) , Y 0 , . . . ,Ŷ i , . . . ,Ŷ j , . . . , Y p ),
for f ∈ Diff 0 (M) and Y 0 , . . . , Y p right invariant vector fields obtained from Y 0 , . . . Y p vector fields on M. Note that we decorate Lie bracket and differential on Diff 0 (M) with the superscript Diff 0 to prevent from confusion with the corresponding notions on M.
We consider the group Ham(M, ω) of Hamiltonian diffeomorphisms of (M, ω). It acts on the right on Diff 0 (M) by ϕ · f := f • ϕ, for f ∈ Diff 0 (M) and ϕ ∈ Ham(M, ω).
One checks this action preserves the symplectic form Ω Diff 0 . The infinitesimal action of a Hamiltonian vector field X H (i.e. X H is defined by ı(X H )ω = dH for H ∈ C ∞ (M)) is then given by
X H · f := f * X H , for f ∈ Diff 0 (M),(3)
which is indeed a section of f * T M. Donaldson [4] obtained a moment map for the above action of Ham(M, ω) on the symplectic manifold (Diff 0 (M), Ω Diff 0 ). We will recover it from our construction of a formal moment map.
Symplectic connections and Diff 0 (M)
In the sequel, we will need to attach a Fedosov star product to a diffeomorphism f ∈ Diff 0 (M). Since one of the ingredients for constructing a Fedosov star product is a symplectic connection, we need a map that sends a diffeomorphism to a symplectic connection.
To obtain such a map, let us recall how the existence of a symplectic connection on any symplectic manifold is settled. First, a symplectic connection on (M, ω) is a connection on T M that leaves ω parallel and has no torsion. Starting from any torsion-free connection ∇ on M (which always exists), one turns it into a symplectic connection ∇. One first define a tensor N ∇ by
ω(N ∇ (X, Y ), Z) := (∇ X ω)(Y, Z) for all X, Y, Z ∈ T M.
Then, one checks by direct computation that the connection
∇ X Y := ∇ X Y + 1 3 N ∇ (X, Y ) + 1 3 N ∇ (Y, X),
for X, Y ∈ X(M), defines a symplectic connection. Any two torsion-free connections ∇ and ∇ differ from each other by a tensor A(·)· ∈ Γ(T M ⊗ S 2 T * M). Recall that if ∇ is symplectic, then ∇ is symplectic if and only if A(·, ·, ·) := ω(·, A(·)·) is a completely symmetric 3-tensor on M. Also, recall that the space of symplectic connections E(M, ω) is an affine space modelled on the space Γ(S 3 T * M) of completely symmetric 3-tensors on M.
Now, consider a symplectic connection ∇ and a diffeomorphism f . One builds the torsion-free connection f * ∇ defined for X, Y ∈ X(M) by
(f * ∇) X Y := f −1 * (∇ f * X f * Y ). Definition 2.1. Choose ∇ ∈ E(M, ω) a symplectic connection, define the map π ∇ π ∇ : Diff 0 (M) → E(M, ω) : f → π ∇ (f ) := ∇ f , where (∇ f ) X Y := (f * ∇) X Y + 1 3 N f * ∇ (X, Y ) + 1 3 N f * ∇ (Y, X) and X, Y ∈ X(M). Remark 2.2.
Similarly, one may also define a map from the space of torsion-free connections with values in the space of symplectic connections.
Our construction of equivariant formal moment map will rely on the next proposition showing the equivariance of π ∇ .
Proposition 2.3. Let ∇ be a symplectic connection, f ∈ Diff 0 (M) and write f * ∇ := ∇ + A f for A f ∈ Γ(T M ⊗ S 2 T * M). Then, writing π ∇ (f ) = ∇ + B f , we have B f (X, Y, Z) = 1 3 ⊕ XY Z ω(X, A f (Y )Z) for X, Y, Z ∈ T M and ⊕ denotes the cyclic sum. Moreover, π ∇ (f • ϕ) = ϕ * π ∇ (f ) for all ϕ ∈ Ham(M, ω). Proof. To compute π ∇ (f ), we first determine N f * ∇ : for X, Y, Z ∈ T M, ω(N f * ∇ (X, Y ), Z) = ((f * ∇) X ω) (Y, Z) = ω(Z, A f (X)Y ) − ω(Y, A f (X)Z) because ∇ is symplectic. Now, B f (X)Y = A f (X)Y + 1 3 N f * ∇ (X, Y ) + 1 3 N f * ∇ (Y, X)
and one checks by direct computation that
ω(X, B f (Y )Z) = 1 3 ⊕ XY Z ω(X, A f (Y )Z). Now, for ϕ ∈ Ham(M, ω), we have ϕ * f * ∇ = ϕ * ∇ + ϕ * A f . Hence, A f •ϕ = A ϕ + ϕ * A f . Remark that ω(·, A ϕ (·)·)
is already completely symmetric as the action of Ham(M, ω) preserves the symplecticity of the connection ∇. By applying the first part of this proposition and the fact that ϕ preserves ω, we have
ω(X, B f •ϕ (Y )Z) = ω(X, A ϕ (Y )Z) + 1 3 ⊕ XY Z ω(ϕ * X, A f (ϕ * Y )ϕ * Z). = ω(X, A ϕ (Y )Z) + ω(X, (ϕ * B f )(Y )Z). It means π ∇ (f • ϕ) = ∇ + A ϕ + ϕ * B f = ϕ * (∇ + B f ) = ϕ * π ∇ (f )
Fedosov star products
We summarise the construction of Fedosov star products [5].
On (M, ω), consider a basis {e 1 , . . . , e 2n } a basis of T x M at x ∈ M for which ω ij := ω(e i , e j ). We consider the dual basis {y 1 , . . . , y 2n } of T * x M. The formal Weyl algebra W x at x ∈ M is the algebra of formal symmetric forms on T x M of the form:
a(y, ν) := ∞ 2k+r=0 ν r a r,i 1 ...i k y i 1 . . . y i k
where a r,i 1 ...i k symmetric in i 1 . . . i k and 2k + r is the total degree, with product • given, for a(y, ν) and b(y, ν) ∈ W x by
(a • b)(y, ν) := exp ν 2 Λ ij ∂ y i ∂ z j a(y, ν)b(z, ν) y=z
We globalise the above over M to get the formal Weyl algebra bundle W := x∈M W x . We also consider differential forms with values in the W by W ⊗ ΛM whose sections are tensors on M that write locally as:
2k+l≥0, k,l≥0,p≥0 ν k a k,i 1 ...i l ,j 1 ...jp (x)y i 1 . . . y i l dx j 1 ∧ . . . ∧ dx jp .
The a k,i 1 ...i l ,j 1 ...jp (x) are symmetric in the i's and antisymmetric in the j's. The space ΓW ⊗ Λ * M is filtered with respect to the total degree
ΓW ⊗ Λ * M ⊃ ΓW 1 ⊗ Λ * M ⊃ ΓW 2 ⊗ Λ * M ⊃ . . . .
Extending fiberwisely the •-product turns ΓW ⊗ Λ * M into an algebra. That is, for a, b ∈ ΓW and α, β ∈ Ω * (M), we define (a ⊗ α)
• (b ⊗ β) := a • b ⊗ α ∧ β. The graded commutator [s, s ′ ] := s • s ′ − (−1) q 1 q 2 s ′ • s where s,
resp. s ′ are of anti-symmetric degree q 1 , resp. q 2 makes W-valued forms a graded Lie algebra.
A symplectic connection ∇ on (M, ω) induces a derivation ∂ of degree +1 (antisymmetric degree) on W-valued forms by :
∂a := da + 1 ν [Γ, a] for a ∈ ΓW ⊗ ΛM, where Γ := 1 2 ω lk Γ k ij y l y j dx i , for Γ k ij the Christoffel symbols of ∇ on a Darboux chart, making ω lk Γ k ij completely symmetric in i, j, l. Setting R := 1 4 ω ir R r jkl y i y j dx k ∧ dx l , for R r jkl := (R(∂ k , ∂ l )∂ j ) r the components of the curvature tensor of ∇, the curvature of ∂ is ∂ • ∂ a := 1 ν [R, a].
To make this connection flat, we consider connections on ΓW of the form
Da := ∂a − δa + 1 ν [r, a],
where r is a W-valued 1-form and δ is defined by
δ(a) := dx k ∧ ∂ y k a = − 1 ν [ω ij y i dx j , a].
The curvature of D is
D 2 a = 1 ν R + ∂r − δr + 1 2ν [r, r] − ω, a . Define δ −1 a pq := 1 p + q y k i(∂ x k )a pq if p + q > 0 and δ −1 a 00 = 0,
where a pq is a q-forms with p y's and p + q > 0. Fedosov showed [5], for any given closed central 2-form Ω, there exists a unique solution r ∈ ΓW ⊗ Ω 1 M with W-degree at least 3 of equation:
R + ∂r − δr + 1 ν r • r = Ω,
and satisfying δ −1 r = 0. Becaue Ω is central for the •-product, it makes D flat. Set D the flat connection obtained as above. Flat sections ΓW D := {a ∈ ΓW|Da = 0} form an algebra for the •-product since D is a derivation. The symbol map is defined by σ : a ∈ ΓW D → a| y=0 ∈ C ∞ (M) [[ν]]. Fedosov showed [5] that σ is a bijection with inverse Q defined by
Q := k≥0 δ −1 (∂ + 1 ν [r, ·]) k .
Hence, the •-product induces a star product * on C ∞ (M) [[ν]], called Fedosov star product.
The following is a technical lemma we will need. Its proof can be found in [6]. In the sequel, to emphasize the dependence of * (resp. r, D and Q) in the choices ∇ and Ω, we will write * ∇,Ω (resp. r ∇,Ω , D ∇,Ω and Q ∇,Ω ) and simply * ∇ (resp. r ∇ ,D ∇ and Q ∇ ) when Ω = 0.
As in [12], our technique relies on a canonical lift of smooth path of symplectic connections and formal series of closed 2-forms to isomorphisms of Fedosov star product algebra, which comes from Fedosov [5].
To state it, we need sections of the extended bundle W + ⊃ W which are locally of the form
2k+l≥0,l≥0 ν k a k,i 1 ...i l (x)y i 1 . . . y i l .
similar to (2.3), with p = 0, but we allow k to take negative values, the total degree 2k + l of any term must remain nonnegative and in each given nonnegative total degree there is a finite number of terms.
Theorem 2.5. Consider smooth paths t ∈ [0, 1] → ∇ t ∈ E(M, ω) and t ∈ [0, 1] → χ t ∈ Ω 2 (M).
Assume that for all t : d dt χ t = dθ t for some smooth path θ t ∈ Ω 1 (M). Then there exists maps B t : ΓW → ΓW defined by
B t a := v t • a • v −1 t
for v t ∈ ΓW + being the unique solution of the initial value problem:
d dt v t = 1 ν h t • v t v 0 = 1 with h t := −(D ∇ t , νχt ) −1 d dt Γ ∇ t + d dt r ∇ t , νχt − νθ t . Moreover, B t (D ∇ 0 , νχ 0 a) = D ∇ t , νχt (B t a) for all a ∈ ΓW so that B t | ΓW D ∇ 0 , νχ 0 : ΓW D ∇ 0 , νχ 0 → ΓW D ∇ t , νχ t
is an isomorphism of flat sections algebras and hence
σ • B t • Q ∇ 0 , νχ 0 : (C ∞ (M)[[ν]], * ∇ 0 , νχ 0 ) → (C ∞ (M)[[ν]], * ∇ t , νχt )(4)
is an equivalence of star product algebras.
The proof can be found in [5]. Similar to [12], h t above depends polynomially on ∇ t , χ t and θ t and their covariant derivatives making the path t → h t smooth. It implies t → v t is smooth as well.
A *-product algebra bundle over Diff 0 (M)
Consider a fixed symplectic connection ∇ on M. We consider the family of Fedosov star products { * ∇ f , νf * χ } f ∈Diff 0 (M ) . Definition 2.6. To the family of Fedosov star products { * ∇ f , νf * χ } f ∈Diff 0 (M ) , we attach a star product algebra bundle V over Diff 0 (M) defined by
V := V(∇) = Diff 0 (M) × C ∞ (M)[[ν]] p → Diff 0 (M),
with fiber p −1 (f ) above f ∈ Diff 0 (M) endowed with the Fedosov star product * ∇ f , νf * χ .
The above bundle depends on the choice of a symplectic connection. Consider two symplectic connections ∇ and ∇ and produce the two star product families { * ∇ f , νf * χ } f ∈Diff 0 (M ) and { * ∇ f , νf * χ } f ∈Diff 0 (M ) as well as the two bundle V(∇) and V( ∇).
We show V(∇) and V( ∇) are isomorphic as * -product algebra bundles. Indeed, we consider the segment t → ∇ t := ∇ + t( ∇ − ∇) of symplectic connections. For all f ∈ Diff 0 (M), we apply Theorem 2.5 to the path t → (∇ t ) f , the constant path of 2-forms t → f * χ and the choice β t = 0, to obtain a * -product equivalence between the fibers of V(∇) and V( ∇) above f .
We go on with the definition of a compatible formal connection D on sections of V. Recall that being a formal connection means that D acts on sections of V by d Diff 0 + β, with β = k=1 ν k β k being a formal series of 1-forms on Diff 0 (M) with values in differential operators on functions of M. The compatibility with respect to the family of star products { * ∇ f , νf * χ } f ∈Diff 0 (M ) is :
D(F * ∇ f , νf * χ G) = D(F ) * ∇ f , νf * χ G + F * ∇ f , νf * χ D(G),
for any sections F, G of V.
Consider a smooth path t → f t ∈ Diff 0 (M) with tangent vector at t given by d dt f t = Y t • f t that is a vector field on M along f t , we will define D through a notion of a parallel lift of the path f t using Theorem 2.5. For this, we use the paths t → ∇ ft and t → f * t χ. A natural candidate for θ t needed in Theorem 2.5 is θ t := f * t ı(Y t )χ, which comes from the computation of d dt f * t χ = df * t ı(Y t )χ. Hence, with these data we obtain equivalences of
* -products σ • B t • Q ∇ f 0 , νf * 0 χ . Definition 2.7. For Y • f ∈ T f Diff 0 (M), with φ Y t the flow of the vector field Y on M, set : • the connection 1-form α ∈ Ω 1 (E(M, ω), ΓW 3 ) by α f (Y • f ) := (D ∇ f , νf * χ ) −1 d dt 0 Γ ∇ φ Y t •f + d dt 0 r ∇ φ Y t •f , ν(φ Y t •f ) * χ − νf * ı(Y )χ , for d dt Γ ∇ φ Y t •f = 1 2 ω lk d dt Γ ∇ φ Y t •f k ij y l y j dx i ,
• the 1-form β with values in formal differential operators:
β f (Y • f )(F ) := 1 ν [α f (Y • f ), Q ∇ f , νf * χ (F )] y=0 , for F ∈ C ∞ (M)[[ν]],
• the formal connection D := d Diff 0 + β.
Proposition 2.8. D is a formal connection on V compatible with the family of Fedosov star product { * ∇ f , νf * χ } f ∈Diff 0 (M ) . Moreover, the parallel transport for D along the path t → f t ∈ Diff 0 (M) is given by the equivalence of star product algebra obtained from Theorem
2.5 with paths t → ∇ ft , t → f * t χ and θ t = f * t ı(Y t )χ, where d dt f t = Y t • f t . Proof.
For the parallel transport property, we consider the equivalence induced by v t as in Theorem 2.5 with the data of the above statement. It means v t is generated by h t with
h t = −(D ∇ f t , νf * t χ ) −1 d dt Γ ∇ f t + d dt r ∇ f t , νf * t χ − νf * t ı(Y t )χ . Starting from F ∈ C ∞ (M)[[ν]]
, seen as an element of the fiber of V above f 0 , we propagate it above the path f t by
F (f t ) := v t • Q ∇ f 0 , νf * 0 χ (F ) • v t y=0
.
We have to check
(D Yt•ft F )(f t ) is zero. That is (D Yt•ft F )(f t ) = d dt v t • Q ∇ f 0 , νf * 0 χ (F ) • v t y=0 + β ft (Y t • f t )(F (f t )), = 1 ν h t , v t • Q ∇ f 0 , νf * 0 χ (F ) • v t y=0 + 1 ν α ft (Y t • f t ), v t • Q ∇ f 0 , νf * 0 χ (F ) • v t y=0 , = 0,
which vanishes by definition of α.
For the compatibility, we refer to the original computation in [1], see also [12] for the corresponding statement.
R(Y • f, Z • f )F := D Y •f (D Z•f F ) − D Z•f (D Y •f F ) − D [Y,Z]•f F,R(Y • f, Z • f )F = 1 ν [R(Y • f, Z • f ), Q(F )] y=0 (5) for R(Y • f, Z • f ) being the 2-form with values in ΓW defined by R f (Y • f, Z • f ) := −νf * (χ(Y, Z)) + d Diff 0 f α(Y • f, Z • f ) + 1 ν [α f (Y • f ), α f (Z • f )], (6)
Moreover,
• R f (Y • f, Z • f ) ∈ ΓW D ∇ f ,νf * χ , • R f (Y • f, Z • f )| y=0 = −νf * (χ(Y, Z)) + O(ν 2 ).
Proof. In Equation (6), the terms involving the connection 1-form α come from standard computation of R. The term −νf * (χ(Y, Z)) doesn't play any role in Equation (5), it is just added to make sure R will take values in a space of flat sections.
To
check R f (Y • f, Z • f ) ∈ ΓW D ∇ f , νf * χ , we compute D ∇ f , νf * χ applied to the RHS of (6). First, because f * (χ(Y, Z)) is a function on M, D ∇ f , νf * χ f * (χ(Y, Z)) = d(f * χ(Y, Z)). Now, we detail the terms of D ∇ f , νf * χ d Diff 0 f α(Y • f, Z • f ).
To do that we use the flows φ Y The last term from the differential of α gives
D ∇ f , νf * χ α · ([Y, Z] • f ) = d dt 0 Γ ∇ φ [Y,Z] t •f + r ∇ φ [Y,Z] t •f , ν(φ [Y,Z] t •f ) * χ − νf * ı([Y, Z])χ (9)
In the computation of
D ∇ f , νf * χ d Diff 0 f α(Y • f, Z • f )
, the terms in Γ and r from the first lines of (7) and (8) are compensated by the terms in Γ and r from (9). The second line of (7) minus the second line of (8) are compensated by
D ∇ f , νf * χ 1 ν [α f (Y • f ), α f (Z • f )],
where we use tha fact that the forms νı(Y )χ and νı(Z)χ are central. What remains is
D ∇ f ,νf * χ R f (Y • f, Z • f ) = −d(f * χ(Y, Z)) − d dt 0 ν(φ Y t • f ) * ı(Z)χ + d dt 0 ν(φ Z t • f ) * ı(Y )χ + νf * ı([Y, Z])χ, = 0.
Finally, because α(·) is of degree at least 3, we have
R f (Y • f, Z • f )| y=0 = −νf * (χ(Y, Z)) + O(ν 2 ).
A formal symplectic form on Diff 0 (M)
Recall that a formal symplectic form on a manifold F is a formal power series of closed 2-forms σ :
= σ 0 + νσ 1 + . . . ∈ Ω 2 (F )[[ν]],
which starts with a symplectic form σ 0 . It is a formal deformation of the symplectic form σ 0 . Acting as in the theory of finite dimensional vector bundles, we consider the trace of the curvature R of V.
A trace for a star product * on a symplectic manifold (M, ω) is a map
tr : C ∞ c (M)[[ν]] → R[ν −1 , ν]] such that tr([F, G] * ) = 0, for all F, G ∈ C ∞ c (M)[[ν]]
. A trace always exits for a given star product on (M, ω) and it is made unique by asking the following normalisation. We ask for the normalised trace to statisfy
tr(F ) = 1 (2πν) m M BF ω m m! , for all F ∈ C ∞ c (U)[[ν]].
The trace can be written as
tr(F ) = 1 (2πν) m M F ρ ω m m! , for ρ ∈ C ∞ (M)[ν −1 , ν]]
, called the trace density. Let us denote by tr * ∇ f , νf * χ the normalised trace of the Fedosov star product * ∇ f , νf * χ and by ρ ∇ f , νf * χ its trace density. Proof. The proof is a standard computation similar to that in [12]. First, one compute directly that d Diff 0 Ω Diff 0 = 0 on right invariant vector fields using the Lemma below which is a particular case of Theorem 3.1 from [8].
Ω Diff 0 (Y • f, Z • f ) := −(2π) m ν m−1 tr * ∇ f , νf * χ ( R f (Y • f, Z • f )| y=0 ), for f ∈ Diff 0 (M) and Y • f, Z • f ∈ T f Diff 0 (M).Lemma 3.4. Let t → f t be a smooth path in Diff 0 (M). Then d dt 0 tr * ∇ f t , νf * t χ (F ) = tr * ∇ f 0 , νf * 0 χ 1 ν [α f 0 ( d dt 0 f t ), Q ∇ f 0 , νf * 0 χ (F )] y=0 .
The fact that Ω Diff 0 is preserved by the action of Ham(M, ω), comes from the naturality of Fedosov construction. Indeed, the pull-back by ϕ ∈ Ham(M, ω) on ΓW ⊗ ΛM maps Fedosov flat connections to Fedosov flat connections as :
ϕ * D ∇ f , νf * χ (ϕ −1 ) * = D ∇ f •ϕ , ν(f •ϕ) * χ ,
where we use ∇ f •ϕ = ϕ * ∇ f from Proposition 2.3. restricts to an isomorphism of flat section algebras ϕ * : ΓW ∇ f , νf * χ → ΓW ∇ f •ϕ , ν(f •ϕ) * χ , where we use ∇ f •ϕ = ϕ * ∇ f from Proposition 2.3. Hence, the normalised traces are related by
tr * ∇ f •ϕ , ν(f •ϕ) * χ = tr * ∇ f , νf * χ • (ϕ −1 ) * .
Also the connection 1-forms are related by
α f •ϕ ((ϕ·) * Y) = ϕ * α f (Y),
for Y ∈ T f Diff 0 (M). From this, we deduce
tr * ∇ f •ϕ , ν(f •ϕ) * χ ( R f •ϕ ((ϕ·) * Y, (ϕ·) * Z)| y=0 ) = tr * ∇ f , νf * χ ( R f (Y, Z)| y=0 ), for all Y, Z ∈ T f Diff 0 (M) which means that Ω Diff 0 is Ham(M, ω)-invariant.
The fact that Ω Diff 0 deforms Ω Diff 0 is a consequence of our computation of the first term of R in Theorem 3.1.
A formal moment map on (Diff 0 (M), Ω Diff 0 )
Let (X, σ) be a manifold equipped with a formal symplectic form σ. Assume there is an action · of a regular Lie group G on X preserving the symplectic form. An equivariant formal moment map is a map
θ : g → C ∞ (X)[[ν]],
for g the Lie algebra of G, such that for all g ∈ G, Y ∈ g and x ∈ X (formal moment map)
ı d dt t=0 exp(tY) · x σ = d X θ(Y)(10)
(equivariance) θ(Ad(g)Y) = (g·) * θ(Y).
Remark 3.5. Regular Lie group means Lie group admitting an exponential map which is not necessarily true for Fréchet Lie group. In our context, the group is the group of Hamiltonian diffeomorphisms with Lie algebra the space C ∞ 0 (M) of functions on M with integral equals to 0. The exponential of F ∈ C ∞ 0 (M) is given by the flow of the Hamiltonian vector field X F . Remark 3.6. This formulation of formal moment map as taking values in formal functions on the manifold comes from the notion of quantum moment maps in deformation quantization [9]. We will use the next two Lemmas.
Lemma 3.8.
[9] Consider a smooth map t ∈ [0, 1] → H t ∈ C ∞ (M), then the derivative of the action of ϕ H· t on ΓW ⊗ ΛM is given by the formula: d dt (ϕ H· t ) * = (ϕ H· t ) * ı(X Ht )D + Dı(X Ht ) +
1 ν −ω ij y i X j Ht + 1 2 (∇ 2 kq H t )y k y q − ı(X Ht )r, · ,
where D is obtained with symplectic connection ∇ and the choice of a series of closed 2-forms.
Lemma 3.9. Let H ∈ C ∞ (M), ∇ ∈ E(M, ω) and f ∈ Diff 0 (M), we have:
Q ∇ f , νf * χ (H) = H − ω ij y i X j H + 1 2 ((∇ f ) 2 kq H)y k y q − ı(X H )r ∇ f , νf * χ + α f (f * X H ).
The proof of the Lemma 3.9 follows by direct computations, using Lemma 3.8 and similar to the corresponding result from [12].
2. 1
1The symplectic structure on Diff 0 (M) We consider the connected component Diff 0 (M) of the group of diffeomorphisms on M. It is a Fréchet manifold modeled on the space X(M) of smooth vector fields on M. At any point f ∈ Diff 0 (M), the tangent space T f Diff 0 (M) is identified to the space Γ(f * T M) of smooth sections of the pullback bundle f * T M. So that a tangent vector at f is a vector field along f .Given Y ∈ X(M), we extend it as the right invariant vector field
Lemma 2. 4 .
4Suppose b ∈ Γ(W ) ⊗ Λ 1 M satisfy Db = 0. Then the equation Da = b admits a unique solution a ∈ Γ(W ), such that a| y=0 = 0, it is given by b = D −1 a := −Q(δ −1 a).
3 A formal moment map picture on Diff 0 (M ) 3.1 The curvature of D The curvature of D evaluated at Y • f and Z • f tangent vector at the point f ∈ Diff 0 (M) is defined by
for a section F of V. In the RHS above we use the natural extension of vector fields along M as right invariant vector fields on Diff 0 (M), as well as the formula for the Lie bracket(2).
Theorem 3 . 1 .
31The curvature of D evaluated at the section F of V and tangent vectors Y • f and Z • f at f ∈ Diff 0 (M) is given by
Consider local equivalences B of * | C ∞ (U )[[ν]] with the Moyal star product * Moyal on U a contractible Darboux chart B : (C ∞ (U)[[ν]], * ) → (C ∞ (U)[[ν]], * Moyal ) so that BF * Moyal BG = B(F * G).
Definition 3 . 2 .
32Define the formal 2-form Ω Diff 0 on Diff 0 (M) by
Theorem 3 . 3 .
33The formal 2-form Ω Diff 0 is a formal symplectic form on Diff 0 (M) deforming Ω Diff 0 . Moreover, the action of Ham(M, ω) on Diff 0 (M) preserves Ω Diff 0 .
Theorem 3 . 7 .
37The map µ : C ∞ 0 (M) → C ∞ (Diff 0 (M))[[ν]] : H → f → (2π) m ν m−1 tr * ∇ f , νf * χ (H) is an equivariant formal moment map for the action of Ham(M, ω) on (Diff 0 (M), Ω Diff 0 ). Moreover, at first order in ν, we recover Donaldson moment map for the action of Ham(M, ω) on (Diff 0 (M), Ω Diff 0 )
t , φ Z t and φ [Y,Z] t of Y , Z and [Y, Z] as vector fields on M as well as the formula for the
∇ φ Z t •f + r ∇ φ Z t •f , ν(φ Z t •f ) * χ , α f (Y • f ) .(8)
differential of forms which is still valid for forms with values in ΓW. We start withSimilarly, we getThe equivariance is immediate from the naturality of the Fedosov construction. The adjoint action by ϕ ∈ Ham(M, ω) on C ∞ 0 (M) is given by the pull-back by ϕ −1 . So that, for all f ∈ Diff 0 (M),To check the formal moment map equation, we proceed as in[12]. One compute, forUsing Lemma 3.4 and after the formulas from Lemma 3.9 and 3.8,.Now, α f (Y • f ) is a 0-form and at y = 0 it remains,By the definition of α, we haveFinally, by Theorem 3.1 giving the formula for R and Equation(3)giving the infinitesimal action of X H on Diff 0 (M), we haveMultiplying both sides of the above equation by the constant (2π) m ν m−1 one obtains the formal moment map equation(10). At first order in ν, one knows the first term of the normalised tracewhich starts by the moment map from[4]. Hamiltonian diffeomorphisms as Hamiltonian automorphisms of the star product[13]. Let H t ∈ C ∞ (M) generating ϕ H· t . Consider the smooth path of connections t → (ϕ H· t ) * ∇ and t → (ϕ H· t ) * χ, for the symplectic connection ∇. Recall that by naturality of the Fedosov construction (ϕ H· t ) * is an isomorphism of flat sections algebra:Then, using Lemmas 3.8 and 3.9, similarly as in[12], one shows:Hence, B t is a Hamiltonian automorphism of * ∇, νχ .Everything we have done in the paper depends on the choice of a symplectic connection. We postpone the analysis of this dependence to a future work.Bibliography
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La Fuente-Gravy, Kähler geometry and deformation quantization with moment maps, ICCM proceedings. A Futaki, L , A. Futaki, L. La Fuente-Gravy, Kähler geometry and deformation quantization with moment maps, ICCM proceedings 2018, 31-66 (2020).
Quantum moment map and obstructions to the existence of closed Fedosov star products. A Futaki, L La Fuente-Gravy, Journ. of Geom. and Phys. 163ArticleA. Futaki, L. La Fuente-Gravy, Quantum moment map and obstructions to the exis- tence of closed Fedosov star products, Journ. of Geom. and Phys. 163, Article 104118 (2021).
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Fuente-Gravy, Futaki invariant for Fedosov star products. L La, Journ. of Sympl. Geom. 175L. La Fuente-Gravy, Futaki invariant for Fedosov star products, Journ. of Sympl. Geom. 17 (5) (2019).
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| [] |
[
"Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data",
"Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data"
] | [
"Amir N Khan \nMax-Planck-Institut für Kernphysik\nPostfach 103980D-69029HeidelbergGermany\n"
] | [
"Max-Planck-Institut für Kernphysik\nPostfach 103980D-69029HeidelbergGermany"
] | [] | We analyze new data from the COHERENT experiment of the coherent neutrino-nucleus scattering to investigate the electromagnetic interactions of neutrinos. With almost double the statistics and precision now, the statistical significance of the observed process has now enhanced to 11.6σ. We derive constraints on the electromagnetic properties of neutrinos using the new COHERENT data. The constraints improve by more than a factor of two compared to the previous bounds. Furthermore, we discuss the unique behavior of the neutrino millicharge at lower energy recoils and show its unique dependence on its interference with the standard model contribution, inverse power of recoil energy and the mass of the target particle in comparison to the other interactions.PACS numbers: xxxxxI. INTRODUCTIONCoherent elastic neutrino-nucleus scattering (CEνNS) is a SM process that was predicted forty years ago [1-4] and was recently observed by the COHERENT experiment[5,6]. The importance of the process ranges from its ability as a precision probe of the SM parameters [1-4, 7-12] to test new interactions at low momentum transfer including its importance for the direction detection of dark matter[13][14][15][16][17][18][19][20][21][22][23][24]. The COHERENT collaboration has updated the result for the aforementioned process by doubling the statistics and the precision by reducing the overall systematic errors to half[25]. In particular, the error on the quenching factor improves from 25% to 4%. Compared to the first result at 6.7σ, the updated significance level has now reached 11.6σ. With this improvement, it is natural to expect a better sensitivity to any new physics or improvement in the limits. To this aim, we analyze the new data to constrain the neutrino electromagnetic properties if they contribute to the CEνNS and derive constraints on neutrino magnetic moment, millicharge, charge radius and neutrino anapole moment using the new COHERENT data.In low energy scattering experiments with low target recoils, the sensitivity to any new physics mainly depends on three factors: (i) whether the new physics interactions interfere with the SM interactions or not, (ii) the proportionality of the new physics coupling strength to the inverse power of the target's recoil energy, and (iii) mass of the target particle. These three factors are | 10.1016/j.nuclphysb.2022.116064 | [
"https://export.arxiv.org/pdf/2201.10578v3.pdf"
] | 255,252,625 | 2201.10578 | 405ce8284f0b7bc3f03b521a792e4a5894a255b7 |
Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data
Amir N Khan
Max-Planck-Institut für Kernphysik
Postfach 103980D-69029HeidelbergGermany
Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data
(Dated: January 9, 2023)PACS numbers: xxxxx
We analyze new data from the COHERENT experiment of the coherent neutrino-nucleus scattering to investigate the electromagnetic interactions of neutrinos. With almost double the statistics and precision now, the statistical significance of the observed process has now enhanced to 11.6σ. We derive constraints on the electromagnetic properties of neutrinos using the new COHERENT data. The constraints improve by more than a factor of two compared to the previous bounds. Furthermore, we discuss the unique behavior of the neutrino millicharge at lower energy recoils and show its unique dependence on its interference with the standard model contribution, inverse power of recoil energy and the mass of the target particle in comparison to the other interactions.PACS numbers: xxxxxI. INTRODUCTIONCoherent elastic neutrino-nucleus scattering (CEνNS) is a SM process that was predicted forty years ago [1-4] and was recently observed by the COHERENT experiment[5,6]. The importance of the process ranges from its ability as a precision probe of the SM parameters [1-4, 7-12] to test new interactions at low momentum transfer including its importance for the direction detection of dark matter[13][14][15][16][17][18][19][20][21][22][23][24]. The COHERENT collaboration has updated the result for the aforementioned process by doubling the statistics and the precision by reducing the overall systematic errors to half[25]. In particular, the error on the quenching factor improves from 25% to 4%. Compared to the first result at 6.7σ, the updated significance level has now reached 11.6σ. With this improvement, it is natural to expect a better sensitivity to any new physics or improvement in the limits. To this aim, we analyze the new data to constrain the neutrino electromagnetic properties if they contribute to the CEνNS and derive constraints on neutrino magnetic moment, millicharge, charge radius and neutrino anapole moment using the new COHERENT data.In low energy scattering experiments with low target recoils, the sensitivity to any new physics mainly depends on three factors: (i) whether the new physics interactions interfere with the SM interactions or not, (ii) the proportionality of the new physics coupling strength to the inverse power of the target's recoil energy, and (iii) mass of the target particle. These three factors are
different for the four types of electromagnetic interactions in the ν−e elastic scattering and in the CEνNS processes, as we will discuss later. For example, the neutrino millicharge is more sensitive to low energy recoils [26][27][28] because it interferes with the SM interactions, and its coupling is proportional to the inverse square of the target recoil energy. We will discuss this aspect in more detail in section V. These kinematical considerations are equally valid for the target recoils in the dark matter scattering [29].
We organize the rest of the paper as follows. In the next section, we discuss the basics of the differential cross-section of the CEνNS in the SM. Then, in Sec. III, we discuss data analysis for the COHERENT setup with the new data. Then, in Sec. IV, we introduce the electromagnetic properties of neutrinos and derive constraints using the new COHERENT data. Next, in Sec.
V, we discuss in detail why millicharge neutrinos are kinematically more special than the other electromagnetic properties of neutrinos. Finally, we provide the conclusion of this work in sec VI.
II. COHERENT ELASTIC NEUTRINO NUCLEUS SCATTERING
At the tree level in the SM, the differential cross-section of the neutrino with flavor 'α' scattering off the spin-0 nucleus of CsI with proton number 'Z' and neutron number 'N ' is given by [1-4, 7, 8],
dσ α dT (E ν , T ) = G 2 F M π Zg V p + N g V n ) 2 1 − T E ν − M T 2E 2 ν F 2 (q 2 ) ,(1)
where 'G F ' is the Fermi constant, 'E ν ' is the energy of the incoming neutrinos, 'T ' is nuclear the recoil energy, q 2 = 2M T is the squared momentum transfer, and 'M ' is the mass of the target nucleus. Here, g V p = (2g V u + g V d ) and g V n = (g V u + 2g V d ), where g V u and g V d are the neutral current coupling constants for the 'up' and 'down' quarks which, in terms of the weak mixing angle 'θ W ' at tree level are given by
g V u = 1 2 − 4 3 sin 2 θ W ,(2)g V d = − 1 2 + 2 3 sin 2 θ W .(3)
We will use sin 2 θ W = 0.23857 ± 0.00005, the low energy (q → 0) value evaluated in MS scheme [30]. Using the low energy value of sin 2 θ W and including the small radiative corrections [30,31], we find the values of the coupling constants g V u = 0.197, g V d = −0.353 for electron neutrinos and g V u = 0.191, g V d = −0.350 for muon neutrinos. In Eq. (1), F (q 2 ) is the nuclear form factor, and we use the Klein-Nystrand form [32] as given in the following
F (q 2 ) = 4πρ 0 Aq 3 [sin(qR A ) − qR A cos(qR A )] 1 1 + a 2 q 2 ,(4)
where ρ 0 is the normalized nuclear number density, A is the atomic number of CsI, R A = 1.2A 1/3 fm is the nuclear radius, and a = 0.7 fm is the range of the Yukawa potential.
III. DATA ANALYSIS
The COHERENT detector receives a prompt signal from the mono-energetic (29.8 MeV) beam of muon-neutrinos (ν µ ) produced from the π + decay at rest (π + → µ + ν µ ) at the Oak Ridge Spallation Neutron Source. Subsequently, continuous fluxes of electron-neutrinos (ν e ) and muonanti-neutrinos (ν µ ) with energy peaks, respectively, around 35 Mev and 52.8 MeV produced in µ + decays (µ + → ν e e +ν µ ) with the characteristic time scale of the muon lifetime, namely 2.2 µ sec, is received. The fluxes are produced from 3.20 × 10 23 protons on target from the liquid mercury. The average production rate of the SNS neutrinos from the pion decay chain is r = 0.0848 neutrinos of each flavor per proton [25].
The detector, located at a distance L = 19.3 m from the source, uses CsI[Na] as a target, where the Na contributes small enough to be neglected. For such a setup, the total number of events of the nuclear recoil in a given energy bin 'i' and neutrino flavor 'α' reads
N i α = N T i+1 T i dT T max 0 dT E max ν E min ν dE ν dσ α dT (E ν , T ) dφ να (E ν ) dE ν E(T )G(T , T ),(5)
where E(T ) is the detection efficiency function, G(T , T ) is the gamma distribution function for the detector energy resolution, T and T denotes the nuclear recoil energy and the reconstructed recoil energy, respectively. Here, N = (2m det /M CsI ) N A is the total number of CsI nucleons,
m det = 14.57 kg, N A is the Avogadro's number, M CsI is the molar mass of CsI, E min ν = M T /2,
M is the mass of the target nucleus, E max ν is the maximum neutrino energy and dφ να (E ν )/dE ν is the flux corresponding to the flavor 'α' [22].
The recent measurement of COHERENT [25] considers the recoiled energy-dependent quenching factor, f q (T ) and measures the energy spectrum in terms of photo-electrons (p.e). Therefore, to calculate the total number of events in a particular bin 'i' of photo-electrons, we use the following relation between the recoil energy and the number of photo-electrons (N p.e )
N p.e. = f q (T ) × T × Y,(6)
where Y = 13.35 photons/keV is the light yield and f q (T ) is taken from [25].
To fit the data of the energy spectrum in Fig. 3 of ref. [25] with the model including the SM and the new physics parameters, we use the following least-square function
χ 2 = 9 i=2 N i obs − N i exp (1 + α) − B i (1 + β) σ i 2 + α σ α 2 + β σ β 2 ,(7)
where N i obs denotes the observed events in the i-th energy bin and σ i is the relevant uncertainty. N i exp is the total number of expected events, which is the sum of the three neutrino flavors as given in eq. (5). B i is the sum of prior predicted beam-related neutron and the neutrino-induced neutron backgrounds in a given energy bin. The first and second penalty terms correspond to the systematic uncertainty of the signal and backgrounds where 'α' and 'β' are the corresponding nuisance parameters. The uncertainty in the signal is σ α = 0.127 and the uncertainty on the total background is σ β = 0.6. The signal uncertainty includes a contribution from the neutrino flux, quenching factor, efficiency, form factor and light yield. We took all information from ref. [25]. One can expect stronger or even weaker constraints with timing information since the only parameter affected by the timing information is the total efficiency. However, since the time and recoil energy is completely uncorrelated and thus efficiencies for the two cases are also uncorrelated [25], our results are thus valid without incorporating the timing information. In order to avoid the possible flavor dependence on the timing information, we restrict our analysis to only flavor-conserving interactions.
IV. ELECTROMAGNETIC INTERACTIONS OF NEUTRINOS IN CEνNS
A. Millicharge neutrinos
The electromagnetic contribution due to the electrically charged neutrinos, parameterized in terms of Q αα , to the SM weak interaction for the coherent neutrino-nucleus (ν − N ) scattering is given by the interactions
L em α = −ie Q αα ν α γ µ ν α + N γ µ N A µ ,(8)
where A µ is mediating electromagnetic field and 'e' is the unit electric charge. Since the electromagnetic interaction terms add coherently to the vector part of the weak interaction, this modifies the weak mixing angle θ W in eq. (3) accordingly as
sin 2 θ W → sin 2 θ W 1 − πα em √ 2 sin 2 θ W G F M T Q αα (9)
where α em is the fine structure constant.
We estimate the statistical significance of the millicharge neutrinos by fitting the two parameters, Q ee and Q µµ . We consider two cases while fitting parameters. First, we fit one parameter at a time and fix the other to zero, and we show the result in the left-side plot of Fig. 1. Next, we fit the two parameters together and show the results in the right-side plot of Fig. 1. One can see from (interference effect) and the dependence on the inverse square of the nuclear recoil energy in the cross-section, the two-parameter fit prefers non-zero values at the best-fit minimum. We obtain the following constraints from the one parameter fits at 90% C.L.,
−0.55 × 10 −7 < Q µµ /e < 0.75 × 10 −7 , −1.10 × 10 −7 < Q ee /e < 3.90 × 10 −7 .(10)
Stronger limits on millicharge neutrinos come from the observational studies [33][34][35][36][37]. The strongest upper limit on the millicharge neutrino is Q ν ≤ 2×10 −15 e from the time arrival dispersion and the energy spread of neutrinos from SN1987A [33]. The laboratory bounds from the ν − e are also several orders of magnitude smaller in size than the bounds of this study [26,[38][39][40][41][42][43][44][45][46].
For instance, the TEXONO experiment derives the limit, Q ν ≤ 2.1 × 10 −12 e. However, we can easily understand this difference from the kinematical considerations, as we will discuss in Sec. V.
However, the robustness of the bounds depends on the experimental details. Exploring how the observational constraints also depend on the kinematic details of the astrophysical environments would be interesting.
B. Neutrino magnetic moment
To understand the importance of interference and the low recoil energy dependence of the millicharge neutrinos, we also compare the neutrino magnetic moment to the same data. In general, for the Majorana (M ) or Dirac (D) neutrinos' couplings to the electromagnetic field strength (F µν ), the magnetic moments appear as [47][48][49][50][51]
L M = − 1 4ν c αL λ M αβ σ µν ν βL F µν or L D = − 1 2ν αR λ D αβ σ µν ν βL F µν ,(11)
where λ X = µ X − i X , which is hermitian for the Dirac neutrinos and antisymmetric for Majorana neutrinos. Only transition magnetic moments are possible for Majorana neutrinos while the flavor diagonal is zero. Because of the unknown final state neutrino flavor in a scattering process, in practice, no distinction between Dirac and Majorana neutrinos is possible. For simplicity, we only consider the flavor diagonal cases for the electron (µ νe ) and muon neutrinos (µ νµ ). In the SM, a non-zero neutrino magnetic moment can arise at the one-loop level, which quantifies as follows, [47]
µ αβ = 3eG F m ν 8 √ 2π 2 ∼ 3 × 10 −19 µ B m ν 1eV(12)
As clear from eq. (11), the helicity of the final state neutrino changes in interaction due to the magnetic moment. Therefore there is no interference with the SM cross-section, and the corresponding contribution adds to the standard model at the cross-section level. We add the following differential cross-section [44] for the neutrino magnetic moment (MM) of neutrino scattering off a spin-0 nucleus to the SM cross-section in eq. (1),
dσ M M α dT (E ν , T ) = πα 2 em µ 2 να m 2 e 1 T − 1 E ν + T 4E 2 ν Z 2 F 2 (q 2 ),(13)
where we take the magnetic moment (µ να ) in units of Bohr's magneton (µ B ), and m e is the electron mass. One can notice that compared to the millicharge of neutrinos as given in eq. (9) in combination with eq. (1), the neutrino magnetic moment has no interference with the SM, and the dependence on the inverse power of the nuclear recoil is only linear in the leading terms.
In this case, we consider two parameters µ νµ and µ νe and fit them to the new COHERENT data [25] using eq. (7), first, with one parameter at a time while keeping the other zero and then fitting two parameters together. The results for both cases are shown respectively in the left-side and right-side plots of Fig. 2. We obtain the following constraints from the one-parameter fits at 90% C.L.,
−0.04 × 10 −8 < µ νµ /µ B < 0.04 × 10 −8 , −0.40 × 10 −8 < µ νe /µ B < 0.40 × 10 −8 .(14)
C. Neutrino charge radius
Radiative corrections induce the neutrino charge radius for neutrinos in the SM. In the general effective electromagnetic vertex of massive neutrinos,νΛ µ νA µ , the neutrino charge radius term is given by [52][53][54], where q is the momentum transfer and F (q 2 ) is a form factor related to the neutrino charge radius
Λ µ (q) = γ µ F (q 2 ) γ µ q 2 r 2 6 ,(15)r 2 ν via r 2 ν = 6 dF ν (q 2 ) dq 2 q 2 =0 .(16)
The SM prediction for the charge radius of neutrino [52][53][54][55][56] is
r 2 να SM = − G F 2 √ 2π 2 3 − 2 ln m 2 α m 2 W ,(17)
where m α is the mass of the charged lepton associated with ν α and m W is the mass of the W ±
In the SM, only flavor diagonal charge radii exist, while in general, transition charge radii are also possible [56]. We consider only the former case whose contribution coherently adds to the SM cross-section. This contribution adds to the coherent cross-section by making the following replacement for the effective weak mixing angle in eq. (3),
sin 2 θ W → sin 2 θ W 1 + πα em 3 √ 2 sin 2 θ W G F r 2 να(19)
Notice that, unlike the millicharge neutrinos in eq. (9), the contribution to the cross-section due to the neutrino charge radius does not directly depend on the nuclear recoil energy and the target mass. Thus one cannot expect enhanced sensitivity at low energy recoils. This was also noted before in refs [26,58]. We fit r 2 νe and r 2 νµ with the new COHERENT data first by taking one parameter at a time and then the two parameters together. We show our results in Fig. 3 and constraints from one parameter-at-a-time at 90% C.L. are in the following.
Λ µ (q) = −γ µ γ 5 F (q 2 ) −γ µ γ 5 q 2 a ν ,(21)
where the form factor 'F (q 2 )' is related to the neutrino anapole moment 'a να ' by the expression,
a ν = − dF ν (q 2 ) dq 2 q 2 =0 .(22)a ν SM = − r 2 ν SM 6 ,(23)
and numerical values accordingly are, a ν eSM = 4.98 × 10 −32 cm 2 , a ν µSM = 2.88 × 10 −32 cm 2 ,
a ν τ SM = 1.80 × 10 −32 cm 2 .(24)
In the case of the CEνNS, one can add the contribution of the neutrino anapole moment by replacing the effective weak mixing angle in eq. (3) as,
sin 2 θ W → sin 2 θ W 1 − πα em 18 √ 2 sin 2 θ W G F a να(25)
Again one can notice that, unlike the neutrino millicharges, the anapole moment does not directly depend on the nuclear recoil energy and the target mass, and one does not expect an enhanced sensitivity at low energy recoils. We fit the parameters a νe and a νµ with the new [66], grand unified theories [67,68] and the extra dimension models [69] do predict the charge quantization. In addition, there are several extensions of the SM that predict new particles with fractional charges [70][71][72][73][74][75]. The fractionally charged particles are also promising candidates for dark matter [75][76][77][78][79][80][81][82][83][84]. Even charges of the SM particles can deviate from the integer multiple of 'e/3', where 'e' is the magnitude of the unit electric charge.
Neutrinos are the favorite candidates for such particles, often called milli-charged neutrinos [85][86][87][88].
The SM is an anomaly-free gauge theory with several accidental global symmetries. In the SM with one generation of fermions, neutrinos are neutral because all the SM anomalies consistently cancel out. However, for the SM with three generations, at least two of the three massless neutrinos in general [87,88]. This dequantization is related to the emergence of anomaly-free gaugeable B − L symmetry [85]. From the theoretical considerations, there is no upper limit available on the millicharge neutrinos. All the known limits are experimental [26,[38][39][40][41][42][43][44][45][46] or observational [33][34][35][36][37]. We plot the second term of eq. (9), writing as
f (Q) = πα em √ 2 sin 2 θ W G F M T Q αα ,(27)
to show the target mass dependence as a function of millicharge with target mass dependence.
The comparison between a CsI nuclear target and any electronic target is shown in the righthand side plot of Fig. 5 We have derived one parameter at-a-time and two-parameter bounds for the two types of neutrino flavors involved in the COHERENT experimental detection. In general, there is an improvement in constraints by a factor of two than previous constraints obtained from the COHERENT old data [44]. Also, the constraints on the electromagnetic interactions of muon neutrino flavor are stronger than those on the electron flavor because of the larger fluxes and, thus, larger statistics of the muon neutrino flavors.
ν τ (COH-Prev) − − − − ν e (Others)
For comparison, we summarize limits from the COHERENT previous data, [5] and neutrinoelectron elastic scattering from several other experiments including reactor, solar, and accelerator in Table I. One can see that as expected the bounds from the COHERENT new data have improved by a factor of two to the previous bounds. On the other hand, as discussed before, the limits from neutrino-electron scattering are stronger by several orders of magnitude in most cases. The limits from the recent XENONnT experiment are currently the strongest among all [89].
We have discussed the importance of millicharge neutrinos in detail. Millicharge neutrinos are important from theoretical and observational points of view. Their observation would be strong evidence for physics beyond the SM and have significant astrophysical implications. It will support neutrinos' Dirac nature and the electric charges' dequantization in the standard model. As shown in Fig. 1, there is a slight preference for the non-zero best value of the neutrino millicharges. This effect is more prominent in the two-parameter case of Fig. 1. This sensitivity can be understood by the unique kinematic behavior of the millicharge neutrino interactions, as discussed in detail. V.
While the interference with the SM and unique dependence of the neutrino millicharge interaction is obvious when we have shown its dependence on the target mass by comparing the electronic recoil in fig. 5. We showed that electronic targets could provide intrinsically stronger limits than nuclear recoils, while their sizes depend on the specific interaction process.
In conclusion, we have obtained new constraints on all electromagnetic interactions of neutrinos using the latest data from the COHERENT experiment. The current and future experiments of the coherent elastic neutrino-nucleus scattering experiments for both spallation neutron or reactor neutrino sources would improve the sensitivity to electromagnetic interaction. In particular, the neutrino millicharge and neutrino magnetic moment are more significant at such lower recoils.
Furthermore, a better understanding of neutrino electromagnetic interactions will help understand other nonstandard neutrino interactions and dark matter in direct detection experiments.
Note Added: When this work was under review, ref. [92] appeared on arXiv which also discusses and constrains the neutrino electromagnetic properties using the same new data set of COHERENT.
ACKNOWLEDGMENTS
The author thanks Kate Scholberg and Dan Pershey for providing helpful information about the data and other details. I also thank Douglas McKay (KU) for his valuable suggestions on the manuscript. The Alexander von Humboldt Foundation financially supported this work.
Fig. 1
1that because of the interplay between the SM and the neutrino millicharge contribution
Fig. 1 .
1Neutrino millicharge (NMC) of the muon and electron flavors from the new COHERENT data for one (left) and two (right) parameter fits. The red star in the right-hand side plot corresponds to the best-fit value. See text for discussion.
Fig. 2 .
2Neutrino magnetic moment (NMM) of the electron and muon neutrino from the new COHERENT data for the one (left) and two (right) parameters fits. The red star in the right-hand side plot corresponds to the best-fit value. See text for discussion.
boson. The numerical values for the corresponding flavor of neutrinos are [52-57] r 2 νe SM = −0.83 × 10 −32 cm 2 , r 2 νµ SM = −0.48 × 10 −32 cm 2 , r 2 ντ SM = −0.30 × 10 −32 cm 2 .
− 0 .
060 × 10 −30 < r 2 νµ /cm 2 < 0.05 × 10 −30 ,−0.67 × 10 −30 < r 2 νe /cm 2 < 0.10 × 10 −30 .(20)D. Neutrino anapole momentIf neutrino carries a non-zero charge radius, it can also have a non-zero anapole moment[55, 59- 65]. It determines the correlation between the spin and charge distributions of neutrinos and has the same dimensions as the charge radius. In the general vertex for electromagnetic interactions, νΛ µ νA µ , the anapole term is defined by[61,62,65]
Fig. 3 .
3Neutrino charge radius (NCR) of the electron and muon neutrino from the new COHERENT data for the one (left) and two (right) parameters fits. The red star in the right-hand side plot corresponds to the best-fit value. See text for discussion.COHERENT data. First, we fit one parameter at a time while fixing the other to zero and then fit the two parameters together. We show our obtained results inFig. 4, and constraints from this analysis in the case of parameter-at-a-time at 90% C.L. are the following.−0.30 × 10 −30 < a νµ /cm 2 < 3.7 × 10 −30 , −0.60 × 10 −30 < a νe /cm 2 < 4.0 × 10 −30 . (26) V. THEORETICAL AND EXPERIMENTAL SIGNIFICANCE OF MILLICHARGED NEUTRINOS The electric charge quantization comes from its empirical observation. The standard model (SM) does not predict the origin of electric charge quantization. However, several theories beyond the SM like those with magnetic monopoles
Fig. 4 .
4Neutrino anapole moment (NAM) of the electron and muon neutrino from the new COHERENT data for the one (left) and two (right) parameters fits. The red star in the right-hand side plot corresponds to the best-fit value. See text for discussion. could be electrically charged. This dequantization leads to the emergence of three gaugeable U (1) symmetries, L e − L µ , L µ − L e and L e − L τ . Only one of the three differences can be anomaly-free and the corresponding difference in each case adds to the hypercharge of the SM. Thus, it leads to the fractional charges of neutrinos and to the dequantization of electric charge at least in the lepton sector [87, 88]. For massive neutrinos, the existence of fractional charges of neutrinos depends on the nature of their masses. If neutrinos are Majorana fermions, then new anomalies arise, but they cancel out, leaving neutrinos neutral [85, 86]. Consequently, the electric charge remains quantized in the minimally extended SM. On the other hand, Dirac neutrinos with three right-handed partners, singlets under SU (3) c × SU (2) L , are electrically charged. These small electric charges of Dirac neutrinos modify the charges of charged leptons and quarks due to their hypercharges' dependence on the right-handed neutrinos' hypercharges. It leads to the dequantization of electric charges
Neutrinos are electrically neutral in the SM. In several extensions of SM with Dirac neutrinos, the electric charges of the neutrino can arbitrarily take any values. Currently, all information about it comes from experiments. No theoretically predicted value is available. More importantly, the limits strongly depend on the process under consideration. Three critical factors contribute to neutrino millicharges. These are the interference of their amplitude with the standard model, inverse power dependence on the recoil energy, and the size of the target mass. On the other hand,
Fig. 5 .
5Effects from interference and kinematics of the process. Left : The ratio between the millicharge, magnetic moment, charge radius, anapole moment cross-sections plus the SM cross-sections and the SM cross-section as a function of the millicharge, magnetic moment, charge radius, and anapole moment at fixed neutrino energy and nuclear recoil energy. Right : The millicharge term in Eq. (9) as a function of Q for the target nuclear and electronic masses. See the text for the discussion. the other electromagnetic properties partially depend on these factors. For instance, there is the interference of the neutrino charge radius and anapole moment with the SM, but there is no inverse recoil energy and target mass dependence. Likewise, for the neutrino magnetic moment, there is no interference with SM and no dependence on the target mass, while there is only one power of inverse recoil energy dependence.We show the millicharge dependence on the interference and the inverse power of the recoil energy for the relevant fixed neutrino energy (30 MeV) and recoil energy(11 keV) in the left-hand side plots ofFig. 5. Here, the ratio between the new physics plus the SM differential crosssection and the SM alone cross-section for the four cases was taken as a function of the neutrino millicharge, magnetic moment, charge radius, and anapole moment. In the region of interest for CEνNS, millicharge interactions compete with the SM cross-section up to 10 −7 e and drop to zero 3.3 × 10 −7 e at the start and start growing afterward. In contrast, the magnetic moment starts deviation from the SM at 10 −9 µ B , the charge radius starts deviation from the SM at 10 −31 cm 2 and anapole moment starts deviation from the SM at around 10 −31 cm 2 .
TABLE I .
Ibecause the SM CEνNS cross-section is directly proportional to the target mass times the sum of the squares of the total number of protons and neutrons. However, the electronic targets still dominate, and there are at least two orders of magnitude intrinsically stronger constraints on millicharge neutrinos in this case than the CEνNS. Notice that this property does not hold for the other three electromagnetic properties because of no inverse dependence on the target mass.Thus, the difference in the limits strongly depends on the difference between the electronic and the nuclear masses, no matter how different the precision of the two types of experiments is.In this work, we have analyzed the new data from the COHERENT experiment to derive limits on four types of neutrino electromagnetic interactions: the neutrino millicharge, magnetic moment, charge radius, and neutrino anapole moment. In the latest update of COHERENT experimental on the detection of coherent neutrino-nucleus elastic scattering, the collaboration has reported almost double statistics and improved the precision to twice their first result. Furthermore, the statistical significance of the observed process has now enhanced to 11.6σ. This improvement motivates the improved constraints on any new physics. Finally, following our previous work, we have analyzed the new data to constrain neutrino electromagnetic interactions.90% C.L. bounds on neutrino magnetic moment, neutrino millicharges, charge radius and anapole
moment from COHERENT new data (COH-2021), COHERENT previous (COH-prev) data [27] and other
laboratory experiments. Bounds for XENONOnT were taken from ref. [89], Borexino from ref. [90], and
solar from ref. [58] while all other bounds for other experiments were taken from ref. [91].
mass in comparison to electric recoils. This effect gets relatively weaker at the cross-section level
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| [] |
[
"B eppoSA X follow -up search for the X -ray afterglow of G R B 970111",
"B eppoSA X follow -up search for the X -ray afterglow of G R B 970111"
] | [
"A St R O N O M Y A N ",
"D A St ",
"R O Ph ",
"Y Sic ",
"S ",
"M Feroci ",
"L A A Ntonelli ",
"M G Uainazzi ",
"J M M Uller ",
"E C Osta ",
"L P Iro ",
"J H Eise ",
"J J M In 't Zand ",
"F Frontera ",
"D D Al F "
] | [] | [] | ium e 4 , L. N icastro 4 , M . O rlandini 4 , E . P alazzi 4 , G . Zavattini 5 , P. G iom m i 2 ,A .N .P arm ar 6 ,A . O w ens 6 ,A .J.C astro-T irado 7 ,M .C .M accarone 8 , and R .C .B utler 9 1 Isti tuto diA stro si ca Spazi al e,C N R ,V i a Fosso delC aval i ere,I-00133 R om a,Ital y 2 B eppoSA X Sci enti c D ata C enter,V i a C orcol l e 19,I-00131 R om a,Ital y 3 Space R esearch O rgani zati on i n the N etherl ands,Sorbonnel aan 2,3584 C A U trecht,T he N etherl ands 4 Isti tuto Tecnol ogi e e Studi o R adi azi oniExtraterrestri ,C N R ,V i a G obetti101,I-40129 B ol ogna,Ital y 5 D i parti m ento diFi si ca,U ni versi t a diFerrara,V i a Paradi so 11,I-44100 Ferrara,Ital y 6 A strophysi cs D i vi si on,Space Sci ence D epartm ent ofESA ,EST EC ,P. O .B ox 299,2200 A G N oordw i jk,T he N etherl ands 7 Laboratori o de A stro si ca Espaci aly Fi si ca Fundam ental ,IN TA ,M adri d,Spai n 8 Isti tuto Fi si ca C osm i ca e A ppl i cazi oniInform ati ca,C N R ,V i a U .La M al fa 153,I-90146 Pal erm o,Ital y 9 A genzi a Spazi al e Ital i ana,V i al e R egi na M argheri ta 202,I-00162 R om a,Ital y | null | [
"https://export.arxiv.org/pdf/astro-ph/9803015v1.pdf"
] | 17,855,312 | astro-ph/9803015 | 97830270f87f5f880f0e156cdf85e311a0ad28e5 |
B eppoSA X follow -up search for the X -ray afterglow of G R B 970111
3 Mar 1998 25. 12. 2021
A St R O N O M Y A N
D A St
R O Ph
Y Sic
S
M Feroci
L A A Ntonelli
M G Uainazzi
J M M Uller
E C Osta
L P Iro
J H Eise
J J M In 't Zand
F Frontera
D D Al F
B eppoSA X follow -up search for the X -ray afterglow of G R B 970111
3 Mar 1998 25. 12. 2021arXiv:astro-ph/9803015v1 A & A m anuscript no. (w i l lbe i nserted by hand l ater) Y our thesaurus codes are: 011 (13.07.1;13.25.1)
ium e 4 , L. N icastro 4 , M . O rlandini 4 , E . P alazzi 4 , G . Zavattini 5 , P. G iom m i 2 ,A .N .P arm ar 6 ,A . O w ens 6 ,A .J.C astro-T irado 7 ,M .C .M accarone 8 , and R .C .B utler 9 1 Isti tuto diA stro si ca Spazi al e,C N R ,V i a Fosso delC aval i ere,I-00133 R om a,Ital y 2 B eppoSA X Sci enti c D ata C enter,V i a C orcol l e 19,I-00131 R om a,Ital y 3 Space R esearch O rgani zati on i n the N etherl ands,Sorbonnel aan 2,3584 C A U trecht,T he N etherl ands 4 Isti tuto Tecnol ogi e e Studi o R adi azi oniExtraterrestri ,C N R ,V i a G obetti101,I-40129 B ol ogna,Ital y 5 D i parti m ento diFi si ca,U ni versi t a diFerrara,V i a Paradi so 11,I-44100 Ferrara,Ital y 6 A strophysi cs D i vi si on,Space Sci ence D epartm ent ofESA ,EST EC ,P. O .B ox 299,2200 A G N oordw i jk,T he N etherl ands 7 Laboratori o de A stro si ca Espaci aly Fi si ca Fundam ental ,IN TA ,M adri d,Spai n 8 Isti tuto Fi si ca C osm i ca e A ppl i cazi oniInform ati ca,C N R ,V i a U .La M al fa 153,I-90146 Pal erm o,Ital y 9 A genzi a Spazi al e Ital i ana,V i al e R egi na M argheri ta 202,I-00162 R om a,Ital y
R ecei ved 13 January 1998;accepted 3 M arch 1998 A bstract. T heB eppoSA X satel l i tehasrecentl y opened a new way towards the sol uti on of the l ong standi ng gam m a-ray bursts'(G R B s) eni gm a,provi di ng accurate coordi nates few hours after the event thus al l ow i ng for m ul ti wavel ength fol l ow -up observati onalcam pai gns.
T he B eppoSA X N arrow Fi el d Instrum ents observed the regi on of sky contai ni ng G R B 970111 16 hours after the burst. In contrast to other G R B s observed by B eppoSA X no bri ght aftergl ow was unam bi guousl y observed.A fai nt source (1SA X J1528. 1+ 1937)i s detected i n a posi ti on consi stentw i th the B eppoSA X W i de Fi el d C am era posi ti on, but unconsi stent w i th the IPN annul us. W hether 1SA X J1528. 1+ 1937 i s associ ated w i th G R B 970111 or not,the X -ray i ntensi ty ofthe aftergl ow i s si gni cantl y l ower than expected,based on the properti es of the other B eppoSA X G R B aftergl ow s. G i ven thatG R B 970111 i sone ofthe bri ghtestG R B sobserved, thi s i m pl i es that there i s no obvi ous rel ati on between the G R B gam m a-ray peak ux and the i ntensi ty ofthe X -ray aftergl ow .
K ey w ords: G am m a-rays:bursts;G am m a-rays:observati on;X -rays:observati on Send o print requests to:feroci @ saturn. i as. rm . cnr. i t
1.Introduction
T he com prehensi on of the nature of the G am m a-R ay B ursts (G R B s) i s a l ong-standi ng probl em of a worl dw i de sci enti c com m uni ty si nce the announcem ent of thei r di scovery (K l ebesadeletal .1973).M any observati onal(Fi shm an & M eegan 1995)and theoreti cal(Lam b 1995;Paczynski1995) e orts di d not succeed i n understandi ng the ori gi n of G R B s. T he l aunch of the B ep-poSA X satel l i te (B oel l a etal .1997a) revol uti oni zed the el d, openi ng a new observati onal w i ndow soon after the G R B event.D ue to i ts G am m a R ay B urst M oni tor (G R B M ,40{700 keV , Frontera etal .1997a;Ferocietal . 1997a) and i ts W i de Fi el d C am eras (W FC s,2{26 keV , Jager etal . 1997) thi s satel l i te i s capabl e of detecti ng G R B s i n the gam m a-ray band and accuratel y l ocal i zi ng them i n X -rays through a coded m ask proporti onal counter.
Fi ve G R B s,am ongst those si m ul taneousl y detected by the G R B M and the W FC s,were prom ptl y anal yzed, al l ow i ng m ul ti wavel ength fol l ow -up observati onalcampai gns. T he rst resul t i s the B eppoSA X di scovery of the X -ray aftergl ow of G R B 970228 (C osta etal . 1997, C osta etal .1997a) and the di scovery ofa rel ated opticaltransi ent by ground-based tel escopes (van Paradi js etal .1997).Furtherresul tshave been achi eved w i th the detecti on of the X -ray aftergl ow s of G R B 970402 (Ferocietal .1997b,Pi ro etal .1997a),G R B 970508 (C osta etal .1997cetal . ,Pi ro etal .1997b) and G R B 971214 (H ei se etal .1997a,A ntonel l ietal .1997).From G R B 970508 an i ndi cati on of an extragal acti c ori gi n has been deri ved through the detecti on of an opti cal transi ent (B ond, 1997; D jorgovski etal . 1997) and the m easurem ent of i ts opti calspectrum (M etzger etal .1997),provi di ng a l owerl i m i t of0. 835 forthe redshi ftofthe possi bl e G R B opti calaftergl ow .
O ne ofthe m ost i ntri gui ng m ysteri es ofG R B em i ttersi spossi bl y sol ved,butthe overal lpi cture i sfarfrom cl ear.In fact,outofthe ve eventsforw hi ch B eppoSA X perform ed rapi d fol l ow up searches ofa X -ray counterpart, one (G R B 970111) has gi ven a resul t that i s si gni cantl y di erent from the other four.T he cel esti all ocati on ofG R B 970111 was observed by B eppoSA X just 16 hours after the G R B event,and no unam bi guous evi dence for an X -ray aftergl ow was found. A new fai nt source(1SA X J1528. 1+ 1937)wasdetected ata ux l evel that i s m uch l ower than that expected on the basi s of the properti esofthe otherG R B sl aterobserved by B ep-poSA X . H ere we present thi s detecti on, di scuss i ts associ ati on w i th G R B 970111 and the di versi ty from the other four B eppoSA X G R B s.
2.G R B M and W FC detection
O n 1997 11 January, 09: 43: 59. 99 U T the G R B M onboard B eppoSA X was tri ggered by an i ntense gam m aray burst, show i ng a m ul ti peak structure and l asti ng 43 s (C osta etal . 1997b). T he peak i ntensi ty was (3: 9 0: 3) 10 6 erg cm 2 s 1 i n the energy range 40{ 700 keV .T hi s G R B wasal so detected by the W FC uni t 2, w i th a si m i l ar ti m e pro l e structure but a l onger durati on (see Fi g. 1). T he 2{10 keV peak ux was (4: 1 0: 7) 10 8 erg cm 2 s 1 .T he uence ofthe event i n 40{700 keV was (4: 14 0: 31) 10 5 erg cm 2 w hi l e i n 2{10 keV i twas(1: 6 0: 1) 10 6 erg cm 2 .In Fi g.1 the gam m a-ray (G R B M ) and X -ray (W FC )l i ghtcurves ofthe event are show n.
G i ven thatG R B 970111 wasone ofthe earl i estX -ray transi entsdetected i n the W FC atthe Q ui ck Look A nalysi s,the posi ti on ofthe event was prom ptl y di stri buted w i th a 10 0 error radi us (C osta etal .1997b),som ew hat worse than that obtai nabl e from the i nstri nsi c capabi li ti es ofthe W FC s.A fter about 20 days a revi sed error box oftheG R B 970111l ocati on w i th a 3 0 errorradi uswas produced,centred ata posi ti on 4 0 . 2 apartfrom the centroi d oftheprevi ousone(i n ' tZand etal .1997 T he l atestW FC errorbox onl y i ncl udesan unknow n X -ray source,1SA X J1528. 1+ 1937.T he fal se col ouri mage obtai ned from the M EC S i s show n i n Fi g. 2. T he i m age show sthe W FC errorbox,i ntersected by the IPN annul us,together w i th the M EC S source errorbox. T he data anal ysi s of the M EC S i m age, perform ed w i th the X im age (G i om m i etal . 1991), gi ves the posi ti on of the new source at R A = 15 h 28 m 09 s : 2 and D ecl . = + 19 37 0 02", w i th a 60" error radi us (90% condence l evel ). T he probabi l i ty that thi s detecti on i s due to a background uctuati on i s of the order of 10 6 .T he count rate of 1SA X J1528. 1+ 1937 i s (1: 8 0: 5) 10 3 countss 1 i n the M EC S (2{10 keV ) and (7 3) 10 4 countss 1 i n the LEC S (0. 1{2 keV ).Taki ng i nto account the vi gnetti ng correcti on for the oaxi s posi ti on, and assum i ng a C rab-l i ke energy spectrum , the above count rates correspond to uxes of (1: 2 0: 3) 10 13 erg cm 2 s 1 i n the 2{10 keV energy range and (8 4) 10 14 erg cm 2 s 1 i n the 0. 1{2 keV energy range.
T he M EC S error box of the new source 1SA X J1528. 1+ 1937 i s al m ost enti rel y contai ned w i thi n the W FC error box of G R B 970111. C onsi deri ng the 99% con dence IPN annul us the source 1SA X J1528. 1+ 1937 i s at a posi ti on onl y m argi nal l y consi stent w i th G R B 970111. T herefore, i f we use the reduced W FC -IPN error box,the upper l i m i t to the 2{10 keV ux of 1: 6 10 13 erg cm 2 s 1 (3 ).
In the context of the possi bl e associ ati on of 1SA X J1528. 1+ 1937 w i th G R B 970111 i t i s i nteresti ng to note that di vi di ng the N FIobservati on i nto three ti m e i nterval sconsi sti ng ofthe rst10 ks,the fol l ow i ng 15 ks,and the l ast 26 ks of exposure ti m e, gi ves the count rates l i sted i n Tabl e 1.Even i f the count rate i s rather l ow , consi deri ng the com bi nati on of the tem poraland positi onalcoi nci dences and the i ndi cati on ofa decayi ng behavi or,then the possi bi l i ty of a random occurrence of 1SA X J1528. 1+ 1937 i n the error box ofG R B 970111 i s hi gher than the 3% deri ved from the source stati sti cs ofthe A SC A G IS (C agnonietal .1997).
4.D iscussion and conclusions
T heB eppoSA X fol l ow -up observati on ofthe errorbox of G R B 970111 was the rst prom pt fol l ow -up observati on ofa G R B ever perform ed by an X -ray satel l i te.B efore B eppoSA X the ti m e-scal e of a possi bl e X -ray em i ssi on from G R B rem nantswascom pl etel y unknow n.T hi s rst basi cal l y non-detecti on, therefore,coul d onl y be i nterpreted as an upper l i m i t to the ti m e-scal e of the decl i ne of an X -ray aftergl ow or to i ts ux. N ow , w i th the detecti on of the X -ray aftergl ow s of G R B 970228, G R B 970402,G R B 970508 and G R B 971214,B eppoSA X has set up a new scenari o i n w hi ch G R B 970111 seem s m i spl aced.A l so the detecti on ofthe X -ray aftergl ow of a G R B (G R B 970828,R em i l l ard etal .1997;M urakam i etal .1997) by the R ossi X T E and the A SC A satel l i tes supportsthe generalfram ework forthe G R B s'aftergl ow bui l t by B eppoSA X . G R B 970228,G R B 970402 and G R B 970828 showed a si m i l ar behavi or,w i th a fadi ng X -ray counterpart conti nuousl y decayi ng from the G R B m ai n em i ssi on i nto the aftergl ow fol l ow i ng an approxi m ate t 1:3 l aw .In the caseofG R B 970228,thespectralanal ysi s (Frontera etal . 1997b) con rm s the conti nui ty between the l atest G R B em i ssi on and the X -ray counterpart detected after few hours. T hi s tem poral behavi our coul d be expl ai ned i n the fram ework of the rebal l m odel (C aval l o & R ees 1978;R ees & M eszaros 1992) as a hi ghl y radi ati ve expansi on ofa rel ati vi sti c shel l .G R B 970508 has show n a X -ray counterpartw hosedecay i sm orecom pl i cated than the above three.T he above m odelcoul d sti l laccountfor thi s di erent behavi or, but i t needs to i nvoke a non-uni form surroundi ng m edi um ,w i th a densi ty scal i ng as r 2 (V i etri1997).
W hether 1SA X J1528. 1+ 1937 i s rel ated to G R B 970111 or not,thi s gam m a-ray burst had a m uch fasterdecay than observed forany ofthe others.In order to m ake thi s cl ear,we com pare a hypotheti c power-l aw decay ofG R B 970111 w i th the\typi cal " power-l aw decay ofG R B 970228 reported i n C osta etal .(1997a).T herefore,i n Fi g.3 we assum e that the new X -ray source i s associ ated w i th the G R B and i m pose a power-l aw decay ofthe aftergl ow starti ng from the W FC m ean ux at a ti m e centred on the G R B X -rays durati on.T he needed power-l aw i ndex i s -1. 5. Tryi ng to extract G R B 970111 from the group as an i ntri nsi cal l y di erentevent,we note thati ts gam m a-ray uence i s about m ore than three ti m es l arger than the l argest of the other three. O n the other hand, even i f thi s G R B i s ofthe \N o H i gh Eenergy" type (that i s,i t show s onl y weak em i ssi on above 300 keV ,Pendl eton et al .1997),the rati o between the X -ray (2{10 keV ) and gam m a-ray (40{700 keV ) uences i s about 4% , to be com pared to 20% (2{10 keV ) for G R B 970228 (Frontera etal .1997b),5% (2{10 keV ) for G R B 970402 (N i castro etal .1997) and 40% (2{26 keV ) for G R B 970508 (Pi ro etal .1997b).G R B 970111 appears therefore as the one (together w i th the A pri l event) w i th the l ess e ci ent l ow X -rays m echani sm for energy rel ease.Furtherm ore, no opti calsourcewasfound i n theW FC errorbox changi ng i ts i ntensi ty m ore than 0. 5 m agni tudes at a l evelof B = 23 and R = 22. 6 from about 19 hours to about one m onth l ater (C astro-T i rado etal .1997;G orosabeletal . 1998).A radi o search at 1. 4 G H z (Frai letal .1997)and at m i l l i m etri c wavel ength (Sm i th etal . 1997) di d not nd a counterpartto 1SA X J1528. 1+ 1937.T hese resul ts support the i dea that the opti cal ,radi o and m i l l i m etri c channel s are une ci ent as wel l .Si nce G R B 970111 was one ofthe bri ghtesteventseverdetected i n gam m a-rays, one m ay concl ude that i ts gam m a-ray channelwas eci ent enough to di ssi pate m ost ofthe energy generated i n the burst.
A l ternati ve i nterpretati ons ofthe l ack ofX /opti cal -/radi o aftergl ow ofthe G R B 970111 m ay be ei thera very rapi dl y evol vi ng aftergl ow ,w i th a decay l aw faster than observed i n theotherB eppoSA X G R B aftergl ow s,orthe absence of an aftergl ow source.T he form er hypothesi s woul d be i n agreem entw i th the m odelby Tavani(1997) ofa decay behavi orrepresented by a powerl aw w i th i ndex 21=8 due to the observati on i n a xed energy band (2{10 keV ) of a synchrotron em i ssi on spectrum w i th a rapi dl y evol vi ng cri ti calfrequency.A l ternati vel y,thel attersi tuati on coul d bedue,asan exam pl e,to thescenari o i n w hi ch the event that caused the G R B occurred i n a regi on i n w hi ch the i nterstel l ar m edi um densi ty i s l ow enough (perhaps the externalregi ons ofa host gal axy) to justi fy the absence ofan externalshock,possi bl y responsi bl e for the aftergl ow em i ssi on i n the other cases (K atz & Pi ran 1997).
A cknow l edgem ents. T hi sresearch i ssupported by the Ital i an Space A gency (A SI) and C onsi gl i o N azi onal e del l e R i cerche (C N R ).B eppoSA X i s a m ajor program of A SI w i th parti ci pati on of the N etherl ands A gency for A erospace Program s (N IV R ).A l lauthors w arm l y thank the extraordi nary team s ofthe B eppoSA X Sci enti c O perati on C enterand O perati on C ontrol C enter for thei r enthusi asti c support to the G R B program .
R eferences
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).T henew posi ti on was R . A . = 15 h 28 m 15 s and D ecl . = + 19 36 0 . 3 (equi nox 2000. 0).Very recentl y theW FC hardwareteam agai n i m proved the i nstrum ent cal i brati on, further reduci ng the error box area to an i rregul ar ci rcl e of 1 0 . 8 radi us (99% con dence) (H ei se etal .1997),centred at R . A .= 15 h 28 m 11 s and D ecl . = + 19 35 0 . 9.T hi s new regi on i scontai ned w i thi n the previ ousone,buti scentred about 1 0 apart. T he Interpl anetary N etwork used the del ay i n the G R B arri valti m es between the i nterpl anetary U l ysses m i ssi on and the C om ptonG R O and B eppoSA X near-ig. 1. B eppoSA X G R B M (40{700 keV ) and W FC (2{26 keV ) l i ght curves ofG R B 970111 Earth satel l i tes(G al am a etal .1997)to obtai n a narrow stri p ofpossi bl e arri valdi recti onsi n the sky.T hi sal l ow s the reducti on ofthe G R B error box to a porti on ofthe W FC errorci rcl e. 3.B eppoSA X detection of 1SA X J1528.1+ 1937 T he earl i est (10 0 ) error box ofG R B 970111 was i m aged w i th the narrow el d X -ray i nstrum ents (N FI) onboard B eppoSA X :the Low Energy C oncentratorSpectrom eter (LEC S,0. 1{10 keV ,Parm ar etal .1997) and the three M edi um Energy C oncentratorSpectrom eters(M EC S,2{ 10 keV ,B oel l a etal .1997b).T hi sTargetofO pportuni ty observati on wasstarted 59, 400 safterthe G R B M tri gger ti m e,from 12 January 02: 14 to 13 January 06: 01 U T ,for a totalnetexposureti m eof52, 139 sw i th theM EC S and 11, 594 s w i th the LEC S (the l atter bei ng operated onl y duri ng satel l i te ni ght-ti m e).A ttheti m ew hen theN FIobservati on wasperform ed the W FC i m proved error box was not avai l abl e and thereforeany sourcei ncl uded i n theerrorbox regi on was a potenti alcounterpart for the G R B 970111.T wo rel ati vel y bri ghtX -ray sourcesweredetected i n the 10 0 error box by theB eppoSA X /N FI(B utl eretal .1997),resol ved i nto three sourcesi n the R O SAT A l lSky Survey(Voges etal .1997).O ne ofthem ,R X J152845+ 1944. 5,was al so i nsi de the earl y i ntersecti on ofthe W FC error box and the IPN errorstri p (H url eyetal .1997).A pecul i arradi o source was detected w i th the V LA (Frai letal .1997)i n a posi ti on coi nci dent w i th thi s X -ray source.T he nal W FC errorbox,however,excl udesthi ssourceaspossi bl e counterpartto G R B 970111.
F ig. 2 .
2Fal se col our i m age of the M EC S observati on of the G R B 970111. T he W FC error box i s show n as an i rregul ar ci rcl e contai ni ng the M EC S error ci rcl e of the X -ray source 1SA X J1528. 1+ 1937.T he l atterl i es outsi de the regi on ofthe W FC error box i ntersected by the IPN annul us.O n the top l eft the source R X J152845+ 1944. 5 i s cl earl y detected
T able 1 .
12{10 keV ux vari ati on of 1SA X J1528. 1+ 1937. T he ori gi n ofthe el apsed ti m e i sthe G R B 970111 tri ggerti
F
ig. 3. X -ray (2{10 keV ) decay l aw of the candi date counterpart of G R B 970111, com pared to G R B 970228. T he dot-dashed and the sol i d hori zontall i nes are the m ean X -ray ux for G R B 970228 and G R B 970111, respecti vel y. T he i ncl i ned dashed l i ne i s the decay l aw suggested by C osta et al . (1997a) for G R B 970228. T he i ncl i ned sol i d l i ne show s the pow er-l aw i ndex, 1. 5, needed for connecti ng the W FC G R B 970111 m ean ux and the 1SA X J 1528. 1+ 1937 ux A l ternati vel y,assum i ng that 1SA X J1528. 1+ 1937 i s not rel ated to G R B 970111 we can deri ve a l ower l i m i t to the power-l aw i ndex by usi ng the upper l i m i t ofthe M EC S ux i n the regi on of sky de ned by the error box,to obtai n a val ue very si m i l arto the 1. 5 val ue gi ven above.
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| [] |
[
"ChromoSkein: Untangling Three-Dimensional Chromatin Fiber With a Web-Based Visualization Framework",
"ChromoSkein: Untangling Three-Dimensional Chromatin Fiber With a Web-Based Visualization Framework"
] | [
"Matúš Talčík ",
"Filip Opálený ",
"Tereza Clarence ",
"Katarína Furmanová ",
"Jan Byška ",
"Barbora Kozlíková ",
"David Kouřil "
] | [] | [] | We present ChromoSkein, a web-based tool for visualizing three-dimensional chromatin models. The spatial organization of chromatin is essential to its function. Experimental methods, namely Hi-C, reveal the spatial conformation but output a 2D matrix representation. Biologists leverage simulation to bring this information back to 3D, assembling a 3D chromatin shape prediction using the 2D matrices as constraints. Our overview of existing chromatin visualization software shows that the available tools limit the utility of 3D through ineffective shading and a lack of advanced interactions. We designed ChromoSkein to encourage analytical work directly with the 3D representation. Our tool features a 3D view that supports understanding the shape of the highly tangled chromatin fiber and the spatial relationships of its parts. Users can explore and filter the 3D model using two interactions. First, they can manage occlusion both by toggling the visibility of semantic parts and by adding cutting planes. Second, they can segment the model through the creation of custom selections. To complement the 3D view, we link the spatial representation with non-spatial genomic data, such as 2D Hi-C maps and 1D genomic signals. We demonstrate the utility of ChromoSkein in two exemplary use cases that examine functional genomic loci in the spatial context of chromosomes and the whole genome. | 10.48550/arxiv.2211.05125 | [
"https://export.arxiv.org/pdf/2211.05125v1.pdf"
] | 253,446,904 | 2211.05125 | 1b5c2a8aff4ca25a5c153e6bcf222cdcfe209e12 |
ChromoSkein: Untangling Three-Dimensional Chromatin Fiber With a Web-Based Visualization Framework
Matúš Talčík
Filip Opálený
Tereza Clarence
Katarína Furmanová
Jan Byška
Barbora Kozlíková
David Kouřil
ChromoSkein: Untangling Three-Dimensional Chromatin Fiber With a Web-Based Visualization Framework
1Index Terms-Biological visualizationchromatin3Dgenomic datainteraction
We present ChromoSkein, a web-based tool for visualizing three-dimensional chromatin models. The spatial organization of chromatin is essential to its function. Experimental methods, namely Hi-C, reveal the spatial conformation but output a 2D matrix representation. Biologists leverage simulation to bring this information back to 3D, assembling a 3D chromatin shape prediction using the 2D matrices as constraints. Our overview of existing chromatin visualization software shows that the available tools limit the utility of 3D through ineffective shading and a lack of advanced interactions. We designed ChromoSkein to encourage analytical work directly with the 3D representation. Our tool features a 3D view that supports understanding the shape of the highly tangled chromatin fiber and the spatial relationships of its parts. Users can explore and filter the 3D model using two interactions. First, they can manage occlusion both by toggling the visibility of semantic parts and by adding cutting planes. Second, they can segment the model through the creation of custom selections. To complement the 3D view, we link the spatial representation with non-spatial genomic data, such as 2D Hi-C maps and 1D genomic signals. We demonstrate the utility of ChromoSkein in two exemplary use cases that examine functional genomic loci in the spatial context of chromosomes and the whole genome.
INTRODUCTION
O RGANIZATION of a genome in three-dimensional (3D) space significantly impacts its function. In eukaryotic cells, the DNA is packed into a micron-sized nucleus with the help of proteins, which together build a so-called chromatin fiber. Chromatin represents a unique combination of two data characteristics. First, the underlying DNA molecule is a linear nucleotide sequence. Second, the fiber folds into a densely packed 3D shape.
The specific spatial configuration of chromatin in a nucleus is not yet fully determined. Thus, it is an object of intense study in biology. Over the years, several methods for revealing the organization of chromatin have been developed. Most prominent is the chromosome conformation capture (3C) suite of experiments [15]. These methods cross-link together segments of the DNA that are in physical proximity. Aggregating the resulting interactions across many cells gives a measure of how likely two genomic parts are mutually interacting. The Hi-C method [22], a variant of 3C, resolves all-to-all interactions between same-sized DNA segments. The output of this method is usually in the form of a contact matrix, see Figure 1. The matrix values can then be used to assemble a prediction of chromatin's conformation in 3D.
While a number of tools feature a 3D view, the 3D aspect is underutilized in genomic visualization. We hypothesize that the lack of tools supporting 3D interaction is caused mainly by two reasons. First, the Hi-C experiment outputs 2D contact maps. Therefore, most tools naturally work with the 2D representation. Second, 3D visualization inherently presents issues like occlusion that make it harder to work with 3D data on a two-dimensional display. Nusrat et al. [28] further remark that visualizing 3D spatial chromatin data suffers from the fact that it only shows a prediction and omits information about the resolved structure's ambiguity.
Despite these claims, certain theories can only be examined in the 3D context. For example, positioning of a gene in a chromosome and its spatial distance to functional landmarks, such as promoters and enhancers, has a direct impact on gene regulation [10]. Certain data characteristics-e.g., density or mutual spatial orientation-are more intuitively obtained from the 3D representation. The examined structure, i.e., chromatin fiber, is by definition a spatial object and, according to recent biological research, it makes sense to examine it in its natural context [4]. We found out that many tools use 3D chromatin models merely for illustration while interaction with the 3D data is severely limited.
To address this gap in available chromatin visualization tools, we set out to design and develop a novel tool that places the 3D representation at the forefront. We focus primarily on two facets: a) performant and high-fidelity visual representations that highlight the spatiality of the underlying data; and b) interaction allowing direct exploration of the 3D chromatin model. We deliver our visualization solution for the web environment to promote easier adoption and collaboration.
In summary, we present the following key contributions:
• An extensive overview of existing genomic tools featuring 3D visualization, reviewing available visual representations, interactions, and analytical features.
• Design and prototypical implementation of a 3Dcentric visualization tool for chromatin data, called ChromoSkein 1 .
• Discussion and demonstration of ways of linking the 3D representation with conventional genomic 1D and 2D visualizations.
• Two exemplary use case scenarios demonstrating the utility of 3D representation in domain-specific analytical tasks.
BACKGROUND & MOTIVATION
Our work is motivated by a specific biological application domain. We, therefore, start with a brief introduction to chromatin conformation research and outline the high-level motivation for developing a novel tool.
Background
While methods for resolving the linear DNA sequence are well established, the acquisition of DNA's threedimensional organization is still challenging. It is known that the double helix is further organized in nucleosomes, which in turn form a chromatin fiber packaged into chromosomes in the cell nucleus.
The organization of chromatin in a cell is the subject of several experimental methods, referred to as chromosome conformation capture (3C) techniques [15]. These methods can indicate the spatial organization of chromatin. In general, 3C techniques are based on measurements of the frequency of interactions between fragments of DNA. Earlier methods could yield results only for a few of these fragments. However, high-throughput variations of 3C methods, such as Hi-C [2], [22], enable biologists to infer the number of interactions between all equally-sized regions of the whole chromatin fiber. The trade-off is a very limited precision. Depending on the experiment's parameters, the size of these regions is in the order of thousands (kilobases, kb) to millions of nucleotides (megabases, Mb). One such group of nucleotides is then referred to as a bin. The output of a Hi-C assay is a 2D contact frequencies matrix, sometimes also called interaction frequencies matrix, that assigns each pair of bins a number proportional to the number of interactions observed between these two bins.
Further analysis of Hi-C matrix data reveals multi-scale organization of genomes. The organization manifests in the Hi-C map as specific patterns, e.g., a chessboard-like pattern signifying A/B compartments [22], or point peaks in distant sequential regions indicating topologically associating domains (TADs), important in gene expression [7].
The contact frequency maps coming from the Hi-C experiment imply a genome's spatial configuration. Computational biologists thus infer the 3D structure using probabilistic algorithms, polymer simulations, or various other methods [29]. The Hi-C matrix values serve as constraints for 1. Available at https://github.com/chromoskein/chromoskein the computation, as illustrated in Figure 1. These methods generate a three-dimensional model prediction, where every bin is assigned a position in 3D space Examples of tools for genome structure prediction are LorDG [41] or 3DMax [30].
Biological Motivation
The three-dimensional chromatin structure can be used to analyze and confirm hypotheses. For example, Stevens et al. [37], who first reconstructed whole-genome models using single-cell Hi-C from haploid mouse cells, examined the 3D model to confirm known spatial features. They found that chromosomes occupy specific areas-territories-in the nucleus. They did so by isolating individual chromosomes and de-emphasizing the rest to indicate the chromosomes' nuclear position.
Similarly, Tan et al. [37] reconstructed whole genome chromatin models but from diploid human cells. They also look at chromosome territories. The shape and mutual arrangement of chromosomes are evident from the 3D model, such as how the chromosomes intermingle. Tan et al. also show how gene-rich chromosomes occupy space closer to the nuclear center while gene-poor chromosomes lie on its periphery. With the 20kb resolution, the 3D model exhibits a hierarchical, fractal organization of chromatin, with regions of high spatial clustering and segregation. Thanks to Tan et al.'s novel method for capturing conformation of diploid cells, the resulting structures showcase differences in the shape of active and inactive X chromosomes. While this can be quantified by, for example, a radius of gyration, this feature can be intuitively identified during the visual exploration of the 3D shape.
The development of ChromoSkein is motivated by these recent investigations of chromatin conformation. Spatial representations clearly play an important role in several stages of the analytical process. We see an opportunity for intuitive visualization and interaction methods to expedite the analysis of 3D chromatin data.
Why Focus on 3D?
Bioinformaticians trained in Hi-C analysis can imply insights from the contact matrix representation. Neverthe-less, they are still limited by their cognitive abilities. Hi-C maps are an abstract representation that captures many relationships between genomic regions. 3D structure prediction allows generating a spatial model that satisfies all the constraints and gives one possible conformation in space. This relieves scientists of having to reconstruct a mental image of the spatial conformation.
Additionally, certain characteristics of a spatial structure cannot be inferred purely from the 2D representation. For example, biologists might be interested in density of functional landmarks (e.g., genes) in a certain nucleus compartment. This metric can only be determined from the modeled 3D structures, either visually or by radius of gyration. Similarly, mutual orientation of elements, e.g., chromosomes, is a relationship indirectly captured in the Hi-C map but observable only once the structure is visualized in 3D.
Finally, Hi-C maps are often quite extensive, occupying gigabytes in storage. At a common 15kb resolution, where one bin corresponds to a sequence of 15 thousand basepairs, a human genome results in approx. 207 thousand bins. Storing the information about observed interactions for all possible pairs of bins thus results in 207 000 2 numbers. As the Hi-C map is symmetric, some viewers only show half of it, i.e., triangular Hi-C map (see Figure 2B), resulting in N 2 /2 numbers to store. However, the asymptotic complexity O(n 2 ) remains the same. The 3D representation, on the other hand, requires only a list of bin positions, reducing the space complexity from O(n 2 ) to O(n). The 3D model thus serves as a more efficient form for storage and transfer.
Naturally, the 3D representation has its disadvantages. As Nusrat et al. [28] note, 3D chromatin models represent only a prediction that satisfies the constraints defined by the Hi-C map. This uncertainty is, however, understood and accounted for by biologists investigating 3D chromatin. 3D visualization also inherently presents issues related to showing 3D phenomena on a 2D display, such as occlusion. Occlusion management, however, has been a topic of research both in general computer graphics and in visualization specifically and we now have at our disposal a number of techniques to deal with this issue. Overall, we do not argue that a 3D representation should completely replace the 2D matrix representation. We do, however, believe that it can serve as a valuable addition and that existing tools underutilize this representation.
RELATED WORK
While building on general molecular visualization, tools visualizing genomic data have developed in their own strand of research. Here we review visualization's role in the investigation of genomes' spatial organization and survey existing visualization tools. Table 1 summarizes the reviewed tools and highlights available and missing functionality.
Marti-Renom and Mirny [25] discuss the challenges and approaches in resolving the biological structure spanning several magnitudes of scales with available experimental and imaging technology. Goodstadt and Marti-Renom [12] further survey means of visualizing existing genomic data types and list available software tools. Waldispühl et al. [44] provide additional insight into the technical challenges related to visualizing 3D genomes.
Initially, chromatin structure was examined purely using Hi-C matrices, i.e., in 2D representation. Juicebox [9] first enabled interactive zooming in the large-scale space of contact frequencies. Its authors leverage a tiling approach inspired by web map services, such as Google Maps. HiGlass [19] further expanded on the idea of zoomable Hi-C maps by implementing additional capabilities, e.g., several linked views and juxtaposing genomic (mostly 1D) data to the Hi-C map. Other visualization types complementing the Hi-C matrices are also common, such as the arc-plot variations used by 3DIV [46].
Most publications on visualizing genomes come from bioinformatics research and therefore largely focus on the applications. Nusrat et al. [28] provide a comprehensive survey of the topic from the visualization research perspective. Recently, L'Yi et al. [23] introduced Gosling, a framework for genomic data visualization that mostly focuses on 2D plots. They leverage a grammar-based approach popularized in the visualization community by Vega-Lite [33].
Working with three-dimensional genomic data presents several additional challenges. Goodstadt and Marti-Renom [13] debate the challenges of visualizing 3D genomes directly. Many domain experts employ existing molecular graphics tools, e.g., PyMol [34] or ChimeraX [31], to perform the analysis of 3D chromatin. These tools often have a decades-long history and over time have developed into colossal suites tailored mostly for the analysis of smaller molecules, such as proteins. Their applicability to 3D genomic data is limited due to a high learning curve and sometimes technical limitations on larger datasets. Furthermore, including other views (e.g., Hi-C matrix) requires external software or extensive scripting.
One of the first tools tailored to the exploration of the genome in its spatial context is Genome3D [1]. The tool allows switching between three levels of scale that correspond to the inherent hierarchy of the genome. GMOL [27] expanded on Genome3D and increased the number of explorable levels to six. Interaction is mostly done through command-line input. Both of these tools were created as desktop programs, limiting wider adoption in research.
Software published as a web application, on the other hand, is instantly available. In genomic research, tools like the UCSC Genome Browser [18] benefited from the decision to target the web platform. Consequently, many tools for exploring 3D genomic data were developed for the web to lower the adoption threshold. 3DGB [3] devises a solution for storage, querying, and mesh-based rendering of 3D genomic data. Li et al. [21] use 3Disease to analyze spatial chromatin rearrangement in cancer and developmental diseases. They employ a plotting library to visualize small genomic sub-parts: TADs. TADs can be more closely examined in TADkit [11]. TADkit is a WebGL-based viewer for the analysis of TADs utilizing TADbit [35] library developed by the same team. CSynth [40] combines 3D structure modeling from contact frequencies with an interactive visualization of the result, allowing human-in-the-loop workflow. Delta [38] features a view able to show 3D conformation of small part of the whole genome.
Many of the mentioned tools begin with a form requiring to specify parts of the dataset, e.g., genomic coordinates range, to visualize and therefore inhibit holistic analysis of Yes Yes * Needs a server for computations or data management. † Restricted to two views. ? Unknown due to unavailability of the tool and/or missing information in the accompanying publication. both local parts as well as global context. Furthermore, while the above-listed tools feature 3D visualization in some form, this aspect is often underutilized and used only for illustration, while the actual analysis is done in other views, e.g., 2D Hi-C contact maps, or 1D feature tracks. We identified four tools where 3D views play a larger role within the analytical workflow and allow performing tasks directly in the 3D representation.
GenomeFlow [42], a continuation of GMOL, offers functionality for modeling and analysis of 3D genomic data. Users can examine predicted 3D structures and augment them by loading additional data, e.g., a list of chromatin loops or gene annotations, which are then overlaid over the 3D model. Interaction with the 3D model itself is, however, limited. GenomeFlow does not allow visibility management and bins cannot be selected from the 3D view. Trieu et al. [42] also do not mention the ability of linked views and how the interaction between the 2D Hi-C maps and 3D predicted structure is coordinated. The tool is implemented as a desktop application and has not been maintained recently which, in our opinion, limits its utility for recent datasets.
Similarly, HiC-3DViewer [8] combines modeling and visualization albeit on web using client-server architecture. To work with the 3D structure, it offers mapping of genomic signals onto the 3D model. The 3D view is complemented with a pop-up Hi-C map and a 1D track panel. Selection in the Hi-C map is reflected in the 3D view, but selection in the opposite direction is not possible. The rendering uses flat colors without shading, limiting the perception of the chromatin's overall shape. The tool does not include cutaways or filtering, leading to high visual complexity.
Nucleome Browser [47] developed by the 4D Nucleosome consortium is the third relevant tool we identified. It combines linked views offering all possible modalities: 1D for genomic signal tracks, 2D for Hi-C contact maps, and 3D for predicted 3D chromatin structure. The overall functionality and interaction inside the 3D view are rudimentary. The browser allows switching between global and local views but only at three granularities. Arbitrary selections are only unidirectional: users can select genomic regions in the 1D or 2D views which are reflected in the 3D, but not the other way around. Only a whole chromosome or a single bin can be selected from the 3D view.
WashU Epigenome Browser [20] is a genomic browser combining and linking different views. Its 3D view implements bidirectional linking in a limited way, allowing only single-point selections. This tool, however, offers a large number of options for annotating the 3D structure. Besides the prevalent coloring by numerical values, it can label the 3D structure with text and glyphs.
Currently, most web-based 3D genome visualizations use WebGL or libraries built on it, e.g., three.js. WebGL lacks in capabilities compared to graphics APIs for native desktops. This limits both the data sizes web viewers can render interactively and the visual fidelity. Furthermore, visualizing large chromatin datasets opens up an issue with occlusion, which inhibits a proper exploration of the 3D dataset. None of the reviewed existing tools include any techniques for occlusion management to allow peeking into a dense 3D dataset, apart from a few tools that implement hiding selected regions. Finally, a lack of interaction options for exploring and manipulating the 3D genome model is prominent across all the surveyed tools. For the most part, the linking functionality is implemented in the 1D-to-3D or 2D-to-3D direction but the other way around is often limited. Users can usually select only singular bins, and have to turn back to 1D and 2D views for advanced selections. This hinders the exploration and prevents reasoning about the observed chromatin structure.
REQUIREMENTS
We base our requirements on discussions with a domain scientist during the initial phases of our collaboration. Furthermore, we consider our investigation of related tools and the exposed feature set limits detailed in the previous section. As a result, we define the following requirements for a novel chromatin visualization tool:
• Efficient 3D rendering: The tool should be able to handle large datasets with sizes in the magnitude of hundreds of thousands of bins and render them in real-time with interactive framerates (30+ FPS).
• Support shape comprehension: The visual representations should highlight the spatiality of data and help users to gain an understanding of the shape and size of the model.
•
Visibility management:
Chromatin models tend to be dense and highly occluded. Biologists need be able to strategically highlight salient parts while removing the occlusion that prevents localizing them.
Only the least amount of information should be removed to preserve context. (1D and 2D) representations. These selections should be mirrored in the remaining views as they are all better suited to different tasks. After localizing a significant part in the 3D view, it may be desired to look at multiple correlating data in several tracks of 1D views.
The requirements served as constraints and guiding principles for the design of our novel tool, which we describe in the following section.
3D-CENTRIC DESIGN OF CHROMOSKEIN
In this section, we describe our design choices and detail the implementation of a 3D-centric chromatin visualization tool, shown in Figure 2, which we call ChromoSkein. We describe how we turn the 3D data into visual representations and how we render and shade the visual marks. Afterward, we discuss interactions with the 3D visualization.
Visual Representations
3D chromatin modeling methods output a set of discrete 3D positions associated with nucleotide bins. The set is ordered, i.e., the order of bins corresponds to the order of individual segments in the DNA sequence. We reviewed representations used in the existing 3D chromatin visualization tools, see Table 1. In the majority of cases, tools include spherical and tubular representations, sometimes combining both in what is known in molecular visualization as balls-and-stick representation. Some tools implement more unconventional marks, such as crosses in Nucleome Browser [47]. Working with large 3D models, visual clutter is of primary concern. For that reason, simple and clear visual marks are preferred. We decided to include three established spatial representations-a spherical and two continuous tubular representations, see Figure 3. The spherical representation is the most straightforward way of displaying the raw modeling data. Each binrepresenting a certain number of nucleotides-is mapped to a sphere. This visual encoding indicates that the positional information is available only at this granularity for this whole genomic segment, i.e., it is impossible to say where each nucleotide lies in 3D space. The lost connectivity information is an undeniable limitation of the spherical representation. Therefore, we also provide two continuous representations. In the first, straight tubular representation, the consecutive bin positions are connected with straight tubes. This representation communicates the connectivity, while the original data points can still be inferred from the positions of the tube joints. However, the results can look unnatural, particularly for low-resolution data, where long straight segments and sharp angles at joints occur. Therefore, the last, smooth tubular representation, uses the bin positions to define a spline that forms the centerline of a tube. This technique produces more visually pleasing and organic-looking results, but it comes at the cost of no longer providing precise information about original bin positions.
We chose these three options mainly because of their simplicity. We specifically decided not to include a ballsand-stick representation: In this visual mapping there are twice as many marks on the screen, which leads to visual clutter without any added benefit.
3D Rendering & Shading
We currently deal with chromatin spatial models containing over tens of thousands of bins: our largest model of Mouse genome comprises of 25 thousand bins. In the future we expect even bigger models, as experimental methods improve in resolution. Such data sizes present a challenge for rendering in real-time, especially in the web environment. To achieve high performance, we avoid polygonal representation and instead use billboards with screen-space raytraced parametric objects. The advantage of this approach is twofold: We lower the number of vertices that need to be updated every frame and we obtain a pixel-perfect objects representation. For the straight tubular representation, we use efficient and visually correct rounded cones described by Groß and Gumhold [14].
To compute the smooth tubular representation precisely, we would need to fit a cubic spline through the data bins. However, to achieve fast rendering, we only approximate the cubic spline with quadratic Bezier curves, as proposed by Truong et al. [43]. To render the tube itself, we use the method by Reshetov and Luebke [32].
When rendering the tubular representations, we estimate the optimal tube thickness based on the proximity of bins. To ensure a reasonable thickness, we limit it to half the distance between two bins so that two bins do not overlap and visibly merge into one. Depending on the used model reconstruction algorithm, the spacing between consecutive bins is not always uniform. Therefore, we use a statistical rule based on the interquartile range (IQR), to bound it between: Q 1 /Q 3 ± 1.5 * IQR, where IQR = Q 3 − Q 1 , Q 1 = lower quartile, Q 3 = upper quartile. The default thickness is set in the middle of this range and users can adjust the value within the range.
When it comes to shading, virtually all the existing chromatin visualization tools we reviewed employ a simple Phong shading model. Some tools even skip all shading and only show the 3D model with flat colors [8]. Phong shading model considers only local illumination omitting shading by neighboring structures. For larger chromatin models, it leads to an ambiguous view where it is impossible to discriminate global features such as holes and crevices. Thus, the comprehension of an overall shape is hindered.
To enhance perception of such spatial features, we shade the scene with real-time screen space ambient occlusion (SSAO) [39]. It can be difficult to configure the SSAO radius parameter for large datasets, as a large radius accentuates only bins deeply inside the structure. In contrast, a small radius highlights only differences between close objects. We rectify this by stacking two results of SSAO computations with small (for near objects) and large (for deeply buried objects) radii. Compared to the Phong shading model, in ChromoSkein we are clearly able to comprehend shape features, such as a hole in the center of a genome occupied in cells by nucleolus. A comparison can be seen in Figure 4.
Interactions With 3D Chromatin
We aim to give bioinformaticians the tools necessary to go beyond static visualization and to support them in analyzing the 3D model directly in its spatial context. Next, we describe two interconnected interactions used to filter, explore, and examine the 3D data.
Selections
One of the key interactions with 3D chromatin required by biologists is the ability to select genomic regions. The selection task is a prominent interaction across all visualization tools [45]. Available genomic tools, for the most part, implemented selections in 1D and 2D representations and highlighted the selected parts in the 3D model. Very few of them allow selecting objects directly in the 3D model. A bioinformatician might wish to select bins based on some spatial feature, e.g., bins close to a surface or, the opposite, deeply nested in the nucleus. Such task is impossible to do in the matrix representation. Therefore, in ChromoSkein, we designed methods for selecting bins in the 3D view.
In the 3D view, the atomic element is a single bin. Thus, we choose to represent selections as a subset of the model bins. We allow users to create an arbitrary number of selections. In ChromoSkein, we implemented three selection tools for the 3D view that together support all possible interactions we identified as critical for the users:
Point selection, Continuous sequence selection, and Sphere selection, all demonstrated in Figure 5.
The simplest way of selecting bins is selecting one bin at a time.
Point selection serves for the definition of small selections of visually prominent bins or fine selection refinements. It is performed by clicking on the given bin in the 3D view. One of the most important features found in 3D chromatin is which bins or genomic loci are located in a neighborhood to a selected location. Therefore, Sphere selection allows selecting bins within a certain spatial distance from a given point or bin. Users can again pick a single bin by clicking on the 3D representation, but in this case, all bins within a spherical radius are selected. The radius is adjustable and the bins within the radius are highlighted upon hover. Finally, biologists are sometimes interested in chromatin loops, where the loop end points are distant in the linear sequence but spatially close. To allow selecting loops in 3D, we use Continuous sequence selection. When two bins are picked in the 3D view, the linear sequence of bins between them is selected.
We display the selections in 3D by coloring the visual marks representing bins. The colors are randomly generated for each track but can be adjusted by the user.
Occlusion Management
Chromatin fiber is tightly packed in a nucleus, resulting in a highly intertwined structure with large parts hidden due to occlusion. Biologists are interested to know, for example, how deeply nested a genomic locus is-the position can affect the function or regulation of genes located in the locus. Therefore, providing bioinformaticians tools to manage occlusion is essential. We include two techniques to filter the ChromoSkein allows users to segment the chromatin model by adding bins to selections tracks. The first tool for filtering the 3D model is by toggling visibility of each such segmentation track. Users can hide any track which leads to hiding of associated bins from the 3D representation. Segmentation tracks can also be loaded from an external file. This allows, for example, loading chromosome segmentation and exploring the model by toggling visibility of individual chromosomes.
Sometimes the focus of the analysis cannot be defined semantically but rather spatially. To support these cases, ChromoSkein provides cuttings planes. Users can add one or more cutting planes either along one of the major axes or an arbitrarily oriented plane spanning from the camera's point of view. The model is then cut by the plane: The primitives building the 3D bin representations are clipped at the intersection and resulting surface holes are filled.
Clipping planes work in tandem with our rendering style supporting depth cues, highlighting both local and global structural features, such as holes nested inside the model. Users can also choose to keep some selections visible at all times which can be useful for studying immersed parts while keeping parts of their surroundings. All these features are presented in Figure 6.
LINKING 3D WITH OTHER DATA
Due to the complexity of investigating organization and function of cell nucleus-brought by the dynamicity and scale range-it is essential that biologists integrate several available modalities, each focusing at different aspects of chromatin fiber. Other genomic data and visualizations have been proposed, as reviewed by Nusrat et al. [28].
In this section, we dive into the design of linking the 3D representation described above with non-spatial data types and their conventional visual representations.
Linked Views
Visualization systems typically connect different data by using multiple linked views [26]. Not all tasks are best performed in 3D view. Additionally, many biologists are already trained on and used to 1D and 2D visualizations. Therefore, we include two conventional genomic visualizations in ChromoSkein. First, we implemented a genomic browser to display 1D genomic signals. Second, we included a distance map to serve as a proxy for any 2D maps, e.g., Hi-C matrix viewer. In ChromoSkein, we couple these two linear data visualizations into a track view, shown in Figure 7.
Both the distance map and the genomic browser are prototypical implementations to complete the feature set required by the bioinformaticians analytical workflow. There are tools that focus more on each modality, e.g., HiGlass for Hi-C maps, but do not include ChromoSkein's capabilities for 3D data.
Track View: Genomic Browser
Genomic browsers are the most common way of working with genomic data. They display data laid out along one axis-usually the horizontal x-axis-annotated by genomic coordinates. Two typical examples of data shown in genomic browsers are genomic signals coming from biological experiments, e.g., ChIP-seq which analyzes protein-DNA interactions [17], and gene annotations marking positions of genes on the DNA sequence.
In ChromoSkein, we differentiate two types of 1D data: segmentations and signals. Segmentations assign genomic regions to a segmentation track while signals assign numerical values to each bin. The signals can be loaded into ChromoSkein in BED format. Segmentation tracks can either come from an external file or result from user selections. We display signal tracks as simple line charts while segmentations are drawn as horizontally stacked bar charts.
Track View: Distance Map
Hi-C experiments typically output 2D matrices with numerical values denoting frequencies of contact between bins. In ChromoSkein, a distance map stands in for a full Hi-C map. The distance map is generated from the 3D model and in principle should contain the same information. 3D reconstruction algorithms are rated according to the difference between the generated distance map and the original Hi-C map. If the reconstruction is good, the two 2D maps should be more or less equal.
We display the distance map as a triangular shape. The horizontal axis shows bins and each triangular field carries the distance information mapped to a color value. Users can zoom and pan around the 2D map. As an optimization, the distance map is calculated on-demand based on which part of the map is zoomed into.
We employ a simple level-of-detail scheme. If the size of a distance map is such that it cannot allocate enough pixels for all bins, a level change is applied. We merge multiple consecutive bins by averaging their positions and calculate the distance between those larger bins. Distance maps can be also generated from custom selections. This can be useful to observe distance relationships of a subset of chromatin.
Interactive Linking
Nusrat et al. [28] mention that views can be weakly, medium, or strongly linked. That simple categorization applies for 1D or 2D views that share reference frame of genomic coordinates. The 3D view steps out of this conformity, which extends the design space of possible linking.
We allow users to create many panels of the two types, i.e., 3D view or track view. Users can synchronize cameras between 3D views. That way they can observe one 3D model with different adjustments, e.g., hidden specific chromosomes. The 2D distance map and 1D genomic tracks in a track view are linked by default: Zooming and panning will be reflected across all tracks in one track view.
The linking between a 3D view and a track view is accomplished through custom selections. In the 3D view biologists select parts that are interesting because of their spatial features. On the other hand, they might also want to let the selections be guided by either the linear data, e.g., signal peaks, or patterns in the 2D map. For that reason, we implemented several selection tools also for the track view.
Annotating the 3D Structure
While linked views are based on the concepts of juxtaposition, superimposing data in the 3D is also possible. This allows biologists to contextualize 1D genomic data in space. Biologists call this annotating the 3D model. There are several ways that the color channel can be used to augment the 3D structure with other information. All types of annotation are demonstrated in Figure 8.
The first type of annotation is coloring the 3D representation based on chromosome segmentation, as seen in several existing 3D chromatin visualization tools. However, other segmentations can also be relevant: e.g., A/B compartments or TADs. In ChromoSkein, we therefore generalize this type of annotation. We allow users to segment the model by selecting its parts using several selection tools or loading the segmentation from an external file. We then color the 3D representation as we described in Section 5.3.1.
However, annotating the 3D model based on segmentation is not the only way to use the color channel. We can also map genomic signals from other biological experiments and superimpose the 1D data onto 3D. These types of data typically have more fine-grained resolution than 3D structures. Therefore, we allow users to choose how multiple values per bin are mapped, i.e., a minimum, maximum, average, or median. In order to color bins, we normalize the data and then apply one of the color maps recommended by the scientific color guide [5].
In some cases, the values might not come from an external file but instead can be computed on-the-fly. Bioinformatics tools often require sophisticated computation and powerful hardware but some algorithms can be implemented and run directly on a client device. This removes the need for running an external tool and importing the results into a visualization system. Users can thus iterate faster and often re-define the model subset meant for computation. As an example, we include computation of Solvent Accessible Surface Area (SASA) in ChromoSkein.
Finally, biologists are often interested in highlighting short segments of the DNA that carry functional elements such as genes. These short segments, called loci (singular locus), generally fall into a range less than a single bin. Therefore, semantically, we can consider them as short segmentations and implement them by selections. However, due to convention, we chose to highlight such loci using markers. Since markers most often span just a single bin, we display them as spheres and make them salient by using a greater radius and a different color, adjustable by the user.
IMPLEMENTATION
ChromoSkein is implemented as a client-only web application, which makes it usable without the need to install desktop software. The omission of a remote server component ensures that potentially confidential data need not be uploaded to a third-party server.
ChromoSkein itself is divided into two main parts: the application itself, written using the popular React frontend framework, and a narrowly focused visualization library written using the WebGPU API [24]. We decided to use WebGPU as it offers low-level access to GPU and addresses many of the issues of WebGL. Note that We-bGPU is currently in development and requires experimental versions of modern browsers, e.g., Google Chrome Canary. The project is open-sourced at https://github. com/chromoskein/chromoskein, along with instructions on how to set up the browser to run our tool.
EXEMPLARY USE CASES
We present two use cases prepared in collaboration with a biologist to demonstrate the capabilities of ChromoSkein. The biologist first performed analysis purely from 2D matrices, pointing out its deficiencies. Afterward, they utilized 3D visualization to gain additional insight beyond what is possible from only 1D and 2D genomic data.
Case study I
In the first case study, we look at chromosomal dynamics from Di Stefano et al. [6], focusing on the 3D model of a region surrounding mouse gene Sox2 and the gene itself. Without the utilization of 3D visualization, one has very limited options for exploring the dynamical behavior of Sox2 gene from pure XYZ coordinates; some of the potential analysis approaches are shown in Figure 9a
-b.
Firstly, the number of pairwise contacts to Sox2 gene was explored as a function of time and number of interactions up to certain distance. In most of the selected distance cutoffs, the number of contacts increases in the first 10 000 steps of dynamical simulation while steadily lowering below starting point until the end of the simulation (step 60 000). This suggests a tight packing of the chromatin fiber around Sox2 gene at the beginning of the simulation where the number of the interactions is higher. As the simulation progresses, the fiber around Sox2 gene will likely unfold.
Secondly, one can monitor the spatial trajectory of Sox2 gene to understand its dynamics and mobility. If the monitored position fluctuates in all three axes uniformly, its mobility is considered isotropic. Constraints in one of the axes hint at the presence of anisotropy. This can be related to the stability of specific conformation, which may play an important role in the regulation of gene transcription.
However, without further visualization and exploration of 3D dynamics, it is not possible to infer further details. Visualization of Sox2 (Figure 9c) trajectory helps to understand these processes better and clarify the hypothesis based on previous analysis (Figure 9a-b). Indeed, it confirms the unfolding of chromatin locus in later stages of simulation (>10 000 s) and the rearrangement of chromatin folding around Sox2. Moreover, Chromoskein can calculate solvent accessible surface area (SASA), which allows further interpretation of un/folding events. Although Sox2 region converges to conformation with fewer interactions (Figure 9a), SASA values of most bins are lower towards the end of the simulation (brighter blue in Figure 9c) than in the initial conformation, making it less accessible to potential transcription factors (specific proteins regulating gene activity). This suggests the tendency for more permanent gene expression of Sox2 gene with less flexible regulation. Fig. 9. a) A number of pairwise contacts to Sox2 as a function of time and distance cutoff. b) Measurement of Sox2 trajectory in 3D space. c) Selected conformations of Sox2 locus at time points 0-30,000-60,000s with Sox2 colored in orange. The chromosomal locus is colored based on solvent accessible surface area (SASA) with dark blue corresponding to a higher value while light blue corresponds to a lower value of SASA. Parts a) and b) were made in R programming language. Fig. 10. a) Distribution of pairwise distances from genes of interest (Lmo7, Neurod6, Rergl, Sox2, and Tet2) to the rest of the chromosome where they are located. b) Number of pairwise contacts for Lmo7/Neurod6/Rergl/Sox2 and Tet2 to the rest of the chromosome where they are located, considering different distance cutoff values (10,20,40). c) 3D model of the whole mouse genome with colored chromosomes where Lmo7/Neurod6/Rergl/Sox2 and Tet2 (colored as orange beads) are present. Numbers 1-3 are showing a specific cross-section through the mouse genome in the z-axis. Number 1 only depicts genes localized on the genome surface, while numbers 2-3 reveal additional genes buried inside the chromatin. Parts a) and b) were made in R programming language.
Case study II
In the second case study, we inspected the same Sox2 gene in the context of whole genome organization, as published by Stevens et al. [36], along with the addition of Lmo7, Neurod6, Rergl, Tet2 genes.
Exploration of Hi-C matrices or single coordinates without 3D visualization will inform us about probabilistic interactions among all the chromosomes but not about their mutual spatial arrangements/colocalization. Distance distribution estimates of a selected gene towards the rest of the chromosome can indirectly inform us about the folding of a region surrounding the gene (Figure 10a-b).
3D visualization of whole genome structure allowed us to uncover that Tet2 and Rergl genes are located on the 'surface' of the genome. Their corresponding chromosomes are likely interacting with so-called nuclear lamina (Figure 10c1-c2). This may suggest genes' participation in specific processes, such as lamina regulation or nuclear transport.
There are many non-genome arrangements that cannot be captured via Hi-C experiment. However, they can manifest themselves within the 3D genome model as cavities or holes; therefore, depth cues and usage of cutting planes can provide us with information about them.
By sliding a cutting plane through the whole genome 3D structure (Figure 10c3), we can uncover the rest of genes, located on chromosomes buried deeply inside the chromatin core. We used the ability to keep chromosomes visible even under the cutting plane to keep the context. We find that Neurod gene is positioned in close proximity to a spatial caveat inside the genome, which is most likely occupied by the nucleolus, suggesting an important regulatory role or potential interaction with rRNA or rDNA.
CONCLUSION
We are currently receiving progressively better predictions of how DNA interacts in the 3D space. Formulating hypotheses about overall chromatin conformation intrinsically requires appropriate visual support, which is lacking in the currently available tools. In our work, we aim to bring the 3D spatial representation of chromatin to the forefront and design a novel tool that would assist experts in several stages of their research. We clearly identified a gap in the available software, where the semantics of the domain problem elicit new methods, making existing molecular visualization tools insufficient.
We believe that our newly proposed ChromoSkein tool will contribute to understanding of the importance of chromatin spatial organization. We foresee that the description of our design choices, the architecture of the tool, along with the availability of the source code, will serve the communities of chromatin experts, software developers, and visualization experts in adopting our solutions.
The described visualization framework gives us a solid basis for further research. Several topics already came up in preliminary discussions with domain experts. First, like other biological phenomena, chromatin is a dynamic structure that is constantly changing. Studying these movements is highly important since chromatin adjusts to the cell cycle and drastically changes its conformation. From the visualization perspective, time series data for 3D models of chromatin present intriguing challenges for which novel solutions will be required. Second, studying the evolutionary changes in chromatin structure and spatial organization among and across species can be naturally supported by comparative visualization. Finally, chromatin research is intrinsically based on integration of experimental methods, data types, and data sources. The same approach is being applied to tooling, now also enabled thanks to modern web technologies. In the future, we foresee a high potential in integrating our tool with applications such as HiGlass, that are targeting to solve one specific problem.
Matúš Talčík is a doctoral student at Masaryk University in Brno, Czech Republic. He studied computer graphics after which he moved on to visualization. He applies his knowledge to make rendering of large biological data sets beautiful and real-time.
Filip Opálený is a doctoral student at Masaryk University in Brno, Czech Republic and a member of Visitlab research laboratory, focusing on vizualization of biological data. Barbora Kozlíková is an Associate Professor at the Faculty of Informatics at Masaryk University in Brno, Czech Republic. She is the head of the Visitlab research laboratory, specializing in the design of visualization and visual analysis methods and systems for diverse application fields, including biochemistry, medicine, and geography. She has published over 70 research papers.
Tereza Clarence
David Kouřil is a postdoctoral researcher at Masaryk University in Brno, Czech Republic. He received his doctoral degree from TU Wien in Vienna, Austria in April 2021. He focuses on three-dimensional biological data and designs novel visualization and interaction methods that support exploration and understanding of the environments that this data represents.
Fig. 1 .
1Overview of reconstruction process: first, contact frequency data are measured using Hi-C experiments. These are used as a constraint in modelling algorithms. After that, chromatin can be studied in 3D.
Fig. 2 .
2Overview of ChromoSkein. A) 3D viewports. A1) Colored by imported chromosomes. Shown with tooltip on the hovered bin. A2) Colored by solvent accessible surface area (SASA) computed by ChromoSkein. A3) With hidden chromosome coloring but with the visible user-made selection of interest. B) Track view with distance map of the 3D structure shown along with chromosome selections track, user-made selection track, and 1D SASA track. C) Configuration panel.
•
Bi-directional linking: Users should be able to select genomic loci in the 3D view or the other
Fig. 3 .
3Three types of visual representations available in ChromoSkein. From left: spherical, straight tubular, and smooth tubular representation.
Fig. 4 .
4Shading greatly influences perception of the overall shape. The standard Phong shading model results in a highly cluttered view (a); SSAO with a small radius highlights local features (b), while a bigger radius accentuates global features (c). We use two SSAO passes with different radii to combine benefits of both (d).
Fig. 5 .
5From
Fig. 6 .
6Demonstration of our cuttings planes. Entire chromatin is cut by a user-defined cutting plane and holes are filled. Two users' selections (chromosomes) are configured to be unaffected by cutting planes. model by adjusting visibility of its parts: one semantic and one based on spatial position.
Fig. 7 .
7Example of a track view containing 4 tracks (from top to bottom): distance map, segmentation into chromosomes (loaded as selections from a file), custom user-made selection in 3D viewport, and 1D track (in this case solvent accessible surface area).
Fig. 8 .
8From top to bottom: chromatin colored by segmentation (chromosomes), genomic signal (solvent accessible surface area), markers.
Postdoctoral researcher at the Center for Disease Neurogenomics at Icanh School of Medicine at Mt Sinai, NY. She received her doctoral degree at the Francis Crick Institute and King's College London, UK in Computational Biology in 2022. Her research focuses on 3D genome function, organization and modelling. Katarína Furmanová is an Assistant professor and member of the Visitlab research laboratory at Masaryk University in Brno, Czech Republic. She obtained her Ph.D. in Computer Graphics in 2019 from the same university. After finishing her Ph.D., she spent one year as a postdoc at Aarhus University in Denmark. Her current research interest involve visualization of medical and biological data. Jan Byška is an Assistant Professor at the Masaryk University in Brno, Czech Republic and a part-time Associate Professor at the University of Bergen, Norway. He is a member of the Visitlab research laboratory, where his work focuses mostly on various challenges in the field of visualization of molecular and time-dependent data.
TABLE 1
1Comparison of 3D visualization genomic tools for three-dimensional datasets.
ACKNOWLEDGMENTSThe presented work has been supported by the Czech Ministry of Education project no. LTC20033. Authors wish to thank Marc Marti-Renom for consultations and Hanka Pokojná for creating ChromoSkein's logo.
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Nucleome Browser: An integrative and multimodal data navigation platform for 4D Nucleome. X Zhu, Y Zhang, Y Wang, D Tian, A S Belmont, J R Swedlow, J Ma, 10.1038/s41592-022-01559-3Nature Methods. 19X. Zhu, Y. Zhang, Y. Wang, D. Tian, A. S. Belmont, J. R. Swedlow, and J. Ma, "Nucleome Browser: An integrative and multimodal data navigation platform for 4D Nucleome," Nature Methods, vol. 19, p. 911-913, 2022. doi: 10.1038/s41592-022-01559-3
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"YOUR CONTRASTIVE LEARNING IS SECRETLY DOING STOCHASTIC NEIGHBOR EMBEDDING"
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"Tianyang Hu \nHuawei Noah's Ark Lab\n\n",
"Zhili Liu \nHuawei Noah's Ark Lab\n\n\nHong Kong University of Science and Technology\n\n",
"Fengwei Zhou \nHuawei Noah's Ark Lab\n\n",
"Wenjia Wang \nHong Kong University of Science and Technology\n\n\nHong Kong University of Science and Technology (Guangzhou)\n\n",
"Weiran Huang \nQing Yuan Research Institute\nShanghai Jiao Tong University\n\n"
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"Huawei Noah's Ark Lab\n",
"Huawei Noah's Ark Lab\n",
"Hong Kong University of Science and Technology\n",
"Huawei Noah's Ark Lab\n",
"Hong Kong University of Science and Technology\n",
"Hong Kong University of Science and Technology (Guangzhou)\n",
"Qing Yuan Research Institute\nShanghai Jiao Tong University\n"
] | [] | Contrastive learning, especially self-supervised contrastive learning (SSCL), has achieved great success in extracting powerful features from unlabeled data. In this work, we contribute to the theoretical understanding of SSCL and uncover its connection to the classic data visualization method, stochastic neighbor embedding (SNE)(Hinton & Roweis, 2002), whose goal is to preserve pairwise distances. From the perspective of preserving neighboring information, SSCL can be viewed as a special case of SNE with the input space pairwise similarities specified by data augmentation. The established correspondence facilitates deeper theoretical understanding of learned features of SSCL, as well as methodological guidelines for practical improvement. Specifically, through the lens of SNE, we provide novel analysis on domain-agnostic augmentations, implicit bias and robustness of learned features. To illustrate the practical advantage, we demonstrate that the modifications from SNE to t-SNE | 10.48550/arxiv.2205.14814 | [
"https://export.arxiv.org/pdf/2205.14814v2.pdf"
] | 249,192,258 | 2205.14814 | 740b343fac0e9bc5b1906c99c1071b67f4c163e1 |
YOUR CONTRASTIVE LEARNING IS SECRETLY DOING STOCHASTIC NEIGHBOR EMBEDDING
Tianyang Hu
Huawei Noah's Ark Lab
Zhili Liu
Huawei Noah's Ark Lab
Hong Kong University of Science and Technology
Fengwei Zhou
Huawei Noah's Ark Lab
Wenjia Wang
Hong Kong University of Science and Technology
Hong Kong University of Science and Technology (Guangzhou)
Weiran Huang
Qing Yuan Research Institute
Shanghai Jiao Tong University
YOUR CONTRASTIVE LEARNING IS SECRETLY DOING STOCHASTIC NEIGHBOR EMBEDDING
Published as a conference paper at ICLR 2023
Contrastive learning, especially self-supervised contrastive learning (SSCL), has achieved great success in extracting powerful features from unlabeled data. In this work, we contribute to the theoretical understanding of SSCL and uncover its connection to the classic data visualization method, stochastic neighbor embedding (SNE)(Hinton & Roweis, 2002), whose goal is to preserve pairwise distances. From the perspective of preserving neighboring information, SSCL can be viewed as a special case of SNE with the input space pairwise similarities specified by data augmentation. The established correspondence facilitates deeper theoretical understanding of learned features of SSCL, as well as methodological guidelines for practical improvement. Specifically, through the lens of SNE, we provide novel analysis on domain-agnostic augmentations, implicit bias and robustness of learned features. To illustrate the practical advantage, we demonstrate that the modifications from SNE to t-SNE
INTRODUCTION
Recently, contrastive learning, especially self-supervised contrastive learning (SSCL) has drawn massive attention, with many state-of-the-art models following this paradigm in both computer vision b;Grill et al., 2020;Zbontar et al., 2021) and natural language processing (Fang et al., 2020;Wu et al., 2020;Giorgi et al., 2020;Gao et al., 2021;. In contrast to supervised learning, SSCL learns the representation through a large number of unlabeled data and artificially defined self-supervision signals, i.e., regarding the augmented views of a data sample as positive pairs and randomly sampled data as negative pairs. By enforcing the features of positive pairs to align and those of negative pairs to be distant, SSCL produces discriminative features with state-of-the-art performance for various downstream tasks.
Despite the empirical success, the theoretical understanding is under-explored as to how the learned features depend on the data and augmentation, how different components in SSCL work and what are the implicit biases when there exist multiple empirical loss minimizers. For instance, SSCL methods are widely adopted for pretraining, whose feature mappings are to be utilized for various downstream tasks which are usually out-of-distribution (OOD). The distribution shift poses great challenges for the feature learning process with extra requirement for robustness and OOD generalization (Arjovsky et al., 2019;Krueger et al., 2021;Bai et al., 2021;He et al., 2020b;Zhao et al., 2023;Dong et al., 2022), which demands deeper understanding of the SSCL methods.
The goal of SSCL is to learn the feature representations from data. For this problem, one classic method is SNE (Hinton et al., 2006) and its various extensions. Specially, t-SNE (Van der Maaten & Hinton, 2008) has become the go-to choice for low-dimensional data visualization. Comparing to SSCL, SNE is far better explored in terms of theoretical understanding (Arora et al., 2018;Linderman & Steinerberger, 2019;Cai & Ma, 2021). However, its empirical performance is not satisfactory, especially in modern era where data are overly complicated. Both trying to learn feature representations, are there any deep connections between SSCL and SNE? Can SSCL take the advantage of the theoretical soundness of SNE? Can SNE be revived in the modern era by incorporating SSCL?
In this work, we give affirmative answers to the above questions and demonstrate how the connections to SNE can benefit the theoretical understandings of SSCL, as well as provide methodological guidelines for practical improvement. The main contributions are summarized below.
• We propose a novel perspective that interprets SSCL methods as a type of SNE methods with the aim of preserving pairwise similarities specified by the data augmentation.
• The discovered connection enables deeper understanding of SSCL methods. We provide novel theoretical insights for domain-agnostic data augmentation, implicit bias and OOD generalization. Specifically, we show isotropic random noise augmentation induces l 2 similarity while mixup noise can potentially adapt to low-dimensional structures of data; we investigate the implicit bias from the angle of order preserving and identified the connection between minimizing the expected Lipschitz constant of the SSCL feature map and SNE with uniformity constraint; we identify that the popular cosine similarity can be harmful for OOD generalization.
• Motivated by the SNE perspective, we propose several modifications to existing SSCL methods and demonstrate practical improvements. Besides a re-weighting scheme, we advocate to lose the spherical constraint for improved OOD performance and a t-SNE style matching for improved separation. Through comprehensive numerical experiments, we show that the modified t-SimCLR outperforms the baseline with 90% less feature dimensions on CIFAR-10 and t-MoCo-v2 pretrained on ImageNet significantly outperforms in various domain transfer and OOD tasks.
PRELIMINARY AND RELATED WORK
Notations. For a function f : Ω → R, let ∥f ∥ ∞ = sup x∈Ω |f (x)| and ∥f ∥ p = ( Ω |f (x)| p dx) 1/p . For a vector x, ∥x∥ p denotes its p-norm, for 1 ≤ p ≤ ∞. P(A) is the probability of event A. For a random variable z, we use P z and p z to denote its probability distribution and density respectively. Denote Gaussian distribution by N (µ,Σ) and let I d be the d×d identity matrix. Let the dataset be D n = {x 1 ,···,x n } ⊂ R d where each x i independently follows distribution P x . The goal of unsupervised representation learning is to find informative low-dimensional features z 1 ,···,z n ∈ R dz of D n where d z is usually much smaller than d. We use f (x) to as the default notation for the feature mapping from R d → R dz , i.e., z i = f (x i ).
Stochastic neighbor embedding. SNE (Hinton & Roweis, 2002) is a powerful representation learning framework designed for visualizing high-dimensional data in low dimensions by preserving neighboring information. The training process can be conceptually decomposed into the following two steps: (1) calculate the pairwise similarity matrix P ∈ R n×n for D n ; (2) optimize features z 1 ,···,z n such that their pairwise similarity matrix Q ∈ R n×n matches P . Under the general guidelines lie plentiful details. In Hinton & Roweis (2002), the pairwise similarity is modeled as conditional probabilities of x j being the neighbor of x i , which is specified by a Gaussian distribution centered at x i , i.e., when i ̸ = j,
P j|i = exp(−∥x i −x j ∥ 2 2 /2σ 2 i ) k̸ =i exp(−∥x i −x k ∥ 2 2 /2σ 2 i ) , (2.1)
where σ i is the variance of the Gaussian centered at x i . Similar conditional probabilities Q j|i 's can be defined on the feature space. When matching Q to P , the measurement chosen is the KL-divergence between two conditional probabilities. Hinton, 2008) modified the pairwise similarity by considering joint distribution rather than conditional, and utilizes t-distribution instead of Gaussian in the feature space modeling. It is worth noting that SNE belongs to a large class of methods called manifold learning . In this work, we specifically consider SNE. If no confusion arises, we use SNE to denote the specific work of Hinton & Roweis (2002) and this type of methods in general interchangeably.
Self-supervised contrastive learning. The key part of SSCL is the construction of positive pairs, or usually referred to as different views of the same sample. For each x i in the training data, denote its two augmented views to be
x ′ i and x ′′ i . Let D ′ n = {x ′ 1 ,···,x ′ n }, D ′′ n = {x ′′ 1 ,···,x ′′ n } and define l(x ′ i ,x ′′ i ) = −log exp(sim(f (x ′ i ),f (x ′′ i ))/τ ) x∈D ′ n ∪D ′′ n \{x ′ i } exp(sim(f (x ′ i ),f (x))/τ ) ,
where sim(z 1 , z 2 ) = ⟨ z1 ∥z1∥2 , z2 ∥z2∥2 ⟩ denotes the cosine similarity and τ is a temperature parameter. The training objective of the popular SimCLR can be written as
L InfoNCE := 1 2n n i=1 (l(x ′′ i ,x ′ i )+l(x ′ i ,x ′′ i )).
Recently, various algorithms are proposed to improve the above contrastive learning. To address the need for the large batch size, MoCo Chen et al., 2020b) utilizes a moving-averaged encoder and a dynamic memory bank to store negative representations, making it more device-friendly. Grill et al. (2020); ; Zbontar et al. (2021); radically discard negative samples in SSCL but still achieve satisfactory transfer performance. Another line of works (Caron et al., 2020;Li et al., 2021;Liu et al., 2022) mines the hierarchy information in data to derive more semantically compact representations. Radford et al. (2021); even extend the contrastive methods to the multi-modality data structure to achieve impressive zero-shot classification results.
Theoretical understanding of SSCL. In contrast of the empirical success, theoretical understanding of SSCL is still limited. While most of theoretical works (Arora et al., 2019;Tosh et al., 2020;HaoChen et al., 2021;2022;Wang et al., 2022;Wen & Li, 2021;Wei et al., 2020;Ji et al., 2021;Ma et al., 2023) focus on its generalization ability on downstream tasks, there are some works studying specifically the InfoNCE loss. One line of works (Oord et al., 2018;Bachman et al., 2019;Hjelm et al., 2018;Tian et al., 2019; understand the InfoNCE loss from mutual information perspective, showing that the negative InfoNCE is a lower bound of mutual information between positive samples. Other works (Wang & Isola, 2020; are from the perspective of geometry of embedding space, showing that InfoNCE can be divided into two parts: one controls alignment and the other prevents representation collapse. In this paper, we study SSCL from the SNE perspective, which, to the best of the authors' knowledge, has no discussion in existing literature. The closest work to ours is Balestriero & LeCun (2022), which proposed a unifying framework under the helm of spectral manifold learning. In comparison, our work focus specifically on the connection between SSCL and SNE.
SNE PERSPECTIVE OF SSCL
A closer look at the training objectives of SNE and SimCLR reveals great resemblance -SimCLR can be seen as a special SNE model. To see this, denote D 2n = D ′′ n ∪ D ′ n as the augmented dataset with index x 2i−1 = x ′′ i and x 2i = x ′ i . If we change the l 2 distance to the negative cosine similarity and let σ 2 i ≡ τ . Admitting similar conditional probability formulation as in (2.1) yields that for i ̸ = j,
Q j|i = exp(sim(f ( x i ),f ( x j ))/τ ) k̸ =i exp(sim(f ( x i ),f ( x k ))/τ ) .
(3.1) By taking
P j|i = 1, if x i and x j are positive pairs 0, otherwise, (3.2)
the SNE objective (2.2) can be written as
2n i=1 2n j=1 P j|i log P j|i Q j|i = n k=1 −log( Q 2k−1|2k )−log( Q 2k|2k−1 ) ,
which reduces to the SimCLR objective L InfoNCE , up to a constant scaling term only depending on n.
Now that we have established the correspondence between SNE and SimCLR, it's clear that the feature learning process of SSCL also follows the two steps of SNE.
(S1) The positive pair construction specifies the similarity matrix P . (S2) The training process then matches Q to P by minimizing some divergence between the two specified by the training objective, e.g., KL divergence in SimCLR.
The main difference between SNE and SSCL is the first part, where the P in SNE is usually densely filled by l p distance, ignoring the semantic information within rich data like images and texts. In contrast, SSCL omits all traditional distances in R d and only specifies semantic similarity through data augmentations, and the resulting P is sparsely filled only by positive pairs as in (3.2). For structurally rich data such as image or text, the semantic information is invariant to a wide range of transformations. Human's prior knowledge of such invariance guides the construction of positive pairs in SSCL, which is then learned by the feature mapping.
Remark 3.1 (SNE vs SSCL). We would like to clarify on the main difference between SNE and SSCL that we focus in this work. Although standard SNE (Hinton et al., 2006) is non-parametric without explicit feature maps, and is optimized for the whole dataset, these are not the defining properties of SNE. SNE can also utilize explicit feature maps and mini-batch training (Van Der Maaten, 2009). On the other hand, SSCL can also benefit from larger/full batches and can also be modified to directly optimize the features z i 's. In this work, we omit these subtleties 1 and focus on the (S1) perspective, which we view as the most significant difference between SNE and SSCL.
ANALYSIS
In this section, to showcase the utility of the SNE perspective, we demonstrate how the feature learning process of SSCL methods, e.g., SimCLR, can become more intuitive and transparent. Specifically, we re-derive the alignment and uniformity principle (Wang & Isola, 2020) as well as provide novel analysis on domain-agnostic augmentations, the implicit bias and robustness of learned features. To aid the illustration, we device toy examples with simulated Gaussian mixture data.
Gaussian mixture setting. Let the data follow d-dimensional Gaussian mixture distribution with m components where P
x ∼ 1 m m i=1 N (µ i ,σ 2 I d )
. The special case with d = 2, m = 5, σ = 0.1 is illustrated in Figure 1(a) with 250 independent samples. To apply contrastive methods, consider constructing positive pairs by direct sampling, i.e., if x is from the first component, then we sample another x ′ ∼ N (µ 1 ,σ 2 I d ) independently as its alternative view for contrast. The negative samples are the same as in standard SimCLR training.
DOMAIN-AGNOSTIC DATA AUGMENTATION
Now that we have established in (S1) that the input space pairwise distance is specified by the data augmentation, a natural question to ask is what are the corresponding induced distances. In this section, we investigate this problem for domain-agnostic data augmentations.
The quality of data augmentation has great impact on the performance of SSCL methods, which reflects people's prior knowledge on the data. However, when facing new data without any domain knowledge, we have to rely on domain-agnostic data augmentations, e.g., adding random noises (Verma et al., 2021), for contrast. We first consider using general random noise augmentation, i.e., for any x ∈ R d , let x ′ = x+δ where δ follows some distribution with density ϕ(x). Then, for any x i , the probability density of having t ∈ R d as its augmented point can be characterized as P t|xi = P(x i and x ′ i = t form a positive pair|x i ) = ϕ(t−x i ). We have the following proposition on Gaussian-induced distance.
Proposition 3.2 (Gaussian noise injection). If the noise distribution is isotropic Gaussian with mean zero, the induced distance is equivalent to the l 2 distance in R d , up to a monotone transformation.
Another popular noise injection method is the mixup (Zhang et al., 2017), where the augmented data are comprised of convex combinations of the training data. For each x i , a positive pair can be constructed from another x j such that x ′ i = x i + λ(x j − x i ) and λ ∈ (0,1) is the hyperparameter usually modeled with Beta distribution. For independent x 1 ,x 2 ∼ P x , denote the convoluted density of λ(x 1 −x 2 ) as p λ (x), which is symmetric around 0. Then, if employing mixup for positive pairs in SSCL, the induced distance can be written as P x1,x2 = P x2,x1 = p λ (x 1 −x 2 ).
Gaussian vs. mixup. Verma et al. (2021) proposed to use mixup when domain-specific information is unattainable and provided supportive analysis on its advantage over isotropic Gaussian noise from the classification generalization error point of view. Through (S1) perspective, we can intuitively explain why data-dependent mixup noises can be potentially better from the perspective of the "curse of dimensionality". Consider the d-dimensional Gaussian mixture setting with m < d separated components. Notice that µ 1 ,··· ,µ m can take up at most (m − 1)-dimensional linear sub-space of R d . Denoted the space spanned by µ i 's as S µ . For the light-tailed Gaussian distribution, and the majority of samples will be close to S µ . Hence, majority of the convoluted density p λ (x) will also be supported on S µ , so does the corresponding P x2,x1 . Thus, the induced distance from mixup will omit irrelevant variations in the complement of S µ and focus on the low-dimensional sub-space S µ where µ i 's actually differ. This effectively reduces the dimension dependence from d to m − 1. In comparison, isotropic Gaussian noise induces l 2 distance for positive pairs with support of R d , which will be much more inefficient, especially when m ≪ d. Since it is well-known that the performance of regression or classification models is strongly influenced by the intrinsic dimension of the input space (Hamm & Steinwart, 2021), keeping the data in a low-dimensional space is preferable.
ALIGNMENT AND UNIFORMITY
Characterizing the learned features of SSCL is of critical importance. Wang & Isola (2020) proposed alignment and uniformity as principles for SimCLR type contrastive learning methods. Such results can be intuitively understood through the perspective of (S1) and (S2). Consider the common case where the feature space is (d z − 1)-sphere. First, (3.2) indicates that only similarities (distances) between positive pairs are non-zero (finite) and all other pairwise similarities (distances) are zero (infinity). Preserving (3.2) requires the features of positive pairs to align (cosine similarity tends to 1) and those of negative pairs to be as distant as possible. If in the extreme case where positive pairs match exactly, i.e., f (x i ) = f (x ′ i ) for any i = 1,···,n, we call it perfect alignment. If perfect alignment is achieved and the features are constrained on the unit sphere, matching (3.2) implies pushing n points on the feature space as distant as possible. Maximally separated n points on a d-sphere has been studied in geometry, known as the Tammes problem (Tammes, 1930;Erber & Hockney, 1991;Melisseny, 1998). We say perfect uniformity is achieved if all the pairs are maximally separated on the sphere. There are some simple cases of the Tammes problem. If d = 2, perfect uniformity can be achieved if the mapped points form a regular polygon. If d ≥ n − 1, the solution can be given by the vertices of an (n − 1)-simplex, inscribed in an (n − 1)-sphere embedded in R d . The cosine similarity between any two vertices is −1/(n − 1) and in this case, L InfoNCE can attain its lower bound 2 . As n → ∞, the point distribution converges weakly to uniform distribution. As can be seen in Figure 1(a, b), perfect alignment and perfect uniformity are almost achieved by standard SimCLR in the Gaussian mixture setting.
As we will demonstrate in Section 3.1.4 that the spherical feature space can be bad for OOD generalization, adopting of the Euclidean space will change the statement of the uniformity property and can also be analyzed from the SNE perspective. Details can be found in Appendix A.5.
IMPLICIT BIAS
Existing theoretical results on SSCL provide justification of its empirical success in classification. However, there is more to it than just separating different classes and many phenomena are left unexplained. Take the popular SimCLR (Chen et al., 2020a) on CIFAR-10 as an example, we can consistently observe that the feature similarities within animals (bird, cat, deer, dog, frog, horse) and within objects (airplane, automobile, ship, truck), are significantly higher than those between animals and objects 3 . This can be viewed as an implicit bias towards preserving semantic information, which might be surprising as we have no supervision on the label information during the training process. However, existing literature on implicit bias is scarce. As advocated in Saunshi et al. (2022), ignoring inductive biases cannot adequately explain the success of contrastive learning. In this section, we provide a simple explanation from the perspective of SNE.
For a more concrete illustration, consider training SimCLR in the Gaussian mixture setting with d = 1, d z = 2, m = 4, µ i = i, and σ = 0.1. Denote the 4 components in ascending order by A,B,C,D. Perfect alignment and uniformity imply that their feature maps (a, b, c, d) on the unit-circle should be vertices of an inscribed square. What left unsaid is their relative order. Clockwise or counter-Clockwise from a, regardless of the initialization, we can observe SimCLR to consistently produce the order a → b→ c→ d. Remark 3.3 (Relative ordering and neighbor-preserving). The order-preserving property showcased with d = 1 is mainly for illustration, as in one-dimension, the neighboring info is simplified as the order, which is much easier to understand. The results remain the same in high dimensions as long as the clusters are well separated with an obvious order of clusters. For instance, some relative orders in Figure 1(a,b) are also stable, e.g., the neighbor of blue will consistently be purple and yellow.
With great resemblance to SNE, SSCL methods also exhibit neighbor-preserving property and we identify it as an implicit bias. Such implicit bias can be universal in SSCL and the phenomenon in Figure A.3 is also a manifestation. In deep learning, the implicit bias is usually characterized by either closeness to the initialization (Moroshko et al., 2020;Azulay et al., 2021), or minimizing certain complexity (Razin & Cohen, 2020;. In the case of SimCLR, we hypothesize the implicit bias as the expected Lipschitz constant, which has deep connections to SNE with uniformity constraint. For a feature map f onto the unit-sphere, define
C(f ) = E x,x ′ ∥f (x)−f (x ′ )∥ 2 ∥x−x ′ ∥ 2 , (3.3)
where the x 1 ,x 2 are independent samples from the data distribution. Definition 3.4 (SNE with uniformity constraint). Assume data x 1 ,···,x n ∈ R d . If the corresponding SNE features z 1 , ··· , z n ∈ R dz are constrained to be the maximally separated n points on the (d z −1)-sphere, we call this problem SNE with uniformity constraint.
The key of SNE is matching the pairwise similarity matrices Q to P . When solving SNE with uniformity constraint, the only thing to be optimized is the pairwise correspondence, or ordering of the mapping. We have the following theorem that links the neighbor-preserving property to C(f ). Theorem 3.5. Let x 1 ,··· ,x n ∈ R d such that ∥x i − x j ∥ 2 > 0 for any i,j and let z 1 ,··· ,z n ∈ R dz be maximally separated n points on the (d z −1)-sphere. Denote P = (p ij ) n×n and Q = (q ij ) n×n as the corresponding pairwise similarity matrices of x i 's and z i 's respectively. Let π denote a permutation on {1,···,n} and denote all such permutations as T . Let Q π as the π-permuted matrix Q and define C 1 (P, Q π ) = i̸ =j q π(i)π(j) p ij and π * = argmin π∈T C 1 (P, Q π ).
2 Notice that in this case, the optimal feature mapping will contain little information of the data, mapping anchor samples to interchangeable points with identical pairwise distances 3 Figure A.3 illustrates the phenomenon. Details can be found in Appendix A.1
Then, π * also minimizes ∥P − Q π ∥ F where ∥ · ∥ F is the Frobenius norm andP = (p ij ) n×n is a (monotonically) transformed similarity matrix withp ij = −1/p ij .
Theorem 3.5 showcases the relationship between minimizing C(f ) and the structure preserving property by considering a special SNE problem, where the pairwise similarity is not modeled by Gaussian as standard. Although
q ij = −∥f (x i ) − f (x j )∥ 2 is unorthodox,
it is reasonable since the larger the distance, the smaller the similarity. We have the following corollary to explain the neighborpreserving property of SSCL and the implicit bias associated with minimizing the complexity C(f ). Corollary 3.6 (Implicit bias of SSCL). When SSCL model achieves perfect alignment and perfect uniformity, if the complexity C(f ) is minimized, the resulting feature map preserves pairwise distance in the input space, resembling SNE with uniformity constraint.
Corollary 3.6 links the implicit bias of SSCL to the SNE optimization with uniformity constraint. In the case of perfect alignment and perfect uniformity, SSCL can be seen as a special SNE problem where the feature z 1 ,···,z n must be maximally separated on the unit-sphere. Recall the 1-dimension Gaussian case. There are in total 3! = 6 different orderings for the 4 cluster means, among which, a → b→ c→ d will give the lowest SNE loss. As can be seen in Figure A.4, both C(f ) and the SNE loss are monotonically decreasing during training for the Gaussian mixture setting.
When the alignment or uniformity is not perfect, the resulting feature mapping can still be characterized via SNE, with the uniformity constraint relaxed as a form of regularization. In our numerical experiments on the CIFAR-10 data, we observe C(f ) to be monotonically decreasing during the training process, supporting our hypothesis. More details can be found in Appendix A.3. Corollary 3.6 sheds light on the implicit semantic information preserving phenomenon shown in Figure A.3, as in the input space, images of dogs should be closer to images of cats, than airplanes.
TARGETING OOD: EUCLIDEAN VS SPHERICAL
Almost all SSCL methods require normalization to the unit-sphere and the similarity on the feature space is often the cosine similarity. In comparison, standard SNE methods operate freely on the Euclidean space. In this section, we show that the normalization can hinder the structure-preserving and there is a fundamental trade off between in-distribution and out-of-domain generalization.
Consider the 2-dimensional Gaussian mixture setting as illustrated in Figure 1(a). Notice that as long as the mixing components are well separated, the learned feature mapping on the sphere will always be the pentagon shape, regardless of the relative locations of the clusters. This is a result of the uniformity property derived under spherical constraint. Distant clusters in the input space will be pulled closer while close clusters will be pushed to be more distant, which results in the trade off between in-distribution and out-of-domain generalization. On one hand, close clusters being more separated in the feature space is potentially beneficial for in-distribution classification. On the other hand, the spherical constraint adds to the complexity of the feature mapping, potentially hurting robustness.
In the Euclidean space, pushing away negative samples (as distant as possible) will be much easier, since the feature vectors could diverge towards infinity 4 and potentially preserve more structural information.
To verify our intuition, we relax the spherical constraint in the Gaussian mixture setting and change the cosine similarity in SimCLR to the negative l 2 distance in R. The learned features are shown in Figure 1(c). Comparing to Figure 1(b), we can get the extra information that the purple cluster is far away to the others. If we introduce a small mean shift to the data, moving the distribution along each dimension by 1, the resulting feature maps differ significantly in robustness. As illustrated in Figure 1(d) vs. (e), the standard SimCLR are much less robust to OOD shifts and the resulting classification accuracy degrades to only 48.4%, while that for the modified SimCLR remains 100%. The same OOD advantage can also be verified in the CIFAR-10 to CIFAR-100 OOD generalization case (details in Appendix C.3 Figure C.8) and large-scale real-world scenarios with MoCo (Chen et al., 2020b) as baseline (details in Section 5).
IMPROVING SSCL BY SNE
The proposed SNE perspective (S1,S2) can inspire various modifications to existing SSCL methods.
In this section, we choose SimCLR as our baseline and investigate three straightforward modifications. For empirical evaluation, we report the test classification accuracy of nearest neighbor classifiers on both simulated data and real datasets. Experiment details can be found in Appendix C.
WEIGHTED POSITIVE PAIRS
In practice, positive pairs are constructed from anchors (training data), by i.i.d. data augmentations, e.g., random resized crop, random horizontal flip, color jitter, etc. Take random crop as an example, pair 1 and 2 may be from 30%, 80% random crops, respectively. Their similarities should not be treated as equal, as in typical SSCL methods. Incorporating the disparity in the data augmentation process is straightforward in the perspective of SNE, where the InfoNCE loss can be naturally modified as
1 2n n i=1 p ii ′ ·(l(x i ,x ′ i )+l(x ′ i ,x i )).
The weight p ii ′ in P can be specified manually to reflect human's prior knowledge. To test out the effect of such modification, we conduct numerical experiments on CIFAR-10 using the standard SimCLR. The weighting scheme is based on the Intersection over Union (IoU) of random resized crops. For each positive pair, let p ii ′ ∝ exp(IoU(x i ,x ′ i )/τ ′ ), where τ ′ > 0 is a hyperparameter (temperature) controlling the strength of the weighting scheme, i.e., the bigger the τ ′ , the closer to the unweighted state. The CIFAR-10 test performance vs. τ ′ is shown in Figure 2(a). The baseline is 80.7% and can be significantly improved to 82.1% if choosing τ ′ = 1.
T-SIMCLR: t-SNE STYLE MATCHING
Most SSCL algorithms differ mainly in (S2), i.e., defining Q and matching it to P , where fruitful results in SNE literature can be mirrored and applied. Now that we have identified the advantage of modeling features in Euclidean spaces in Section 3.1.4, the most promising modification that follows is to introduce t-SNE to SimCLR. Since we are learning low-dimensional features from high-dimensional data, preserving all pairwise similarities is impossible and the features tend to collapse. This is referred to as the "crowding problem" in Van der Maaten & Hinton (2008) (see Section 3.2 therein). t-SNE utilizes the heavy-tail t-distribution instead of the light-tail Gaussian, to model Q and encourage separation in feature space. Correspondingly, the training objective L InfoNCE can be modified as
1 n n i=1 −log 1+∥f (x i )−f (x ′ i )∥ 2 2 /(τ t df ) −(t df +1)/2 1≤j̸ =k≤2n (1+∥f ( x j )−f ( x k )∥ 2 2 /(τ t df )) −(t df +1)/2 , (4.1)
where t df is the degree of freedom for the t-distribution. Besides substituting the cosine similarity to the l 2 distance, the key modification is the modeling of feature space similarity Q, from Gaussian to t-distribution as suggested by Van der Maaten & Hinton (2008) to avoid the crowding problem and accommodate the dimension-deficiency in the feature space. We call the modified method t-SimCLR and we expect it to work better, especially when the feature dimension is low, or in the OOD case. Figure 2(b) shows the comparison between SimCLR and t-SimCLR on CIFAR-10 with different feature dimensions, where t-SimCLR has significant advantages in all cases and the smaller the d z , the larger the gap. Without decreasing the standard d z = 128, t-SimCLR improves the baseline from 80.8% to 83.9% and even beats it using only d z = 8 with accuracy 81.7%. Remark 4.1 (Degree of freedom). Standard t-SNE utilizes t-distribution with t df = 1, to better accommodate the extreme d z = 2 case. In practice, t df can vary and as d z increases, larger t df might be preferred. We recommend using t df = 5 as the default choice. The performance of t df vs d z can be found in Appendix C, as well as discussion on the fundamental difference between t df and τ .
LARGE SCALE EXPERIMENTS
In this section, we apply the same modifications proposed in Section 4.2 to MoCo-v2 (Chen et al., 2020b), as it is more device-friendly to conduct large scale experiments. We name our model t-MoCo-v2. Both models are pre-trained for 200 epochs on ImageNet following the setting of Chen et al. (2020b). The linear probing accuracy of t-MoCo-v2 on ImageNet is 67.0%, which is comparable to the MoCo result 67.5%. With the same level of in-distribution classification accuracy, we conduct extensive experiments to compare their OOD performance. The results in Table 1 and 2 suggest that our modification significantly improves the domain transfer and the OOD generalization ability without sacrificing in-distribution accuracy.
Domain Transfer. We first conduct experiments on the traditional self-supervision domain transfer benchmark. We compare MoCo-v2 and t-MoCo-v2 on Aircraft, Birdsnap, Caltech101, Cars, CIFAR10, CIFAR100, DTD, Pets, and SUN397. We follow transfer settings in Ericsson et al. (2021) to finetune the pre-trained models. The results are reported in Table 1. Our model t-MoCo-v2 surpasses MoCo-v2 in 8 out of 9 datasets, showing a significantly stronger transfer ability. Notice that our model is pre-trained with 200 epochs, surprisingly, compared with the original MoCo-v2 model pre-trained with 800 epochs, the fine-tuning results of t-MoCo-v2 are still better on Birdsnap, Caltech101, CIFAR100, and SUN397.
Out-of-domain generalization. As illustrated in Section 3.1.4, standard SSCL methods, e.g., SimCLR, MoCo, etc., could suffer from OOD shift. To demonstrate the advantage of our modification, we investigate the effectiveness of our method on OOD generalization benchmarks: PACS Li et al. . We follow the standard way to conduct the experiment, i.e., choosing one domain as the test domain and using the remaining domains as training domains, which is named the leave-one-domain-out protocol. As can be seen in
DISCUSSION
This work proposes a novel perspective that interprets SSCL methods as a type of SNE methods, which facilitates both deeper theoretical understandings and methodological guidelines for practical improvement. More interpretations of SSCL from preserving the distance between distributions can Mialon et al. (2020). On the other hand, standard SNE methods can also borrow existing techniques in SSCL to improve their performance on more complicated data, e.g., incorporating data augmentations instead of or on top of pre-defined distances. In this sense, by choosing feature dimension to be 2, various SSCL methods can also be used as data visualization tools Damrich et al., 2022). Specifically on CIFAR-10, standard t-SNE can barely reveal any clusters while our t-SimCLR with d z = 2 produces much more separation among different labels. More details can be found in Appendix C.7.
Appendix
A TECHNICAL DETAILS
A.1 IMPLICIT BIAS OF SIMCLR ON CIFAR-10. Recall the domain-agnostic data augmentation process. For any x i , the probability density of having t ∈ R d as its augmented point can be characterized as P t|xi = P(x i and x ′ i = t form a positive pair |x i ) = ϕ(t−x i ). For isotropic Gaussian densities with mean 0 and covariance matrix σ 2 I, ϕ(t − x i ) ∝ exp(−∥t−x i ∥ 2 2 /2σ 2 ), which is monotonic with the l 2 distance between t and x i . In the Gaussian mixture setting, the feature extractor is a fully connected ReLU network. Besides C(f ), we also evaluate the popular sum of squared weights. The observations on SimCLR are listed as below:
A.3 INVESTIGATIONS ON C(f ).
• The expected Lipschitz constant C(f ) is small in initialization. It first increases (till around 100 iterations) and then consistently decreases. This empirically supports the implicit bias towards minimizing C(f ).
• C(f ) and the sum of squared weights share very similar patterns. • The SNE loss is non-increasing, as if we are doing stochastic neighbor embedding using l 2 -distance.
In the CIFAR-10 case, the feature extractor is ResNet-18 plus a fully-connected projection layer. The output from ResNet-18 is usually called representation (512 dimensional) and is utilized for downstream tasks while the projection (128 dimension) is used for training. Such a representationprojection set up is common in SSCL. Ma et al. (2023) aimed to decipher the projection head and revealed that the projection feature tends to be more uniformly distributed while the representation feature exhibits stronger alignment. Besides C(f ), we also evaluate the l 2 -norm of the representation. The observations for SimCLR and t-SimCLR on CIFAR-10 are summarized as below:
Iterations C(f) • C(f ) for the projection layer shares similar patterns as in the Gaussian mixture case, first increase and then decreases. However, C(f ) for the representation layer monotonically decreases.
SNE Loss
• C(f ) for the projection layer and the l 2 -norm in the representation layer share almost identical patterns. • Comparing SimCLR, both the the calculated C(f ) and l 2 -norm are much smaller for t-SimCLR.
In conclusion, on one hand, our empirical results demonstrate that the complexity of the feature extractor C(f ) does decrease during training and seem to be implicitly minimized. On the other hand, its trend is shared with other more popularly used complexity measurements.
A.4 PROOF OF COROLLARY 3.6
In this section, we illustrate with rigor how the hypothesized implicit bias can give rise to structurepreserving property of SSCL. Corollary 3.6 states that minimizing the (Lipschitz) complexity of the feature mapping will also result in the best match between P and Q (under permutation). To provide more theoretical insight, we present the following lemma in the simpler vector-matching case. Lemma A.1. Let 0 < x 1 < ··· < x m and 0 < y 1 < ··· < y m be two real-valued sequences, normalized such that
m i=1 x 2 i = m i=1 y 2 i = 1.
Consider a permutation π of {1, ··· , m} and denote all such permutations as T . Then
argmin π∈T m i=1 y π(i) x i = argmin π∈T m i=1 (x i −y π(i) ) 2 := π * ,
where π * (i) = i for all i = 1,···,m.
Proof. By the rearrangement inequality, we have
m i=1 y π(i) x i ≥ m i=1 y i x i . Similarly, m i=1 (x i −y π(i) ) 2 = m i=1 x 2 i + m i=1 y 2 i −2 m i=1 x i ·y π(i) ≥ 2−2 m i=1 x i ·y i . C(f) ||f||2 3URMHFWLRQ&I 5HSUHVHQWDWLRQ&I 5HSUHVHQWDWLRQ||f||2
(a) SimCLR on CIFAR-10. Lemma A.1 gives a vector-version illustration of our Corollary 3.6, stating that minimizing the expected derivative (to zero) of the mapping function f , i.e., i f (x i )/x 1 leads to preserving the norm difference of the input vector and output vector.
C(f) ||f||2 3URMHFWLRQ&I 5HSUHVHQWDWLRQ&I 5HSUHVHQWDWLRQ||f||2 (b) t-SimCLR on CIFAR-10.
Next, we provide the proof of Theorem 3.5.
Proof of Theorem 3.5. Straightforwardly, we can write
∥P −Q π ∥ F = i̸ =j 1 p ij +q π(i)π(j) 2 = i̸ =j 1 p 2 ij + i̸ =j q π(i)π(j) 2 +2 i̸ =j q π(i)π(j) p ij = 2C 1 (P,Q π )+ i̸ =j 1 p 2 ij + i̸ =j q ij 2
Thus, minimizing C 1 (P,Q π ) also minimizes ∥P −Q π ∥ F . Theorem 3.5 is a straightforward generalization of Lemma A.1. Next, we provide proof for Corollary 3.6, restated below.
Proof of Corollary 3.6. Recall the SimCLR loss
L InfoNCE = 1 2n n i=1 (l(x i ,x ′ i )+l(x ′ i ,x i )), where l(x i ,x ′ i ) = −log exp(sim(f (x i ),f (x ′ i ))/τ ) x∈Dn∪D ′ n \{xi} exp(sim(f (x i ),f (x))/τ ) .
Without loss of generality, let τ = 1. Notice that l(x i , x ′ i ) is monotonically decreasing as sim(f (x i ),f (x ′ i )) increases, due to the monotonicity of function x x+c with respect to x > 0 for any c > 0. Hence, in order for L InfoNCE to be minimized, perfect alignment is required, i.e., f (x i ) = f (x ′ i ) for any i = 1,...,n.
With perfect alignment achieved, L InfoNCE only concerns the pairwise similarity between negative samples f (x i )'s, which can be simplified as L InfoNCE ≥ L uniform where
L uniform = 1 n n i=1 −log e e+ j̸ =i exp(sim(f (x i ),f (x j ))) ≥ log 1 n n i=1 1+ 1 e j̸ =i exp(sim(f (x i ),f (x j ))) ≥ log 1+ 1 n·e 1≤i̸ =j≤n exp(sim(f (x i ),f (x j ))) .
L uniform can be minimized by mapping x i 's as distant as possible, hence the connection to Tammas problem and the uniformity principle.
With sufficient capacity of the feature mapping f , the SimCLR loss can be minimized to its (empirical) global minima. However, such f is not unique since L InfoNCE is invariant to permutations of mapping relationships from x i to f (x i ). If f * n further minimizes C(f ) on the sample level, i.e.,
f * n := argmin f C n (f ) = argmin f 1≤i̸ =j≤n ∥f (x i )−f (x j )∥ 2 ∥x i −x j ∥ 2 ,
Then, f * n also solves a type of SNE problem with uniformity constraint (3.4) as stated in Theorem 3.5. To see this, if we define q ij = −∥f (x i )−f (x j )∥ 2 and p ij = −∥x i −x j ∥ 2 , which is reasonable since the larger the distance, the smaller the similarity, we can directly apply the results in Theorem 3.5. Remark A.2. As can be seen from Theorem 3.5 and the proof of Corollary 3.6, we showcase the relationship between minimizing C(f ) and structure preserving property by considering a special SNE problem, where the pairwise similarity is not modeled by Gaussian as standard, hence the word "resembling" in Corollary 3.6. Although q ij = −∥f (x i )−f (x j )∥ 2 is unorthodox, it is reasonable since the larger the distance, the smaller the similarity. If we consider the SNE method as in Hinton et al. (2006), our proof does not go through directly and demands more complicated analysis. However, our results are still valid in connecting the complexity of the feature map to the pairwise similarity matching.
Our statement in Corollary 3.6 requires perfect alignment or perfect uniformity. When the assumptions are not perfectly met, we can still obtain insights for the resulting feature mapping. Alignment and uniformity (Wang & Isola, 2020) is not the whole story of contrastive learning, and our identified structure-preserving property implicitly induced by complexity minimization provides an other angle of the learning process. From this perspective, contrastive learning can be thought of as a combination of alignment and SNE with uniformity constraint. In Figure A.3, while obtaining approximate alignment and uniformity, the feature mapping also preserves the relative relationships of the clusters (labels).
A.5 ALIGNMENT AND UNIFORMITY OF T-SIMCLR
Due to the change of training objective, we may want to reevaluate the properties of the learned feature from t-SimCLR. We will show that alignment still hold while uniformity is changed (to infinity).
Let us consider a compact region Ω ⊂ R d and x i ∈ Ω. Let t be the transformation such that the augmented data point x ′ i = t(x i ) is still in Ω. Wang & Isola (2020) showed that the contrastive loss can be decomposed into the alignment loss and the uniformity loss. Zimmermann et al. (2021) further showed that the contrastive loss converges to the cross-entropy between latent distributions, where the underlying latent space is assumed to be uniform, and the positive pairs are specified to be an exponential distribution. In this section, we show a parallel result, which states that in the population level, the t-SNE loss is the cross-entropy between two distributions of generating positive pairs. Theorem A.3. Let H(·,·) be the cross entropy between distributions. Let p(x) be the density of x, p(·|x) be the conditional density of generating a positive pair, and define
q f (x ′ |x) = C f (x) −1 p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 , with C f (x) = Ω p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dx ′ .
Then, we have
E x∼p(x) (H(p(·|x),q f (·|x)) = L a (f )+L u (f ), (A.1)
which corresponds to the population-level t-SimCLR loss where
L a =E x∼p(x) E x∼p(x ′ |x) log(1+∥f (x)−f (x ′ )∥ 2 2 ), L u =E x∼p(x) logE x∼p( x) (1+∥f (x)−f ( x)∥ 2 2 ) −1 . Proof. Note that H(p(·|x),q f (·|x)) =− Ω p(x ′ |x)log p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dx ′ +logC f (x) = Ω p(x ′ |x)log(1+∥f (x)−f (x ′ )∥ 2 2 )dx ′ − Ω p(x ′ |x)log(p(x ′ ))dx ′ +log Ω p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dx ′ = Ω p(x ′ |x)log(1+∥f (x)−f (x ′ )∥ 2 2 )dx ′ − Ω p(x ′ |x)log(p(x ′ ))dx ′ +logE x ′ ∼p(x ′ ) (1+∥f (x)−f (x ′ )∥ 2 2 ) −1 .
Taking expectation with respect to x leads to
E x∼p(x) H(p(·|x),q f (·|x)) =E x∼p(x) E x ′ ∼p(x ′ |x) log(1+∥f (x)−f (x ′ )∥ 2 2 )+E x∼p(x) logE x∼p( x) (1+∥f (x)−f ( x)∥ 2 2 ) −1 − Ω Ω p(x)p(x ′ |x)log(p(x ′ ))dx ′ dx =L a (f )+L u (f )−C p , where C p = Ω Ω p(x)p(x ′ |x)log(p(x ′ ))dx ′ dx = Ω Ω p(x,x ′ )log(p(x ′ ))dx ′ dx does not depend on f . E x∼p(x) H(p(·|x),q f (·|x)) = Ω p(x) 1 p(x) Ω p(x,x ′ )log p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dx ′ dx − Ω Ω p(x)p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dxdx ′ = Ω Ω p(x,x ′ )log p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dx ′ dx − Ω Ω p(x)p(x ′ ) 1+∥f (x)−f (x ′ )∥ 2 2 dxdx ′ .
This finishes the proof.
In Theorem A.3, L a is the alignment loss and L u is the uniformity loss. The decomposition is much more natural for t-SimCLR as opposed to that in L InfoNCE , mainly due to the change from conditional to joint distribution when modeling the pairwise similarity. Furthermore, if the t-SimCLR loss is minimized, we must have p(·|x) = q f (·|x), provided f has sufficient capacity. Note that if p(·|x) = q f (·|x), then P j|i and Q j|i are perfectly matched, which indicates that we obtain a perfect neighbor embedding. Theorem A.3 implies that the optimal feature mapping f * satisfies p(·|x) = q f * (·|x), which further implies that for any x ∈ Ω,
C f * (x) −1 p(x ′ ) 1+∥f * (x)−f * (x ′ )∥ 2 2 ∝ C(x) −1 p(x ′ |x) ⇔ C f * (x) −1 1 1+∥f * (x)−f * (x ′ )∥ 2 2 ∝ C(x) −1 p(x,x ′ ) p(x)p(x ′ ) , (A.2)
where C(x) = p(x ′ |x)dx ′ . Unlike the usual normalized SimCLR, t-SNE does not assume any special structure on f (e.g., ∥f ∥ 2 = 1), thus f can go to infinity. Comparing to the finite sample t-SimCLR loss, the population version is trickier to analyze. This is because for a given point x ′ , it can be an augmented sample of some x (with probability p(x ′ |x)), or a negative sample of x (when we treat x ′ as another sample point). This reflects the essential difficulty between population and finite samples in contrastive learning, not only for t-SimCLR.
For clustered data, (A.2) provides two important messages, provided that the augmentation is not too extreme and the augmented sample x ′ stays in the same cluster as the original x. On one hand, when x 1 and x 2 belongs to different clusters, the joint density p(x = x 1 ,x ′ = x 2 ) will be very small, close to zero, which indicates that ∥f * (x 1 )−f * (x 2 )∥ 2 is very large, tending to infinity. On the other hand, for x 1 and x 2 belonging to the same cluster, p(x = x 1 ,x ′ = x 2 ) will be relatively large. Hence, the features of the same cluster will stay close. Overall, we will observe similar clustered structure in the feature space. This is confirmed in the Gaussian mixture setting in Figure 1(c), in which case, the problem can be oversimplified as mapping 5 points in R 2 to the unit-circle.
B CONNECTION TO DISTANCE BETWEEN DISTRIBUTIONS
Through the lens of stochastic neighbor embedding, the feature learning process of SSCL methods can be seen as minimizing certain "distances" between distributions in different dimensions. Ideally, the feature should preserve the distributional information about the data. Since the data and the feature do not lie in the same metric space, quantitatively measuring their distributional distance is difficult. Fortunately, there are existing tools we can utilize, specifically, Gromov-Wasserstein distance (Mémoli, 2011;Salmona et al., 2021).
Let X , Z be two Polish spaces, each endowed respectively with probability measures p x and p z . Given two measurable cost functions c x : X × X → R, c z : Z × Z → R, and D : R × R → R, the Gromov-Wasserstein distance can be defined as
GW p (p x ,p z |c x ,c z ) := inf π∈ (px,pz) X 2 ×Z 2 D(c x (x,x ′ ),c z (z,z ′ )) p dπ(x,z)dπ(x ′ ,z ′ ) 1/p ,
where (p x ,p z ) denotes all the joint distributions in X × Z such that the marginals are p x and p z . Typically, D(c x ,c z ) is chosen to be |c x −c z | and c x (x,x ′ ) is usually chosen to be ∥x−x ′ ∥ p . The key idea of the Gromov-Wasserstein distance to circumvent the dimension mismatch is to change from comparing marginal distribution to pairwise distributions, which is very similar to the SNE objective. Consider Monge's formulation of the optimal transportation problem and let z = f (x). By choosing c z (z i ,z j ) = log( Q j|i ) with Q specified as in (3.1), c x (x i ,x j ) = P j|i with P specified as in (3.2) and letting D(c x ,c z ) = c x (log(c x )−log(c z )), we have
GW 1 (p x ,p f (x) ) ≤ E x,x ′ (D(c x (x,x ′ ),c z (f (x),f (x ′ )))),
where the right hand side recovers the expected InfoNCE loss. Hence, the SNE perspective can also be viewed as minimizing the Gromov-Wasserstein distance between p z and p x .
It is worth noting that such an interpretation only relates to contrastive learning, not including generativebased self-supervised learning methods such as Masked AutoEncoder (MAE) . to the original size), horizontal flip, color jitter (randomly change the brightness, contrast, saturation and hue of an image). To illustrate the natural weighting scheme in Section 4.1, we considered random resized crop and specifies the weights by the IoU (intersection over union) of the positive pair. In particular, two augmented images are created from an anchor image. Each augmentation crops a rectangular region of the image, denoted by r 1 ,r 2 respectively, and their IoU is defined by the area of intersection r 1 ∩r 2 divided by the area of the union r 1 ∪r 2 . The IoU is always between 0 and 1. In our experiment, we chose the default settings and Figure C.7 illustrates the IoU histogram of 1000 constructed positive pairs.
C.3 DEGREE OF FREEDOM IN t-SIMCLR
Feature dimension efficiency in OOD case. To further investigate the generalization ability of SSCL methods, we devise a challenging setting where the model is trained on CIFAR-10 and tested on CIFAR-100 classification. In this case, we evaluate the effect of increasing feature dimensions in the projection layer, as an extension on the CIFAR-10 in-distribution case. The results are shown in Figure C.8, where there are two things to note:
• The gain of extra dimensions in the OOD case does vanish later than that in the in-distribution case.
• The advantage of SimCLR vs. t-SimCLR is very significant with around 10% improvement when d = 128 using nearest neighbor 5 classification, indicating that t-SimCLR produces better separated clusters.
Relationship between t df and d z . The larger the degree of freedom t df , the less heavy-tail the t-distribution. As d z decreases, the crowding problem becomes more severe and as recommended by (Van der Maaten & Hinton, 2008), a smaller t df tends to work better. We evaluate the sensitivity of t df (1, 5, 10) under different choices of d z (1,2,4,8,16,32,64,128) in CIFAR-10 and the results are reported in Figure C.9. As can be seen, when d z is small (1,2,4,8), t df = 1 outperforms. Comparing t df = 5 and t df = 10, the two perform similarly when d z is large (16,32,64,128) but the smaller t df = 5 yields better accuracy when d z = 1,2,4.
Tuning temperature vs. tuning t df . As illustrated in Section 4.2, when the feature space dimension is low, the heavy-tailed t-distribution is a better choice than Gaussian to alleviate the crowding problem. Even though tuning the temperature of L InfoNCE , i.e., making τ larger, can also have the effect of making the distribution less concentrated (τ can be seen as the standard deviation), tuning temperature and tuning t df are fundamentally different. The former is controlling how fast does the similarity Q i,j decays as the distance between z i and z j increases, while the latter serves as a scaling factor, offering constant level modification of the scheme. In our experiments with SimCLR vs t-SimCLR on CIFAR-10, temperature is tuned as a hyperparameter. The difference in τ can never make up to the difference between the baseline SimCLR and t-SimCLR. We found τ = 0.5 to work better for the base SimCLR while larger τ works better with our t-SimCLR. We recommend τ = 5 as the default choice.
C.4 IMAGENET PRE-TRAINING
To show the ability for large scale domain transfer and OOD generalization, we conduct experiments on ImageNet pre-training based on MoCo-v2 with its official implementation 6 . We follow most of their settings, e.g, data augmentation, 200 epochs pre-training, and optimization strategy, etc. The loss is modified according to Section 4.2 and batch normalization is applied along every dimension. We grid search the t df and τ with range {2, 5, 10, 15} and {0.2, 2, 5, 10} respectively. Finally we choose t df = 10 and τ = 5 to be the optimal hyperparameters. We use this pre-train model as initialization for domain transfer and OOD experiments.
C.5 DOMAIN TRANSFER
We compare MoCo-v2 pre-trained with 800 / 200 epochs and t-MoCo-v2 on Aircraft, Birdsnap, Caltech101, Cars, CIFAR10, CIFAR100, DTD, Pets, and SUN397 in Table C.3. We follow the transfer settings in Ericsson et al. (2021) to finetune the pre-trained models. For datasets Birdsnap, Cars, CIFAR10, CIFAR100, DTD, and SUN397, we report the top-1 accuracy metric, while for Aircraft, Caltech101, and Pets, we report the mean per-class accuracy metric. We also follow Ericsson et al. (2021) to split each dataset into training, validation, and test sets. On each dataset, we perform a hyperparameter search as follows.
(1) We choose the initial learning rate according to a grid of 4 logarithmically spaced values between 1×10 −4 and 1×10 −1 ; (2) We choose the weight decay parameter according to a grid of 4 logarithmically spaced values between 1×10 −6 and 1×10 −3 , plus no weight decay;
(3) The weight decay values are divided by the learning rate; (4) For each pair of learning rate and weight decay, we finetune the pre-trained model for 5000 steps by SGD with Nesterov momentum 0.9, batch size of 64, and cosine annealing learning rate schedule without restarts. As can be seen in Table C.3, our t-MoCo-v2 with 200 epochs even outperform the baseline with 800 epochs on average.
C.6 OOD GENERALIZATION
To demonstrate the advantage of our modification, we also compare MoCo-v2 pre-trained with 800 / 200 epochs and t-MoCo-v2 on OOD generalization benchmarks: PACS Li et al. (2017), VLCS Fang et al. (2013), Office-Home Venkateswara et al. (2017). We follow the standard way to conduct the (2021). We adopt the leave-one-domain-out cross-validation setup in DomainBed with 10 experiments for hyperparameter selection and run 3 trials. As can be seen in Table C.4, our t-MoCo-v2 with 200 epochs even significantly outperform the baseline with 800 epochs for all of the three datasets.
C.7 SSCL INSPIRED DATA VISUALIZATION
t-SNE (Van der Maaten & Hinton, 2008) and its variants are designed for data visualization. However, for more complicated data, such as colored images, the results are not satisfactory. Using standard t-SNE, the 2D visualization of the 50K training images of CIFAR-10 (labels denoted as 0, 1,...,9) can be seen in Figure C.10, where different labels are hardly separated. The poor performance of t-SNE on CIFAR-10 can be traced back to the poor distance choice on images, i.e., l 2 -norm. Inspired by the success of SSCL for natural images, t-SNE can potentially be improved by incorporating data augmentations.
In light of our perspective (S1), t-SNE can take advantage of the distance specified with (3.2) and the resulting model is essentially our t-SimCLR with feature dimension 2. The visualization from t-SimCLR is shown in Figure C.11, which is much more separated (the nearest neighbor classification accuracy on CIFAR-10 test data is 56.6%). By choosing the feature dimension to be 2, various SSCL methods can also be made into data visualizing tools. In Figure C.12, we visualize the outcome from SimCLR (the nearest neighbor classification accuracy on CIFAR-10 test data is 24.8%). ; Damrich et al. (2022) where they focused specifically on data visualization and stochastic neighbor embedding.
Similar investigations have been carried in
Figure 1 :
1Gaussian mixture setting with 5 components. (a) illustration of data with 250 samples. (b) learned features by standard SimCLR with normalization (cosine similarity) to 1-sphere. (c) learned features by modified SimCLR without normalization (l 2 similarity). (d, e) feature mapping of the two methods in case of OOD mean shift. The linear classification accuracy is 48.4% in (d) and 100% in (e).
vs. t-SimCLR.
Figure 2 :
2Nearest neighbor classification test accuracy on CIFAR-10 with ResNet-18 after 200 epochs pre-training. (a) "N/A" stands for the baseline SimCLR. The x-axis is the temperature for IoU weighting scheme. (b) Comparison between SimCLR and t-SimCLR with different feature dimensions.
Remark 4 . 2 (
42Training epochs). For the CIFAR-10 experiments, we reported the results of ResNet-18 after 200 training epochs, similar to the setting ofYeh et al. (2021). We also conducted 1000-epoch experiments and found that our modifications provide consistent improvements throughout the training process, not in terms of speeding up the convergence, but converging to better solutions. Details can be found in Appendix C.1 and Figure C.6.
( 2017 )
2017,VLCS Fang et al. (2013), Office-HomeVenkateswara et al. (2017)
Figure A. 3
3plots the cosine similarity heat map of learned features from SimCLR on CIFAR-10 dataset. To calculate the similarity of class A (figures denoted by a i ) to class B (figures denoted by b i ), we first calculate the mean of b i asb. Then, we sum up i sim(a i ,b) and plot is with colors. Hence, the similarity matrix shown inFigure A.3 is not symmetric.
Figure A. 3 :
3Cosine similarity heat map of learned features from SimCLR on CIFAR-10 dataset. The darker the color, the larger the similarity.A.2 PROOF OF PROPOSITION 3.2
Figures A. 4
4and A.5 illustrate the evolution of different complexity measurements during the training process under the Gaussian mixture setting and the CIFAR-10 respectively.
Figure A. 4 :
4Empirical evaluation on the complexity of the learned feature mapping during training under the Gaussian mixture setting. Two complexity measurements are considered, i.e., C(f ) as in (3.3) and the SNE loss as in (2.2). The SNE loss here only serves as in indicator for how well the pairwise distances are preserved. The training objective is the standard InfoNCE loss. The SNE loss decreases quickly until in the first 100 iterations and then stays flat.
Figure A. 5 :
5Empirical evaluation on the complexity of the learned feature mapping during training on CIFAR-10. Two complexity measurements are considered, i.e., C(f ) as in (3.3) and l 2 -norm. Specifically, we calculate the expected Lipschitz constant on both the representation layer (512dimensional) and the projection layer (128-dimensional). Figure (a) and (b) show the trends (along the 200 training epochs) for SimCLR and t-SimCLR respectively.
Figure C. 6 :
6Nearest neighbor test accuracy vs. training epochs. SimCLR and t-SimCLR share similar trends and convergence speed.
Figure C. 7 :Figure
7The histogram of IoUs for 1000 constructed positive pairs in CIFAR-10. The empirical distribution is almost symmetric around 0.C.8: Extension onFigure 2(b). Nearest neighbor classification accuracy for SimCLR vs. t-SimCLR on both CIFAR-10 (in-distribution) and CIFAR-100 (out-of-distribution) using different feature dimensions.
Figure C. 9 :
9Nearest neighbor classification accuracy on CIFAR-10 for t-SimCLR using different feature dimensions and different degrees of freedom (t_df).
Figure C. 10 :
1050K CIFAR-10 training images visualization in 2D with t-SNE.
Figure C. 11 :
1150K CIFAR-10 training images visualization in 2D with the default t-SimCLR.
Figure C. 12 :
1250K CIFAR-10 training images visualization in 2D with the SimCLR.
The overall training objective for SNE is Significant improvements have been made to the classic SNE. Im et al. (2018) generalized the KL-divergence to f -divergence and found that different divergences favors different types of structure. Lu et al. (2019) proposed to make P doubly stochastic so that features are less crowded. Most notably, t-SNE (Van der Maaten &inf
z1,···,zn
n
i=1
n
j=1
P j|i log
P j|i
Q j|i
.
(2.2)
Table 1 :
1Domain transfer results of vanilla MoCo-v2 and t-MoCo-v2.Method
Aircraft Birdsnap Caltech101 Cars CIFAR10 CIFAR100 DTD
Pets SUN397 Avg.
MoCo-v2
82.75
44.53
83.31
85.24
95.81
72.75
71.22 86.70
56.05
75.37
t-MoCo-v2
82.78
53.46
86.81
86.17
96.04
78.32
69.20 87.95
59.30
77.78
Table 2 ,
2our t-MoCo-v2 indicates significant improvement over MoCo-v2. Both experiments indicate our modification exhibits substantial enhancement for domain transfer and OOD generalization ability. Similar to domain transfer scenario, compared with the original MoCo-v2 model pre-trained with 800 epochs, t-MoCo-v2 is better on all of the three datasets. More experiment details, including detailed comparisons, are in Appendix C.
Table 2 :
2OOD accuracies of vanilla MoCo-v2 and t-MoCo-v2 on domain generalization benchmarks.Method
PACS VLCS Office-Home Avg.
MoCo-v2
58.5
70.4
36.6
55.2
t-MoCo-v2
61.3
75.1
42.1
59.5
be found in Appendix B. Our analysis has limitations and the insights from SNE are not universally
applicable for all SSCL methods, e.g., Zbontar et al. (2021); Yang et al. (2021) don't fit in our
framework. However, this work is an interesting addition to existing theoretical works of SSCL and
more investigations can be made along this path. While there are various extensions of the classic SNE,
in this work, as a proof of concept, we mainly showcased practical improvements from t-SNE. We expect
more modifications can be developed by borrowing advances in the SNE literature, e.g., changing to
f -divergences (Im et al., 2018) or consider optimal transport Bunne et al. (2019); Salmona et al. (2021);
Table C .
C3: Domain transfer results of vanilla MoCo-v2 and t-MoCo-v2.Table C.4: OOD accuracies of vanilla MoCo-v2 and t-MoCo-v2 on domain generalization benchmarks. Method PACS VLCS Office-Home Avg. experiments, i.e., choosing one domain as the test domain and using the remaining domains as training domains, which is named the leave-one-domain-out protocol. The top linear classifier is trained on the training domains and tested on the test domain. Each domain rotates as the test domain and the average accuracy is reported for each dataset in Table C.4. On each dataset, we perform a hyperparameter search following DomainBed Gulrajani & Lopez-PazMethod
Aircraft Birdsnap Caltech101 Cars CIFAR10 CIFAR100 DTD
Pets SUN397 Avg.
MoCo-v2 (800 epochs)
83.80
45.51
83.01
86.18
96.42
71.69
71.70 89.11
55.61
75.89
MoCo-v2 (200 epochs)
82.75
44.53
83.31
85.24
95.81
72.75
71.22 86.70
56.05
75.37
t-MoCo-v2 (200 epochs)
82.78
53.46
86.81
86.17
96.04
78.32
69.20 87.95
59.30
77.78
MoCo-v2 (800 epochs)
58.9
69.8
41.6
56.8
MoCo-v2 (200 epochs)
58.5
70.4
36.6
55.2
t-MoCo-v2 (200 epochs)
61.3
75.1
42.1
59.5
All the contrastive losses are written in full batches for simplicity in this work as we focus on analyzing the optimal solutions of SSCL methods rather than the optimization process.
In practice, various regularization, e.g, weight decay, are employed and the resulting features will be bounded.
When evaluating by training linear classifiers for 100 epochs, the accuracy for SimCLR is 46.4% and that for t-SimCLR is 48.14% (averaged over 3 replications).
https://github.com/facebookresearch/moco
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Learning a parametric embedding by preserving local structure. Laurens Van Der Maaten, Artificial intelligence and statistics. PMLRLaurens Van Der Maaten. Learning a parametric embedding by preserving local structure. In Artificial intelligence and statistics, pp. 384-391. PMLR, 2009.
Visualizing data using t-sne. Laurens Van Der Maaten, Geoffrey Hinton, Journal of machine learning research. 911Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.
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Unsupervised feature learning via non-parametric instance discrimination. Zhirong Wu, Yuanjun Xiong, X Stella, Dahua Yu, Lin, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionZhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3733-3742, 2018.
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Lewei Yao, Runhui Huang, Lu Hou, Guansong Lu, Minzhe Niu, Hang Xu, Xiaodan Liang, Zhenguo Li, Xin Jiang, Chunjing Xu, arXiv:2111.07783Filip: Fine-grained interactive language-image pre-training. arXiv preprintLewei Yao, Runhui Huang, Lu Hou, Guansong Lu, Minzhe Niu, Hang Xu, Xiaodan Liang, Zhenguo Li, Xin Jiang, and Chunjing Xu. Filip: Fine-grained interactive language-image pre-training. arXiv preprint arXiv:2111.07783, 2021.
. Chun-Hsiao Yeh, Cheng-Yao Hong, Yen-Chi Hsu, Tyng-Luh Liu, Yubei Chen, Yann Lecun, arXiv:2110.06848arXiv preprintDecoupled contrastive learningChun-Hsiao Yeh, Cheng-Yao Hong, Yen-Chi Hsu, Tyng-Luh Liu, Yubei Chen, and Yann LeCun. Decoupled contrastive learning. arXiv preprint arXiv:2110.06848, 2021.
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which computes the cosine similarities in the embedding space between the test image and its nearest neighbors, and make the prediction via weighted voting. We train each model with batch size of 256 and 200 epochs for quicker evaluation. For t-SimCLR. He, C.1 CIFAR-10 SETTINGS CIFAR-10 (Krizhevsky, 2009) is a colorful image dataset with 50000 training samples and 10000 test samples from 10 categories. We use ResNet-18. 2016) as the feature extractor, and the other settings such as projection head all follow the original settings of SimCLR. without specifying otherwise, we grid search the t df and τ with range {1, 2, 5, 10} and {1, 2, 5, 10} respectivelyC.1 CIFAR-10 SETTINGS CIFAR-10 (Krizhevsky, 2009) is a colorful image dataset with 50000 training samples and 10000 test samples from 10 categories. We use ResNet-18 (He et al., 2016) as the feature extractor, and the other settings such as projection head all follow the original settings of SimCLR (Chen et al., 2020a). To evaluate the quality of the features, we follow the KNN evaluation protocol (Wu et al., 2018). which computes the cosine similarities in the embedding space between the test image and its nearest neighbors, and make the prediction via weighted voting. We train each model with batch size of 256 and 200 epochs for quicker evaluation. For t-SimCLR, without specifying otherwise, we grid search the t df and τ with range {1, 2, 5, 10} and {1, 2, 5, 10} respectively.
Ablation of training epochs We also run the SimCLR and t-SimCLR experiments in the more standard 1000 epochs setting. For SimCLR, we use batch size of 512, learning rate of 0.3, temperature of 0.7, and weight dacay of 0.0001. For t-SimCLR, we use batch size of 512, learning rate of 0.8, temperature of 10, weight dacay of 0.0002, and t df = 5. The nearest neighbor accuracy for SimCLR is 87. 2% vs. that for t-SimCLR is 88.8%.Ablation of training epochs We also run the SimCLR and t-SimCLR experiments in the more standard 1000 epochs setting. For SimCLR, we use batch size of 512, learning rate of 0.3, temperature of 0.7, and weight dacay of 0.0001. For t-SimCLR, we use batch size of 512, learning rate of 0.8, temperature of 10, weight dacay of 0.0002, and t df = 5. The nearest neighbor accuracy for SimCLR is 87.2% vs. that for t-SimCLR is 88.8%.
IMAGE AUGMENTATION When processing images, several popular augmentations are usually adopted (following the setting in. e.g., random resized crop (crops a random portion of image and resize it. SimCLR Chen et al.C.2 IMAGE AUGMENTATION When processing images, several popular augmentations are usually adopted (following the setting in SimCLR Chen et al. (2020a)), e.g., random resized crop (crops a random portion of image and resize it
| [
"https://github.com/facebookresearch/moco"
] |
[
"Particle-in-cell Modeling of Electron Beam Generated Plasma",
"Particle-in-cell Modeling of Electron Beam Generated Plasma"
] | [
"Shahid Rauf [email protected] \nApplied Materials, Inc\n3333 Scott Blvd95054Santa ClaraCaliforniaUSA\n",
"D Sydorenko \nUniversity of Alberta\nT6G 2E9EdmontonAlbertaCanada\n",
"S Jubin \nPrinceton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA\n",
"W Villafana \nPrinceton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA\n",
"S Ethier \nPrinceton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA\n",
"A Khrabrov \nPrinceton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA\n",
"I Kaganovich \nPrinceton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA\n"
] | [
"Applied Materials, Inc\n3333 Scott Blvd95054Santa ClaraCaliforniaUSA",
"University of Alberta\nT6G 2E9EdmontonAlbertaCanada",
"Princeton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA",
"Princeton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA",
"Princeton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA",
"Princeton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA",
"Princeton Plasma Physics Laboratory\n100 Stellarator Rd08543PrincetonNew JerseyUSA"
] | [] | Plasmas generated using energetic electron beams are well known for their low electron temperature (Te) and plasma potential, which makes them attractive for atomic-precision plasma processing applications such as atomic layer etch and deposition. A 2d3v particle-in-cell (PIC) model for an electron beam-generated plasma in Argon confined by a constant applied magnetic field is described in this article. Plasma production primarily occurs in the path of the beam electrons in the center of the chamber. The resulting plasma spreads out in the chamber through non-ambipolar diffusion with a short-circuit effect allowing unequal electron and ion fluxes to different regions of the bounding conductive chamber walls. The cross-field transport of the electrons (and thus the steady-state characteristics of the plasma) are strongly impacted by the magnetic field. Te is anisotropic in the electron beam region, but low and isotropic away from the plasma production zone. The plasma density increases, and the plasma becomes more confined near the region of production when the magnetic field strengthens. The magnetic field reduces both electron physical and energy transport perpendicular to the magnetic field. Te is uniform a global model for an N2 ebeam plasma, which highlighted that gas dissociation is weaker and ion to excited state density ratio higher in the ebeam plasma compared to RF plasmas. Petrov and colleagues[11,21,22]have developed a model which combines on-axis solution of the Boltzmann equation for the electron energy distribution (EED) with a fluid model for the bulk plasma. These authors have examined control of the EED, the influence of magnetic fields on plasma properties, and the effects of N2 and SF6 addition to Ar on charged species concentrations and Te. Petrov et al. also compared modeling results to experiments with generally good agreement observed. Rauf et al. [18] described a hybrid 3-dimensional (3D) model for ebeam plasmas where a Monte Carlo model was used for plasma generation by the electron beam. Levko and Raja examined the influence of SF6 addition to Ar on the stability of electron beam driven discharge using a 1D PIC model [23]. Huang et al. described a 1D PIC model for an electron beam plasma [24]. They found that the electron energy distribution is Druyvesteyn-like with a high energy tail. Most of the multi-dimensional models for electron beam plasmas are fluid-based and use fluid assumptions about charged species transport in the magnetized plasma. Classical transport equations describing the effect of magnetic field on electron transport coefficients [25] are often used in these models. The work in Ref.[18] however illustrates that such models need significant adjustments to be able to match experiments. Plasma physical and energy transport in this lowpressure magnetized plasma appears to be non-classical. To self-consistently understand the characteristics of ebeam plasma, we simulated them using a particle-in-cell (PIC) based kinetic model. The PIC modeling results are discussed in this article, which are currently being theoretically analyzed to understand electron transport and develop models for electron transport coefficients. This detailed physics analysis will be discussed in a subsequent publication.5This article is organized in the following manner. The computational model is described in Sec. 2. Modeling results are discussed in Sec. 3 and Sec. 4 includes a summary.II. COMPUTATIONAL MODELThe simulations in this article have been done using EDIPIC, a 2-dimensional (2d3v) electrostatic particle-in-cell (PIC) modeling code. EDIPIC is open source and available on GitHub with the necessary documentation [26]. It uses a standard explicit leap-frog algorithm in Cartesian geometry, with the Boris scheme for particle advance [27]. The electrostatic field is obtained from the Poisson's equation solved using the PETSc library [28]. The code includes a Monte-Carlo model of elastic, inelastic, and ionization electron-neutral collisions. Cross-sections for collisions with Argon neutrals used in the simulations described below are from Ref. [29]. EDIPIC can simulate crucial atomistic and plasma-surface interaction processes needed for simulations of partially ionized plasmas, including, but not limited to, the secondary electron emission induced by electrons and ions. The Langevin model of Coulomb collisions for electrons is also implemented. EDIPIC has been verified in several international benchmarks [30, 31]. EDIPIC is written in Fortran 90 and parallelized using Message Passing Interface (MPI).Good scalability of up to 400 CPU cores has been demonstrated. The code is equipped with numerous, diverse diagnostic capabilities, including but not limited to, the phase-space data and ion and electron velocity distribution functions, as well as spectral analysis procedures required to study wave propagation in plasmas.Electron temperature can be anisotropic in the plasma, in particular along the path of the electron beam. We, therefore, compute the components of the temperature along each coordinate direction using [32]: | 10.1088/1361-6595/acd3a9 | [
"https://export.arxiv.org/pdf/2302.14121v1.pdf"
] | 257,233,099 | 2302.14121 | 80a9473b30eabe758e74ebfa3c2f40801673fa17 |
Particle-in-cell Modeling of Electron Beam Generated Plasma
Shahid Rauf [email protected]
Applied Materials, Inc
3333 Scott Blvd95054Santa ClaraCaliforniaUSA
D Sydorenko
University of Alberta
T6G 2E9EdmontonAlbertaCanada
S Jubin
Princeton Plasma Physics Laboratory
100 Stellarator Rd08543PrincetonNew JerseyUSA
W Villafana
Princeton Plasma Physics Laboratory
100 Stellarator Rd08543PrincetonNew JerseyUSA
S Ethier
Princeton Plasma Physics Laboratory
100 Stellarator Rd08543PrincetonNew JerseyUSA
A Khrabrov
Princeton Plasma Physics Laboratory
100 Stellarator Rd08543PrincetonNew JerseyUSA
I Kaganovich
Princeton Plasma Physics Laboratory
100 Stellarator Rd08543PrincetonNew JerseyUSA
Particle-in-cell Modeling of Electron Beam Generated Plasma
1
Plasmas generated using energetic electron beams are well known for their low electron temperature (Te) and plasma potential, which makes them attractive for atomic-precision plasma processing applications such as atomic layer etch and deposition. A 2d3v particle-in-cell (PIC) model for an electron beam-generated plasma in Argon confined by a constant applied magnetic field is described in this article. Plasma production primarily occurs in the path of the beam electrons in the center of the chamber. The resulting plasma spreads out in the chamber through non-ambipolar diffusion with a short-circuit effect allowing unequal electron and ion fluxes to different regions of the bounding conductive chamber walls. The cross-field transport of the electrons (and thus the steady-state characteristics of the plasma) are strongly impacted by the magnetic field. Te is anisotropic in the electron beam region, but low and isotropic away from the plasma production zone. The plasma density increases, and the plasma becomes more confined near the region of production when the magnetic field strengthens. The magnetic field reduces both electron physical and energy transport perpendicular to the magnetic field. Te is uniform a global model for an N2 ebeam plasma, which highlighted that gas dissociation is weaker and ion to excited state density ratio higher in the ebeam plasma compared to RF plasmas. Petrov and colleagues[11,21,22]have developed a model which combines on-axis solution of the Boltzmann equation for the electron energy distribution (EED) with a fluid model for the bulk plasma. These authors have examined control of the EED, the influence of magnetic fields on plasma properties, and the effects of N2 and SF6 addition to Ar on charged species concentrations and Te. Petrov et al. also compared modeling results to experiments with generally good agreement observed. Rauf et al. [18] described a hybrid 3-dimensional (3D) model for ebeam plasmas where a Monte Carlo model was used for plasma generation by the electron beam. Levko and Raja examined the influence of SF6 addition to Ar on the stability of electron beam driven discharge using a 1D PIC model [23]. Huang et al. described a 1D PIC model for an electron beam plasma [24]. They found that the electron energy distribution is Druyvesteyn-like with a high energy tail. Most of the multi-dimensional models for electron beam plasmas are fluid-based and use fluid assumptions about charged species transport in the magnetized plasma. Classical transport equations describing the effect of magnetic field on electron transport coefficients [25] are often used in these models. The work in Ref.[18] however illustrates that such models need significant adjustments to be able to match experiments. Plasma physical and energy transport in this lowpressure magnetized plasma appears to be non-classical. To self-consistently understand the characteristics of ebeam plasma, we simulated them using a particle-in-cell (PIC) based kinetic model. The PIC modeling results are discussed in this article, which are currently being theoretically analyzed to understand electron transport and develop models for electron transport coefficients. This detailed physics analysis will be discussed in a subsequent publication.5This article is organized in the following manner. The computational model is described in Sec. 2. Modeling results are discussed in Sec. 3 and Sec. 4 includes a summary.II. COMPUTATIONAL MODELThe simulations in this article have been done using EDIPIC, a 2-dimensional (2d3v) electrostatic particle-in-cell (PIC) modeling code. EDIPIC is open source and available on GitHub with the necessary documentation [26]. It uses a standard explicit leap-frog algorithm in Cartesian geometry, with the Boris scheme for particle advance [27]. The electrostatic field is obtained from the Poisson's equation solved using the PETSc library [28]. The code includes a Monte-Carlo model of elastic, inelastic, and ionization electron-neutral collisions. Cross-sections for collisions with Argon neutrals used in the simulations described below are from Ref. [29]. EDIPIC can simulate crucial atomistic and plasma-surface interaction processes needed for simulations of partially ionized plasmas, including, but not limited to, the secondary electron emission induced by electrons and ions. The Langevin model of Coulomb collisions for electrons is also implemented. EDIPIC has been verified in several international benchmarks [30, 31]. EDIPIC is written in Fortran 90 and parallelized using Message Passing Interface (MPI).Good scalability of up to 400 CPU cores has been demonstrated. The code is equipped with numerous, diverse diagnostic capabilities, including but not limited to, the phase-space data and ion and electron velocity distribution functions, as well as spectral analysis procedures required to study wave propagation in plasmas.Electron temperature can be anisotropic in the plasma, in particular along the path of the electron beam. We, therefore, compute the components of the temperature along each coordinate direction using [32]:
2 along the magnetic field lines and slowly decreases perpendicular to it. Electrons are less energetic in the sheath regions where the sheath electric field repels and confines the low-energy electrons from the bulk plasma. Even though electron and ion densities are similar in the bulk plasma due to quasi-neutrality, electron and ion fluxes on the grounded chamber walls are unequal at most locations. Electron confinement by the magnetic field weakens with increasing pressure, and the plasma spread out farther from the electron beam region.
I. INTRODUCTION
Sub-nm scale precision is increasingly required in many critical plasma processing applications used for fabricating leading-edge microelectronics devices [1,2]. Precise control of ion energy and ion / radical composition are necessary during critical plasma processes to achieve the requisite atomic-scale fidelity. Two applications that will benefit from such precise control are atomic layer etch [3,4] and atomic layer deposition [5], which attempt to remove or add the material atomic layer by atomic layer while leaving the underlying material undamaged. Previous studies have shown that ions with energy higher than a few eV can alter or damage the sub-surface material [2,6,7]. True atomic precision plasma processing is therefore difficult to achieve using radio frequency (RF) plasmas (e.g., inductively and capacitively coupled plasmas) where ion energies are generally greater than 10 eV. Electron beam-generated plasmas (hereafter referred to as ebeam plasmas) have been shown to have characteristics suitable for atomic precision plasma processing [8]. With high ion density, low electron temperature (Te) and ion energies less than 5.0 eV in plasmas of molecular gases [9,10], ebeam plasmas can deliver a large flux of low energy ions to adjacent substrates. Significant amount of research has been reported on the characterization of ebeam plasmas [9 -12] as well as materials processing using these plasmas [2, 13 -16]. Work has also been done on developing industrial scale ebeam plasma processing systems [17,18]. Several authors have previously described models for magnetized ebeam plasmas in the literature. Fernsler et al. [19] developed model for a large area ebeam plasma and used this model to identify the plasma properties needed for the resulting sheet plasma to efficiently reflect microwaves. They correctly predicted that the plasma is generated more efficiently by energetic electron beam compared to RF, leading to a lower Te in ebeam plasmas. Lock et al. [20] described where the subscript denotes species, is the mass of the particle, ; , , is the velocity in the , , and directions, �⃗ = � �⃗ − ⟨� �⃗ ⟩ is the particle random velocity, and angular brackets 〈•〉 denote averaging over particles. We also calculate components of the heat flow vector defined as
[33]:
: , , = � : , , �,
where is the density of species s. These quantities are used in the discussion below.
III. COMPUTATIONAL RESULTS
2-dimensional simulations of the electron beam generated plasma (ebeam plasma) have been done in Cartesian geometry for the plasma reactor configuration shown in Fig. 1. As the plasma system is symmetric around x = 0, only ½ of the plasma is simulated with appropriate symmetry boundary conditions imposed at x = 0. 2 keV electrons are launched in the z-direction from a 2.8 mm wide perforated metal window at the bottom. The electron beam width and current quoted in this paper are for ½ of the window from 0 -2.8 mm. All surfaces of the plasma chamber are assumed grounded. A z-directed uniform magnetic field is applied across the plasma. All simulations have been done for 0.8 ms, and the plasma properties reach steady state in this time for the conditions considered.
Before we examine the effect of magnetic field and gas pressure on the characteristics of the ebeam plasma, we first describe plasma properties for the following operating conditions: 20 mT gas pressure in Ar, 100 G magnetic field, 12.5 mA/m electron beam current (1/2 width), and 2 keV beam electron energy. The steady-state plasma potential, source of electrons and Ar + ions, electron density (ne), and electron temperature (Tex) are shown in Fig significantly with increasing magnetic field. This is expected as the plasma is primarily produced by the beam electrons, which remain confined for the range of magnetic field strength considered.
As the magnetic field is increased, the gradient of electron density perpendicular to the magnetic field increases. This is due to reduced cross-field transport of electrons compared to electron transport along magnetic field lines. As a result of better electron confinement, the maximum value of ne is found to increase with magnetic field strength. We will look closely at the profile of ne perpendicular to the electron beam in Fig. 5. As discussed earlier, Tex is mostly uniform in the z-direction except near the boundaries where the kinetic energy of the trapped electrons is converted to potential energy as they are repelled by the electric field in the sheath region. Peak
Tex increases slightly with increasing magnetic field and the gradient of Tex across the magnetic field increases slightly as well, which is explained later using the results in Fig. 6.
To quantitatively understand how ne and Tex are impacted by the magnetic field, we have plotted ne and Tex at mid-height (averaged between z = 3.5 -5.5 cm) as a function of x in Fig. 5.
Both the absolute and normalized ne are shown. Peak ne clearly increases and ne decays faster in the x-direction with increasing magnetic field. Also, as mentioned earlier, peak Tex increases slightly with magnetic field. Although difficult to discern in Fig. 4, Tex decays faster in the xdirection when the magnetic field is stronger. It is clear from the results in Figs. 3 -5 that electron physical transport is more strongly impacted by the magnetic field than electron energy transport.
Although the electrons collide with the background gas as they spread in the chamber, they lose relatively little energy in these collisions due to the mass difference between electrons and neutral Argon atoms. Consequently, the electron temperature drops very gradually across the magnetic field.
To help further explain the trends of ne and Txe with the magnetic field, we have plotted the flux to the right wall is smaller at a larger magnetic field due to better plasma confinement.
Plasma steady-state characteristics are also sensitive to gas pressure due to its influence on collisional processes and charged species transport. We next examine the effect of varying the gas pressure from 10 -40 mTorr with a fixed 200 G magnetic field. Other operating conditions are 6.25 mA/m beam electron current (1/2 width) and 2 keV beam electron energy. The steady-state electron density ne is shown in Fig. 8 for different pressures. As the neutral gas density increases proportional to the pressure, beam electrons collide more often with the background gas.
Consequently, more charged species are produced and ne increases with increasing pressure.
Charged species transport (drift and diffusion) is also slower along the magnetic field lines at higher pressure, which further helps increase ne. The electron density ne decreases faster in the xdirection at lower pressure due to better confinement by the magnetic field, which can be seen in Fig. 10(b).
The effect of pressure on steady-state Tex is shown in Fig. 9. We comment on the effect of pressure on the anisotropy of electron temperature when the results in Figs. 10 To quantitatively examine the effect of gas pressure on ne and Tex, we have plotted ne and
Tex at mid-height (averaged between z = 3.5 -5.5 cm) as a function of x in Fig. 10. Both the absolute and normalized ne are shown. The electron density increases with pressure, but the gradient of the electron density is shallower at higher gas pressure. At the lowest pressure considered, 10mTorr, the density is peaked sharply within 1 cm of the beam, plateauing at a lower value across the remaining distance. The trends vs. pressure are related to more intense plasma production and enhanced electron diffusion across the magnetic field lines at higher pressures.
The peak Tex increases considerably with gas pressure, but gas pressure only has a minor impact on the gradient of Tex.
To help explain the trends of ne and Txe with gas pressure, we have plotted the electron velocity and z-component of the electron temperature Tze in Fig. 11 increase with higher pressure (Fig. 11(d)) indicating that electrons transport is also altered along the magnetic field direction. The vze plot has been truncated at ±240 m/s to highlight the velocity in the plasma bulk and vze is significantly larger in the sheath regions.
To examine charged species' transport out of the plasma chamber, we have plotted the electron and ion fluxes to the walls in Fig. 12
IV. CONCLUSIONS
Plasmas generated using energetic electron beams (ebeam plasmas) are well known in the literature for their low Te and plasma potential. The resulting low ion energy makes them attractive for plasma processing applications requiring atomic-scale precision, such as atomic layer etch and deposition. A 2-dimensional particle-in-cell (PIC) model for ebeam plasmas confined using a constant magnetic field is described in this article. It is found that plasma production primarily occurs in the path of the magnetically confined beam electrons. The resulting plasma spreads out in the chamber through separate mechanisms for electrons and ions, as the ambipolar electric field is short-circuited by the conducting chamber walls. Plasma transport is strongly impacted by the magnetic field. Te is observed to be anisotropic (Tez > Tex) in the beam region, but it is low and isotropic away from the region of plasma production. Charged species densities increase and become better confined near the plasma production region as the magnetic field becomes stronger.
The magnetic field reduces both electron physical and energy transport perpendicular to the magnetic field. Te is uniform along the magnetic field lines and slowly decreases perpendicular to it. Electrons are less energetic in the sheath regions, where the sheath electric field pushes back the low energy electrons from the plasma. Even though electron and ion densities are similar in the bulk plasma due to quasi-neutrality, electron and ion fluxes on the grounded chamber walls are unequal at most locations. Electron flux exceeds the ion flux near the region of plasma production.
away from the electron beam area. The ions are observed to leave on the sidewall too, where the electron flux is negligible. Electron confinement by the magnetic field weakens with increasing pressure, and the electrons spread out farther from the region of production. Te becomes less anisotropic in the region of production at higher gas pressure with more collisions. Te is observed to increase with increasing pressure, which is attributed to electron energy thermalizing better in the production region at higher pressure.
These PIC simulations have been done to self-consistently understand electron transport in the low-pressure magnetized ebeam plasmas. The modeling results are reported in this article. We are currently theoretically analyzing these results to understand charged species transport and how electron transport coefficients scale with magnetic field and gas pressure. Results from this theoretical analysis will be presented in an upcoming publication.
V. AUTHOR DECLARATION
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflict of interest
The authors have no conflicts to disclose.
. 2 .
2The beam electrons are launched in the z-direction and the applied z-directed magnetic field keeps them well confined near the center of the chamber. Consequently, plasma production primarily occurs in a narrow region near the chamber center. The beam electrons lose some energy along their trajectory as they collide with the background gas. The slower electrons near the top wall ionize the background gas at a slightly higher rate due to the higher ionization cross-section at lower energy. The electrons and ions generated during the ionization collisions spread out fairly symmetrically around z = 4.5 cm, despite the slightly skewed source. The steady-state electron density ne (and consequently the ion density since quasi-neutrality is maintained) has the typical diffusion-like profile with a peak at the chamber center and lower density near the walls. However, as discussed later in the article, the magnetic field reduces electron transport perpendicular to the magnetic field, which significantly impacts the density profile. The electron velocity distribution function (EVDF) is anisotropic in the beam region, and we have computed the components of Te in the different directions separately. Outside of the beam region (x > 2.8 mm), Tex ~ Tey ~ Tez. However, in the path of the energetic electron beam, the electron temperature is anisotropic with Tez > Tex. We have plotted the x-component of electron temperature Tex inFig. 2(c). The electron temperature is more uniform both along and across the magnetic field lines compared to ne, so electron energy spreads efficiently at these low-pressure conditions. Tex is uniform in the bulk plasma along the magnetic field axis but decreases slowly in the x-direction. The electrons are less energetic near all surfaces where the sheath electric field repels the low-energy electrons from the bulk plasma and keeps them confined. The plasma potential is 4-5 times Tex, and is not as uniform in the bulk plasma as Tex.We next consider the effect of applied magnetic field strength on plasma properties at 20 mTorr. For all these simulations, beam electrons are launched with 2 keV energy and the beam current launched from the beam inlet is 12.5 mA/m (1/2 width). The magnetic field is varied between 50 -200 G. The effect of magnetic field strength on steady state ne and Tex are shown in Figs. 3 and 4, respectively. While not shown in this article, the electron source does not change
x-directed mean electron velocity vex and heat flux Qxe in Figs. 6(a) and 6(b) as a function of the magnetic field at 20 mTorr. vxe and Qxe have been averaged over z = 3.5 -5.5 cm. The z-directed mean electron velocity vze has been plotted inFig. 6(c) as a function of the magnetic field. vze is averaged from x = 2.5 -3.5 cm and plotted as a function of z. There are no beam electrons in the region where vze has been plotted inFig. 6(c). As the magnetic field is increased, electrons are better confined, and their diffusion across the magnetic field lines is impeded. Therefore, vxe decreases with increasing magnetic field. Also due to better electron confinement, less electron energy leaves the beam region in the x-direction at the higher magnetic field. Heat flux also decreases more quickly in the x-direction at a stronger magnetic field as electron energy can escape more easily in the z-direction. Classical transport theory would indicate that electron transport should not be impacted by the magnetic field along the direction of the magnetic field. However, as shown inFig. 6(c), vze is a strong function of magnetic field and is lower in the bulk plasma at higher magnetic field. The vze plot has been truncated at ±240 m/s to highlight the velocity in the plasma bulk. vze is significantly larger in the sheath regions.Even though equal number of electrons and ions leave the plasma region at the walls in steady state, their fluxes don't have to balance at each location on the wall as the walls are conductive. Thus, the ambipolar electric field which would result in ambipolar diffusion is effectively short-circuited[34]. To illustrate the effect of magnetic field on charged species transport out of the plasma, we have plotted the ion and electron fluxes at the top, bottom, and right walls inFig. 7. These fluxes have been averaged over 5 cells adjacent to the walls (1.07 mm). At the top and bottom walls, the electron current is high in the beam regions due to the beam electron current, so we have not shown electron flux for x < 2.8 mm. Closer to the region of plasma production, more electrons are observed to leave at the walls compared to ions. As we move to larger x, the electron flux falls off more quickly due to electron confinement by the magnetic field.At a certain x-location, the ion flux surpasses the electron flux. This transition occurs at a smaller x for larger magnetic field due to better electron confinement. More electrons are observed to leave at the top surface compared to the bottom, particularly at smaller x near the region of plasma production. Although the ions are not magnetized for the magnetic field strengths considered, the ion density matches that of electrons throughout the bulk of the plasma to maintain quasineutrality.Only in the sheath regions near the walls do the ion and electron densities diverge. One can observe in Figs. 7(a) and 7(b) that the ion flux is higher near x = 0 for a larger magnetic field due to higher ion density. However, the ion flux decreases more quickly in the x-direction at the larger magnetic field. Virtually no electrons leave at the right wall for the conditions examined. Ions do exit there, but their flux is smaller on the sidewall compared to the top and bottom surfaces. Ion
and 11 have been presented. However, peak Tex increases considerably with pressure. For the range of gas pressures modeled, Tex is reasonably uniform in the z-direction except for a sharp drop near the boundaries where the kinetic energy of the trapped electrons is converted to potential energy as they are repelled by the sheath electric field.
as a function of gas pressure at 200 G magnetic field. In Figs. 11(a) -(c), the velocity and temperature have been averaged over z = 3.5 -5.5 cm and plotted as a function of x. In Fig. 11(d), vze has been averaged from x = 2.5 -3.5 cm and plotted as a function of z. This vze vs. z plot is outside the beam region where electrons are not being produced. Because of their low density, contribution of high energy beam electrons to the reported velocity and temperature is negligible. As the beam electrons have high velocity in the z-direction, the velocity of newly created electrons and scattered beam electrons is initially biased in the z-direction. Subsequent collisions generally increase the x-component of the velocity. The electrons collide more frequently with the background gas at higher gas pressure, so vxe increases with increasing gas pressure. This increase in vxe with pressure comes at the expense of vze, and as shown in Fig. 11(c), vze is significantly lower in the beam region at higher pressure. Based on Figs. 10(c) and 11(b), even though Tex ~ Tez in the region outside the beam, Te is anisotropic in the beam region. Electrons get heated in the beam region when the beam electrons collide with the background gas. Heating in the initial collisions is biased in the z-direction due to the z-directed beam electrons. As there are more collisions at 40 mTorr, more energy gets transferred in the x-direction and anisotropy in Te decreases. vze in the bulk plasma is observed to
as a function of pressure. Electron flux has not been shown in the beam region (x < 2.8 mm) as it is dominated by the beam electrons. We find that even though an equal number of electrons and ions leave the plasma in steady state, electron and ion fluxes are not equal at different locations on the wall. Near the beam region at both the top and bottom surfaces, the electron flux is higher than the ion flux as electrons are confined by the magnetic field and can leave more easily in the z-direction. Crossing a certain x location, which increases with pressure, the ion flux becomes larger than the electron flux at both the top and bottom surfaces. On the sidewall, only ions exit, and the electron flux is negligible for the range of pressures examined.
23 .
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Steady-state (a) electron production rate Se, (b) electron density ne, (c) electron temperature Te, and (d) potential. These results are at 20 mTorr gas pressure, 100 G magnetic field, 12.5 mA/m beam electron current, and 2 keV beam electron energy. 3. Steady-state electron density ne at 20 mTorr gas pressure for (a) 50, (b) 100, (c) 150 and (d) 200 G magnetic field. Other operating conditions are 12.5 mA/m beam electron current, and 2 keV beam electron energy. 4. Steady-state electron temperature Te at 20 mTorr gas pressure for (a) 50, (b) 100, (c) 150 and (d) 200 G magnetic field. Other operating conditions are 12.5 mA/m beam electron current, and 2 keV beam electron energy. 5. The effect of magnetic field on (a) electron density ne (b) normalized ne, and (c) electron temperature Te at the mid-plane. Operating conditions in the Ar plasma are 20 mTorr gas pressure, 12.5 mA/m beam electron current, and 2 keV beam electron energy. 6. The effect of magnetic field on (a) x-directed velocity of electrons vxe (b) electron heat flux in the x-direction Qxe, and (c) z-directed velocity of electrons vze. Operating conditions in the Ar plasma are 20 mTorr gas pressure, 12.5 mA/m beam electron current, and 2 keV beam electron energy. 7. The effect of magnetic field on the fluxes of electron and Ar + ions to the (a) top (b) bottom, and (c) right surfaces. Operating conditions in the Ar plasma are 20 mTorr gas pressure, 12.5 mA/m beam electron current, and 2 keV beam electron energy. and 2 keV beam electron energy. 9. Steady-state electron temperature Te at 200 G magnetic field for (a) 10, (b) 20, (c) 30 and (d) 40 mTorr gas pressure. Other operating conditions are 6.25 mA/m beam electron current and 2 keV beam electron energy. 10. The effect of gas pressure on (a) electron density ne (b) normalized ne, and (c) electron temperature Te at the mid-plane. Operating conditions in the Ar plasma are 200 G magnetic field, 6.25 mA/m beam electron current, and 2 keV beam electron energy. 11. The effect of gas pressure on (a) x-directed velocity of electrons vxe, (b) z-directed temperature of electrons Tze, (c) z-directed velocity of electrons vze plotted as a function of x. (d) z-directed velocity of electrons vze plotted as a function of z. Operating conditions in the Ar plasma are 200 G magnetic field, 6.25 mA/m beam electron current, and 2 keV beam electron energy. 12. The effect of gas pressure on fluxes of electron and Ar + ions to the (a) top (b) bottom, and (c) right surfaces. Operating conditions in the Ar plasma are 200 G magnetic field, 6.25 mA/m beam electron current, and 2 keV beam electron energy.
Figure 1
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| [
"https://github.com/PrincetonUniversity/EDIPIC-2D,"
] |
[
"IMPACT OF EXPLAINABLE AI ON COGNITIVE LOAD: INSIGHTS FROM AN EMPIRICAL STUDY",
"IMPACT OF EXPLAINABLE AI ON COGNITIVE LOAD: INSIGHTS FROM AN EMPIRICAL STUDY"
] | [
"Lukas-Valentin Herm [email protected] \nUniversity of Würzburg\nWürzburgGermany\n"
] | [
"University of Würzburg\nWürzburgGermany"
] | [] | While the emerging research field of explainable artificial intelligence (XAI) claims to address the lack of explainability in high-performance machine learning models, in practice, XAI targets developers rather than actual end-users. Unsurprisingly, end-users are often unwilling to use XAI-based decision support systems. Similarly, there is limited interdisciplinary research on end-users' behavior during XAI explanations usage, rendering it unknown how explanations may impact cognitive load and further affect end-user performance. Therefore, we conducted an empirical study with 271 prospective physicians, measuring their cognitive load, task performance, and task time for distinct implementationindependent XAI explanation types using a COVID-19 use case. We found that these explanation types strongly influence end-users' cognitive load, task performance, and task time. Further, we contextualized a mental efficiency metric, ranking local XAI explanation types best, to provide recommendations for future applications and implications for sociotechnical XAI research. | 10.48550/arxiv.2304.08861 | [
"https://export.arxiv.org/pdf/2304.08861v1.pdf"
] | 258,187,065 | 2304.08861 | b872482af407059ae02b9458522cc39bf74bf0f2 |
IMPACT OF EXPLAINABLE AI ON COGNITIVE LOAD: INSIGHTS FROM AN EMPIRICAL STUDY
Lukas-Valentin Herm [email protected]
University of Würzburg
WürzburgGermany
IMPACT OF EXPLAINABLE AI ON COGNITIVE LOAD: INSIGHTS FROM AN EMPIRICAL STUDY
1 Research PaperExplainable Artificial IntelligenceCognitive LoadEmpirical Study
While the emerging research field of explainable artificial intelligence (XAI) claims to address the lack of explainability in high-performance machine learning models, in practice, XAI targets developers rather than actual end-users. Unsurprisingly, end-users are often unwilling to use XAI-based decision support systems. Similarly, there is limited interdisciplinary research on end-users' behavior during XAI explanations usage, rendering it unknown how explanations may impact cognitive load and further affect end-user performance. Therefore, we conducted an empirical study with 271 prospective physicians, measuring their cognitive load, task performance, and task time for distinct implementationindependent XAI explanation types using a COVID-19 use case. We found that these explanation types strongly influence end-users' cognitive load, task performance, and task time. Further, we contextualized a mental efficiency metric, ranking local XAI explanation types best, to provide recommendations for future applications and implications for sociotechnical XAI research.
Introduction
Due to recent advances in computing, the spectrum of potential use cases for the application of artificial intelligence (AI) is constantly expanding, enabling end-users to rely almost solely on data-driven decision support systems (DSS) (Berente et al., 2021. That is, integrating AI into information systems forms intelligent systems to enhance end-users' and organizations' effectiveness (Gregor andBenbasat, 1999, Herm et al., 2022). In this context, AI refers to an abstract concept mimicking human cognitive abilities through the application of mathematical and statistical algorithms, to generate (i.a.) machine learning (ML) models capable of automatically finding nonlinear relationships within data. So, decision knowledge is derived without the need for explicit programming (Goodfellow et al., 2016, Russell andNorvig, 2021). Research has focused on overcoming algorithmic constraints by increasing the decision complexity of ML algorithms, resulting in ML applications capable of outperforming domain experts even in complex and high-stakes use cases , McKinney et al., 2020. Furthermore, a subclass of ML algorithms, called deep learning (DL), uses deep neural network architectures to achieve unsurpassed performance. In turn, the inner decision logic of these models is no longer traceable by humans, which reduces end-users' willingness to use these AIbased DSSs; thus, their overall acceptance is decreased, potentially leading to algorithm aversion (Berger et al., 2021.
To address this issue, the research stream of explainable AI (XAI) has developed approaches to overcome the lack of traceability while maintaining the performance of these black-box models . However, as all that glitters is not gold, these approaches are mostly mathematically driven models that provide technical explanations, as opposed to addressing the actual end-users of the system with a sound explanatory scope. That is, recent XAI approaches have mainly been designed by developers for developers (Arrieta et al., 2020, van der Waa et al., 2021. Consequently, first research endeavors emerged proposing research agendas (Laato et al., 2022), first-hand end-user evaluations , Shin, 2021, and design knowledge (Herm et al., 2022, Meske and). Yet, it is not completely apparent how an end-user's heuristic mental model behaves, in terms of different perception factors, when operating within a use-case (Laato et al., 2022). Unsurprisingly, IS research calls for further examinations of AI-based explanations from a sociotechnical perspective (Gregor andBenbasat, 1999, Herm et al., 2023), which also interfere with the research streams of human-computer interaction and cognitive science (Langer et al., 2021, Liao andVarshney, 2022). Crucially, an explanation is a social and cognitive process of knowledge transfer from an XAI-based DSS to the end-user (Miller, 2019). It is unclear how end-users perceive these explanations, as increased cognitive load may be imposed when end-users rely on them to solve real-world tasks (Hudon et al., 2021). Furthermore, it is unknown whether this increased cognitive load affects end-user performance or the time required to solve a task (Hemmer et al., 2021). Consequently, XAI explanations should be perceived as mentally efficient to prevent end-users from feeling overwhelmed, stressed, and unable to perform well (Buçinca et al., 2020, Paas et al., 2016. Complicating matters further, due to the increasing attention to XAI in research and practice, numerous XAI applications are being developed, creating an XAI jungle from which to select an appropriate XAI approach (Das andRad, 2020, Dwivedi et al., 2022). That is, organizations must determine how XAI explanations affect end-user behavior and what type of explanation should be used to form a sound DSS application within intelligent systems (Gregor and Benbasat, 1999). Hence, Mohseni et al. (2021) proposed an initial systematization that groups XAI explanation types in an implementation-independent manner, providing a foundation for future research and further facilitating generalizable findings that can be transferred to any type of XAI application in practice. Following Mohseni et al. (2021)'s systematization, we contribute to IS research (Gregor andBenbasat, 1999, Meske and by comparing these explanation types in an end-user-centered manner. Therefore, we conduct an empirical study in the field of medicine using COVID-19 X-ray images to measure end-users' cognitive load. Similarly, we benchmark end-users' task performance and time required to solve the task. Lastly, we combine these findings to put the mental efficiency metric of Paas et al. (2016) into the context of sociotechnical XAI research. To summarize our research intent, we propose the following research question (RQ):
RQ: Do XAI explanation types affect end-users' cognitive load and what are the ramifications for task performance and task time?
The remainder of this paper is organized as follows: Section 2 presents the theoretical foundations, related work, and our measurement model. Section 3 describes the research methodology by following Müller et al. (2016) and the applied study design. Section 4 presents the data analysis, including demographic data, descriptive statistics, and hypotheses testing. Then, Section 5 discusses the findings to answer our RQ, derives implications for research and practice, and describes the study's limitations and recommendations for future research. Finally, Section 6 summarizes our research findings by drawing conclusions.
Theoretical Foundation
(Explainable) Artificial Intelligence
Artificial Intelligence. Following Berente et al. (2021), AI can be envisioned as an arbitrary frontier of computational advancements that mimics human-like or superhuman intelligence, enabling DSSs to assist end-users in accomplishing any task. A DSS employs ML models to enable these artificial cognitive capabilities. Here, ML is an umbrella term that encompasses mathematical and statistical algorithms used to automatically infer decision knowledge using historical data (Goodfellow et al., 2016). To this end, recent research has developed increasingly complex algorithms with high predictive power, making the models' rationale less tractable . Unsurprisingly, research has derived the performance-explainability trade-off, where inherently understandable models have been proposed to have the lowest performance and -conversely -DL models to have the highest performance . Here, DL is subsumed under the umbrella term ML and refers to a deep neural network architecture with decision logic that is no longer comprehensible to humans . DL applications can generate promising outcomes, even in high-stakes use cases (e.g., medicine) where a wrong decision could cost human lives (Dwivedi et al., 2022, McKinney et al., 2020. However, this may reduce end-users' willingness to use the system as non-traceability could lead to ambiguity and uncertainty in task solving (Epley et al., 2007). Alternatively, end-users may not be allowed to use the system due to regulations, such as the General Data Protection Regulation (GDPR) (Goodman and Flaxman, 2017). Explainable Artificial Intelligence. In response, the multidisciplinary research stream of XAI has emerged. Its objective is to develop transfer techniques that make these opaque black-boxes comprehensible to users while preserving the predictive power of the underlying DL model (Arrieta et al., 2020. Thus, post-hoc explainability methods have been developed for specific types of ML models (model-specific) or a subset of them (model-agnostic); for different dataset formats (e.g., images, text, or tabular); and for different task types (e.g., classification or regression). They can also be distinguished by the nature of their explanatory scope -either explaining predictions for individual observations (local) or explaining the ML model's inner decision logic (global) (Das andRad, 2020, Speith, 2022). This results in a plethora of explanation possibilities for depicting a rationale. In conjunction, countless distinct XAI applications have been developed in practice, creating an XAI jungle from which to choose and thus complicating the development process. Therefore, Mohseni et al. (2021) systematized these explanation types in an application-independent fashion. Table 1 summarizes these explanation types, their respective descriptions, and an exemplary excerpt of XAI's implementation jungle for each explanation type:
Type 1 Description 1 Exemplary Implementations 2
How
Holistic representation of how the ML model's inner decision logic operates -global explanation type.
ProfWeight, SHAP, DALEX, Saliency
How-To
Hypothetical adjustment of the ML model's input yielding a different output (counterfactual explanation) -local explanation type. DiCE, KNIME, PDP
What-Else
Representation of similar instances of inputs that result in similar ML model outputs (explanation by example) -global explanation type.
SMILY, Alibi
Why Description of why a prediction was made by informing which input features are relevant to the ML model -local explanation type.
SHAP, LIME, ELI5, Anchor
Why-Not Description of why an input was not predicted to be a specific output (contrastive explanations) -local explanation type. CEM, ProtoDash Table 1.
Description and Implementation of XAI Explanation Types
In practice, this XAI jungle is exacerbated by developers primarily designing these XAI implementations for developers without prioritizing the actual end-users (van der Waa et al., 2021). As first research endeavors target the end-user of an XAI-based DSS, these interdisciplinary research outcomes must be incorporated into practical applications to design valuable explanations for end-user (Arrieta et al., 2020). Following the explanation theory of Miller (2019), a useful explanation is defined as a social and cognitive process of knowledge transfer from an XAI-based DSS to the end-user. Thus, if an explanation is perceived as inadequate, contradicts an end-user's mental model, or does not appeal to their emotions or beliefs, trust issues can occur and user acceptance may be reduced, leading to algorithm aversion (Berger et al., 2021, Shin, 2021. Following recent IS research, a mental model defines any type of mental representation used to encode beliefs, facts, and knowledge when conceptualizing cognitive processes (Bauer et al., 2023). In this sense, the extent to which these explanation types affect end-users' cognitive load is unknown, which is an essential factor in the design and development of appropriate XAI implementations .
Cognitive Load Measurement
Although the human cognitive system can be considered an information-processing engine, its capacity is limited when using information systems. Providing too much or distracting information in an instructional design can lead to a high cognitive load for the end-user (Bahari, 2023). The cognitive fit theory (CFT) (Vessey, 1991) posits the relationship between a task and the required information presentation (i.e., the type of XAI explanation), where an inappropriate explanation type leads to poor end-user task performance. Moreover, end-users are unlikely to have a solid understanding of the instructional design or to build a representative mental model of the task problem (Simon, 1955). This leads to them feeling overburdened, stressed, and incapable of performing sound decision-making (Anderson et al., 2020, Paas et al., 2004. Therefore, cognitive research developed a computational approach that combines mental effort, task performance, and task time into a quantitative variable called mental efficiency to classify the goodness of instructional design with respect to end-users' information processing to prevent excessive mental workload in complex cognitive tasks (Paas et al., 2016). Accordingly, XAI explanations should require an appropriate level of cognitive load to represent the model's decision and facilitate seamless knowledge transfer to the end-user. Paired with an appropriate level of task performance and task time, the high mental efficiency of an XAI explanation constitutes a well-designed XAI-based DSS , Hudon et al., 2021. While cognitive load is a multifaceted construct comprising various components, cognitive science research has developed several approaches for measuring it. Objective measures exist, such as eye activity, along with subjective measures, such as self-reported mental effort (Schmeck et al., 2015). While the former focuses on the identification of unconscious factors among participants, the latter targets conscious factors. Accordingly, both approaches behave complementary (Tams et al., 2014).
Preliminaries and Research Gap
To investigate the extent to which cognitive load from the perspective of XAI explanations has already been researched, we conducted a structured literature review according to Webster and Watson (2002). We focused on the information systems-related databases ScienceDirect, AIS eLibrary, and Web of Science, as well as the computer science-related databases ACM Digital Library and IEEE Xplore. Specifically, we used the following search term: "((expla* | interpreta*) AND (explainable artificial intelligence | artificial intelligence | deep learning | machine learning | AI | XAI)) AND (cognitive load | mental load | mental effort | mental workload | cognitive capacity)". Without restricting our search in terms of (journal) rankings, we identified n = 2,814 publications as potentially relevant. Hence, we consider publications that examine or discuss the effects of XAI explanations (packages) on end-users' cognitive load as relevant. This results in n = 17 publications after performing an abstract, keyword, and full-text analysis. Theoretical Considerations. Most publications (n = 12) have merely centered the theoretical relevance of cognitive load (e.g., Herm et al., 2021) and assumed that reduced cognitive load positively affects end-user performance (e.g., Hemmer et al., 2021) and assists end-users to solve the task faster (Bertrand et al., 2022). It is also hypothesized that increased problem complexity might be perceived as cognitively demanding (Cai et al., 2019). Similarly, research suggests that increased cognitive load reduces enduser trust in the system (e.g., Sultana and Nemati, 2021). In this context, research has derived tentative design principles (Fahse et al., 2022a) or design frameworks that assume reduced information granularity diminishes cognitive load (Barda et al., 2020). Empirical Research. Only scarce research (n = 5) has focused on testing XAI's cognitive load. These contributions have mainly compared a single XAI implementation package or single XAI explanation type with a black-box implementation (e.g., Abdul et al., 2020) under simplified conditions, such as proxy tasks (Buçinca et al., 2020). From that, these contributions provide first evidence, that increased explainability will reduce end-user's cognitive load (Kulesza et al., 2013). In addition, research has focused on the connection between the end-user's cognitive load and their confidence or trust (Davis et al., 2020, Karran et al., 2022, implying that increased cognitive load slightly negatively affects perceived confidence and trust. Research Gap. In summary, this sparse stream of research contains merely a handful of theoretical and empirical contributions. Regarding the former, theoretical considerations already hypothesize that use case complexity may affect end-users' cognitive load, which in turn affects task performance, task time, and trust. Concerning the latter, previous empirical contributions have mainly examined the cognitive load of end-users on a single XAI explanation package or type. Most strikingly, there is currently no research contribution that examines multiple implementation-dependent XAI explanation types simultaneously to provide conceptual guidance for a domain-independent XAI-based DSS application. Furthermore, while research has emphasized the potential impact of cognitive load on end-user task performance and time to solve a particular task, empirical evidence is lacking. As a result, to the best of our knowledge, we are the first to use these preliminary results to perform a holistic empirical cognitive load evaluation of these implementation-independent XAI explanation types and their impact on task performance and task time.
Measurement Model
To conduct our research, we derive and test hypotheses to investigate the cognitive load of the aforementioned explanation types and their effects on task performance and task time. Beyond this hypotheses testing, the findings are then used as input for the mental efficiency metric of Paas et al. (2016) to enable a summative evaluation (cf. Table 2). In the following, we describe the derivation of the hypotheses for our RQ and provide an overview of the measurement model.
Figure 1. Measurement Model
In line with the CFT, we derive a group structure consisting of one independent variable and three dependent variables. The independent variable is the type of XAI explanation, while the dependent variables are mental effort, task performance, and task time. Here, the independent variable represents the choice of XAI explanation types to provide reasoning for the DL model's decision logic. The dependent variable of mental effort, defined as the total sum of cognitive processing that a human is engaged in, indicates the perceived level of cognitive load required to comprehend the provided XAI explanation for task solving Pérez-Fuster, 2019, Paas andVan Merriënboer, 1993). Similarly, the dependent variable of task performance results from the end-user's ability to use the provided XAI explanation to solve a task within a use case. Finally, the dependent variable of task time results from the time required by an end-user to solve a task when using an XAI explanation. First, we assume that assisting an end-user with any type of XAI explanation would reduce the mental effort required to comprehend an ML model's reasoning for a classification (Mohseni et al., 2021). This is because these explanations pinpoint towards relevant parts of the observation for the model's classification, compared with end-users who have to figure this out for themselves . Therefore, we propose the following hypothesis: H1: Any type of XAI explanation reduces mental effort compared with no explanation. Second, while research suggests that XAI explanations differ in terms of their perceived explainability , we assume that this degree of explainability is in line with the perceived mental
XAI Explanation Type
Computational Variables End-User Perception effort required to comprehend the reasoning of an ML model. That is, while explanation types such as Why and Why-Not explanations are local explanations -and therefore have a more straightforward explanatory fashion and scope than global explanation types (e.g., How) (Buçinca et al., 2020, Speith, 2022) -we hypothesize that variations in explanatory scope and style would result in a different level of required mental effort for each XAI explanation type. Thus, we formulate the following hypothesis: H2: Each type of XAI explanation differs in terms of mental effort. Third, providing information that requires a high cognitive load may overwhelm people during task solving, resulting in weak task performance (Hemmer et al., 2021). This may be the case when too much information is presented in a complex scenario, wherein humans are either incapable of comprehending all of it or deliberating among the levels of relevance within it (Hudon et al., 2021). Bringing this into an XAI perspective, we hypothesize that XAI explanation types that require less mental effort would improve end-user task performance. Therefore, we propose the following hypothesis: H3: Less mental effort when using XAI explanations leads to improved end-user task performance. Fourth, in research, cognitive load is considered as the number of items processed within a limited time period (Leppink et al., 2014); thus, it impedes any other cognitive tasks or activities (Barrouillet et al., 2007). That is, tasks that require a relatively significant amount of time to solve are perceived as requiring increased mental effort, resulting in a linear relation Pérez-Fuster, 2019, Otto andDaw, 2019). Hence, we hypothesize that XAI explanations that require less mental effort would help end-users to solve tasks faster than explanations that demand more mental effort. Therefore, we formulate the following hypothesis: H4: Less mental effort when using XAI explanations leads to reduced end-user task time.
3
Research Design
Methodology Overview
We follow the methodology of Müller et al. (2016) to ensure the rigor of our research. This involves a four-step process, namely 1) RQ, 2) data collection, 3) data analysis, and 4) results interpretation. Figure 2 presents an overview of the research design, followed by descriptions of the four research steps.
Figure 2.
Overview of the Research Design 1) RQ: Based on the structure literature review, we found that only a handful of contributions have investigated various XAI explanation types in a holistic and implementation-independent manner. Moreover, research assumes that these explanations differ in their cognitive load. Building on this knowledge gap, we derived hypotheses to investigate this assumption and further demonstrate whether this also affects task performance and task time. We use these findings to calculate the mental efficiency of these XAI explanation types. 2) Data collection: We use a publicly available dataset of COVID-19 chest X-ray images, a DL model, and the aforementioned XAI explanation types to test our hypotheses through a user-based study. 3) Data analysis: From the analysis, we derive a knowledge base for our research. 4) Results interpretation: Ultimately, we answer our RQ and derive implications for research and practice.
Data Collection
XAI-Explanations
Research Question
Scenario:
Data Analysis
Why-Not Baseline What-Else
Why How
Mental Effort
Results Interpretation
How-To
Userbased Study
Survey Design
To answer our RQ, we focused on a high-stakes use case from medicine. Specifically, we used chest Xray images of COVID-19-infected and healthy humans (Tawsifur et al., 2022) to train a DL model -a convolutional neural network (CNN) -consisting of 11 layers, which yields a classification accuracy of 96.43% on the validation set. Then, we had the CNN classify several images of infected and healthy humans and enriched the images with the explanation style of the aforementioned XAI explanation types from Section 2.1. Following the study design of Herm et al. (2023), we chose a within-subjects design for our study. First, we asked participants about their demographics, introduced the high-stakes use case, and described how the XAI-based DSS operates, enabling them to put themselves in the position of a physician deciding on a patient's well-being. Subsequently, we asked each participant to perform one assignment for every explanation type: Within each assignment, they received an input image of a chest X-ray, the corresponding XAI augmented image (XAI explanation), and a comprehensive description of the XAI explanation. For each explanation type, we designed two variants, one image with an infected chest and one for healthy patients. Only one variant is shown at a time (evenly and randomly distributed). Using the provided XAI explanation, each participant was asked to classify whether the depicted chest is infected with COVID-19 or not. Then, they were asked to rate the mental effort required for this classification task on a seven-point Likert scale (extremely low to extremely high).
Using their classification, we measured their (task) performance (correct or incorrect) and clocked the required time to complete the task (task time). Both measurements were performed for every assignment and every participant. An example of the study design for the explanation type Why is presented in Figure 3. See Herm (2023) for the complete questionnaire.
Figure 3. Example of the Study Design
To avoid bias, we did not present the performance metrics of the used CNN (performance bias); did not use colors nor representations of XAI implementation packages (e.g., SHAP) to avoid confirmation bias; avoided learning effects through randomization; and only provided a comprehensive description for the XAI explanation to avoid forcing anchoring bias. Additionally, we incorporated several mechanisms, including attention checks, to ensure the validity of responses. Furthermore, we asked an XAI researcher to appraise our study design and a physician to review whether the classification tasks are equally difficult. Also, we conducted a preliminary study to test its validity. As we focused on the actual enduser of an XAI-based DSS, we targeted novice users in terms of AI experience. That is, we focused on prospective physicians currently enrolled as medical students, since experienced physicians might exhibit bias toward XAI-based DSS, and moreover, we wanted to focus on the future healthcare workforce , Logg et al., 2019.
Is this chest diseased with Covid-19?
Please use the information provided above to solve this task.
Yes No
Rate your perceived level of mental effort during this task.
Description of Explanation:
In the center section, the system's decisionmaking process is explained. Here, the light gray area with black border represents the area that the system considers relevant to the overall classification of Covid-19 or no Covid-19. The rest of the image is not considered as relevant.
Input Image: Explanation:
Data Analysis
Survey Overview and Demographics
To recruit our participants, we used the Prolific.co platform, where we offered a monetary incentive of £10 per hour. Using this platform, we were able to specify and address our target group of prospective physicians (Peer et al., 2017). For this purpose, we gathered feedback from n = 271 participants. Since we performed several validation checks, such as randomly completed questionnaires, time-based outliers, lazy patterns, and control questions, we considered feedback from n = 246 participants to be optimal for our study. Among these, n = 130 participants were female, n = 115 were male, and n = 1 was diverse. Since we targeted enrolled students, n = 12 participants were younger than 20 years, n = 193 were 20−30 years old, and n = 41 were older than 31 years. They were located in Europe (n = 125), North America (n = 64), or Africa (n = 50). Regarding AI experience, n = 95 had no experience, while n = 112 had fewer than 2 years, and only n = 39 had more than 2 years. Further, n = 84 participants had less than 2 years of experience in medicine, n = 101 had 2−5 years, and n = 61 had more than 6 years.
Data Results
First, we provide an overview of the results and their distribution for the dependent variables for each explanation type (cf. Table 2). Subsequently, we utilize the findings to test our hypotheses in Table 3. Descriptive Statistics. Table 2 highlights the results of the dependent variables: First, the mental effort findings, including medians and deviations, are plotted in Figure a). Second, the total numbers of correct and incorrect answers are presented in Figure b). Here, the average task performance is calculated by the ratio between correct and incorrect answers per type. Third, an overview of the distribution and kernel density of the time required per task is provided in Figure c). These results are also summarized in tabular form. Thereon, we calculate the mental efficiency of the explanation types (Paas et al., 2016).
c) Task Time a) Mental Effort b) Task Performance
Correct n=120
Correct n=134
Correct n=160
Correct n=167
Correct n=214
Correct n=199
False n=126
False n=112
False n=86
False n=79
False n=32 Mental Effort. The absence of an XAI explanation (Baseline) led to the highest required mental effort in this study (median = 6). The local explanations Why (median = 2) and Why-Not (median = 3) required the least mental effort to solve the task. By contrast, the global explanation How (median = 5) and the How-To explanation (median = 5) required the most mental effort across all XAI explanation types. Within this range, providing multiple images for a task to indicate similar examples (What-Else, median = 4) was rated as requiring moderate mental effort. Task Performance. Regarding task-solving performance, without any XAI explanation (Baseline), the participants solved approximately 49% of the tasks correctly. Consistent with the mental effort results, using the explanations Why (87%) and Why-Not (81%) led to the highest task performance. When participants were supplied with a global explanation (How), their task performance increased slightly (55%) compared with no XAI explanation. Finally, the explanation types How-To (65%) and What-Else (68%) were in the middle of this comparison. Task Time. When participants did not use XAI explanations (Baseline), they took the longest time on average (72.59 sec) to solve a task. By contrast, the explanations Why (34.50 sec) and Why-Not (38.92 sec) almost halved the elapsed time. We noticed that these explanation types exhibited a high density around this meantime compared with explanations the What-Else or How-To. In this respect, the mean task times of the How-To (49.84 sec), How (51.68 sec), and What-Else (60.10 sec) explanations were much closer to the baseline than those of the Why and Why-Not explanations. Mental Efficiency. Since an ME above null would indicate that the end-users' performance was higher than expected compared with the mental effort invested (Paas et al., 2016), the explanations Why (0.34) and Why-Not (0.23) can be considered highly efficient. By contrast, the most mental effort was required to solve a task when no XAI explanation (Baseline, −0.34) or How explanation (−0.15) was presented. The explanations How-To (−0.11) and What-Else (−0.08) also performed slightly better in this calculation but still yielded negative values. Hypotheses Testing. To test our hypotheses (cf. Section 2.4), we follow Motulsky (2014) and apply different testing methods for H1−H4 depending on the type of test case, as demonstrated in Table 3. For each hypothesis, the results are plotted and then the statistical method, resulting p-value, and corresponding decision of acceptance or rejection are provided below.
H. Description
Test p-Value 1,2 Dec. 3
H1
Mental effort of every XAI explanation is lower than baseline. Kruskal-Wallis Cf. Figure a)*** Acc.
H2
Mental effort of every XAI explanation differs. Friedman 3.48e-15*** Acc.
H3
Decreased mental effort results in increased task performance. Spearman 0.024** Acc.
H4
Decreased mental effort results in decreased task time. Spearman 2.78e-12*** Acc.
Legend: 1) * <0.10, ** <0.05, *** <0.001; 2) for H1, each test yielded high significance; 3) decision of acceptance (acc.) or rejection (rej.) of the hypothesis. Table 3.
Results of Hypotheses Testing
Using the results in Table 3, we decide whether to accept or reject our hypotheses as follows: First, to test whether providing an XAI explanation reduces the mental effort required to solve a task compared with no explanation (Baseline) (H1), we performed five Kruskal-Wallis tests that compared each XAI explanation with our baseline individually. This yielded highly significant results for each comparison, which confirm H1. Second, to investigate whether, due to the different explanatory scopes, each XAI explanation led to different levels of perceived mental effort, we performed a Friedman test and compared all types. Since this procedure revealed highly significant results, we accept H2. Third, to test whether using XAI explanations perceived as less demanding in terms of mental effort led to higher task performance (H3), we performed a Spearman correlation test to find an association between these two dependent variables. We found evidence of a significant correlation and thus accept H3. Finally, to test whether lower mental effort also correlates with lower task time (H4), we performed a Spearman correlation test to detect an association between mental effort and task time. We obtained a highly significant correlation, confirming H4.
5
Results Interpretation
Discussion of Results
To address our RQ, we interpret the results presented in Section 4.2 compromising end-users' cognitive load, task performance, and task time. Subsequently, we discuss the computed metric mental efficiency (cf. Table 2) to combine the findings of these dependent variables. (Karran et al., 2022) already assumed that any type of XAI explanation assists the end-user in solving tasks, thereby reducing the required cognitive effort, as any type of explanation helps to render the end-user's mental model more congruent with the task problem compared with no explanation. We support this assumption through H1. Conversely, unstable explanations can influence the mental model. Thus, explanations are likely to impact end-user trust, especially when abductive reasoning is engaged (e.g., in complex or high-stake use cases) (Lakkaraju and Bastani, 2020). Hence, providing explanations to end-users encourages them to simplify their mental model based on the information supplied and potentially rely solely on the ML model's rationale, which could theoretically lead to mispredictions (Janssen et al., 2022). As previous work (e.g., Buçinca et al., 2020) has already assumed that end-users perceive explanations to be individually demanding due to variations in the amount of information available and the style of explanation, we found that each XAI explanation type differs in terms of mental effort (H2). These results are reinforced, as end-users reasoning can be distinguished into a rational or an intuitive cognitive process, emphasizing a salience features evaluation or a systematic evaluation (Hamilton et al., 2016). In this regard, local explanations, namely the explanation types Why and Why-Not, yielded the lowest median and standard deviation among our results concerning mental effort. While this is consistent with Weerts et al. (2019)'s empirical study, which tested a local explanation using the SHAP package, Herm et al. (2021) also found that using this package can lead to misinterpretation due to confusing color palettes or additional information. Combining our research and related studies, we expected the use of color-free Why or Why-Not explanations to impose the least mental effort on end-users. By contrast, the XAI explanation How returned the highest mental effort score of our study and ranked close to the baseline. In research, this type of explanation is highly debated as it provides the most information compared with other types; hence, it can be presumed to have the highest explanatory scope (Hudon et al., 2021). Still, it could possibly also overwhelm non-ML experts (Fürnkranz et al., 2020). Comparing our findings with Herm et al. (2023)'s explainability evaluation and their assumption that explainability is concomitant with cognitive load, we observe tendencies indicating a correlation between the two factors. That is, when comparing explainability and mental effort, comparative results emerged for the Baseline, How, How-To, and Why explanation types. In turn, we identified differences for Why-Not and What-Else explanation types. Here, end-users perceive What-Else explanations as more explainable (presumably) due to their information scope, but requiring increased mental effort to comprehend, which is congruent with the assumption of Miller (2019). Still, in this clinical context, distinct requirements for the explanatory scope and domain-specific regulations necessitate a detailed level of granularity (Ghanvatkar and Rajan, 2022). Also, contrary to the research of Herm et al. (2023), the local How-To explanation demanded an increased mental effort compared to the global What-Else explanation. Reflecting Sultana and Nemati (2021), we surmised that this was due to the complexity of our task, which might be different with fewer features or image segments. Given the broad distribution of task time when using the What-Else explanation, we assumed that mental effort also depends on whether end-users grasp or struggle with this type of explanation. Still, researchers argued that this type is relatively facile to realize and its application merits prior training of end-users (Kim et al., 2016). Lastly, considering local explanation types (Why, Why-Not) perceived best, these explanation types might cause difficulties, as end-users tend to rely on features that are highlighted by the explanations (Bauer et al., 2023). Hence, guidelines are required to ensure the application of XAI in high-stakes use cases (Kloker et al., 2022). Impact of Cognitive Load on Task Performance. As we obtained significant results for a linear correlation between perceived mental effort and task performance (H3), this denotes a general surplus in end-user task performance. Yet, our results are consistent with Fahse et al. (2022b)'s and Hemmer et al. (2021)'s assumptions that cognitive load and task performance are diametrically related. However, we recognize some relative outliers. In particular, one might expect the responses rated "extremely low" in terms of mental effort to have produced the best task performance results; however, we found moderate to high relative task performance. This could be due to the relatively small sample size, and an outlier could skew the results. In addition, a related study already found that participants become negligent when a task is not mentally demanding (so-called "cognitive underload"), and thus, errors accumulate (Lavie, 2010). Nevertheless, when explanations require less cognitive load, the end-user's mental model is more capable of retrieving information and recognizing new circumstances more quickly (Abdul et al., 2020). In this regard, these results should be taken with a grain of salt as we targeted novice medical end-users who were unlikely to have actively used an XAI-based DSS before. These results may change once end-users are taught how to use these types of systems or use them more frequently due to the iterative learning process (Engström et al., 2017). Thus, the explanations What-Else or How may be favored due to their increased information scope but cease to overwhelm eligible participants. Impact of Cognitive Load on Task Time. Given that research has previously assumed a linear relationship between perceived mental effort and task time (Bertrand et al., 2022, Leppink andPérez-Fuster, 2019), highly significant results also emerged for H4. Here, the task time per level of mental effort was consistent with the results and the corresponding mental effort medians. The high density within the Why and Why-Not explanation types indicates general straightforward intelligibility for novice end-users. Surprisingly, considering this linear relationship, the global What-Else explanation was perceived as less demanding, yet participants were able to solve our tasks faster with the local How-To explanation. We attribute this to the nature of the explanation, as participants might not have used such support before. Further, Buçinca et al. (2020) argue that increased task time results from end-users' commitment to comprehend the provided explanation, as they may not trust the AI's recommendations. Conversely, a comparatively low task time could indicate over-reliance on explanations. In research, the task time factor is highly controversial: Liao and Varshney (2022) stated that in the absence of time pressure, more complex explanations should be preferred as an end-user is able to iteratively discover new relationships within the explanations. Contrary, in real-world applications, a thorough evaluation process is temporarily infeasible (Shaft and Vessey, 2006). However, research has already discussed that increased task time may impact end-user satisfaction (Hsiao et al., 2021). Mental Efficiency Ramifications. Local explanation types perform best regarding mental efficiency, resulting in a positive value. Therefore, end-users employing Why and Why-Not explanations exceed the performance-mental effort ratio, leading to a higher-than-expected result (Paas et al., 2016). However, compared to the cognitive load results of What-Else explanations, these explanations are about the participants' expected value. That is, while our findings imply relatively high mental effort, this mental efficiency result hints at relatively high end-user commitment levels, which could be consistent with the perceived explainability results of Herm et al. (2023). Comparatively, there is limited research evaluating XAI explanation metrics that incorporate end-user understanding (Gentile et al., 2021). Merely Ghanvatkar and Rajan (2022) and Fahse et al. (2022b) derived metrics to measure a person's effectiveness. Since we target cognitive load, we also transfer the cognitive load into the context of XAI. Accordingly, we distinguish as follows: Ghanvatkar and Rajan (2022) consider layer-wise relevance propagation (global explanation) to be the most effective as it provides the utmost information. However, this can also be critical as end-users may be unable to complete a task when overloaded. Conversely, for domain-specific requirements (e.g., in a clinical context), XAI explanations mandate a certain level of information, raising the importance to focus on effectiveness rather than efficiency. With this in mind, while one should not rely solely on effectiveness or efficiency, the trade-off should be determined based on the use case at hand (Forsythe et al., 2014).
Impact of XAI Explanation Types on Cognitive Load. Recent research
Implications, Limitations, and Future Research
Theoretical Implications. While research (e.g., Buçinca et al., 2020, Hudon et al., 2021 has only partly investigated the cognitive load of singular XAI implementations, holistic comparisons of distinct implementation-independent XAI explanation types are lacking. This is especially critical when considering potential bias, which may confuse end-users or even force them to make erroneous decisions (Nourani et al., 2022). From a theoretical perspective, we have contributed to the existing body of human-technology interaction knowledge, one of the cores in IS research (Riefle and Benz, 2021), by researching XAI's cognitive load and related effects on task performance and task time to ultimately derive a mental efficiency metric for the evaluation of XAI explanations. To best of our knowledge, we are the first to place this type of metric into the context of XAI and thus also take the end-user's mental model into account. Likewise, by directly comparing task performance and task time to cognitive load, we contribute to this relatively sparse body of knowledge in XAI research.
Here, we demonstrate that XAI explanations are essential for recommendation-based decision support because they reduce cognitive load, increase task performance, and reduce task time. Consequently, local explanations perform best in terms of mental efficiency. Following the ongoing (IS research) debate on the selection of explanation types (Gregor and Benbasat, 1999, we therefore provide initial insights on cognitive load for implementation-independent XAI explanation types. Drawing on this, this jigsaw piece contributes to the overall puzzle of the end-user's heuristic mental model. Although we did not measure trust and reliance during the experiment, we can identify some tendencies that could indicate end-user over-reliance especially on more straightforward explanations (e.g., Why). Thus, while Miller (2019) posits four requirements for the goodness of an explanation, our research indicates that focusing on causal reasoning and selective representation likely facilitates misclassification when the AI's recommendation is inaccurate. This may also be related to the type of end-users, as we focus on young professionals who tend to use the XAI-based DSS for support and pattern learning. In contrast, experts are prone to focus on using these explanations for verification (Gregor and Benbasat, 1999) and AI-experienced individuals have more reservations about AI explanations . In this regard, our results may differ as we focus on additional enduser groups, which means that the role of explanations may vary (Bauer et al., 2023). As recent research has shown that providing an explanation has a positive impact on trust and attitudes toward an AIenabled DSS , this end-user over-reliance can lead to unwarranted trust that results in automation bias, even in high-risk use cases (Jacovi et al., 2021). Therefore, cognitive forcing strategies should accompany the utilization of XAI-based explanations. Although recent IS research calls for a paradigm shift in XAI, proposing the application of hypothesisdriven support instead of recommendation-driven support to accommodate the end-user's cognitive process (Miller, 2023); the evaluation of explanations remains critical to ensure appropriate knowledge transfer of inferred evidence for an end-user action. Moreover, this approach forces end-users to be more committed, which increases their cognitive load and consequently emphasizes the need for mentally efficient explanations. To this end, we further advance the theoretical debate through the provision of a sociotechnical metric to evaluate XAI explanation types. As our results can be considered as a cognitive load-centered starting point for the discussion on the role of explanations in IS research, it currently lacks longitudinal analysis to determine additional aspects such as learning effects. This includes a combined study of other factors such as trust, acceptance, and satisfaction, which appear to be essential to understand the end-users' heuristic behavior. Ultimately, the benefits of providing XAI-based explanations in DSS will facilitate the integration of ML algorithms into organizational information systems, thus embedding the potentials of AI into intelligent systems (Gregor andBenbasat, 1999, Wanner et al., 2022). Practical Implications. In the early days of XAI research, XAI was seen as the silver bullet for enduser adoption of AI in any use case (Goebel et al., 2018); however, we found significant differences in perceived cognitive load, task performance, and required task time among the XAI explanation types. Thus, several considerations must be made: First, our research identified that developers of recent XAI implementations (cf. Table 1) must reconsider their applications, building upon our results, with respect to sociotechnical factors (e.g., cognitive load) and redesign them to match end-users' mental model. Second, we demonstrated that not every explanation type is appropriate for every situation; thus, practitioners must determine an appropriate explanation based on various factors, such as performance constraints, time constraints, or use case requirements. Third, it should be considered that explanations are usually a simplification of the ML model's rationale, and therefore, they are unlikely to contain the entire decision logic, which may include bias, adversarial attacks, or open Pandora's box due to the nonapplicability of the GDPR (Slack et al., 2021). That is, relying on inherently explainable ML models could be essential once a defined performance threshold is fulfilled (Rudin, 2019). Limitations and Future Research. Like any empirical study, ours has its limitations. We focused our fundamental XAI research on implementation-independent explanation types to derive unbiased insights for further XAI development, which must be translated into concrete and use case-specific applications. Specifically, researchers and practitioners must integrate these insights into their XAI explanations and then re-evaluate their improved artifacts. Our research could also be expanded as follows: Following the triangulation approach of Tams et al. (2014), future research should validate our findings by using further complementary measurement approaches such as eye-tracking studies and electroencephalograms to identify how end-users behave when using these explanation types. Second, our study should be expanded by examining additional factors to determine a holistic understanding of an end-user's behavior. Third, given the assumption of a tendency toward over-reliance within our study, future research should carry out dedicated research to examine potential trust miscalibrations. This includes investigating whether end-users are able to detect erroneous recommendations from the XAIbased DSS. Fourth, while we focused on a representative use case from the medical field, future research should leverage our findings to conduct further studies in other high-stakes use cases and with different types of end-user groups. However, our results lay the foundation for the end-user-centered design of XAI explanations and the derivation of design principles for XAI-based DSSs .
Conclusion
AI is emerging as a frontier of computational advances for mimicking or surpassing human intelligence. However, in high-stake decision-making use cases, the models' internal decision logic hinders the use of DL-based applications due to being incomprehensible to end-users and thus reducing their willingness (Berente et al., 2021. XAI has gained momentum by making these black-boxes understandable while maintaining the predictive power of the underlying model . Despite the proliferation of XAI applications, actual end-users are not sufficiently addressed (van der Waa et al., 2021). Unsurprisingly, using these systems for high-stakes use cases will likely result in overwhelmed and stressed end-users, which might not perform well due to high cognitive load (Hudon et al., 2021). In this regard, actual user-centered XAI research is relatively limited (Laato et al., 2022).
To address this knowledge gap on end-users' cognitive behavior, we used COVID-19 X-ray images to conduct an empirical study, thereby investigating how distinct implementation-independent XAI explanation types affect end-users' cognitive load, task performance, and the time required to solve a task. Combining our results, we calculate the mental efficiency of these explanation types. This facilitates an in-depth empirical study and thus, the derivation of implications for future research and practice. In doing so, we contributed to the current body of XAI knowledge to surmount the "inmates running the asylum" situation (Miller, 2019) in sociotechnical XAI research.
!
!"#$ &'()*(+",-' × ! !"#$ !.+' $ ! +',!"/ '))*(! √& adapted from Paas et al. (2016), mental effort and task performance standardized and task time standardized and reversed scale applied for computation.
)
Hypothesis 1 & 2 c) Hypothesis 4 b) Hypothesis 3
Table 2 .
2Descriptive Results of Cognitive Load Questionnaire
0
50
100
150
200
250
Baseline
How
How−To What−Else
Why
Why−Not
XAI Explanations
Task Performance [Total Amount]
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"Jinghui Cheng \nMcGill University\n\n",
"Jin L C Guo \nMcGill University\n\n",
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] | [
"McGill University\n",
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"McGill University\n",
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] | [] | Sketching out Graphical User Interface (GUI) layout is part of the pipeline of designing a GUI and a crucial task for the success of a software application. Arranging all components inside a GUI layout manually is a time-consuming task. In order to assist designers, we developed a method named GUILGET to automatically generate GUI layouts from positional constraints represented as GUI arrangement graphs (GUI-AGs). The goal is to support the initial step of GUI design by producing realistic and diverse GUI layouts. The existing image layout generation techniques often cannot incorporate GUI design constraints. Thus, GUILGET needs to adapt existing techniques to generate GUI layouts that obey to constraints specific to GUI designs. GUILGET is based on transformers in order to capture the semantic in relationships between elements from GUI-AG. Moreover, the model learns constraints through the minimization of losses responsible for placing each component inside its parent layout, for not letting components overlap if they are inside the same parent, and for component alignment. Our experiments, which are conducted on the CLAY dataset, reveal that our model has the best understanding of relationships from GUI-AG and has the best performances in most of evaluation metrics. Therefore, our work contributes to improved GUI layout generation by proposing a novel method that effectively accounts for the constraints on GUI elements and paves the road for a more efficient GUI design pipeline. | 10.48550/arxiv.2304.09012 | [
"https://export.arxiv.org/pdf/2304.09012v1.pdf"
] | 258,187,119 | 2304.09012 | 8bab2f5c38ebf846a8e5cbb189b8a151359a15fe |
The 36th Canadian Conference on Artificial Intelligence GUILGET: GUI Layout GEneration with Transformer
June 2023
Andrey Sobolevsky *[email protected]
McGill University
Guillaume-Alexandre Bilodeau
McGill University
Jinghui Cheng
McGill University
Jin L C Guo
McGill University
Polytechnique Montréal
McGill University
The 36th Canadian Conference on Artificial Intelligence GUILGET: GUI Layout GEneration with Transformer
MontréalJune 2023This article is © 2023 by author(s) as listed above. The article is licensed under a Creative Commons Attribution (CC BY 4.0) International license (https://creativecommons.org/licenses/by/4.0/legalcode), except where otherwise indicated with respect to particular material included in the article. The article should be attributed to the author(s) identified above.Graphical User InterfaceGUI arrangement graphsdeep learningtrans- formergenerative modelGUI layout
Sketching out Graphical User Interface (GUI) layout is part of the pipeline of designing a GUI and a crucial task for the success of a software application. Arranging all components inside a GUI layout manually is a time-consuming task. In order to assist designers, we developed a method named GUILGET to automatically generate GUI layouts from positional constraints represented as GUI arrangement graphs (GUI-AGs). The goal is to support the initial step of GUI design by producing realistic and diverse GUI layouts. The existing image layout generation techniques often cannot incorporate GUI design constraints. Thus, GUILGET needs to adapt existing techniques to generate GUI layouts that obey to constraints specific to GUI designs. GUILGET is based on transformers in order to capture the semantic in relationships between elements from GUI-AG. Moreover, the model learns constraints through the minimization of losses responsible for placing each component inside its parent layout, for not letting components overlap if they are inside the same parent, and for component alignment. Our experiments, which are conducted on the CLAY dataset, reveal that our model has the best understanding of relationships from GUI-AG and has the best performances in most of evaluation metrics. Therefore, our work contributes to improved GUI layout generation by proposing a novel method that effectively accounts for the constraints on GUI elements and paves the road for a more efficient GUI design pipeline.
Introduction
The design of Graphical User Interface (GUI) is an important aspect that affects the success of many software applications. The first step for the GUI designers is often sketching out the interface layout with wireframes, based on design constraints such as users' needs and software requirements [1]. These GUI layouts define the visual arrangement of elements in a user interface, such as buttons, text fields, and containers. Creating these layouts and variations of them manually, however, can be time-consuming. In this paper, we explore an automated technique that can support designers creating GUI layouts. In our approach, we capture the design constraints that the designers have to consider through GUI arrangement graphs (GUI-AGs) and generate graphical user interface layouts from those constraints. GUI-AGs specify elements required in the UI design and the relationships among them, as illustrated in Figure 1. We use GUI-AGs because they can be used to describe the requirements with an explicit definition of components that are part of the screen and the definition of the visual relations between those components. These graph models offer the flexibility to automatically create layout variations by modifying relations in the graph.
There are several technical challenges for achieving automatically generatation of GUI layouts from GUI-AGs. One challenge is to accurately capture the logical and semantic relationships between GUI elements, such as hierarchical structures and functional dependencies. Another challenge is to generate visually appealing and functional layouts that adhere to design principles and constraints [2]. In addition, the generation process should be efficient to be used in design workflow. To address these challenges, we propose a transformer-based arXiv:2304.09012v1 [cs.CV] 18 Apr 2023 approach for generating GUI layouts from GUI-AGs. Transformer networks have recently achieved state-of-the-art results in a wide range of natural language processing and computer vision tasks. They are particularly well-suited for generating GUI layouts from GUI-AGs, as they can effectively capture the dependencies between elements in the GUI-AG and generate GUI layouts that reflect these dependencies. Our transformer-based model takes as input a series of tokens that express the GUI-AG relationships; it then outputs a realistic GUI layout. We demonstrate the effectiveness of our approach through a series of experiments on real-world datasets [3,4] using metrics specific to this task. Results indicated that our approach produces the most relevant GUI layouts in regard to specified requirements.
Our contributions can be summarized with the following:
• We propose a new transformer-based method to generate GUI layouts from GUI-AGs that takes into account the GUI design constraints. Our experiment demonstrates that it better captures the intended GUI layout compared to previously proposed methods [5,6]; • We introduce new loss functions and metrics for quantitative measurement of the quality of generated GUI layouts; Those metrics can be used for future work on similar tasks. • To encourage reproduction or replication of our study, the source code of our experiment is publicly available with a pre-trained model at https://github.com/ dysoxor/GUILGET. Figure 1. GUI layout generation goal is to generate a realistic GUI layout (b) from a given GUI-AG (a).
Related Work
GUI generation is an emerging but yet under-explored computer vision application, especially for GUI generation based on GUI-AGs. First studies on generating GUI designs automatically adopt generative adversarial network, such as GUIGAN [7] and GANSpiration [8]. However, those approaches respectively do not generate new components style and do not consider any specification of the design -what components should be included or the relation between them. GUIGAN is based on a sequence of subtrees, which are hierarchical tree structures made of components, which is used as input and produces GUI by reusing different components based on their style without having the ability to produce new components. On the other hand, GANSpiration produces new design examples from existing screenshots or random vectors from the latent space representing the screenshots. To the best of our knowledge, no existing GUI generation methods explicitly consider the specifications about the content, including the relationship between components. However, there are several works focusing on image and layout generation for more general domains that can be inspirational for GUI generation [9,10]. For example, PasteGAN [9] generates images from scene graphs and image crops by taking into account the semantic relationships between objects and their visual appearance. Another method, proposed by Zhao et al. [10], uses a transformer to generate image tokens from a given scene graph and then decodes those tokens into a plain image using a VQGAN decoder [11]. In comparison, our work aims to produce first GUI layouts, which is usually the first step in the GUI design process, rather than directly generating the GUI design. Our work can be used as a part of a GUI design generation pipeline.
Our concept of GUI-AG is inspired by scene graph, a widely used structure in computer graphics. The task of generating layouts from scene graphs is studied in computer graphic, but also requires knowledge from natural language processing (NLP) to process the textual input and understand the interactions between each element of the scene graph. Scene graphs are commonly used to represent the structure and relationships between graphical elements in a scene, and layout generation aims to determine the position and size of these elements in the final layout.
There have been several approaches proposed for generating layouts from scene graphs in the literature. For example, SG2IM by Johnson et al. [5] adopted a graph convolutional network (GCN) to process the scene graph. This approach, however, is unable to capture semantic equivalence in graphs. The work of Herzig et al. [12] tackle this issue by learning canonical graph representations from the input scene graph, and it allows to represent more complex scene graphs. Another way of processing scene graphs is by using transformer-based model such as the ones presented in [6,13]. Transformers have shown great performances in processing textual components and learning relations between them. The method presented by Gupta et al. [13] uses self-attention to learn contextual relationships between layout element. In practice, it takes layout elements as input and produces the next layout elements as output. So it has no constraints and the model produces random new layouts. On the other hand, in the work LayoutTransformer by Yang et al. [6], the transformer-based model takes as input constraints in the form of scene graphs and predicts the layout based on relationships.
Many works have been done to generate layouts with different type of input from scene graphs. LayoutVAE [14], a variational autoencoder (VAE)-based framework for generating stochastic scene layouts, is a model that takes as input a set of labels representing the entities that must appear in the layout. It does not take into account any positional constraints, neither the amount of each label as with scene graphs. The Neural Design Network (NDN) [15] combines a GCN with a conditional VAE to generate layouts based on constraints given as input. In NDN, the scene graph is generated based on the amount of components there are and some desired positional constraints. Another work uses a generative adversarial network (GAN) [16] to generate layouts from an input of randomly-placed 2D elements. It uses self-attention modules to refine their labels and geometric parameters jointly to produce a layout.
In contrast to previous works, we propose a method that takes into account the specifications of a UI design using GUI-AGs that are considering the amount of each UI component and component position relative to other components.
Method
Our work is based on the use of transformers [17] that have been shown to be stateof-the-art in multiple natural language processing tasks. A transformer uses the concept of self-attention, which allows to give different weights depending on the part of the input to make predictions. GUI-AGs can be then viewed as a natural language input to the transformer since it is a logical sequence of relationships. Figure 2. Architecture of our model with three main components. Obj/Rel Predictor P takes as input an embedding e P which is a concatenation of several embeddings that describe different information about the given GUI-AG, and it produces contextual features f . Layout Generator G takes as input contextual features f , predicted sizes s and predicted positions p to translate it into a layout-aware representation c and bounding boxes b. Layout Refiner R uses co-attention module with predicted bounding boxes b and layout-aware representation c to improve the layout.
Our proposed method is based on the LayoutTransformer (LT-net) [6], a transformerbased model that aims to generate diverse layouts from scene graphs of images. It consists of (1) an object/relation predictor P that encodes the input scene graph into contextual features f using transformer encoder layers, (2) a layout generator G that, from contextual features f , generates bounding boxes b with distributions matching a learned Gaussian distribution model and layout-aware representations c with transformer decoder layers, and (3) a layout refiner R made of a co-attention module. Our architecture is presented in Figure 2. Compared to LT-net, we introduced several improvements for GUI layout generation that will be presented in the following. GUIs are made of two types of components: (1) widgets (e.g. button, pictogram, text), which are leaf nodes in a GUI-AG representation and do not contain any other component, and (2) spatial layouts (e.g. container, list item, toolbar), which are intermediate nodes that allow to organize the structure of widgets [7]. This tree structure does not exist in images and it changes the way GUI-AGs should be modeled and processed compared to scene graphs for images. In contrast to our work, most of current works that are done in layout generation for GUI ignore the tree structure and only consider widgets to remove the complexity of organizing those widgets inside layouts.
Building a GUI arrangement graph
GUI-AGs are directed graphs made of relationships that are triplets of subject-predicateobject [5]. A GUI does not require as much variety of relations and objects as images. However, it has complexity since GUI layouts have more rules and principles to be learned than in a layout from an image [2]. To train a neural network model, GUI-AGs from ground truth layouts are required. We define five types of possible predicate specifying the relationship among GUI components: left, right, top, bottom and inside. We build a GUI-AG from a layout by parsing a layout description. First, to reduce the size of the GUI-AG in order to have a smaller input in the transformer, we keep randomly only one inside relation within a group of components that are children to the same parent. This step is beneficial due to memory limitations when the input is too large. To illustrate this step, in Figure 3 we notice that CONTAINER [2] has two children, DATE PICKER [3] and NAVIGATION [6], but only one inside relationship is kept as input to the model and the other one is removed. This processing method does not lose information since all components that do not have an inside relationship are considered at the same level and are implicitly inside the parent component. We also randomly choose a sequence of components inside the layout and determine relationship between each pair of components within the sequence. More precisely, as shown in Figure 3, DATE PICKER [3] and NAVIGATION [6] are inside the same parent; in this situation, we randomly choose the sequence in the set of children [NAVIGATION [6], DATE PICKER [3]] and then add the relation between each component of the sequence to form triplets ([NAVIGATION [6], below, DATE PICKER [3]] in this case). By doing so, we get a simplified input that captures most information from the graph, as we will see in subsection 3.2. It is to note however, that we may miss some relationships. For example, if there are three components (a,b,c) and we keep only two relationships (a-b, b-c), the relation between a-c can be uncertain in some cases. Finally, in GUI-AGs, the relationships are reversible; e.g., if in the layout from Figure 3 the NAVIGATION [6] is below DATE PICKER [3], the reverse, DATE PICKER [3] is above NAVIGATION [6], must also be true. Hence, we can keep only one relationship between them.
Object/Relation Predictor
To construct the input for our transformer model, we first convert the GUI-AG into a sequence of relationship triplets s i . We refer to this sequence of relationships as S = {s 1 , s 2 , ...s T } where T is the number of relationships. Relationships are separated by a special token SEP, and a special token CLS is used at the beginning of the entire sequence S.
Instead of using directly the sequence S as input in the Object/Relation Predictor, we embed s 1:T into e P 1:T , which allows to take into consideration different features: word embedding e w 1:T that allows to identify the class of the object (e.g. "button" or "container") or the relation (e.g. "inside" or "right), object ID embedding e o 1:T to differentiate instances of the same object, relationship ID embedding e s 1:T to separate each relationship, type of word embedding e t 1:T to identify parts of a relationship (subject = 1, predicate = 2, object = 3) for example the part of the the relationship "button inside container" are "subject predicate object" and instead of writting the plain text, we rather associate an ID to each part of the relationship to differentiate them, and parent ID embedding e p 1:T , which is a feature that allows for each of the component to know its parent. Those features are concatenated to form the input embedding for the object/relation predictor. It is given by The contextualized feature vectors f 1:T describe objects, relations and their context with features from the input. In order to capture conceptually diverse embedding and exploiting the co-occurrence among objects, predicates and parents, we follow the technique used in BERT [18] and mask randomly words from the input that must be predicted by the object/relation predictor. The size vectors s 1:T are predictions of the bounding boxes size for each object made by the object/relation predictor. It is used as indicator later in the GUI layout generator to generate final bounding boxes size. The position vectors p 1:T are predictions of bounding boxes position for each object.
Finally, in order to compute the objective function to train this part of the network, we need to predictê P 1:T from the features f 1:T using a single linear layer. Indeed, since there is not ground truth for f 1:T , we predictê P 1:T from f 1:T and match it with e P 1:T to minimize the reconstruction error. The objective function for training the module is composed of cross-entropy losses L predSem , given by
L predSem = CrossEntropy(e P t ,ê P t ),(3.2)
for the matching word, object ID, type of word, parent ID which are all extracted from the input GUI-AG, and regression losses given by
L predBox = Regression(s t ,ŝ t ) + Regression(p t ,p t ),(3.3)
which are computed on predicted positionsp 1:T and sizesŝ 1:T with their corresponding ground truth positions and sizes. The total loss of the predictor L pred is a combination of the two losses and given by
L pred = L predSem + L predBox . (3.4)
Layout generator
The goal of the layout generator module is to produce layout-aware representations c 1:T and bounding boxesb 1:T . This module is made of transformer decoder layers and interprets jointly and sequentially contextual features f 1:T , predicted bounding box sizes s 1:T and predicted bounding box positions p 1:T that are computed by the object/relation predictor module. The three given inputs are concatenated and expressed as e G 1:T . After that it is translated into diverse bounding box outputb 1:T . A bounding box is described by its topleft corner position in a 2D Cartesian coordinate system and its size in terms of width and height, i.e.,b t = (x t , y t , w t , h t ), and there is a bounding box produced for each subject, predicate and object. The bounding box of the predicate is the difference between the position of the object and the one of the subject, i.e.,b t = (x t+1 − x t−1 , y t+1 − y t−1 ).
To produce sequentially the bounding boxesb t , the features from the input e G t are also concatenated with the previously produced bounding boxb t−1 . This input is not directly translated into the bounding box but to a layout-aware representation c t that is used to model a distribution in order to use Gaussian Mixture Models (GMM) [19]. We use this instead of directly predicting the bounding box from layout-aware representation in order to have a generative ability. Given a bounding box distribution, the bounding boxb t will be sampled from the posterior distribution p θt (b t |c t ) knowing c t . It can be described as follows:
p θt (b t |c t ) = K i=1 π i N (b t ; θ t,i ), (3.5)
where i indicates the i-th distribution out of K multivariate normal distributions, θ t,i are the parameters of each distribution defined by (µ x t,i , µ y t,i , σ x t,i , σ y t,i , ρ xy t,i ) where µ, σ and ρ denote respectively the mean, standard deviation and the correlation coefficient, π i is a magnitude factor, and N is the multivariate normal distribution.
To define the objective function of the generator L gen , we start by defining the box reconstruction loss L box , which maximizes the log-likelihood of the generated GMM to fit the training data where the ground-truth bounding boxes are denoted as b t = (x t , y t , w t , h t ).
L box = − 1 K log( K i=1 π i N (b t ; θ t,i )). (3.6)
To avoid the over-fitting with this loss function, the GMM distributions are fitted to a multivariate normal distribution Q using a Kullback-Leibler (KL) divergence loss:
L KL = K i=1 D KL (P i ||Q i ). (3.7)
Finally, a relation consistency loss L rel is also used since the two previous losses focuses only on the bounding boxes. It is given by: 8) where N denotes the number of relationships in S. It calculates the Mean Square Error (MSE) between the box disparity of the relation we get, i.e.b rel t which is the predicted bounding box for the predicate, and the corresponding box disparity we calculate from the object and the subject ∆b t = (x t+1 − x t−1 , y t+1 − y t−1 ). The layout generator is trained by minimizing the weighted sum of losses using L gen = λ box L box + λ KL L KL + λ rel L rel , (3.9) where λ box , λ KL and λ rel are weighting factors for each corresponding loss.
L rel = 1 N (∆b t −b rel t ) 2 ,(3.
Layout refiner
Since the bounding boxes are generated sequentially, they require refinement in the layout in order to consider the semantic c 1:T and the bounding boxb 1:T . This is done in the layout refiner using the Visual-Textual Co-Attention (VT-CAtt) [6], which predicts the residual ∆b 1:T for updating the bounding boxes. The objective function of this module L ref is defined with multiple losses. The first one is a regression loss L reg between predicted bounding boxes b 1:T by the layout refiner and the ground truth bounding boxes b 1:T . Another new and taskspecific loss that we implemented is the overlap between children loss L CC , which aims to minimize the overlap of components that share the same parent in the GUI interface. This is specific to design principles in GUI since we do not want the components to overlap because it will hide some components on the final interface. This loss is given by
L CC = C 1 ∩ C 2 min(C 1 , C 2 )
, (3.10) where C designate the area of a children. Another principle to follow is that a children must be inside its parent. To enforce this principle, we define the overlap between children and parent loss L CP and we express it as
L CP = 1 − C ∩ P C , (3.11)
where C is the area of the children and P is the area of its parent that is defined in the input GUI-AG. The objective function on which the layout refiner is trained is a weighted sum of those losses, that is
L ref = λ reg L reg + λ CC L CC + λ CP L CP .
(3.12) where λ reg , λ CC and λ CP are weighting factors for each corresponding loss.
Experiments
In our experiment, we validated our proposed method by generating layouts from GUI-AGs. We aim to evaluate how close the generated layouts are to the ground truth layouts based on the graphs associated to them.
Dataset
We tested our method on the CLAY dataset [3], which is a UI design dataset. UI layouts in RICO dataset [4] are often noisy and have visual mismatches hence CLAY is a dataset that improves RICO by denoising UI layouts. It contains 59,555 human-annotated screen layouts, based on screenshots and layouts from RICO. A total of 24 component categories (e.g. Image, button, text) are available in layouts and 5 predicate categories (above, below, right, left, inside) are considered in GUI-AGs.
By observing data from the CLAY dataset, we decided to remove several irrelevant GUIs and their layouts. Firstly, we removed GUIs that contain two or less types of components. Those screenshots are usually not representing an application GUI but rather GUIs with, for instance, full screen image and video screenshot that do not contain useful information for our task, as there is a lack of component and interactions (predicates) between components. Then, we also removed screens that contain only a navigation bar or popup for example, for the same reason. Also usually in the dataset there are several screenshots associated to the same application but some contain only the navigation bar and others have the navigation bar and also some content. So we removed the former to avoid overfitting. In practice, we achieved that by removing GUIs in which components cover less than 25% of the total area of the screen.
Evaluation metrics
To evaluate the generated layouts with respect to the ground truth, we used the following metrics.
CP Inclusion (CPI) is a metric that captures the overlapping of children with its parent. The metric is computed as 1 − L CP (see Equation 3.11). Hence, the goal of this metric is to indicate if the generated GUI layouts tend to satisfy the UI design principle that states that children must be fully inside its parent as we see in Table 1 with ground truth data.
CC Separation (CCS) is another metric that aims to evaluate if the UI design principles tend to be satisfied. It is computed by 1 − L CC (see Equation 3.10), so the metric measure the ratio of components that does not overlap between each other and in the same time share a common parent.
Alignment [15] metric evaluates an important design principle that is components must be either in center alignment or in edge alignment (i.e. left-, right-, bottom-or top-aligned). We computed alignment with the following:
1 − 1 N C d i min j,i =j {min(l(c d i , c d j ), m(c d i , c d j ), r(c d i , c d j ), t(c d i , c d j ), v(c d i , c d j ), b(c d i , c d j ))}, (4.1)
where N C is the number of components, c d k is the k th component of the d th layout and l, m, r, t, v and b are alignment functions where the distance between the left, horizontal center, right, top, vertical center and bottom are measured, respectively.
W bbox is the similarity between bounding box properties (x lef t , y top , w, h) distribution of the generated GUI layouts and the ground truth GUI layouts. This is computed using Wasserstein distance and inverting it to be a similarity, between 0 and 1, by subtracting the maximum possible distance by the actual distance and normalizing the value. It is a way to measure if the generated bounding boxes are as diversified as in the ground truth data.
GUI-AG Correctness (GUI-AGC) computes the average number of correct relationships that appears in the generated GUI layout. In practice, we compare the input GUI-AG with the corresponding generated GUI layout and count the number of satisfied relationships divided by the total number of relationships. [3]. 3000 layouts are generated with each method and are compared using metrics presented in subsection 4.2. For SG2IM [5] as well as for LayoutTransformer [6], we only consider the parts of the architectures responsible to generate the layout. Table 1 gives the results of our method compared to SG2IM [5] and to LayoutTransformer [6]. As we can observe, there are several metrics where our model gives the best results but not with all metrics. If we compare our model to SG2IM, we can see that in terms of CCS we do not get as good results. However, we notice that CPI and GUI-AGC are the worst for SG2IM. In particular, GUI-AGC shows that only 36.9% of the relations given as input to SG2IM are satisfied in the output, which means that the design constraints are not met with this method. Moreover, we can see with this model that the CCS is high while CPI is low, which means that SG2IM does not organize components inside the layout but uses the whole screen, as it is also shown in the results from Figure 5, which makes it easier to get a high CCS since it does not learn the inside constraint. It is easy not to have overlap between child components in that case. In other words, it is not meaningful to have a high CCS if CPI and GUI-AGC are not also high. The LayoutTransformer model has more understanding of predicates but it is still worse than with our model. We can also see that there is a negative correlation between CCS and CPI -if the model learns to place components inside its parent, there are more possible overlaps between components inside a layout. We want both of these metrics to be similarly high to respect both of those GUI design constraints as we can observe it in the GT data, where all metrics related to GUI design constraint are close to 1. W bbox metric is similar for all of the models which is understandable since our model and LayoutTransformer model generate bounding boxes based on distribution of bounding boxes from the training GUI layouts. On the other hand, SG2IM is not a generative model and aims to predict bounding boxes which leads to learn the most common sizes and positions for different types of component. In order to see if the screen category has an impact on the performance, we conducted an experiment where we compute each evaluation metric for each category separately. Figure 4 (a) summarizes the results. Overall, our model yields similar performances among different app categories. This is a conclusive result which shows that our model is not biased toward certain types of screen categories and is able to produce equally good GUI layout for any of the category.
Quantitative results
Metrics
CPI ↑ CCS ↑ Alignment ↑ W bbox ↑ GUI-AGC ↑ GT
Influence of UI category
Influence of UI complexity
Similarly as with the previous experiment, we want to understand the influence of the UI complexity (indicated by the number of unique component types in the UI [8]) on the performance. Figure 4 (b) shows that with smaller number of unique component types most of the evaluation metrics are better; in other words, our model achieved better performances when the UI is less complex. Particularly, the GUI-AGC metric is inversely proportional to the number of unique component types in the UI. For both CCS and CPI metrics, these results are expected due to the fact that with more diverse components that have various sizes and position standards, it becomes harder to organize all elements inside spatial layouts. There are however two metrics performing equally well over all number of unique component types, which are the Alignment and W bbox metrics. This shows that our model succeeds to always align components whether the complexity is low or high, and generated bounding boxes have almost the same similarity in distribution with the ground truth distribution. The same input is given for the three models. The input from the first row has a low complexity with 3 unique component types while the second input has a larger complexity with 5 unique component types.
number of components. The results of Figure 5 are aligned with the quantitative results from subsection 4.3. Indeed, we can see that SG2IM has a poor semantic understanding of relations for both cases in Figure 5; it also struggles to place components inside their parent as we can observe in the second row from Figure 5 where all components that are supposed to be inside the CONTAINER [3] are not and the container which is supposed to be below the MAP [2] is actually entirely inside it instead. We can note however that the sizes of bounding boxes and their alignments are realistic. The LayoutTransformer shows a better understanding of relations but is not able in the second case to place components inside its parent as exemplified by BUTTON [4] and BUTTON [5] that are outside the CONTAINER [3] in the second row from Figure 5. In contrast, our model respects all the given constraints and placed correctly buttons inside the container. Also, GUILGET generates plausible bounding boxes even though the generated layouts are not aligned in the way it is in the ground truth. However, information from GUI-AG is not complete enough to reproduce the same alignment. Adding global positioning constraints on components to the GUI-AG could be an interesting avenue to investigate.
Conclusion
This work propose a transformer-based model that generates a GUI layout from a given GUI-AG. Our approach is the state-of-the-art in quantitative performance across several metrics and in visual quality. We saw that using attention provides a higher performance than using graph convolution network in capturing semantic of the GUI-AG. The new components from our model compared to LayoutTransformer bring also more understanding in GUI layout constraints. This work also introduce new loss functions and evaluation metrics specific to this task of GUI layout generation. Future work is to generate GUI from the layouts to complete the GUI design pipeline.
Figure 3 .
3Heuristic process of modeling GUI-AG (c) based on given layout (b) and associated screenshot (a). All nodes represent components and all arrows represent possible relationships. We will keep only one inside relation among all children from a component (in blue) randomly, while the others are not used as input (in gray). The other relations are only possible between components that are inside the same parent and we keep only one relation between them.
/relation predictor learns to produce three different outputs: (1) the contextualized feature vectors f 1:T , (2) the size vectors s 1:T , and (3) the position vectors p 1:T .
Figure 4 .
4Quantitative evaluation on different screen categories (a) from CLAY dataset[3]. Evaluation metrics are applied on all 27 screen categories separately. (b) shows the influence of number of unique type components on evaluation metrics.
Figure 5
5shows two examples produced by our model, LayoutTransformer, and SG2IM. Examples were chosen manually based on number of unique component types inside the GUI layout. We show results for low complexity (3-4 unique component types) and medium complexity (5-7 unique component types). We do not show GUI With large complexity (8 or more unique component types) as it is harder to analyze visually because of the larger
Figure 5 .
5Qualitative comparison between our model, LayoutTransformer and SG2IM.
Table 1. Quantitative evaluation on CLAY datasetdata
1.0
0.987
1.0
-
-
SG2IM [5]
0.191
0.974
0.997
0.81
0.369
LayoutTransformer [6] 0.392 ± 0.001 0.805 ± 0.002
0.998 ± 2E-5
0.834 ± 1E-4 0.797 ± 0.001
GUILGET (ours)
0.592 ±0.001 0.623 ± 0.002 0.9983 ± 2E-5 0.811 ± 8E-5
0.868 ±0.001
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| [] |
[
"Periodic stellar variability from almost a million NGTS light curves",
"Periodic stellar variability from almost a million NGTS light curves"
] | [
"1★Joshua T Briegal \nAstrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n",
"Edward Gillen \nAstrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n\nAstronomy Unit\nMary University of London\nMile End RoadE1 4NSLondonQueenUK\n",
"† Didier Queloz \nAstrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n",
"Simon Hodgkin \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RiseCB3 0HACambridgeUK\n",
"Jack S Acton \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"David R Anderson \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"David J Armstrong \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Matthew P Battley \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Daniel Bayliss \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Matthew R Burleigh \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"Edward M Bryant \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Sarah L Casewell \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"Jean C Costes \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n",
"Philipp Eigmüller \nInstitute of Planetary Research\nGerman Aerospace Center\nRutherfordstrasse 212489BerlinGermany\n",
"Samuel Gill \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Michael R Goad \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"Max-Imilian N Günther \nDepartment of Physics\nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n\nEuropean Space Agency (ESA)\nEuropean Space Research and Technology Centre (ESTEC)\nKeplerlaan 12201 AZNoordwijkThe Netherlands\n",
"‡ ",
"Beth A Henderson \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"James A G Jackman \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nSchool of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA\n",
"James S Jenkins \nFacultad de Ingeniería y Ciencias\nNúcleo de Astronomía\nUniversidad Diego Portales\nAv. Ejército 441SantiagoChile\n\nCentro de Astrofísica y Tecnologías Afines (CATA)\nCasilla 36-DSantiagoChile\n",
"Lars T Kreutzer \nAstrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n\nDepartment of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nCB3 OWACambridgeUK\n\nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476GolmGermany\n",
"Maximiliano Moyano \nInstituto de Astronomía\nUniversidad Católica del Norte\n0610, 1270709Angamos, AntofagastaChile\n",
"Monika Lendl \nObservatoire de Genève\nUniversité de Genève\n51 Chemin Pegasi1290SauvernySwitzerland\n",
"Gareth D Smith \nAstrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n",
"Rosanna H Tilbrook \nSchool of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"Christopher A Watson \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK\n",
"Richard G West \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Peter J Wheatley \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n"
] | [
"Astrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK",
"Astrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK",
"Astronomy Unit\nMary University of London\nMile End RoadE1 4NSLondonQueenUK",
"Astrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK",
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RiseCB3 0HACambridgeUK",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK",
"Institute of Planetary Research\nGerman Aerospace Center\nRutherfordstrasse 212489BerlinGermany",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Department of Physics\nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"European Space Agency (ESA)\nEuropean Space Research and Technology Centre (ESTEC)\nKeplerlaan 12201 AZNoordwijkThe Netherlands",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"School of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA",
"Facultad de Ingeniería y Ciencias\nNúcleo de Astronomía\nUniversidad Diego Portales\nAv. Ejército 441SantiagoChile",
"Centro de Astrofísica y Tecnologías Afines (CATA)\nCasilla 36-DSantiagoChile",
"Astrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK",
"Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nCB3 OWACambridgeUK",
"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476GolmGermany",
"Instituto de Astronomía\nUniversidad Católica del Norte\n0610, 1270709Angamos, AntofagastaChile",
"Observatoire de Genève\nUniversité de Genève\n51 Chemin Pegasi1290SauvernySwitzerland",
"Astrophysics Group, Cavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUK",
"School of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Centre for Exoplanets and Habitability\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK"
] | [
"MNRAS"
] | We analyse 829, 481 stars from the Next Generation Transit Survey (NGTS) to extract variability periods. We utilise a generalisation of the autocorrelation function (the G-ACF), which applies to irregularly sampled time series data. We extract variability periods for 16, 880 stars from late-A through to mid-M spectral types and periods between ∼ 0.1 and 130 days with no assumed variability model. We find variable signals associated with a number of astrophysical phenomena, including stellar rotation, pulsations and multiple-star systems. The extracted variability periods are compared with stellar parameters taken from Gaia DR2, which allows us to identify distinct regions of variability in the Hertzsprung-Russell Diagram. We explore a sample of rotational main-sequence objects in period-colour space, in which we observe a dearth of rotation periods between 15 and 25 days. This 'bi-modality' was previously only seen in space-based data. We demonstrate that stars in sub-samples above and below the period gap appear to arise from a stellar population not significantly contaminated by excess multiple systems. We also observe a small population of long-period variable M-dwarfs, which highlight a departure from the predictions made by rotational evolution models fitted to solar-type main-sequence objects. The NGTS data spans a period and spectral type range that links previous rotation studies such as those using data from Kepler, K2 and MEarth. | 10.1093/mnras/stac898 | [
"https://arxiv.org/pdf/2203.15894v1.pdf"
] | 247,792,838 | 2203.15894 | 5ae70110a7505c9c17624382606ccddab75432f7 |
Periodic stellar variability from almost a million NGTS light curves
2021
1★Joshua T Briegal
Astrophysics Group, Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUK
Edward Gillen
Astrophysics Group, Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUK
Astronomy Unit
Mary University of London
Mile End RoadE1 4NSLondonQueenUK
† Didier Queloz
Astrophysics Group, Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUK
Simon Hodgkin
Institute of Astronomy
University of Cambridge
Madingley RiseCB3 0HACambridgeUK
Jack S Acton
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
David R Anderson
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
David J Armstrong
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Matthew P Battley
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Daniel Bayliss
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Matthew R Burleigh
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Edward M Bryant
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Sarah L Casewell
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Jean C Costes
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT7 1NNBelfastUK
Philipp Eigmüller
Institute of Planetary Research
German Aerospace Center
Rutherfordstrasse 212489BerlinGermany
Samuel Gill
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Michael R Goad
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Max-Imilian N Günther
Department of Physics
Kavli Institute for Astrophysics and Space Research
Massachusetts Institute of Technology
02139CambridgeMAUSA
European Space Agency (ESA)
European Space Research and Technology Centre (ESTEC)
Keplerlaan 12201 AZNoordwijkThe Netherlands
‡
Beth A Henderson
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
James A G Jackman
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
School of Earth and Space Exploration
Arizona State University
85287TempeAZUSA
James S Jenkins
Facultad de Ingeniería y Ciencias
Núcleo de Astronomía
Universidad Diego Portales
Av. Ejército 441SantiagoChile
Centro de Astrofísica y Tecnologías Afines (CATA)
Casilla 36-DSantiagoChile
Lars T Kreutzer
Astrophysics Group, Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUK
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
CB3 OWACambridgeUK
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 114476GolmGermany
Maximiliano Moyano
Instituto de Astronomía
Universidad Católica del Norte
0610, 1270709Angamos, AntofagastaChile
Monika Lendl
Observatoire de Genève
Université de Genève
51 Chemin Pegasi1290SauvernySwitzerland
Gareth D Smith
Astrophysics Group, Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUK
Rosanna H Tilbrook
School of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Christopher A Watson
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT7 1NNBelfastUK
Richard G West
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Peter J Wheatley
Department of Physics
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Centre for Exoplanets and Habitability
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Periodic stellar variability from almost a million NGTS light curves
MNRAS
0002021Accepted XXX. Received YYY; in original form ZZZPreprint 31 March 2022 Compiled using MNRAS L A T E X style file v3.0 † Winton Fellow 2 J. T. Briegal et al. ‡ Juan Carlos Torres Fellow § ESA Research FellowHertzprung-Russell and colour-magnitude diagrams -stars:rotation - stars:activity -stars:variables:general -methods:data-analysis -techniques:photometric
We analyse 829, 481 stars from the Next Generation Transit Survey (NGTS) to extract variability periods. We utilise a generalisation of the autocorrelation function (the G-ACF), which applies to irregularly sampled time series data. We extract variability periods for 16, 880 stars from late-A through to mid-M spectral types and periods between ∼ 0.1 and 130 days with no assumed variability model. We find variable signals associated with a number of astrophysical phenomena, including stellar rotation, pulsations and multiple-star systems. The extracted variability periods are compared with stellar parameters taken from Gaia DR2, which allows us to identify distinct regions of variability in the Hertzsprung-Russell Diagram. We explore a sample of rotational main-sequence objects in period-colour space, in which we observe a dearth of rotation periods between 15 and 25 days. This 'bi-modality' was previously only seen in space-based data. We demonstrate that stars in sub-samples above and below the period gap appear to arise from a stellar population not significantly contaminated by excess multiple systems. We also observe a small population of long-period variable M-dwarfs, which highlight a departure from the predictions made by rotational evolution models fitted to solar-type main-sequence objects. The NGTS data spans a period and spectral type range that links previous rotation studies such as those using data from Kepler, K2 and MEarth.
INTRODUCTION
Many of a star's physical properties can be inferred from its brightness variations over time. This variability can arise from a number of mechanisms, either intrinsic to the star through changing physical properties of the star and its photosphere, or through external factors such as orbiting bodies and discs. The rotation of magnetically active stars will also cause visible brightness changes. Eyer & Mowlavi (2008) categorises a large number of distinct variability types which range in period from milliseconds to centuries and in amplitude from a few parts-per-million (ppm) to orders of magnitude in the most explosive forms of variability.
Stellar rotation can be measured through photometric observation, as magnetic surface activity such as spots and plages cause photometric brightness fluctuations over time that is modulated by both the rotation of active regions across the star, as well as active region evolution. Constraining stellar rotation rates is important, as this provides insight into the angular momentum of the star. Skumanich (1972) first hypothesised that a star's rotation rate could be age dependant, obtaining the empirical relation between rotation period rot and age : rot ∝ 0.5 . Knowing a star's age is fundamental to fully understanding its evolutionary state, and so being able to infer this property from an observable quantity such as rotation would greatly improve our understanding of stars in the local neighbourhood. In Barnes (2003) a semi-empirical model for deriving stellar ages from colour and rotation period was suggested, and the term 'gyrochronology' was coined. This model was subject to further improvements in Barnes (2007), a model which is commonly still used to age Solar-type and late-type main-sequence stars. These models work especially well for stars older than the age of the Hyades cluster, by which time we expect the initial angular momentum of stars to have little effect on the rotation period, and the angular momentum evolution to follow a Skumanich law (Kawaler 1988). For low mass stars, it is widely accepted that latetime angular momentum loss will be governed by magnetised stellar winds which depend on magnetic field topology and stellar mass (Booth et al. 2017). For young stars (< 10 Myr) angular momentum evolution may be dependant on magnetic coupling between the star and disc. Studies of pre-main-sequence stars in young clusters such as T-Tauri stars in the Taurus-Auriga molecular cloud (Hartmann & Stauffer 1989) or in NGC 2264 (Sousa et al. 2016) show high levels of short period (< 10 day) photometric variability, but objects with circumstellar discs present appear to rotate slower than those without, highlighting the effect of star-disc coupling on angular momentum evolution.
Understanding a star's activity is important for exoplanet surveys. Not only is stellar activity a large source of noise in both transit and RV surveys (e.g. Queloz et al. 2001;Haywood et al. 2014;Dumusque et al. 2017), but stellar activity may also influence the potential habitability of orbiting planets. Stars that rotate rapidly, for example, often display higher flare rates than their more slowly rotating cousins, and these flares can be important for potential exoplanet habitability. On the one hand, flares can erode exoplanet atmospheres and modify their chemistry (e.g. Segura et al. 2010;Seager 2013;Tilley et al. 2019), while on the other, they can help initiate prebiotic chemistry and seed the building blocks of life (Ranjan et al. 2017;Rimmer et al. 2018), which may be especially important for M dwarf systems.
The angular momentum of a host star and its planets are intrinsically linked. Gallet et al. (2018) demonstrate that tidal interactions between a host star and a close-in planet can affect the surface rotation of the star. They observe a deviation in rotation period from the expected magnetic braking law during the early MS phase of low-mass stars in the Pleiades cluster, which the authors attribute to planetary engulfment events. Conversely, angular momentum transfer through tidal interactions must be considered in the context of stellar spin-down through magnetic braking. The analysis by Damiani & Lanza (2015) demonstrates that to accurately model tidal dissipation efficiency and orbital migration the stellar angular momentum loss through magnetised stellar winds must be accounted for.
Large scale photometric variability studies have recently allowed for data-driven analysis of stellar variability in extremely large samples. Stellar clusters allow studies of groups of stars with similar formation epochs and evolutionary conditions, so historically have been targeted by systematic surveys. These observations have come from ground-based surveys such as Monitor (Hodgkin et al. 2006;Aigrain et al. 2007) with observations of NGC 2516 ), SuperWASP (Pollacco et al. 2006) with observations of the Coma Berenices open cluster (Collier Cameron et al. 2009) and HATNet (Bakos et al. 2004) with observations of FGK Pleiades stars (Hartman et al. 2010). Recently, NGTS (Wheatley et al. 2018) observed the ∼ 115 Myr old cluster Blanco 1, and a study by Gillen et al. (2020a) demonstrated a well-defined singlestar rotation sequence which was also observed by KELT (Pepper et al. 2012) and studied in Cargile et al. (2014). In both of these works a similar sequence was observed for stars in the similarly aged Pleiades, indicating angular momentum evolution of mid-F to mid-K stars follows a well-defined pathway which is strongly imprinted by ∼ 100 Myr.
As part of the transient search conducted by the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee et al. 2014), a catalogue of observed variable stars has been compiled. This catalogue contains variability periods and classifications for 687,695 objects 1 taken from a series of publications entitled 'The ASAS-SN catalogue of variable stars' (e.g. Jayasinghe et al. 2018Jayasinghe et al. , 2021. Such catalogues are not focused on specific clusters or stellar types, but provide a broad view of different forms of stellar variability.
Space missions have allowed wide-field photometric variability surveys of stars with high precision and continuous time coverage. CoRoT (Auvergne et al. 2009), Kepler (Borucki et al. 2010), the extended Kepler mission (K2; Howell et al. 2014) and TESS (Ricker et al. 2014) have provided a wealth of stellar photometric data, which in turn has been the subject of extensive rotation studies (Ciardi et al. 2011;Basri et al. 2011;Affer et al. 2012;McQuillan et al. 2014;Davenport & Covey 2018;Canto Martins et al. 2020;Gordon et al. 2021), revealing large scale trends in stellar variability periods. In particular, studies by McQuillan et al. (2014) and Davenport & Covey (2018) demonstrated a distinct bi-modal structure in the rotation periods of main-sequence stars with respect to colour. Gordon et al. (2021) followed up these studies with analysis of data from the K2 mission, hypothesising the bi-modal structure arises from a broken spin-down law, caused by an internal angular momentum transfer between the core and convective envelope. Further details of this model are discussed in Section 5.
The Next Generation Transit Survey (NGTS; Wheatley et al. 2018) is a ground-based wide-field photometric survey achieving routinely milli-magnitude range photometric precision with 12second sampling cadence and long observation baselines, typically 250 nights of data per target field. The primary science goal of NGTS is to extend the wide-field ground-based detection of transit-1 Accessed on 09/11/2021 ing exoplanets to at least the Neptune size range. Such high precision photometry lends itself well to ancillary stellar physics such as cluster rotation analysis (Gillen et al. 2020a) or stellar-flare detection and characterisation (Jackman et al. 2019). Ground-based observation adds extra layers of difficultly in variability studies when compared to space telescope data, as we must consider irregular sampling and telluric effects. In this study, we employ a generalisation of the autocorrelation function (the G-ACF) which applies to this irregular sampling. We elect to use an autocorrelation function to extract variability as this has proven to be successful for extracting stellar variability by McQuillan et al. (2013) & Angus et al. (2018 and for NGTS data in Gillen et al. (2020a). An Autocorrelation Function (ACF) also allows better detection of pseudo-periodic and phase-shifting variability often seen in young, active stars in comparison to more rigid variability extraction techniques such as Lomb-Scargle periodograms (Lomb 1976;Scargle 1982).
The paper is organised as follows. In section 2 we discuss the Next Generation Transit Survey and the data used, and in section 3 we outline the methods used in this study to extract rotation periods. Our results are summarised in section 4, with a discussion of these results in section 5 and a brief summary of our findings in section 6.
THE NEXT GENERATION TRANSIT SURVEY (NGTS)
NGTS is an array of twelve 20cm telescopes based at ESO's Paranal Observatory in Chile. Each telescope is coupled to a 2K × 2K e2V deep-depleted Andor IKon-L CCD camera with 13.5 m pixels, corresponding to an on-sky size of 4.97". The data for this study were taken with the array in survey mode, where each telescope observes a sequence of survey fields (generally 2 per night), each field having an on-sky size of ∼ 8 deg 2 . These fields are spaced such as one field rises above 30°the previous field sets below 30°. This typically results in approximately 500 hours coverage per field spread over 250 nights.
Fields are selected based on the density of stars, the proportion of dwarf stars, the ecliptic latitude and proximity to any bright or extended objects. Fields are typically selected with ≤ 15,000 stars brighter than an band magnitude of 16, of which ≥ 70% are dwarf stars. These fields will be more than 20°from the Galactic plane. Fields within 30°of the ecliptic plane are also avoided due to the Moon affecting readings during about three nights per month.
In this study, we use the final light curves, associated metadata and quality flags of the standard NGTS pipeline as described in Wheatley et al. (2018). The data for each field is reduced and photometric measurements are made on source apertures to assemble a light curve per target star. As a part of the pipeline, these light curves are passed through a custom implementation of the SysRem algorithm (Tamuz et al. 2005) which removes signals common to multiple stars arising from various sources including the instrument, reduction software and the atmosphere.
Light Curve Extraction
Photometric light curves are extracted for all sources detected within each NGTS field. Source detection is done using the module in (Irwin et al. 2004) to generate an object list that is cross-matched against a number of catalogues. NGTS generates its own input source catalogue, as explained in Section 5 of Wheatley et al. (2018). This source catalogue is cross-matched against a number of external catalogues. Cross-matching is done in position, as well as in colour and separation to limit spurious matches. This allows flagging of potential unresolved binaries in NGTS apertures.
A soft-edged circular aperture with a radius of 3 pixels (15 arcsec) is placed over each of these sources and placed in pixel coordinates using per-image astrometric solutions to account for radial distortion in addition to the autoguiding system. The sky background is estimated using bilinear interpolation of a grid of 64 × 64 pixel regions for which the sky level is determined using a k-sigma clipped median. These raw light curves are then passed through the detrending pipeline described in section 2.2 (Section 6 of Wheatley et al. (2018)).
Systematics Correction
In order to correct first-order offsets common to all light curves the detrending algorithm calculates a mean light curve for all objects to be used as an artificial 'standard star'. This detrending algorithm is based on the SysRem algorithm first described in Tamuz et al. (2005) and adapted from the WASP project (Collier Cameron et al. 2006). Of note, this approach may not fully remove systematic trends which correlate with Moon phase and sidereal time. In particular, Moon phased signals will show artefacts of imperfect background subtraction and any non-linearity in the detectors.
Data Selection
The NGTS pipeline provides flags per image and per timestamp per object light curve which we use to pre-process light curves for variability analysis. These flags alert us to bad-quality data points as a result of pixel saturation, blooming spikes from nearby bright sources, cosmics and other crossing events (including weather and laser guide stars) and any sky background changes. We removed any flagged data point from our light curves, and additionally checked if the majority of the light curve had been flagged (> 80% of data points). If this was the case, we removed the objects from processing.
We clipped our flux data to remove any points lying further than 3 median-absolute-deviations (MAD) from the median to remove any outliers not caught by the NGTS pipeline flags. This cut may remove some variability signals such as long-period eclipsing binaries where the variability is a small fraction of the phase curve. Manual inspection of a single field confirmed this was not the case, however, this cannot be guaranteed for all fields processed. Finally, in order to speed up data processing, we binned our light curve into 20-minute time bins. This reduced the number of data points to process per light curve from 200,000 to roughly 10,000. The G-ACF computation time scales as O ( 2 ) for data points with lag time steps, so reducing the number of timestamps in our light curve significantly improved processing time with a caveat that we will be unable to detect any periods below 40 minutes. For this study that is focused on longer period variability, this limit is not of concern.
We removed 6 fields identified as containing large open cluster populations. This study will focus on stars in the field and this avoids contamination of large numbers of young variable stars in known open clusters. Removing these 6 fields left a total of 829, 481 light curves to process. The positions of the 94 NGTS fields in RA and Dec used in this study are shown in Figure 1. In this Figure, we plot the Kepler and K2 field centre pointings, as well as the position of the galactic plane.
The 94 fields used in this study were observed for an average of 141 nights during different observation campaigns (lasting an average of 218 days) between September 2015 and November 2018. The shortest observational baseline for this data set was 84 days and the longest 272 days. 73 of the 94 fields had observational baselines over 200 days. We detected periodic variability in light curves spanning 8 < NGTS < 16 mag with 50% (90%) of our detections being brighter than 13.5 (15.4) mag.
PERIOD DETECTION
The period detection pipeline is outlined in the flowchart in Figure 2. Further details of each step are given in the subsequent sections.
The open-source code of the periodicity detection pipeline can be found on GitHub 2 .
Generalised Autocorrelation Function (G-ACF)
The G-ACF is essentially an extended and generalised form of the standard auto-correlation function (ACF) which can be applied to any time series, regardless of sampling. A complete and detailed mathematical description of this algorithm is available in a separate paper by Kreutzer et al. [submitted]. This generalisation is done by (a) generalising the lag time to a generalised lagˆwhich is a continuous variable within the range of our time series and (b) defining a selection functionˆand a weight functionˆ.
Taking a standard definition of the ACF (e.g. Shumway & Stoffer 2006):
( ) := 1 max − ∑︁ =1 ( − ) × ( + − ),(1)
where denotes the mean of the time series values and the normalisation is the total sum of squares := ∈ ( − ) 2 .
We can generalise this to:
ˆ;ˆ,ˆ := 1 ∑︁ ∈ +ˆ≤max( ) ( ) − × (ˆ( +ˆ)) − ׈ |ˆ( +ˆ) − ( +ˆ)| ,(2)
The weight function,T
he weight function,ˆ, should be a functionˆ: [0, ∞) → [0, 1] withˆ(0) ≡ 1 to ensure that for a regular time series the G-ACF reduces back to the standard ACF. One such example is a rational weight function such aŝ
( ) = 1 1 + , > 0, ≥ 0,(3)
in which is the time difference between the time label in the original time series and the lagged time series mapped by the selection functionˆ. The parameter was taken as the median value of the time series (as a time difference from the first data point), as prescribed in Kreutzer et al. [submitted]. We experimented with two different weight functions and elected to use the rational function as the final extracted periods were not dependant on this choice and this function is very simple. We used the minimum gap between time stamps as our lag resolution (time steps in generalised lag,ˆ); this was 20 minutes as we bin the data prior to analysis (Section 2.3).
Fast Fourier Transform (FFT)
In order to extract a period from the G-ACF, we elected to use a Fast Fourier Transform (FFT; Cooley et al. 1969). Extracting periods from an ACF can be done in a number of ways, most simply by selecting the first (or largest) peak in the ACF (e.g. as in McQuillan et al. 2014). This can lead to inaccuracies, in particular for weaker signals as this relies on the first peak being prominent in the ACF. We elected to use an extraction method that relies on the periodicity of the ACF, and the regular sampling of the G-ACF lends itself to an FFT. Other more complex methods such as fitting a damped harmonic oscillator to the ACF have been used previously (Angus et al. 2018). This in general did not alter extracted periods enough to warrant the additional complexity for such an exploratory work. We also experimented with using fewer ACF peaks rather than the entire signal in order to refine the period, but again the additional complexity was deemed unnecessary for a large scale rotation study. The FFT is a robust and well-documented method of extracting periodic signals. In this study we used the implementation in the numpy.fft package (Harris et al. 2020). We calculated the FFT with a padding factor of 32, to allow precise resolution of peaks in the Fourier transform. As phase information is lost in taking the ACF of the initial data, a real Fourier transform is sufficient.
To extract the most likely frequencies, we searched for peaks in the Fourier transform. A peak is defined as the central point in a contiguous sequence of 5 points which monotonically increases to the peak, followed by a monotonic decrease from the peak. Additionally, the amplitude of a peak must be greater than 20% of the highest peak in the periodogram to be included. Here an automated cut was made -any Fourier transforms with more than 6 peaks were removed as noise. This threshold was selected based on a manual vetting process for one NGTS field (10,000 objects) which demonstrated that for these objects with 'noisy' Fourier transforms less than 1% had genuine periodic signals. Removing these objects entirely greatly reduced the number of false positives extracted without removing many 'real' signals. 63% of processed objects were flagged as having no significant periodicity based on this FFT check.
Long Term Trend Assessment
A time baseline of ∼ 250 days allows for the extraction of periodic signals up to ∼ 125 days long. Signals longer than this may be present in the data, however observing one or fewer complete variability cycles cannot definitively characterise a periodic signal. This variability may not be periodic, but rather a long term trend in the data arising from instrumental or telluric changes over these timescales. These objects may still contain interesting periodic variability at a shorter timescale, so by detecting and removing a long term trend we can more accurately calculate the period and amplitude of this variability.
If the most significant peak in the FFT (see Section 3.4) was at a period greater than half the length of the signal baseline it was flagged as a long term trend. When this occurred we computed a high-pass filter for the signal by calculating the median flux at each time step in a rolling window which is 10% of the time extent of the light curve. This captures any long term behaviour without removing any shorter period variability. We divided this median filter from our signal and re-ran the cleaned light curve back through the signal detection pipeline. If no signal of interest was detected at this stage (either we found noise or residuals of our median filter), the object was flagged as having a long term trend and removed from processing.
Moon Signal Assessment
During initial testing of the period extraction algorithm, it was noted that a large number of periods between 27 and 30 days were identified by the period search algorithm. Upon closer inspection, these periods had very similar phases and could be split into two groups of signal shapes. The two signal shapes, when phase folded on a new Moon epoch, appeared as a slight increase or decrease in flux at 0.5 phase, i.e. full Moon. This was accompanied by an increase in scatter in the flux measurements at full Moon. Examples of contaminated signals are shown in Figure A1.
We fitted a model to these Moon correlated noise signals ('Moon signals') and flagged and removed any objects which fitted the expected trend. A detailed description of the model and removal process is given in Appendix A.
Alias Checks
As we are using an FFT to extract periodicity from our G-ACF, we are prone to aliasing. Aliasing is a well known and well-described problem in signal processing, and if the true frequency of the signal and the sampling frequency are known it is trivial to calculate the frequency of aliases as
alias = true ± · sampling (4)
where is an integer. We define period as the inverse of frequency, i.e. = 1 . In the case of ground-based observation, the most common sampling period will be 1 day. In addition, although the background correction should remove this, there will remain residuals of the brightness trend expected throughout the night's observation.
Although the sampling of the G-ACF is regular, the sampling of the inputted light curve will affect the shape of the G-ACF. We thus expect peaks in the FFT associated with 1-day systematic signals, as well as the true signal aliased with the 1-day sampling.
For each light curve we first removed any periods arising from the 1-day sampling. We removed periods within 5% of 1 day, as well as within 5% of integer multiples of 1 day in period and integer multiples of 1 / day in frequency. We then assessed whether groups of periods were aliases of one another with respect to common sampling periods using basic graph theory. We construct a graph of frequencies connected by the standard alias formula in Equation 4, using sampling periods of one day, 365.25636 days (one year), 27.32158 days (Lunar sidereal period) and 29.53049 days (Lunar synodic period). Each vertex in the graph represents an FFT peak frequency, with connections (edges) made if two frequencies can be related to one another through Equation 4 given one of our sampling frequencies. Note we considered aliases arising from both the synodic and sidereal Lunar period, however, given the 5% tolerance used for assessing similarity, these two sampling frequencies connected the same frequencies in the majority of examples.
For each connected sub-graph (i.e. a group of frequencies connected by the same sampling aliases) we determined the frequency for which the phase folded light curve had the lowest spread in flux and took this to be the correct period. We calculated the 5 th − 95 th percentile spread in flux within bins of 0.05 width in phase and then calculated the average of these values weighted by the number of points within each flux bin. In addition to the FFT peak periods, we also checked the RMS of twice and half the periods, as in some cases we found twice the FFT peak period was the correct period. This was assessed by-eye initially, and appeared to be much more common for short period objects due to aliasing from the 1-day sampling. This same approach was taken by McQuillan et al. (2013), however, we elected to automate the process rather than by-eye confirmation.
Further Signal Validation
Due to the ground-based nature of NGTS, some fields were not continuously observed for the entirety of the field time-baseline. As a result of bad weather and technical downtime, there were gaps in observations lasting several weeks for a number of the fields used in this study. In these cases it is no longer correct to use the entire time baseline as a cut-off for robust periods. Instead, we elected to find the longest period of continuous observation within these fields and remove any periods greater than half this time length. We define a period of continuous observation as a period in which there are no observation time gaps of greater than 20% of the entire field baseline. For our 250-night observation baseline, this equates to gaps of 50 days or longer. This removed 907 detected periodic signals from 11 different fields, and manual inspection of the removed signals confirmed that many of the removed detections were systematic periods arising from the long sampling gaps, rather than astrophysical variability.
Additionally, a number of detected periodic signals with unphysically large amplitudes were detected. On inspection it appears these signals were incorrectly processed by the NGTS pipeline, resulting in non-physical flux values. In our final sample, we elected to remove any signals with a relative amplitude > 1.0. This removed 58 signals, and manual inspection of the removed signals confirmed the majority of signals removed were non-physical; especially for the largest amplitude signals. The cut-off was chosen empirically based on the signal amplitude distribution of our sample.
Our initial search resulted in 17,845 periodic detections. Removing 907 long term trends left 16,938 detections. Finally, removing 58 unphysically large amplitude signals resulted in 16,880 detections.
Cross-matching with Gaia DR2 & TICv8
In order to assess our variability period sample within a meaningful scientific context, we elected to use Gaia Data Release 2 (DR2, Gaia Collaboration et al. 2018a) for cross-matching and to identify the nature of corresponding objects and their stellar parameters. The NGTS database contains cross-matching information with many external catalogues, including Gaia DR2. Detail on how the crossmatches are found is given in Section 5 of Wheatley et al. (2018) and briefly in Section 2.1 of this paper.
As an extension of the Gaia DR2 data, the most recent Tess Input Catalogue (TICv8, Stassun et al. 2019) contains Gaia DR2 data relevant to this study plus additional calculated values and cross-match data. These include more accurate calculated distances from Bailer-Jones et al. (2018) and reddening values which have been used to calculate absolute magnitudes.
More recently, the Gaia Early-DR3 (Gaia Collaboration et al. 2020) contains improved precision on the astrometric fits to many objects from Gaia DR2, however as we are using many derived parameters from external catalogues we elected to continue to use the DR2 parameters throughout this study.
Extinction Correction
In the final data products, we assess variability in the context of the colour-magnitude diagram which requires the calculation of absolute magnitudes. In order to be as accurate as possible, we combined Gaia G magnitudes ( ) with distance estimates and accounted for extinction. We used the per-object reddening values from TICv8, multiplied by a total-to-selective extinction ratio of 2.72 to account for the Gaia G-band extinction ( ). Further details on how the reddening values and the total-to-selective extinction ratio were calculated can be found in Section 2.3.3 of Stassun et al. (2019). Our final value for absolute magnitude was calculated using the formula:
= − 5 log 10 (distance) + 5 − .(5)
RESULTS
Using the G-ACF period extraction pipeline, we derived variability periods for 16, 880 stars observed with NGTS. A subset of these results is shown in Table 2, along with positions and cross-match data. The format of the results table is shown in Table 1.
Periodicity in Colour-Magnitude Space
Figure 3 shows our variability sample in colour-magnitude space, commonly known as a Hertzsprung-Russell (HR) Diagram or a colour-magnitude diagram (CMD). Table 3 details the breakdown of outputs from the pipeline. Once cross-matched with TICv8, we were left with a total of 16, 880 variable light curves from the initial sample of 829, 481 light curves. This gives a final detection percentage of 2.04%. The detection percentage varies in colour-magnitude space as shown in Figure 3a, highlighting potential regions of increased variability or increased sensitivity of NGTS and the signal detection pipeline. All conversions between eff , − and − in the following sections are calculated using relations defined in the 'Modern Mean Dwarf Stellar Colour and Effective Temperature Sequence' (Pecaut & Mamajek 2013) 3 , interpolated using a univariate cubic spline. The isochrones in the HR diagrams are taken from PARSEC v1.2S (Bressan et al. 2012). We elected to use these isochrones as they have been proven to fit the Gaia DR2 main sequence well in Gaia Collaboration et al. (2018b). We produce isochrones using PARSEC v1.2S, selecting the Gaia DR2 passbands from Evans et al. (2018) 4 . The isochrone at 1 Gyr gives a good indication of where the main sequence lies, with the earlier age isochrone at 10 Myr indicating locations on the HR diagram of potentially younger stellar populations. We note, as shown in Gillen et al. (2020b), that the PARSEC v1.2 models appear to be less reliable at pre-main-sequence ages, but should be sufficient for their indicative use in this study. Figure 3a highlights regions of interest in terms of detection percentage. Additionally, Figure 3b shows the number of detections in each bin. Where detection percentage approaches 100% this is often indicative of a single variable object falling in this colourmagnitude bin. As in Gaia Collaboration et al. (2019), we identify distinct regions of variability within the HR diagram and suggest the types of variable objects which may lie at each location.
The region at the top of the main sequence ( − ∼ 0.4, ∼ 1.0) reveals a high proportion of variable objects. We also see a region of increased variability at the 'elbow' of the main sequence and the Red-Giant Branch (RGB) ( − ∼ 1.5, ∼ 4). These objects may be young, massive objects with high levels of activity.
In Figure 3c we plot the median period in each colourmagnitude bin. Of particular interest, we see distinct regions of different variability periods on the HR diagram. There is a region of short median period at the top of the main sequence ( − ∼ 0.4, ∼ 1.0). Typical spot-driven photometric modulation will not be present on these hotter, radiative stars. The majority of variability seen in this region likely arises from pulsations. There may also be a number of magnetic OBA or chemically peculiar Ap stars within this region. In these stars, photometric brightness fluctuations are seen as a result of fossil magnetic fields imprinting chemical abundance inhomogeneity on the stellar surface (Sikora et al. 2019;David-Uraz et al. 2019). These targets are prime candidates for future spectropolarimetric observations (e.g. Grunhut et al. 2017).
A large number of the longest period variability signals lie on the RGB ( − 1.0, 2.0. These signals could indicate extremely slowly rotating large stars or other photometrically varying sources such as giant star pulsations.
We also see a clear trend of increasing period as we move perpendicular down towards the main sequence along the Hayashi tracks (Hayashi 1961). There are potentially a number of effects at play here: 1) We would expect a population of equal mass binary stars with short rotation periods to lie 0.75 in absolute magnitude above the main sequence, contributing to the shorter median period in this range.
2) We would also expect a population of young stars to lie in this region of colour-magnitude space. In particular, we see short period objects which lie between the 10 Myr and 1 Gyr isochrones.
In this region of the HR diagram potentially lie pre-main-sequence (PMS) Young Stellar Objects (YSO) such as T-Tauri stars with protostellar debris discs, which we expect to have shorter rotation periods than main-sequence stars of the same mass (colour). The median period observed for the bulk of main-sequence objects is 20 to 30 days, as expected.
We plot detection percentage vs. luminosity in
We use the radii values provided by TICv8. These radii values are either taken from pre-existing dwarf catalogue values (from Muirhead et al. 2018), or when these are not available (as is the case for a large majority of the NGTS sources) they are calculated from distance, bolometric corrections, G magnitude and a preferred temperature. Full details of this calculation are given in Stassun et al. (2018). eff values come from spectroscopic catalogues where available, otherwise they are derived from the de-reddened − colour. As expected, we recover a much higher fraction of variable signals from more luminous stars, with up to 15% of the most luminous objects in our sample having detectable variability signals. These objects will correspond to luminous giant stars, where we would expect large-amplitude variability arising from pulsations. The lowest number of variable objects coincides with the peak in the number of objects (at 1.5-2.5 ), where we detect variability in < 2% of objects. We also observe an increase in detection percentage for the faintest objects. Here we should expect to be observing cooler dwarf stars and young stars which generally have higher levels of magnetic activity and could lead to increased detection of photometric variability. Additionally, close binaries may appear more luminous than single stars and from their position above the main sequence in the HR diagram (Figure 3 (a)) appear to have a higher detection percentage than equivalent single stars. Given the width of the luminosity bins used is larger than the expected luminosity increase from a single star to an equal luminosity binary (0.2 dex, a factor ∼ 1.6 in luminosity), this will not have a large effect on the plotted distribution.
We assessed the distribution of detection percentage against on-sky RA and Dec for our population. The distribution of detection percentage for field stars did not appear to have any obvious correlation with on-sky position.
Example Variability Signals
We show six examples of variability signals in Figure 5. A table of stellar parameters for each object is included for reference. We selected the included objects to demonstrate a small selection of the variability we are able to extract from NGTS light curves. The stars are selected to have a range of spectral types, and demonstrate variability with different periods, amplitudes and signal shapes. In particular, using the object numbering as in Figure 5 (1 to 6, top to bottom): 1) An extremely short period, semi-detached eclipsing binary. This object lies above the main sequence, as expected for a nearequal mass binary system.
2) A typical short period pulsation signal from an RR-Lyrae object.
3) A candidate young stellar object (YSO). Objects above the main sequence with periods of 1 to 10 days are excellent YSO candidates, suitable for follow-up infrared and spectroscopic observations. 4) An example of a variable red-giant star. These are stars such as Cepheids, semi-regular variables, slow irregular variables or smallamplitude red-giants. 5) A main-sequence late-G dwarf star, with small amplitude 20to 30-day variability. 6) A long period M-dwarf.
Within the observed G-ACF signals we see artefacts arising from 1-day sampling aliases. These aliases are particularly relevant for signals of period < 1 day, where it was necessary to perform the additional verification steps outlined in Section 3.7.
Cross-matching with previous catalogues
We cross-matched our NGTS variability periods with photometric variability catalogues in the literature. The ASAS-SN variability catalogue is a large catalogue of photometric variability. We took the latest available data, containing 687,695 variable stars from Jayasinghe et al. (2018) through to Jayasinghe et al. (2021) 5 . We cross-matched our catalogue with the ASAS-SN catalogue, matching on TICv8 ID and Gaia DR2 ID. We found 2,439 matches with periods in both catalogues. A period-period comparison is shown in the left panel of Figure 6. The majority (about 1,500 stars) had similar periods from both catalogues. For approximately 750 stars, the periods differed by a factor of 2. This was most common for eclipsing binary targets in which the primary and secondary eclipses were of similar depths, and either the NGTS or ASAS-SN period was half the correct period. Periods with large discrepancies appear to be long term trends within the NGTS or the ASAS-SN data masking any shorter-term variability, or period aliasing resulting from the 1-day sampling seen in both surveys. The NGTS period extraction pipeline will not return periods close to 1 day or multiples thereof to reduce the number of systematic false positive detections. We see a number of periods in the ASAS-SN catalogue falling on exact fractions of 1 day, resulting in the 'stripes' of periods seen in the lower right of the Figure. We see structures within the period-period diagram resulting from objects for which the NGTS and ASAS-SN detections are aliases of one another with respect to 1-day sampling. Equation 4 can be used to calculate these connections and relations of the form ASASSN = 1 sampling ± 1 NGTS (7) are shown in Figure 6. Three obvious sets of aliased periods exist which trace these relations, accounting for approximately 114 matches. We see two sets of related periods arising from 1-day sampling, with the same double phase folding for eclipsing binaries resulting in the set of periods approaching 2 days. There is also a small group of periods connected by aliases arising from 2-day sampling, however, the form of the relation is not shown in the Figure. We were able to find three cross-matches with the MEarth rotation catalogue from Newton et al. (2018). Of these, NGTS was able to extract a short 0.4 day rotation period for an object which not present in the MEarth catalogue (NG1444-2807.12982). The two other objects (NG1214-3922.6732 & NG0458-3916.13434), NGTS detected a near 100 day period, similar to MEarth. The length of these periods would require extended observation from either survey to improve the accuracy as both surveys were only able to observe two to three complete variability cycles.
A variability study was conducted as part of the Gaia Data Release 2 (DR2, Gaia Collaboration et al. 2018a), where photometric time-series data was processed to detect and classify variable sources (as described in Holl et al. 2018). Photometric time series from Gaia are sparsely sampled and not optimised to detect photometric variability, so may produce an incorrect period. We crossmatched 126 objects against the rotation period database provided by the Gaia Collaboration on VizieR 6 , these period comparisons are shown in the right panel of Figure 6. For 60 of the 126 periods 5 The full catalogue is available at https://asas-sn.osu.edu/ variables 6 https://vizier.cds.unistra.fr/viz-bin/VizieR-3? -source=I/345/rm that differed, we phase folded the NGTS data on both periods and manually inspected which phase fold appeared to be favourable. The NGTS period was favoured in the majority of cases through visual inspection. As expected for space-based data we do not see any aliasing artefacts in the Gaia periods as in the cross-matching with ASAS-SN. This is a clear demonstration that the NGTS period recovery pipeline is well suited to deal with aliases arising from 1-day sampling Finally, we cross-matched our sample with the variability catalogue from Canto Martins et al. (2020), which searched for rotation periods in 1000 TESS objects of interest. We found six objects in both catalogues by matching on TIC id. These come from three different results tables from Canto Martins et al. (2020): TIC 14165625 and 77951245 contain 'unambiguous rotation periods', TIC 100608026 and 1528696 contain 'dubious rotation periods' and TIC 150151262 and 306996324 contained no significant variability in the TESS data. Manual inspection of these objects confirmed the NGTS light curves contained variability at the reported period from this study. For TIC 14165625, the reported TESS period was approximately half the NGTS period, and for TIC 77951245 the reported periods were similar (5.8 days and 5.4 days for NGTS and TESS, respectively), although the phase fold on the NGTS data was cleaner using the NGTS period.
Although a large number of photometric variable stars are known in the Kepler field, we are unable to cross-match with these catalogues as we do not observe this part of the sky. Additionally, we do not attempt to cross-match with small catalogues and papers reporting detections of individual variable objects. Two large variability catalogues we do not attempt cross-matches with are The Zwicky Transient Facility (ZTF) catalogue of periodic variable stars (Chen et al. 2020) or the catalogue of variable stars measured by the Asteroid Terrestrial-impact Last Alert System (ATLAS) (Heinze et al. 2018). The ZTF catalogue contains 4.7 million candidate variables and the ATLAS catalogue 621,702 candidate variables. Both surveys target much fainter objects than NGTS: the brightest candidates in both surveys are approximately as bright as the faintest objects observed by NGTS (Masci et al. 2019;Tonry et al. 2018).
Due to the small overlap in brightness and a large number of candidates in each catalogue, we elected not to perform a cross-match. Further cross-matching with smaller catalogues is possible, as we provide the position in RA and Dec, as well as TICv8 and Gaia DR2 identifiers (where available) for all 16, 880 variable sources.
Period ranges of interest
We break our results down into unevenly spaced intervals in variability period in order to assess how samples of similar variability periods are distributed in colour-magnitude space in Figure 7. This reveals more information than Figure 3 as we are able to probe into the high-density main sequence. We have selected the period ranges empirically taking into account the sampling gaps at 14 and 28 days arising from Moon contaminated signals.
The majority of the shortest period variability lies at the top of the main sequence. This could be indicative of -Scuti, RR-Lyrae or rapidly oscillating Ap stars in the instability strip. Typically, RR-Lyrae type objects lie in this region at the lower end of the instability strip and pulsate with periods of less than 1 day. The peak density for less evolved stars is above the main sequence at this period range. Between 1 and 10 days, we would expect to observe the rotation of YSOs such as T-Tauri stars or young main-sequence stars (e.g. as seen in Gaia Collaboration et al. 2019). We may also observe short-period binary star systems at this period range, which would also lie above the main sequence on the HR diagram. In the period range 3 to 14 days we continue to see a peak density above the main sequence, though the bulk moves towards later spectral types compared to the very short periods.
Between 16 and 26 days, we see the peak density move towards the main sequence as well as a distinct lack of objects above the main sequence. At > 30 days, we start to see detections into the RGB as well as more M-type stars. We would expect giant, evolved stars to have longer period rotation or pulsations. Moving from between 32 and 50 day to > 50 day periods we see the bulk of objects move further up the RGB and down the main sequence towards cooler temperatures and redder colours.
Period-colour distribution
We plot our variability periods against colour in Figure 8 and see a number of prominent features. Most striking is the high density of stars known in the literature as e.g. the 'I-Sequence' (Barnes 2003) or the 'Ridge' (Kovács 2015) spanning a period range from 4 − 40 days and − 0.75 − 3.5. The shape of this envelope has been empirically defined by Angus et al. (2019), using a broken power-law gyrochronology model calibrated against the ∼ 800 Myr old Praesepe cluster.
We see a large number of long-period (> 40 days) objects between − of ∼ 0.7 to ∼ 1.4. We would expect a higher density of detections at this colour range due to the high-density main-sequence turnoff and red clump, as shown in Figure 3(b). Older main-sequence stars at this colour range may exhibit long period rotational modulation. The Cepheid instability strip lies within this colour range, and we would expect to see long-period oscillations from evolved stars driven by the mechanism (Saio 1993).
Far below the I-sequence we see a high density of much shorter period, high amplitude variability amongst hot objects at − ∼ 0.5 → 1.5, and Period < 1 day. This population corresponds to the top of the main sequence on an HR diagram.
We see two distinct groups of objects in period range shorter than 1 day, trending to short periods with increasing colour index ( − 0.75 → 1.5). The two distinct groups are from the same region of the HR diagram -the equal-mass binary main sequence. The light curves showed distinct eclipsing binary signals (as seen in object 1 in Figure 5), however, the longer period branch contained light curves phase folded on the correct period and in the shorter period branch light curves phase folded on half this period. This is an artefact of the RMS minimisation step described in Section 3.7. For eclipsing binaries with slightly different primary and secondary eclipse depths the full period will show a 'cleaner' phase folded light curve with separate primary and secondary eclipses. In comparison, for an equal depth binary the phase folded light curve will have a similar RMS if folded on the correct period or half the period, with the primary and secondary plotted over one another in phase space.
Finally, we observe an increasing upper period envelope with increasing colour for − > 1.5. We see a number of objects with − > 2.5 having variability periods up to and exceeding 100 days. These objects are discussed in detail in Section 5.2.
Period Bi-modality
Within the I-sequence envelope we see a hint of a region lacking in periodic signals between ∼ 3500K and ∼ 4500K ( − 2.5-1.5) and ∼ 15-30 days. This gap has been the topic of extensive discussion in recent papers (such as McQuillan et al. (2014); Davenport & Covey (2018)), and although faint, we do observe this gap in this ground-based data set. This gap has previously been fitted using a gyrochrone, roughly following a eff 1/2 relation (Davenport & Covey 2018), as well as an empirical model using a similar eff 1/2 relation (Gordon et al. 2021).
To demonstrate the gap is present in our data we conduct the same analysis as in Figure 3 of Davenport & Covey (2018). We subtract model periods calculated with a 600 Myr gyrochrone defined in Meibom et al. (2011) from our periods. We constrain our data set to objects such that 1.4 < − < 2.2 to avoid the gyrochrone crossing the Moon signal sampling gaps. In Figure 9 we observe a dearth of objects along the gyrochrone, demonstrating the same gap as in the Kepler field is present within the NGTS data.
In Figure 10 we separate our sample into three sub-samples based on a bi-modality gap model and empirical short-period lower limit from Gordon et al. (2021). We observe how far these objects lie in absolute magnitude from an approximate main-sequence isochrone defined at 1 Gyr with Solar metallicity (Δ ), as plotted in Figure 3. We use this to assess where the three sub-samples lie on the CMD, to ascertain if they arise from distinct stellar populations in terms of colour and intrinsic brightness. We elect to remove potentially evolved stars, giants and sub-giants to ensure the models from Gordon et al. (2021) and Angus et al. (2019) which are fitted to main-sequence stars from Kepler and K2 are applicable. We use the code described in Huber et al. (2017) and Berger et al. (2018). The code gives crude evolutionary states for stars based on temperature and radius, with the models derived from Solar-type stars. We remove objects with the 'subgiant' or 'RBG' flags.
We define our 3 sub-samples using a number of model constraints in period-colour space. We use the fifth-order polynomial model defined in Angus et al. (2019) to constrain the long-period upper envelope of stars, and the edge-detection based fit from Gordon et al. (2021) to constrain the short-period lower envelope. We calculate the upper and lower edge of the gap using the model defined in Gordon et al. (2021), and select stars from our I-sequence envelope on either side of this branch. This model was only defined for 0.8 < − < 1.05, so we only use objects within this bound to define the sub-samples. Our third sub-sample is defined as all objects below this boundary in period and will consist of stars not included in the Kepler and K2 data sets which fall well below the well defined I-sequence in period. The model fits used in this section are detailed in Appendix B and plotted in Figure 10a.
The histograms in Δ plotted in Figure 10b show two similar single-peaked distributions from our two longer period sub-samples, and a distinct double-peak distribution for the shorter period subsample. We note that this second peak lies approximately 0.75 magnitudes above the peaks of the two longer period sub-samples which could indicate a population of binary objects which is not present in the upper two sub-samples. This confirms our previous observation from the HR diagram: a group of very short period objects just above the main sequence, which could correspond to a sample heavily contaminated by binary sources. The two longer period subsamples appear to have by-eye similar distributions of Δ , which leads us to believe the two branches are drawn from similar stellar populations in terms of colour, intrinsic brightness and multiplicity.
DISCUSSION
Comparison to similar studies
The NGTS data set demonstrates that we are able to use groundbased photometry to conduct stellar variability studies matching the scale of space-based data. In contrast to, for example, the Kepler data set used by McQuillan et al. (2014) and Davenport & Covey (2018), NGTS sources are not pre-selected. This provides a much more representative sample of field stars which is demonstrated in the much higher number of objects which lie away from the highdensity I-sequence envelope of stars in period-colour space. Objects which lie within the I-sequence will encompass a selection of stars most likely to be main-sequence, single objects similar to the Kepler input catalogue. We overlay data from the Kepler rotation study by McQuillan et al. (2014) with our data in Figure 11. In particular, we see a high density of objects at − ∼ 1.0 with periods longer than roughly 40 days not present in the Kepler data set. These objects lie in the RGB and AGB on the HR diagram, so will be giant objects which have not been removed from the NGTS study. We also see a large number of objects with much shorter periods than the I-Sequence envelope. These objects lie above the main sequence on the CMD and will be either short-period binary sources or potential YSOs.
In addition to finding astrophysical signals of interest, we were also able to observe systematic periodicity down to amplitudes of 0.3%.
This study highlights the power of ground-based photometric surveys in terms of the size and precision of the data set. We are able to extract a data set which rivals that of the Kepler and K2 missions, with a much longer baseline (in the case of K2) and a much greater range of pointings (in the case of Kepler). As a corollary, this study also serves as an exercise that ground-based photometric data may prove more difficult to analyse systematically than spacebased data due to increased sources of noise and aliasing. We note a lower recovery rate of periodic signals than other studies. McQuillan et al. (2014) found variability in 25.6% of their ∼ 130, 000 objects, Gordon et al. (2021) found variability in almost 13% of their 69, 000 objects, and NGTS was able to find variability in about 2% of 829, 481 objects. We note that 21% of all objects were flagged as having signals arising from Moon contamination, our largest source of systematic noise in the study.
The combination of a relatively long baseline (∼ 250 days) and multiple pointings (94 used in this study) allows the NGTS data set to probe out to reasonably long period regimes ( 0.1-130 days) and across a range of spectral types (late-A to mid-M).
Long Period M-Dwarfs
Previous studies such as Newton et al. (2018) have used targeted ground-based photometry to extract very long period variability for M dwarfs. We also observe these extremely long periods (> 100 days) in our M-dwarf sample. Figure 8 shows an upwards trend in period in the mid-M dwarf sample at eff < 3500 . In order to provide a useful comparison to the MEarth rotation study, we also assessed this trend for just dwarf stars (as defined by evolstate). Our sample contains 751 non-evolved, dwarf objects with variability periods with Gaia − > 2.21, which is the bluest limit of the MEarth rotation study catalogue.
In this study, the fields chosen had at most a 250-day time series, which allows us to robustly extract periods up to roughly 125 days in length. Newton et al. (2018) observed periods up to 140 days long for some of these objects, hypothesising that an upper limit close to this period would occur through Skumanich-like angular momentum loss for stars of the ages observed in the local thick disc. Using the Skumanich 1/2 relation and taking the age of the local thick disc to be 8.7 ± 0.1 Gyr (Kilic et al. 2017) we calculate the longest Skumanich relation period to be approximately 145 days. The NGTS rotation periods qualitatively agree with the distribution of rotation periods seen in M dwarfs by Newton et al. (2018), however, we reach the period limit of the NGTS data just shy of the ∼ 140 day limit in the MEarth detections. It is interesting to note the Skumanich relation still appears to hold from the longest period objects across samples, even into the fully convective M-dwarf population for which the physics of spin-down is not fully understood. Further observations of much older open clusters could shed light on this interesting long-period M-dwarf sample, and observations with much longer time baselines would allow us to probe into period regimes where spin-down could be more efficient than the Skumanich relation. We note that current photometric space missions such as TESS (Ricker et al. 2014) may be useful to shed light on this long term variability across the sky, but only at the ecliptic poles where objects will be observed for up to 1 year continuously, with a one year gap before another year of continuous observation. Most of the sky will only be observed for 28 days at a time, meaning a maximum of 14 day periods could be reliably extracted. This NGTS study overlaps both the Kepler rotation period data and the MEarth rotation period data, allowing more robust comparisons to be made between the two previously disjoint samples. The NGTS data set provides a broad view into stellar rotation, targeting similar Solar-type stars as observed by Kepler, as well as more diverse populations across the HR diagram and across a range of pointings.
Period Bi-modality
We continue the ongoing discussion regarding the rotation period gap (McQuillan et al. 2014;Davenport & Covey 2018;Reinhold et al. 2019;Reinhold & Hekker 2020;Angus et al. 2020;Gordon et al. 2021), including the first ground-based data set to have observed this feature in period temperature space. Although the gap is not as clear as in the space-based data, we align models from a number of previous works to a region of lower density in the NGTS data, as shown in Figure 9.
By utilising empirical models from previous studies on Kepler and K2 data, we separated our sample into three sub-samples: this is seen in Figure 10. Within the two upper sub-samples, we see the highest period objects are on average further above the main sequence in than the lower period objects. This effect has been previously observed, as Davenport & Covey (2018) saw a small increase in period as we move up in magnitude from the main sequence, but not as far as to be influenced by large numbers of binary objects. We note, similar to the Davenport & Covey (2018) study that we do not account for metallicity or age when considering the distance from a Solar metallicity defined main-sequence isochrone at 1Gyr. Metallicity has been shown to affect the amplitude of variability signals and additionally may lead to observational biases whereby for a given mass, higher metallicity stars' variability is more easily detected (See et al. 2021). There is also the possibility of contamination by lower mass-ratio binary systems. Further observations of open clusters with defined stellar ages and a tight single-star main sequence may afford more conclusive evidence towards this period gradient across the main sequence. Such studies have been conducted on open clusters across a large range of ages such as Blanco 1 (∼ 100 Myr) (Gillen et al. 2020a), Praesepe (∼ 800 Myr) (Rebull et al. 2016(Rebull et al. , 2017, Ruprecht 147 (∼ 3 Gyr) (Gruner & Barnes 2020) and M67 (∼ 4 Gyr) (Barnes et al. 2016).
The two sub-samples do not appear to be significantly contaminated by multiple systems and arise from similar locations on the HR diagram. Combined with the knowledge that these objects are from a range of pointings, this supports the conclusion of Gordon et al. (2021) that these two sub-samples do not derive from two distinct star formation epochs.
A broken spin-down law as discussed in Gordon et al. (2021) would be explained well by our data, including the possibility that the (very few) objects observed within this gap are currently transitioning between the two longer period sub-samples. In this broken spin-down law, the angular momentum change of the star will deviate from the expected 1/2 relation proposed by Skumanich (1972) due to the transfer of angular momentum between the envelope and the core. Prior to this transfer of angular momentum, the core and envelope are decoupled, resulting in the expected 1/2 spin-down of the envelope but with a rapidly rotating core which will then reduce or even stop the spin-down once the core and envelope recouple. This model has been suggested to fit Kepler data in addition to K2 data (Angus et al. 2020;Gordon et al. 2021), and theorists such as Lanzafame & Spada (2015) and later Spada & Lanzafame (2020) have incorporated these effects into stellar evolution models which have been shown to fit observed cluster data of different ages. The proposed models include a two-zone model of internal stellar coupling, with a parameter describing the mass dependence of the coupling. The recent analysis of the ∼ 3 Gyr old open cluster Ruprecht 147 by Gruner & Barnes (2020) demonstrates that the model from Spada & Lanzafame (2020) incorporating internal angular momentum transfer is best suited to model the rotational evolution of stars redder than K3 in comparison to more naive gyrochronology models.
Another suggestion for the origin of this gap comes from an analysis by Reinhold et al. (2019) and Reinhold & Hekker (2020) of K2 data. In their proposed model, the gap arises from objects in which the photometric variability arising from spots and faculae is of similar magnitude, thus cancelling out resulting in lower amplitude variability that is correspondingly harder to detect. They observed a slight decrease in signal amplitude on either side of the gap in period, and hypothesised objects of this period could exhibit spot-faculae photometric cancellation. We do not observe such an obvious decrease in signal amplitude in our full sample, and when considering a smaller range of amplitudes more aligned with the K2 sample we again did not see this amplitude gradient. This may be attributed to NGTS photometry being less precise than Kepler, and a small change on a signal of 1% amplitude may not be detectable. To accurately determine the dominant surface feature of a star requires observations of spot-crossing events during planetary transits or Doppler images, neither of which are appropriate for follow-up from a large-scale photometric study.
CONCLUSIONS
In this study we extract robust variability periods for 16, 880 stars out of 829, 481 stars observed with the Next Generation Transit Survey (NGTS), based in Paranal, Chile. This is the largest groundbased systematic photometric variability study conducted to date with such precise and high-cadence photometry and highlights both the advantages of such studies as well as the challenges. Using precise ground-based photometry, plus a generalisation of the autocorrelation function to irregularly sampled data, we are able to detect variability amplitudes down to levels of 0.3%. The contamination of signals by systematics demonstrates that using ground-based photometry requires further thought than using much cleaner spacebased data in order to avoid false positives arising through aliases. The most common source of aliases arose from Moon contaminated signals as well as aliasing from the 1-day periodic sampling intrinsic to ground-based observations. We demonstrate we are able to overcome these limitations and produce robust variability signals across our sample.
In comparison to previous large-scale stellar variability studies, we note that with NGTS we are able to observe across the Southern sky (in comparison to Kepler's single pointing, as in Mc-Quillan et al. (2014) and Davenport & Covey (2018)). We do not pre-select our targets as is the case for Kepler and K2, and so we are able to observe variability across a more varied stellar sample. In particular, we extract long term variability periods for a population of cool dwarfs, similar to a population observed by Newton et al. (2018) using MEarth. This is made possible through our longer observation baseline than space-based missions such as K2. This large population, sampled across the sky over a long (250 day) baseline allows this study to connect previous space-based studies on main-sequence, predominantly Solar-type stars with ground-based M-dwarf studies, which were previously unconnected.
Within the bulk of our rotation period 'I-Sequence', we observe a gap between 15 and 25 days, first observed by McQuillan et al. (2014), and later studied in detail by Davenport & Covey (2018), Reinhold et al. (2019), Reinhold & Hekker (2020), Angus et al. (2020) and Gordon et al. (2021). Using models from Gordon et al. (2021), Angus et al. (2019) and Meibom et al. (2011) we are able to demonstrate that the gap is present in our data set, and also show that the two sub-samples of main-sequence objects above and below this gap appear to arise from similar stellar populations on the CMD which are not contaminated by high levels of binarity. This supports the hypothesis of a broken spin-down model as proposed by Lanzafame & Spada (2015) and Spada & Lanzafame (2020) rather than distinct populations of star formation.
We also conclude that although a large population study of field stars is useful for assessing trends in the wider stellar population, without well-defined ages of target stars it is difficult to confirm angular momentum models. We suggest that studies of open clusters with well-defined ages and tight rotation sequences such as the recent study by Gruner & Barnes (2020) will yield the most conclusive evidence towards how stellar angular momentum evolves over the lifetime of a star. Additionally, we observe a number of interesting non-main-sequence populations, including a small population of objects which lie well above the main sequence with short rotation periods. Follow-up observations of these targets would allow us to ascertain whether these stars are young, single stars such as T-Tauri objects, or multiple star systems. This data set presents a wealth of additional data with many avenues for follow-up science. These include both continued systematic variability analysis of the NGTS data and also more in-depth analysis of interesting sub-populations of variable objects not explored in this cardinal NGTS variability study. We fit for the 3 parameters flux 0 , flux 1 and turnover. This model fit is assessed by checking the following criteria, with an example shown in Figure A2.
• Is the model turnover point at the expected point in phase? (between 0.2 and 0.8 in half-phase).
• Is there a flux RMS increase after the model turnover point?
• Is there a noticeable (i.e. > 1 ) change in flux from new to full Moon?
• Is there any missing data at full Moon?
If 3 or more of these criteria are met, the object is flagged as Moon contaminated and removed from the processing.
APPENDIX B: MODEL PARAMETERS
In Figure 10a we use empirical models defined in Angus et al. (2019) and Gordon et al. (2021). In this section, we provide the model equations and the parameters used.
B1 Angus model
We use the Praesepe-calibrated gyrochronology relation defined in Angus et al. (2019). The mathematical form of this fifth-order polynomial relationship is given in Equations B1 & B2 below for two different − regimes:
log 10 ( rot ) = log 10 ( ) + for stars with − > 2.7. Here rot is the rotation period in days, and is age in years. We use the best-fit coefficients from Table 1 of Angus et al. (2019).
B2 Gordon Model
We use the K2 calibrated model from Gordon et al. (2021) to define the upper and lower edges of the bi-modality gap seen in the Isequence envelope. The gap edges are fitted using a function of the form:
= ( − − 0 ) + ( − − 0 ) 1/2 (B3)
where is the rotation period in days. This equation is defined empirically for K2 stars with 0.8 < − < 1.05. We use the best fit coefficients defined in Table 7 of Gordon et al. (2021).
The lower edge of the K2 sample from Gordon et al. (2021) used an edge-detection method, and as such no parametric model form was given. We instead define our lower edge by eye, taking the edge-detection fit line from the Gordon et al. (2021) paper. This paper has been typeset from a T E X/L A T E X file prepared by the author.
Figure 1 .
1An ICRS plot of the position of the 94 NGTS fields used in this study (solid dark blue squares). The Kepler and K2 fields are included as blue and orange squares, respectively, as well as the galactic plane as a thick grey line.
Figure 2 .
2A schematic of the period detection pipeline, per NGTS light curve. * refers to the median absolute deviation (MAD). mean of the time series values set.3.2 The selection function,The selection functionˆprovides a mapping between time labels within the original time series and the lagged time series at each lag time step. The most natural choice of a selection function would be to select the point closest in time within the original time series for each point in the lagged time series. Further details of this selection function, including a cartoon outlining the method, are detailed in Kreutzer et al.[submitted].
empirical detection percentage per bin. This is defined as the ratio of the number of detected periodic signals to all observed objects per bin. 0 detections within bins are coloured grey. number of objects with detected variability within each colour-magnitude bin.
) The median variability period within each colour-magnitude bin.
Figure 3 .
3Binned colour-magnitude (HR) diagrams of the NGTS variability sample. PARSEC v1.2 (Bressan et al. 2012) Solar metallicity isochrones of ages 10 Myr and 1 Gyr are included as solid black and orange lines, respectively.
Figure 4 .
4Histogram of the empirical detection percentage (left y-axis) for all sources against luminosity, as well as the luminosity distribution for all observations (right y-axis).
Figure 4 .
4Luminosity values are taken from TICv8 (Stassun et al. 2019), calculated using Equation 6.
Figure 5 .
5Example variable star signals across the HR diagram. From left to right: A table of stellar parameters. The NGTS light curve, binned to 20 minutes. The G-ACF of the light curve. The light curve phase folded on the extracted period, each successive period is coloured according to a perceptually uniform sequential colourmap. The position of each star on the HR diagram is shown, the numbered labels 1 to 6 correspond to the stars top to bottom. Solar metallicity PARSEC isochrones of ages 10 Myr and 1 Gyr are included as solid black and orange line on the HR diagram, respectively.
Figure 6 .
6NGTS variability periods from this study compared with ASAS-SN periods (left) and Gaia (right). Lines of equal period from both surveys are plotted in light grey, and for the ASAS-SN comparison lines showing periods differing due to incorrect phase folding by a factor two shorter or longer are also plotted in light grey. The red dashed lines and associated equations indicate relations between periods arising from 1-day sampling. Light grey dotted horizontal lines in the left-hand figure and corresponding periods indicate where ASAS-SN has recovered periods corresponding to exact fractions of a day.
Figure 7 .Figure 8 .Figure 9 .
789HR diagrams for the NGTS variability sample broken down into period ranges. Periods in the sample range from ∼ 0.1 to 130 days. The colour bar indicates the percentage of all variable objects across all period ranges which lie in this specific colour-magnitude-period bin. The sum of each bin across all 5 subplots will equal 100%. Solar metallicity PARSEC isochrones of ages 10 Myr and 1 Gyr are included as solid black and orange lines, respectively. Effective temperature and Gaia − colour against period for 16, 880 stars. The colour indicates the 5 th − 95 th percentile spread of the signal in relative flux. To aid the eye, horizontal strips indicate regions of period space likely affected by systematics arising from the Moon or the 1-day sampling alias, with multiples of these periods more transparent. Distribution of the distance from a 600 Myr gyrochrone of the log periods for stars 1.4 < − < 2.2. We see two peaks in the distribution, with a reduced number of rotation periods along the model gyrochrone (grey vertical line). The range of distances from the model to the Moon and half Moon period are included to demonstrate the lower density of objects does not arise from a gap due to the Moon.
− colour vs. period. Three sub-samples spanning the observed period gap (above, below and significantly below the gap) are defined using models (see legend) and coloured blue, green and orange, respectively. Note we do not plot the large model uncertainties defined for theAngus et al. (2019) model for stars outside the range 0.56 < − < 2.7 (0.31 < − < 1.17). of distance in Gaia from a main-sequence isochrone (1 Gyr, Solar metallicity) for our three sub-samples (coloured as in panel(a)).
Figure 10 .Figure 11 .
1011Panel (a): Period-colour diagram of our sample, with three sub-samples defined by empirical models from Gordon et al. (2021) and Angus et al. (2019). Panel (b): Histograms of the magnitude difference in each of the three sub-samples from a main-sequence isochrone. Effective Temperature vs Period data compared for this study (NGTS data, green circles), McQuillan et al. (2014) (Kepler data, grey squares) and Newton et al. (2018) (MEarth data, blue squares).
Figure A1 .Figure A2 .
A1A2Two examples of typical Moon tainted signals. For each object, the light curve is phase folded on the expected Moon period and epoch. 0.0 & 1.0 phase are at new Moon, 0.5 phase is at full Moon. We see an example of an over-corrected signal with a typical decrease in flux at full Moon (top). An under-corrected signal demonstrates the opposite trend (bottom). Both signals exhibit an increase in scatter at full Moon, with an otherwise fairly flat light curve. The three-parameter Moon model fit is used to assess if a signal is contaminated by the Moon. The flux data is phase folded on the period of the Moon and then again in half such that 0.0 in phase corresponds to new Moon and 1.0 in phase corresponds to full Moon. Where = flux 1 − flux 0 1 − turnover = flux 1 −
Table 1 .
1Variability periods, amplitudes, positions and catalogue cross-match identifiers for all variable objects in the NGTS data set(table format).Table 2. A sample of variability periods, amplitudes, positions and catalogue cross-match identifiers in the NGTS data set. A number of catalogue cross-match columns have been excluded for publication clarity. The full table will is available at CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via https://cdsarc.unistra.fr/viz-bin/cat/J/MNRAS, or as supplementary material.Column Format Units Label
Description
1
A18
-
NGTS_ID
NGTS source designation
2
F9.5
deg
NGTS_RA
Source right ascension (J2000)
3
F9.5
deg
NGTS_DEC
Source declination (J2000)
4
F8.5
mag
NGTS_MAG
NGTS I-band magnitude
5
F9.5
days
PERIOD
Extracted variability period
6
F7.5
-
AMPLITUDE
5-95 percentile relative flux
7
I19
-
GAIA_DR1_ID
Cross-matched Gaia DR1 identifier
8
I19
-
GAIA_DR2_ID
Cross-matched Gaia DR2 identifier
9
I10
-
TIC_ID
Cross-matched Tess Input Catalogue (v8) identifier
10
A16
-
TWOMASS_ID Cross-matched 2MASS identifier
11
A19
-
WISE_ID
Cross-matched WISE identifier
12
A10
-
UCAC4_ID
Cross-matched UCAC4 identifier
NGTS ID
NGTS RA NGTS Dec NGTS Mag
Period Amplitude
Gaia DR2 ID
TICv8 ID
NG0613-3633_231
94.88721
-35.20762
14.77188 117.30427
0.07218 2885392740653834368 124854845
NG0613-3633_234
91.91176
-35.20084
15.86231 128.42220
0.18731 2885953869540806656 201389809
NG0613-3633_235
94.93884
-35.20675
12.91320 117.53460
0.04857 2885392878092780544 124854842
NG0613-3633_262
94.95269
-35.20598
14.51873 109.77205
0.11461 2885392225257749760 124854841
NG0613-3633_481
93.77213
-35.22205
13.69049
0.29365
0.13175 2885521658392050944 124689517
NG0613-3633_598
93.31896
-35.22787
11.48757
92.88398
0.00860 2885530999944081792 201530507
NG0613-3633_773
95.01907
-35.23832
12.55225 110.36974
0.07016 2885380160692365824 124855736
NG0613-3633_1101
95.06110
-35.25333
15.16207 128.42220
0.22943 2885381333220681216 124855723
NG0613-3633_1181
95.06864
-35.25766
13.60488 100.74969
0.08537 2885380577306436736 124855720
NG0613-3633_1479
95.12023
-35.27187
14.86311 100.46635
0.25487 2885380439867479040 124922604
Table 3 .
3A table of the output states of the 829, 481 NGTS objects analysed by the signal detection pipeline. Note a further 907 objects were removed due to large observation gaps in a number of fields, and an additional 58 with spuriously large amplitudes resulting in a final total of 16, 880 variability periods (see Section 3.8).Output State
Count
% of total % of detections
Bad Data
43,358
5.227
-
Noisy FFT
528,105
63.667
-
Moon
175,565
21.166
67.043
Alias
57
0.007
0.022
Long Term Trend
64,551
7.782
25.018
Periodic Signal
17,845
2.151
6.916
MNRAS 000, 1-20(2021)
https://github.com/joshbriegal/periodicity_detection
A more recent version of the table including Gaia DR2 colours is maintained at http://www.pas.rochester.edu/~emamajek/EEM_dwarf_ UBVIJHK_colors_Teff.txt 4 using the CMD 3.4 input form at http://stev.oapd.inaf.it/ cgi-bin/cmd MNRAS 000, 1-20 (2021)
ACKNOWLEDGEMENTSWe thank the anonymous referee for constructive suggestions and comments that improved this paper. BasedDATA AVAILABILITYThe NGTS data underlying this article are publicly available through the ESO Archive in line with the NGTS data publication policy.The variability period results produced in this article are available as supplementary material and at CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via https://cdsarc. unistra.fr/viz-bin/cat/J/MNRAS.APPENDIX A: DETAILED MOON SIGNAL ANALYSISIn order to systematically detect Moon contaminated signals (for example as shown inFigure A1), we fit a model to the flux data, phase folded on the expected Moon period for each NGTS field. The expected Moon period is calculated from a scaled expected Moon brightness curve, calculated as a product of the on-sky separation of the field from the Moon and the Moon illumination fraction, = (1 + cos ( ℎ ))/2. ℎ is the Moon phase angle defined for a time and ephemeris. For most fields, this gave a period of approximately 28.5 days, between the synodic and sidereal periods as expected.The model is a simple three-parameter, piecewise model described inEquation A1, where the parameter is the location in half phase ∈ [0, 1]. flux 0 0 ≤ ≤ turnover + turnover < ≤ 1 (A1)
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"Marlin B Schäfer \nMax-Planck-Institut für Gravitationsphysik\nAlbert-Einstein-Institut\nD-30167HannoverGermany\n\nLeibniz Universität Hannover\nD-30167HannoverGermany\n",
"Ondřej Zelenka \nFriedrich-Schiller-Universität Jena\nD-07743JenaGermany\n\nMichael Stifel Center Jena\nD-07743JenaGermany\n",
"Alexander H Nitz \nMax-Planck-Institut für Gravitationsphysik\nAlbert-Einstein-Institut\nD-30167HannoverGermany\n\nLeibniz Universität Hannover\nD-30167HannoverGermany\n",
"He Wang \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Shichao Wu \nMax-Planck-Institut für Gravitationsphysik\nAlbert-Einstein-Institut\nD-30167HannoverGermany\n\nLeibniz Universität Hannover\nD-30167HannoverGermany\n",
"Zong-Kuan Guo \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Zhoujian Cao \nDepartment of Astronomy\nBeijing Normal University\n100875BeijingChina\n",
"Zhixiang Ren \nPeng Cheng Laboratory\n518055ShenzhenChina\n",
"Paraskevi Nousi \nDepartment of Informatics\nAristotle University of Thessaloniki\nGR-54124ThesssalonikiGreece\n",
"Nikolaos Stergioulas \nDepartment of Physics\nAristotle University of Thessaloniki\nGR-54124ThessalonikiGreece\n\nGSI Helmholtz Center for Heavy Ion Research\nPlanckstraße 164291DarmstadtGermany\n",
"Alexandra E Koloniari \nDepartment of Physics\nAristotle University of Thessaloniki\nGR-54124ThessalonikiGreece\n",
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"Gabriele Vedovato \nINFN\nSezione di Padova\nI-35131PadovaItaly\n",
"Sergey Klimenko \nDepartment of Physics\nUniversity of Florida\nPO Box 11844032611-8440GainesvilleFLUSA\n",
"Tanmaya Mishra \nDepartment of Physics\nUniversity of Florida\nPO Box 11844032611-8440GainesvilleFLUSA\n",
"Bernd Brügmann \nFriedrich-Schiller-Universität Jena\nD-07743JenaGermany\n\nMichael Stifel Center Jena\nD-07743JenaGermany\n",
"Elena Cuoco \nEuropean Gravitational Observatory (EGO)\nI-56021Cascina, PisaItaly\n\nScuola Normale Superiore\nPiazza dei Cavalieri 7I-56126PisaItaly\n\nINFN\nSezione di Pisa\nLargo Bruno Pontecorvo, 3I-56127PisaItaly\n",
"E A Huerta \nData Science and Learning Division\nArgonne National Laboratory\n60439LemontIllinoisUSA\n\nDepartment of Computer Science\nUniversity of Chicago\n60637ChicagoIllinoisUSA\n",
"Chris Messenger \nSchool of Physics and Astronomy\nSUPA\nUniversity of Glasgow\nG12 8QQGlasgowUnited Kingdom\n",
"Frank Ohme \nMax-Planck-Institut für Gravitationsphysik\nAlbert-Einstein-Institut\nD-30167HannoverGermany\n\nLeibniz Universität Hannover\nD-30167HannoverGermany\n"
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"Department of Astronomy\nBeijing Normal University\n100875BeijingChina",
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"Department of Informatics\nAristotle University of Thessaloniki\nGR-54124ThesssalonikiGreece",
"Department of Physics\nAristotle University of Thessaloniki\nGR-54124ThessalonikiGreece",
"GSI Helmholtz Center for Heavy Ion Research\nPlanckstraße 164291DarmstadtGermany",
"Department of Physics\nAristotle University of Thessaloniki\nGR-54124ThessalonikiGreece",
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"Department of Physics\nAristotle University of Thessaloniki\nGR-54124ThessalonikiGreece",
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"Department of Physics\nUniversity of Florida\nPO Box 11844032611-8440GainesvilleFLUSA",
"Friedrich-Schiller-Universität Jena\nD-07743JenaGermany",
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"Data Science and Learning Division\nArgonne National Laboratory\n60439LemontIllinoisUSA",
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"Max-Planck-Institut für Gravitationsphysik\nAlbert-Einstein-Institut\nD-30167HannoverGermany",
"Leibniz Universität Hannover\nD-30167HannoverGermany"
] | [] | We present the results of the first Machine Learning Gravitational-Wave Search Mock Data Challenge (MLGWSC-1). For this challenge, participating groups had to identify gravitational-wave signals from binary black hole mergers of increasing complexity and duration embedded in progressively more realistic noise. The final of the 4 provided datasets contained real noise from the O3a observing run and signals up to a duration of 20 seconds with the inclusion of precession effects and higher order modes. We present the average sensitivity distance and runtime for the 6 entered algorithms derived from 1 month of test data unknown to the participants prior to submission. Of these, 4 are machine learning algorithms. We find that the best machine learning based algorithms are able to achieve up to 95% of the sensitive distance of matched-filtering based production analyses for simulated Gaussian noise at a false-alarm rate (FAR) of one per month. In contrast, for real noise, the leading machine learning search achieved 70%. For higher FARs the differences in sensitive distance shrink to the point where select machine learning submissions outperform traditional search algorithms at FARs ≥ 200 per month on some datasets. Our results show that current machine learning search algorithms may already be sensitive enough in limited parameter regions to be useful for some production settings. To improve the state-of-the-art, machine learning algorithms need to reduce the false-alarm rates at which they are capable of detecting signals and extend their validity to regions of parameter space where modeled searches are computationally expensive to run. Based on our findings we compile a list of research areas that we believe are the most important to elevate machine learning searches to an invaluable tool in gravitational-wave signal detection. | 10.1103/physrevd.107.023021 | [
"https://export.arxiv.org/pdf/2209.11146v1.pdf"
] | 252,439,082 | 2209.11146 | b27ceebb19b782c8b04fe4ab007d0f0558f503b8 |
MLGWSC-1: The first Machine Learning Gravitational-Wave Search Mock Data Challenge
Marlin B Schäfer
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
D-30167HannoverGermany
Leibniz Universität Hannover
D-30167HannoverGermany
Ondřej Zelenka
Friedrich-Schiller-Universität Jena
D-07743JenaGermany
Michael Stifel Center Jena
D-07743JenaGermany
Alexander H Nitz
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
D-30167HannoverGermany
Leibniz Universität Hannover
D-30167HannoverGermany
He Wang
Institute of Theoretical Physics
CAS Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
100190BeijingChina
Shichao Wu
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
D-30167HannoverGermany
Leibniz Universität Hannover
D-30167HannoverGermany
Zong-Kuan Guo
Institute of Theoretical Physics
CAS Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
100190BeijingChina
Zhoujian Cao
Department of Astronomy
Beijing Normal University
100875BeijingChina
Zhixiang Ren
Peng Cheng Laboratory
518055ShenzhenChina
Paraskevi Nousi
Department of Informatics
Aristotle University of Thessaloniki
GR-54124ThesssalonikiGreece
Nikolaos Stergioulas
Department of Physics
Aristotle University of Thessaloniki
GR-54124ThessalonikiGreece
GSI Helmholtz Center for Heavy Ion Research
Planckstraße 164291DarmstadtGermany
Alexandra E Koloniari
Department of Physics
Aristotle University of Thessaloniki
GR-54124ThessalonikiGreece
Anastasios Tefas
Department of Informatics
Aristotle University of Thessaloniki
GR-54124ThesssalonikiGreece
Department of Physics
Aristotle University of Thessaloniki
GR-54124ThessalonikiGreece
Nikolaos Passalis
Department of Informatics
Aristotle University of Thessaloniki
GR-54124ThesssalonikiGreece
Francesco Salemi
Dipartimento di Fisica
Università di Trento
I-38123Povo, TrentoItaly
INFN
Trento Institute for Fundamental Physics and Applications
I-38123Povo, TrentoItaly
Gabriele Vedovato
INFN
Sezione di Padova
I-35131PadovaItaly
Sergey Klimenko
Department of Physics
University of Florida
PO Box 11844032611-8440GainesvilleFLUSA
Tanmaya Mishra
Department of Physics
University of Florida
PO Box 11844032611-8440GainesvilleFLUSA
Bernd Brügmann
Friedrich-Schiller-Universität Jena
D-07743JenaGermany
Michael Stifel Center Jena
D-07743JenaGermany
Elena Cuoco
European Gravitational Observatory (EGO)
I-56021Cascina, PisaItaly
Scuola Normale Superiore
Piazza dei Cavalieri 7I-56126PisaItaly
INFN
Sezione di Pisa
Largo Bruno Pontecorvo, 3I-56127PisaItaly
E A Huerta
Data Science and Learning Division
Argonne National Laboratory
60439LemontIllinoisUSA
Department of Computer Science
University of Chicago
60637ChicagoIllinoisUSA
Chris Messenger
School of Physics and Astronomy
SUPA
University of Glasgow
G12 8QQGlasgowUnited Kingdom
Frank Ohme
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
D-30167HannoverGermany
Leibniz Universität Hannover
D-30167HannoverGermany
MLGWSC-1: The first Machine Learning Gravitational-Wave Search Mock Data Challenge
We present the results of the first Machine Learning Gravitational-Wave Search Mock Data Challenge (MLGWSC-1). For this challenge, participating groups had to identify gravitational-wave signals from binary black hole mergers of increasing complexity and duration embedded in progressively more realistic noise. The final of the 4 provided datasets contained real noise from the O3a observing run and signals up to a duration of 20 seconds with the inclusion of precession effects and higher order modes. We present the average sensitivity distance and runtime for the 6 entered algorithms derived from 1 month of test data unknown to the participants prior to submission. Of these, 4 are machine learning algorithms. We find that the best machine learning based algorithms are able to achieve up to 95% of the sensitive distance of matched-filtering based production analyses for simulated Gaussian noise at a false-alarm rate (FAR) of one per month. In contrast, for real noise, the leading machine learning search achieved 70%. For higher FARs the differences in sensitive distance shrink to the point where select machine learning submissions outperform traditional search algorithms at FARs ≥ 200 per month on some datasets. Our results show that current machine learning search algorithms may already be sensitive enough in limited parameter regions to be useful for some production settings. To improve the state-of-the-art, machine learning algorithms need to reduce the false-alarm rates at which they are capable of detecting signals and extend their validity to regions of parameter space where modeled searches are computationally expensive to run. Based on our findings we compile a list of research areas that we believe are the most important to elevate machine learning searches to an invaluable tool in gravitational-wave signal detection.
We present the results of the first Machine Learning Gravitational-Wave Search Mock Data Challenge (MLGWSC-1). For this challenge, participating groups had to identify gravitational-wave signals from binary black hole mergers of increasing complexity and duration embedded in progressively more realistic noise. The final of the 4 provided datasets contained real noise from the O3a observing run and signals up to a duration of 20 seconds with the inclusion of precession effects and higher order modes. We present the average sensitivity distance and runtime for the 6 entered algorithms derived from 1 month of test data unknown to the participants prior to submission. Of these, 4 are machine learning algorithms. We find that the best machine learning based algorithms are able to achieve up to 95% of the sensitive distance of matched-filtering based production analyses for simulated Gaussian noise at a false-alarm rate (FAR) of one per month. In contrast, for real noise, the leading machine learning search achieved 70%. For higher FARs the differences in sensitive distance shrink to the point where select machine learning submissions outperform traditional search algorithms at FARs ≥ 200 per month on some datasets. Our results show that current machine learning search algorithms may already be sensitive enough in limited parameter regions to be useful for some production settings. To improve the state-of-the-art, machine learning algorithms need to reduce the false-alarm rates at which they are capable of detecting signals and extend their validity to regions of parameter space where modeled searches are computationally expensive to run. Based on our findings we compile a list of research areas that we believe are the most important to elevate machine learning searches to an invaluable tool in gravitational-wave signal detection.
I. INTRODUCTION
The first gravitational-wave (GW) observation on September 14, 2015 [1] achieved by the LIGO and Virgo Collaboration [2,3] started the era of GW astronomy. During the first observing run (O1) two more GWs from coalescing binary black holes (BBHs) were detected. The second observing run (O2) saw O(10) additional confident BBH detections as well as the first detection of a binary neutron star (BNS) merger [4][5][6][7][8][9]. The third observing run (O3) was split into two parts, O3a and O3b.
During O3a a further O(40) BBHs as well as a second BNS merger were reported [10][11][12]. O3b added another O(40) BBH events as well as finding the first two confident detections where the component masses are consistent with the merger of a neutron star black hole system (NSBH) [13,14]. The fourth observing run (O4) is scheduled to begin in spring of 2023 and is expected to significantly increase the volume from which sources can be detected [15,16].
GW signals are commonly identified in the background noise of the detectors using matched filtering [13,[17][18][19]. Matched filtering compares pre-computed models of expected signals, known as templates, with the data from the detectors [20]. When a model matches the data to a pre-defined degree and data-quality requirements are met, a candidate detection is reported. Loosely modelled searches [21,22], which look for coherent excess power in multiple detectors, are also employed by the LIGO-Virgo-KAGRA collaboration (LVK) to find potential signals.
The rate of detections has drastically increased from O1 to O3. This increase was enabled by continued detector upgrades at the two advanced LIGO observatories in Hanford and Livingston [2], as well as sensitivity improvements for the advanced Virgo detector [3]. With the entry into service of Kagra [23] a fourth observatory joined the network of ground based GW-detectors towards the end of O3. The rate of detections is expected to further increase during O4 as the sensitivity of the detectors improves and the volume from which sources can be detected grows.
With an increasing rate of detections, it is likely that systems with unexpected physical properties will be observed more frequently in the future. Optimally searching for these is a challenge for matched filtering based searches, where the computational cost scales linearly with the number of templates used. The inclusion of effects such as precession, eccentricity, or higher order modes requires millions of templates to not miss potential signals [24][25][26] and thus are computationally prohibitive, especially when real-time alerts should be issued. Loosely modeled searches are inherently capable of detecting arbitrary sources at a fixed computational cost but are prone to miss more signals due to their lower sensitivity in parameter regions where accurate models exist.
In recent years, machine learning has been applied in many scientific fields to enable or improve research into computationally expensive topics [27]. Some examples include the prediction of protein structure used in pharmaceutical studies [28], improvements to material composition and synthesis [29], or event reconstruction at the Large Hadron Collider [30]. There is also ongoing research into using neural networks to discover closed form expressions from raw data [31] or optimizing machine learning algorithms to take advantage of physical symmetries of the underlying problem [32][33][34].
More relevant to this work, machine learning algorithms have also started to be explored as alternative algorithms for many GW data-analysis tasks. These include detector glitch classification [35][36][37], parameter estimation [38][39][40][41][42], continuous GW detection [43][44][45][46][47][48][49], enhancements for existing pipelines [50][51][52][53][54][55][56][57], surrogate waveform models [58][59][60], as well as various signal detection algorithms [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80]. For a summary of many methods we refer the reader to [81,82]. In this work we focus solely on detection algorithms for BBH GW signals, which have been the most commonly observed type of sources to date [10][11][12]. These signals are the easiest to detect for machine learning algorithms due to their short duration.
Many of the works considering the usage of machine learning for GW signal detection are difficult to crosscompare. Most algorithms target different datasets and derived metrics are often motivated more by machine learning practices than by state-of-the-art GW searches. It is, therefore, hard to pinpoint exactly how capable machine learning search algorithms currently are and where the main difficulties arise. To achieve the goal of an objective characterization of machine learning GW search capabilities, a common ground for comparison is required.
Here we present the results of the first Machine Learning Gravitational-Wave Search Mock Data Challenge (MLGWSC-1). In an attempt to provide a common ground of comparison for different algorithms and in preparation of O4, we have calculated sensitive distances from 6 different submissions calculated on datasets of one month duration to collect and compare a suite of searches. We want to motivate the utilization of machine learning based searches in a production setting by providing a definitive resource to allow for easy comparison between different algorithms, be it machine learning based, matched filtering based, or completely unmodeled. This challenge is the first of its kind 1 and hopefully more will be held in the future, expanding to more difficult scenarios.
The mock data used in this challenge consists of 4 datasets containing noise of increasing realism and signals with increasing complexity for the two detectors LIGO Hanford and LIGO Livingston [2]. The final dataset challenges participants to identify GWs from spinning BBHs with a duration of up to 20 s added to real detector noise from O3a. The signals also take precession effects and higher order modes into account.
Submissions are evaluated on mock data of one month duration for each of the four datasets. We calculate sensitive distances for each algorithm and estimate the computational efficiency based on the runtime. The final dataset should provide an accurate picture of the possible real-world performance these algorithms can achieve. However, we note that direct comparison of the runtime performance of the different algorithms is complicated by differing hardware usage and optimization.
We find that machine learning algorithms are already competitive with state-of-the-art searches on simulated data containing injections drawn from the limited parameter space covered by this challenge. The most sensitive machine learning algorithm manages to retain ≥ 93% of the sensitive distance measured for the Py-CBC pipeline [14] on Gaussian background data down to a false-alarm rate (FAR) of 1 per month. For higher FARs the separation between the approaches generally shrinks.
Most machine learning searches, as tested here, are less sensitive on real noise than on simulated data. The traditional algorithms handle this transition better. As a consequence, the most sensitive machine learning algorithm retains ≥ 70% of the sensitive distance of the PyCBC search down to a FAR of 1 per month. However, the sensitivity achieved of machine learning algorithms on real data is still substantial and shows that they are capable of rejecting non-Gaussian noise artifacts without any hand-tuned glitch classification.
From the evaluation of the different datasets we conclude that the main difficulties for current machine learning algorithms are the ability to analyze the consistency of detected signals between detectors and the maximum duration of signals that can be detected. Solving these issues would allow for better performance at FARs < 1 per month and enable a fast detection of potentially electromagnetic bright sources such as BNS or NSBH mergers.
All code used in this challenge is open source and available at [84]. Therein we also collect the individual submissions by groups that have given their consent, provide the analysis results, and make available all plots used in this paper for all submissions.
This paper is structured as follows. In section II we provide the details on the challenge, the datasets, as well as the evaluation process. All submissions are briefly introduced in section III. The results of the challenge and a brief discussion can be found in section V. We conclude and give an outlook into possible future work in section VI.
II. METHODS
All submissions described in section III are evaluated on the same datasets, and all machine learning submissions are evaluated under the same conditions. Below we describe the provided material from the challenge, the requirements for the submitted algorithms, as well as the evaluation process.
A. Challenge Resources
In this challenge participants are asked to identify GW signals submerged in detector noise. To provide grounds of comparison, all submissions are evaluated on the same datasets. To allow for optimization of the submitted algorithms for the task at hand, participants had access to code that allowed them to generate arbitrary amounts of data equivalent to that used during the final evaluation of this challenge. All code used for data generation and algorithm evaluation is open source and can be found at [84].
In particular, participants had access to the code that was used to generate the final challenge sets, but not the specific seed that was used. The specifics of the datasets are described in section II B. They were also provided with the code that was used to generate the metrics we provide in this paper. Details on the metrics can be found in section II C.
B. Test Data
The challenge provides a script to generate semicontinuous raw test data for any of the four datasets described below. It allows the user to choose a specific seed and a total duration of the output data. The code subsequently generates up to three files; the first containing pure noise, the second containing the same noise with injected GW signals, and the third containing the parameters of the injected signals.
The files containing the pure noise and the noise with additive signals are of the same structure. They are HDF5 [85] files with two groups named "H1" and "L1" containing data from the two detectors LIGO Hanford and LIGO Livingston, respectively. Each group consists of N HDF5-datasets, each holding the detector data of a single segment, as well as information on the GPS starting time of the segment, and its sampling rate. Each segment has a minimum duration of 2 h, is sampled at 2048 Hz, and contains continuous data. The files also contain information on the meta-data used to create the file. This meta-data is removed in the final challenge sets.
We chose to split data into smaller segments of uncorrelated noise for two reasons. First, real detectors are not equally sensitive for months at a time and data quality differs to an extent where certain data cannot be used for analyses. As such, any algorithm should be able to handle gaps in the data. Second, the noise characteristic varies over time. Segmenting simulated data allows us to easily incorporate different models for the power spectrum over the duration of the data. Subsequently, the noise model can be increased in complexity for the four datasets.
Minimal pre-processing is done on the data that is handed to the submitted algorithms. We only apply a low-frequency cutoff of 15 Hz which is used to enable a reduction in file size for real-detector data that has to be downloaded. The low-frequency cutoff reduces the dynamic range of the data, which allows us to scale the data and cast it to lower numerical precision. Any other pre-processing is left to the algorithms and is factored into the performance evaluation. The scaling is inverted during data loading.
A larger index of the dataset signifies a greater complexity and realism of the dataset. Participants may choose to optimize for any of the 4 datasets but are only allowed to submit a single algorithm, which is subsequently tested with all 4 datasets. We do this to test the ability of the search to generalize to slightly varying conditions.
Many parameters of the injected signals are drawn from the same distributions irrespective of the dataset. A summary of these distributions can be found in Ta-
Parameter
Uniform distribution Coalescence phase Φ0 ∈ (0, 2π) Polarization Ψ ∈ (0, 2π) Inclination cos ι ∈ (−1, 1) Declination sin θ ∈ (−1, 1) Right ascension ϕ ∈ (−π, π) Chirp-Distance d 2 c ∈ 130 2 , 350 2 Mpc 2 ble I. All signals are generated using the waveform model IMRPhenomXPHM [86] with a lower frequency cutoff of 20 Hz. The waveform model was chosen for its ability to simulate both precession and higher-order modes. The merger times of two subsequent signals are seperated by a random time between 24 s to 30 s to avoid any overlap. We apply a taper to the start of each waveform. In Figure 1 we show an overview of the intrinsic parameters used in this challenge and compare it to the parameter space searched by state of the art searches [13,14].
Dataset 1
The noise from the first dataset is purely Gaussian and simulated from the PSD model aLIGOZeroDetHighPower [87] for both detectors. This means that the PSD used to generate the data contains no sharp peaks originating from factors such as the power grid, is the same for all segments, and is known to the participants.
Injected signals are non-spinning and no higher-order modes are simulated. The component masses are uniformly drawn from 10 M to 50 M . We enforce the condition that the primary mass has to be equal or larger than the secondary mass. With this mass range, at a lower frequency cutoff of 20 Hz, and for non-spinning systems the signal duration is on the order of 1 s.
The first dataset represents a solved problem, as it has already been excessively studied in the past [61,63,74]. It is meant as a starting point where people new to the field can refer to existing literature to get off the ground initially. We expected many of the algorithms to perform equally well on this set.
The final challenge set for dataset 1 was generated with the seed 1 068 209 514 and a start time of 0.
Dataset 2
The noise for the second dataset is also purely Gaussian and simulated. However, in contrast to the first dataset the PSDs were derived from real data from O3a and as such contain power peaks at certain frequencies and are noisy. We generated a total of 20 PSDs for each detector. The PSDs used to generate the noise are randomly chosen from these lists and as such are unknown to the search algorithm. The lists themselves are known to the participants. The PSDs in both detectors are independent of each other but do not change over time.
Signals are now allowed to have a spin aligned with the orbital angular momentum with a magnitude between −0.99 and 0.99. Additionally, the mass range is adjusted to draw component masses from the range 7 M to 50 M . This change increases the maximum duration of the signals at a lower frequency cutoff of 20 Hz to ≈ 20 s. No higher-order modes are simulated for this dataset and due to the aligned spin requirement no precession effects are present in the waveform.
The second dataset was intended to pose a considerable increase in difficulty to the first dataset. Using an unknown PSD which was derived from real data requires participants to estimate it during the analysis, if the algorithm requires it. However, we expected that increasing the signal duration to up to 20 s would be the more prominent reason for an increase in difficulty as many previous machine learning algorithms have had trouble when dealing with large inputs [39,46,68,88]. Finally, we did not expect a large increase in the difficulty of the dataset due to the inclusion of aligned spins.
The final challenge set for dataset 2 was generated with the seed 2 743 406 703 and a start time of 0.
Dataset 3
The noise for the third dataset is also simulated and purely Gaussian. The increase in difficulty of the noise comes from varying the PSDs over time. Instead of choosing a single random PSD from the list of 20 PSDs per detector described in section II B 2 and generating all noise with that one PSD, the PSD for dataset 3 is randomly chosen for each segment.
The mass range from 7 M to 50 M and subsequently the maximum signal duration of 20 s is unchanged compared to section II B 2. However, instead of requiring the spins to be aligned with the orbital angular momentum, their orientation is isotropically distributed with a magnitude between 0 and 0.99. As a consequence, precession effects are now present in the waveforms. Additionally, we also model all higher-order (l, m)-modes available in IMRPhenomXPHM, which are:
The main challenge of this dataset was intended to be the inclusion of precession effects. While these are not as impactful for short duration, high mass systems, they can substantially alter the signal morphology for lower mass systems. Adding higher-order modes can also substantially increase signal complexity. Both of these effects are currently not modeled in any production search relying on accurate signal models, as their inclusion requires an increase in size of the filter bank to include millions of templates [24,25]. As such, we expected many if not all of the submitted algorithms to struggle with this dataset. On the other hand, any machine learning based algorithm that operates successfully on this dataset may motivate the utilization of machine learning in production searches in the future by extending the searchable parameter space. The final challenge set for dataset 3 was generated with the seed 470 182 217 and a start time of 0.
Dataset 4
Dataset 4 is the only dataset that contains real detector noise obtained from the Gravitational Wave Open Science Center (GWOSC) [89]. All noise was sampled from parts of O3a that had the "data" quality flag and none of the flags "CBC CAT1", "CBC CAT2", "CBC HW INJ", or "BURST HW INJ" were active. We consider only segments where the data from both LIGO Hanford and LIGO Livingston clear the above conditions and excluded 10 s around any detection listed in GWTC-2 [10]. Afterwards we discarded any segments shorter in duration than 2 h. To allow for different noise realizations, we shift the data from LIGO Livingston by a random time from 0 s to 240 s while keeping the data from LIGO Hanford fixed. The time shifts are independent for each segment and to avoid any possible overlap between neighbouring segments, we consider each segment on its own.
To reduce the amount of data that has to be downloaded by participants we pre-selected the suitable parts of the O3a data. We then applied a low frequency cutoff of 15 Hz and scaled the data by a factor of ≈ 2 69 . Finally, the data was converted to single precision and stored in a compressed format. This allowed us to provide a download link to a single file of 94 GB size containing enough data to generate up to 7 024 699 s≈81 d of coincident real noise for both detectors. The data was scaled by the constant factor to avoid the loss of dynamic range due to the conversion from double precision to single precision. When generating test data, the data is converted back to double precision and the scaling is inverted. The code used to downsample the data is also open source and available at [84].
The signals are generated equivalently to the signals in dataset 3, i.e. masses are uniformly drawn from 7 M to 50 M , spins are isotropically distributed with a magnitude from 0 to 0.99, and all higher-order modes available in IMRPhenomXPHM are generated. Consequently, precession effects are simulated.
This dataset is intended to be indicative of a real-world application of the search in parameter regions which are currently sparsely searched. Given that many machine learning searches have proven to generalize well from Gaussian noise to real detector noise at higher FARs in the past [62,64,66,67] we expected that machine learning algorithms that do well on dataset 3 will also be competitive for dataset 4. However, it was expected that handling short glitches may prove difficult for certain searches, especially those focusing most on the merger and ringdown.
The final challenge set for dataset 4 was generated with the seed 2 514 409 456 and a start time of 0.
C. Evaluation
All submissions are evaluated on the challenge sets, which are generated with a seed unknown to the participants at the time of submission. The evaluation is run on the Atlas computing cluster at the Albert-Einstein-Institut (AEI), Hannover. Groups that submitted an algorithm had no direct access to the evaluation stage 2 and final results presented in this work were only communicated back to the groups after the submission deadline had passed.
We compute two metrics for every submission and dataset. These are the wall-clock time required by the algorithm at hand to analyze one month of data as well as the sensitive distance of the search as a function of the false-alarm rate. In essence, the sensitivity as a function of the false-alarm rate is a receiver operating characteristic (ROC) curve that factors in the varying signal strengths of the injected GWs. It is a common measure of search sensitivity for production GW-searches [90] and thus allows for easy comparisons. We do not compute the ROC curve directly, for two reasons. First, it requires the number of a negative samples in the data. Since our data is continuous and the evaluation is left to the groups, defining a negative sample is not possible. Second, the ROC curve can be changed by choosing a different signal population. For instance, the ROC curve can be driven to zero by choosing a population of signals that are excessively far from the detectors. The sensitive distance normalizes the data by the injected population.
For the calculation of the sensitive distances we use two challenge sets for each of the 4 datasets. The first contains pure noise and we will call it the background set from here on out. The second contains the same noise as the background set but adds GW signals into it. This second set will be called the foreground set from here on out. As described in section II D any search algorithm is expected to process these files and return lists of events, where an event is a combination of a GPS time, a ranking statistic-like quantity, and a value for the timing accuracy. We will call these events background or foreground events when they have been derived from the background or foreground set, respectively. For the remainder of this section we will refer to the ranking statistic-like quantity simply as ranking statistic, to simplify our statements.
To calculate the sensitivity as a function of the falsealarm rate, we need to determine the false-alarm rate as a function of the ranking statistic. Next we can also determine the sensitivity as a function of the ranking statistic. Finally, we can combine the two, by evaluating both at the same values of the ranking statistic.
We use the ranking statistic of all background events as points where both the FAR as well as the sensitivity is evaluated. Each of these is certain to be a false positive and thus ensures that the FAR is unique at each threshold, as long as the search does not return identical ranking statistics for multiple background events.
To calculate the FAR at a given ranking statistic we count the number of background events with a ranking statistic greater than this threshold. We, subsequently, turn that into a rate by dividing the number of falsepositives by the duration of the background data, i.e. 2 592 000 s. With N FP,R the number of false-positives at a given ranking statistic R and T the time spanned by the background set, the FAR F can be calculated by
F = N FP,R T .(1)
The sensitive volume of a search at FAR F can be calculated by [90]
V (F) = dxdΛ (F; x, Λ) φ (x, Λ) ,(2)
where x are the spatial coordinates of the injection, Λ are the injection parameters, (F; x, Λ) is the efficiency of the search at FAR F, and φ (x, Λ) is the distribution of the injection parameters x and Λ.
When injections are performed uniformly in volume up to a maximum distance d max , Equation 2 can be approximated by [90]
V (F) ≈ V (d max ) N I,F N I ,(3)
where V (d max ) is the volume of a sphere with radius d max , N I,F is the number of found injections at a FAR of F, and N I is the total number of injections performed. An injection is found if there is at least one foreground event that is within ±∆t of the injection, where ∆t is the time variance assigned to the event by the search algorithm. The number of found injections at a given FAR considers only those foreground events where the ranking statistic assigned to the specified event is greater than the ranking statistic corresponding to the FAR. In machine learning terms Equation 3 is the recall at a given threshold on the network output multiplied by the volume of a sphere with radius d max , assuming that each injection corresponds to exactly one true positive. However, the injections in the datasets are not performed uniformly in volume, as we sample over the chirpdistance instead of the luminosity distance. The chirpdistance is given by [91]
d c = d M c,0 M c 5/6 ,(4)
where d is the luminosity distance,
M c = (m 1 m 2 ) 3/5 /(m 1 + m 2 ) 1/5
is the chirp-mass, and M c,0 =1.4/2 1/5 M is a fiducial chirp-mass used as a basis for calculation. Note that in contrast to [91] we use the luminosity distance instead of the effective distance as our basis.
When sampling the injections from the distributions defined in Table I using the chirp-distance, effectively the maximum luminosity distance d is selected based on the chirp-mass; the smaller the chirp mass, the smaller the maximum luminosity distance at which injections are placed. This allows us to increase the number of detectable low mass systems and, subsequently, make statistically meaningful statements about the sensitivity for these systems without requiring a large increase in the amount of data that needs to be analyzed. However, when considering a fixed chirp mass, injections are still placed uniformly within that sphere of the adjusted maximum luminosity distance. In Equation 3 we assumed that each injection was placed uniformly within the volume spanned by the sphere with volume V (d max ). To adjust it for sampling over luminosity distance we have to factor in that the probed distance depends on the selected chirp mass. We, therefore, find
V (F) ≈ V (d max ) N I N I,F i=1 V d c,max Mc,i Mc,0 5/6 V d c,max Mc,max Mc,0 5/6 ,(5)
where M c,i is the chirp mass of the i-th found injection, d c,max is the upper limit on the injected chirp distances, and M c,max is the upper limit on the injected chirp masses. This expression can be simplified to yield
V (F) ≈ V (d max ) N I N I,F i=1 M c,i M c,max 5/2 ,(6)
which is the formula we use to estimate the sensitive volume of a search algorithm. Instead of quoting the volume directly we convert it to the radius of a sphere with the corresponding volume and quote that instead. We also measure the time the algorithm requires to evaluate an entire month of test data. Since all machine learning search algorithms are running on the same hardware these values can be used to compare the speed of the different analyses on the given hardware. For a summary of the available hardware resources please refer to Table II. However, we expect the computational time to be dominated by pre-processing steps, which can in theory be heavily optimized. For this challenge, though, we did not expect many submissions to invest resources into optimizing their pre-processing and thus advise the reader to not overemphasize the provided numbers.
All runtimes are measured twice; once for the foreground set and once for the background set. In both cases the wall-time that has passed between calling the executable and it returning is measured.
D. Submission Requirements
All submissions are provided with the path to a single file containing the input data they have to process. In particular they have to be able to read HDF5 files, the structure of which is detailed in section II B. Importantly, no pre-processing other than the introduction of a low frequency cutoff of 15 Hz has been applied to the data. All other pre-processing has to be performed by the algorithms themselves. In addition to the path to the input data, each algorithm is provided with a second path at which it is expected to store a single HDF5 file. This file has to contain three one-dimensional datasets of equal size named "time", "stat", and "var".
The "time" dataset is expected to contain the GPS times at which the algorithm predicts a GW signal to be present. These are compared to the injection times to determine which injections were found, which were missed, and how many false positives the analysis produced.
The "stat" dataset is expected to contain a rankingstatistic like quantity for every GPS time in the "time" dataset. Here, ranking-statistic like quantity means a value where larger numbers indicate a higher degree of believe for the search to have found a GW signal. Having a ranking-statistic like quantity associated to all candidate detections enables us to assign a statistical significance to any event.
The "var" dataset is expected to contain the estimated timing accuracy of the search algorithm for all GPS times in the "time" dataset. This value determines the window around the GPS time returned by the search within which an injection has had to be made in order to consider the detection a true positive and the injection to be found. This value may be constant for all times at which the search expects to have seen a signal. We allowed searches to specify this value themselves, as we felt it to be unsuitable for a signal detection challenge to require a fixed timing accuracy. In principle, this freedom can be abused by choosing an accessively high value of ∆t and claiming all events as true positives. However, all groups have chosen values on similar scales and more importantly far shorter than the average separation of two injections.
Throughout the paper, we will refer to events returned by the search. By that we mean a single tuple (t, R, ∆t) contained in the "time", "stat", and "var" datasets, re- spectively.
To be able to execute all algorithms without major problems, we ask participants to either provide a single executable that can be run on the Linux command-line utilizing only the provided software stack or to provide a singularity image that we can execute. In both cases the algorithms have to accept two positional command line arguments; the path to the input data file and the path at which the output file should be stored. The main Python packages available to submitted executables are listed in Table III, for a full list refer to [84].
Each algorithm is executed by hand and closely monitored by the organization team of the challenge. Participants are not allowed to directly tune or influence the final evaluation.
To ensure that participants have submitted the correct version of their algorithm and to make sure that their algorithm behaves as expected on the evaluation hardware and software, all algorithms are first evaluated on a validation set which is generated equivalently to the final test set. The results on this validation set are then communicated back to the submitting group. Once the group has approved that their algorithm performs within the expected margin of error, the algorithm is applied to the real challenge sets. These challenge sets are the same for all participants and were kept secret until the deadline for final submissions had passed.
Since multiple members of the organization team have submitted algorithms to this challenge, the challenge datasets were only generated after the submission deadline had passed. The script to generate test data provides an option to use a random seed. This option was used to generate the final challenge datasets and ensures that no submission had knowledge of the challenge set prior to the submission deadline.
We allowed all participants to retract their submissions at any point prior to the final publication of our results. This means that participants were allowed to retract their submissions even after they were informed about the performance of their algorithm on the final challenge sets and after they have seen the performance of other entries. No group made use of this freedom and retracted their submission after results were internally published.
III. SUBMISSIONS
In this section we briefly introduce the different algorithms. For more details on the individual submissions we refer the reader to the original works cited within each subsection. The subsections are titled by the group name and are given in order of registration to the challenge.
All algorithm preparation was performed by the individual groups using their own available hardware resources. This crucially includes training of machine learning algorithms, for which no resources were provided by the organizers of this challenge. There were no strict requirements to submit algorithms that are based on machine learning techniques. We even encouraged the submission of a few traditional algorithms to quote a point of reference. However, the available resources detailed in section II C for evaluation of the test sets are tailored to suit the needs of machine learning algorithms.
A. MFCNN 3 The submission of the MFCNN group is based on the works from He et al. [92]. The authors of [92] refer to the model as matched-filtering convolutional neural network (MFCNN). MFCNN is a semi-coherent search model. The basic idea of the model is to use waveform templates as learnable weights in neural network layers. Analogously to the standard coincident matched-filtering searches the output of each matchedfiltering layer is maximized and normalized in the unit of matched-filtering SNRs for each GW detector. However, triggers are not generated on a single detector. The remaining part of the neural network is a usual convolutional neural network that is employed afterwards to jointly analyze the output from all detectors. Finally, a SoftMax function is applied to evaluate the confidence score of a GW signal being present in the GW detector network. The architecture was designed to take the advantages of both matched-filtering and convolutional neural networks and combine them to search for real GW events in GWTC-1 [5]. To adapt to this challenge, the source code [93] of the submission was translated from the MXNet framework [94] used in the original work to a PyTorch [95] implementation.
The training data for the model is generated by the code that generates dataset 4. The training data are input into the model directly with none of the usual preprocessing such as band-pass or whitening, which is consistent with the original work [92]. In fact, the model is equipped with a whitening layer to estimate the power spectrum for each input data. The main modification used in this challenge is to randomly sample 25 templates in the first matched-filtering layer from the same parameter space used in dataset 4 of this challenge. It performs significantly better than the original gridded and fixed template configuration. The subsequent convolution network of the model is constructed using the current excellent lightweight models MobileNetV3 [96] which give state-of-the-art results in major computer vision problems. The submission uses curriculum learning, during which the model is trained with decreasing multiples of signal amplitude. The multiplicative factor is lowered from 50 to 1 until convergence. Multiple models were randomly initialized and trained on a NVIDIA Tesla V100 GPU, from which the best was chosen for this submission.
To search for triggers and evaluate the performance of the model, a sliding window approach is implemented. The evaluation data is divided into overlapping segments corresponding to the input size of the model. Subsequently, all segments are passed through the model resulting in a sequence of predictions and a table of SNR peaks from the 25 sorted matched filters. The step size is 1 second and a threshold of 0.5 is set on the network output as in [92]. The "time"-, "var"-and "stat"-dataset of the output file described in section II D are derived from the table of SNR peaks associated with directly filtering the templates with the data. The GPS time and time variance of each trigger are designated as the median value and the interquartile range of SNR peaks from the nearby segments, respectively. We count the coincident SNR peaks between two detectors to quantify the ranking-statistic. Other experiments are still in progress and are supposed to be published alongside further details in a standalone paper.
The final version of the algorithm submitted by the MFCNN group was provided after the submission deadline had past. A vital flaw in their original contribution was discovered and was allowed to be fixed.
B. PyCBC 4 The PyCBC submission is based on a standard configuration of the PyCBC-based archival search for compact-binary mergers [14]. The search infrastructure was used, in addition to cWB, for the first detection of gravitational waves, GW159014 [1], in production analyses by multiple groups to produce gravitational-wave catalogs [13,14] and targeted analyses [97]. A similar low-latency PyCBC-Live analysis is also based around the same toolkit [18,98]. The analysis uses matched filtering to identify candidate observations in combination with a bank of predetermined waveform templates that correspond to the expected gravitational-wave signals [20]. Matched filtering is known to be the optimal linear filter for stationary, Gaussian noise. To account for the potential non-Gaussian noise transients [99][100][101], each candidate and the surrounding data are checked for consistency with the expected signal [102,103]. In addition, the properties of candidates, such as their time of arrival, amplitude, and phases in each detector are checked for consistency with an astrophysical population [104].
The empirically measured noise distribution and the consistency with the expected gravitational-wave signal are combined to calculate a ranking statistic for each potential candidate [104,105]; this ranking statistic is used as the "stat" value of dataset output, along with its associate trigger time in "time". The "var" dataset is set to a constant of 0.25 s. Two template banks are used for the submitted results. For dataset 1, a template bank of non-spinning waveform templates, using the IMRPhenomD [106] model, is created using stochastic placement. Datasets 2, 3, and 4 were evaluated with a common template bank that includes templates that account for spin which is aligned with the orbital angular momentum. Furthermore, only the dominant mode of the gravitational-wave signal was used and effects such as precession were not accounted for. In both cases, the mass boundaries of the template bank conform to the challenge set parameters.
The final version of the algorithm submitted by the PyCBC group was provided after the submission deadline had past. A vital flaw in their original contribution was discovered and was allowed to be fixed. Furthermore, the PyCBC submission strictly speaking uses a different algorithm for dataset 1 than for all other datasets, as the template banks are not the same. The change in template banks was accepted, as this work does not focus on a runtime analysis.
C. CNN-Coinc
5 This submission is based on the works from Gabbard et al. [63] and Schäfer et al. [74]. It utilizes the network architecture presented in [63] with a prepended batchnormalization layer [107]. As such the network processes 8 192 input samples, which corresponds to 4 s at a sampling rate of 2 kHz. The network is trained only once and applied to the data from both detectors individually. Afterwards the outputs are correlated to find coincident events as detailed in [74]. The source code for training the network and applying it to test data of the format used in this challenge is open source and can be found at [108]. The algorithm was designed to enable an easy and efficient estimation of the search background by applying time shifts between the individual detectors data.
While this feature cannot be utilized in this challenge, the original paper [74] highlights the advantages of this approach.
The network is trained on parts of the real O3a noise from the Hanford detector as provided in this challenge. Signals are generated using the waveform approximant IMRPhenomXPHM [86] from the same parameter distribution used in datasets 3 and 4 in this challenge. Merger times of the signals are varied between 2.9 s to 3.1 s from the start of the input window of the network. The signals are pre-whitened by one of the provided Hanford PSDs used in datasets 2 and 3. Noise samples are nonoverlapping parts taken from the real noise data provided by this challenge, where each segment is whitenened by an estimate of the PSD on that segment. The network was trained for 100 epochs using the loss and optimizer settings provided in [74] on a single NVIDIA RTX 2070. The epoch with the greatest binary accuracy on a single training run was chosen for this challenge.
During evaluation the network is applied to the challenge-data using a sliding window approach. Each data segment is whitened by an estimate of the PSD of that segment obtained by Welch's method [20,109]. All data is whitened before the network is applied for computational efficiency. Subsequently, the network is applied to the data via a sliding window with a step size of 204 samples ≈ 0.1 s. Afterwards a threshold of 3.4 is applied on the unbounded Softmax replacement (USR) output, which was introduced in [73]. Coincident events are calculated using the same procedure and parameters as outlined in [74]. The "time"-and "stat"-dataset of the output file described in section II D list the coincident event times and ranking statistic values, respectively. The time variance of the "var"-dataset is set to a constant value of 0.3 s.
D. TPI FSU Jena
6 This submission closely followed the method of [73], which is itself based on [63], with several modifications to adapt to the specifics of the challenge. The core of the algorithm is a convolutional neural network that accepts a 2 × 2048 input tensor corresponding to 1 second of data from 2 detectors sampled at 2048 Hz. Its architecture is derived from that of [73] and deviates from the original network by a larger size of the individual layers and a doubled number of convolutional layers. These modifications are the result of a hyperparameter variation experiment which found these settings to be optimal. A standalone publication on this submission giving further details on the methodology is in preparation. The final layer of the network is a Softmax layer over two inputs 6 The corresponding authors for the TPI FSU Jena submission are Ondřej Zelenka, Bernd Brügmann, and Frank Ohme.
which is used for training and removed using the USR [73] during evaluation. The network is trained on a dataset constructed by whitening a randomly chosen part of the real noise file and slicing it to produce 1-second noise samples and injecting whitened IMRPhenomXPHM-generated BBH waveforms into half the noise samples at SNRs uniformly drawn between 7 and 20. The waveform parameters are drawn from the same distributions as are used in dataset 4 of this challenge. The training dataset consists of 10 6 samples and the validation set of 2 · 10 5 samples.
During evaluation, each segment in the input file is whitened separately using the estimated PSD and sliced into 1-second segments at 0.1-second spacing. These are fed to the network with the USR applied. First-level triggers are selected by applying a threshold of -8, which are then clustered into events. For each event, the "time" and "stat" in the output file are the values of the highest ranking statistic first-level trigger of each cluster, and "var" is set to 0.2 seconds. The algorithm is implemented using the PyTorch framework [95] and spawns child processes to whiten individual segments. The network evaluation is performed by the parent process.
E. Virgo-AUTh
7 This submission is based on a simple per-dataset binary classification scheme. Interestingly, it was found that training a model only on dataset 2 or only on dataset 4 can yield impressive results on the other datasets as well. Specifically, training samples from dataset 2 can generalize well to dataset 3 and 1 and not so well on dataset 4, whereas training samples from dataset 4 can generalize well on datasets 1, 2 and 3. Thus, training samples were only generated from dataset 4. An adaptive normalization mechanism [110] was used instead of batch normalization as the first layer, to handle non-stationary timeseries. For the neural network architecture a deep, ResNet-like model [111] with a depth of 54 layers was used.
One week of training data per dataset was generated and the generated injection parameters were used to construct all corresponding waveforms. This amounted to about 600k background segments of duration 1.25 s with a stride of 2 s between, i.e. the next sample starts 0.75 s after the end of the previous one, and about 580k waveforms, of which 300k were used for the injections. For validation, one day of data was used, resulting in about 86k noise segments and 3.2k waveforms. The noise segments and waveform segments are combined online during training, in a static manner, both for the training and for the validation sets. The input samples are whitened before feeding them to the classifier. The PSD is computed online per batch of 4.25 s with a stride of 3.1 s, and each 1.25 s segment inside this duration is whitened with the same PSD. To increase speed, the Welch method for computing the PSD was implemented in PyTorch [95] and whitening is implemented as the first layer of the final detection module. Notably, this approach of computing the PSD for every 4.25 s and whitening each 1.25 s segment in a sliding window manner was found to be faster than using a precomputed PSD for every 1.25 s (about 40% faster for one month of data). After whitening, the first and last 0.125 s (0.25 s total) are removed from each sample.
The best results were obtained with a ResNet-52 type network. A Deep Adaptive Input Normalization (DAIN) layer [110] was used as the first layer after whitening, to handle distribution shifts that may be present. The final output is binary, i.e., noise plus waveforms or noise only, and the objective function used was a regularized binary cross entropy. The "var" parameter is set to 0.3 s, as the network predictions are high even when the time of coalescence is slightly outside the preset range. The "stat" parameter is set to the network confidence, i.e., a value in the [0, 1] interval corresponding to the probability that a waveform is present. Finally, 0.125 s are added to the expected time of coalescence to account for the time lost in the whitening process.
A standalone publication on the methods used in this submission is in preparation.
F. cWB
8 Coherent WaveBurst (cWB) is a waveform modelagnostic search pipeline for GW signals based on the constrained likelihood method [112][113][114]. The cWB pipeline has been used for the analysis of scientific data collected by the LIGO-Virgo detectors, targeting detection of signals from generic GW sources, including the compact binary mergers [13].
The cWB algorithm identifies the excess-power events in the time-frequency domain representation of strain data from multiple detectors [22,115]. For each event, the cWB pipeline reconstructs the GW waveforms and estimates summary statistics which describe generic properties of the events like the coherence across the detector network, signal strength, and the time-frequency structure.
Recently, a boosted decision-tree algorithm, eXtreme-Gradient Boost (XGBoost) [116], was adopted and implemented within the cWB framework to automate the signal-noise classification of the cWB events [54]. Two types of input data are used for the supervised training: signal events (from simulations) and noise events (from background estimations). For each of those, a subset of cWB summary statistics is fed to XGBoost as input features to train a signal-noise model. As in [54], the detection statistic for the machine learning-enhanced cWB algorithm is defined by:
η r = η 0 · W XGB ,(7)
where, η 0 is cWBs ranking statistic, and W XGB is the penalty factor calculated by XGBoost ranging between 0 (noise) and 1 (signal). This methodology has been recently used in the full reanalysis of publicly available strain data from Advanced LIGO's Hanford and Livingston third observational run [56]: the machine learning-enhanced cWB outperforms the standard human-tuned signal-noise classification used for detection of the compact binary coalescences in the O3 run.
For this study, we chose to use machine learningenhanced cWB; however, cWB typically rejects weak candidate triggers (i.e., with FAR 1 per year) at early production stages. Moreover, the whole workflow is optimized for a trigger production which saturates at FAR ≈ 30 to 50 per month. Therefore, we modified cWB to increase the event production rate by almost 2 orders of magnitude: the result is a cWB with sub-threshold capabilities, able to speed up computation and reduce memory allocations.
While trying to provide the most "generic" result for this study, it was decided to re-use the XGBoost model which was developed for [56]: it should be noted that the model was trained on noise and signal events sets that differ substantially from those adopted for the data sets prepared for MLGWSC-1. The noise backgrounds for dataset 3 and dataset 4 appear to be significantly quieter than O3. Also, the signals were drawn from a spin-aligned stellar-mass BBHs population model with different component mass ranges [117] and with SEOB-NRv4 waveforms [118]. The above-mentioned detection statistic, η r , is used as the "stat" value of dataset output, along with its associated trigger peak-time in "time". The "var" dataset is set to a constant of 0.25 s.
The results from the cWB group were provided after the submission deadline had passed. The group assured that no tuning to the challenge set was performed.
IV. DATA RELEASE
We provide all source code as well as the evaluation results for all submissions at [84]. The repository contains all code accessible to the participants of the challenge, which most importantly includes a script to generate data and one to produce the sensitivity statistics we provide in section V. The repository also contains code for basic visualization as part of the "contributions" folder. Adaption of these scripts were used to create the graphics in this paper. The challenge used the code of release 1.3 of the repository.
Alongside the code provided by the challenge organizers we publish the source code that was used to run the contributions for the groups PyCBC, CNN-Coinc, TPI FSU Jena, and Virgo-AUTh in the "submissions" folder of [84]. The submission code for the MFCNN group can be found at [93].
All analysis output files for all submissions created by our analysis are also publicly available and are stored in the "results" folder in [84]. For each group we make available the raw output on the foreground and the background for all 4 datasets. Additionally, all timing information is available. The exception is the cWB group, for which only results on datasets 3 and 4 are available.
The repository [84] also contains plots used in this paper for all groups, including versions we have not shown here. They can be found in the "plots" folder.
V. RESULTS AND DISCUSSION
In this section we provide the results of our evaluation process described in section II for all 6 submissions. We calculate and discuss sensitive distances, found-missed plots, and runtimes to provide a quantitative comparison between the different submissions. We specifically focus on the difference between machine learning and traditional algorithms and reason where the core differences in performance arise.
The four datasets we use in this study were chosen to answer different questions and serve different purposes. Dataset 1 was meant as an entry point to the challenge that represents a largely solved case [61,63,74]. We expected most submissions to perform very similarly on this dataset. The second dataset was intended as the first major step in difficulty. We expected its main challenge to be the longer duration of the injected signals, as many machine learning algorithms target shorter durations and struggle with large analysis segments [68,88]. Dataset 3 includes precession and higher order mode effects in the injected signals that traditional, modeled searches are not optimized for 9 [24][25][26]. We wanted to test if machine learning algorithms could get closer in performance, or even outperform, the traditional searches in these regions. The intention of dataset 4 was to provide a challenge that is representative of a realistic search on real detector data and a limited parameter space. The data contains non-Gaussian noise artifacts, that can mimic GW signals [120][121][122][123], which are strongly suppressed by sophisticated algorithms in traditional searches [17,90,122]. Most machine learning algorithms that target real noise do not make use of such 9 A full search of the entire O3 data that includes higher order modes has been performed in [119].
noise-mitigation strategies and instead rely solely on the ability of the machine learning algorithm to identify noise artifacts. This approach was reported to be effective for higher FARs in the past [62,64,66,67] and we were, therefore, expecting relatively minor difference between dataset 3 and dataset 4. Furthermore, most traditional algorithms use matched filtering, which is only proven to be optimal for signal recovery when the noise is stationary and Gaussian. Since neither of the two assumptions are true for real detector data, we were also interested to test if machine learning algorithms can perform better than these searches by learning a better noise representation.
A. Sensitivities
In this subsection we discuss the sensitive distances of the different submissions, which are a measure for how many sources can be detected at any given level of certainty, i.e. at a particular FAR. They are the core metric to determine the quality of any search. We focus on the low FAR region and truncate the plot at a FAR of 10 3 per month. We chose this cutoff for two reasons. First, to function as a standalone search, algorithms may only report events with low FARs. State of the art pipelines send out alerts only when the FAR is smaller than O (1) per month [98]. Second, for high FARs a non-negligible number of detections originate from false associations. This means that a large number of triggers that originate from random noise coincidences are close enough to an injection to be counted as true positives.
Since all machine learning submissions chose to optimize for dataset 4, results on all prior sets also test the capability of generalizing to different signal (sub)populations. Dataset 3 is a special case, as it uses the same distribution to draw the parameters of the injected signals as dataset 4. It, therefore, differs only in the noise contents and is a good test of the performance difference of different algorithms between simulated and real noise.
The results of this challenge are summarized in Figure 2 and Table IV Table IV we give the numeric values for the sensitive distances at three selected FAR values of 1, 10, and 100 per month for all submissions and datasets. We also provide information on the wall-clock time used to evaluate the different sets. Due to time constraints, we only show sensitivity curves for dataset 3 and 4 for the submission from the cWB group. We also note that PyCBC used a different template bank to analyze dataset 1 than for the remaining three datasets.
We find that the machine learning algorithms from the TPI FSU Jena group presented in section III D and the Virgo-AUTh group presented in section III E are very close in sensitivity for datasets 1, 2, and 3. The submission from the TPI FSU Jena group reaches a slightly higher sensitive distance at all FARs for all of these three datasets. However, the Virgo-AUTh submission retains ≥ 90% of the sensitive distance achieved by the TPI FSU Jena submissions for FARs ≥ 2 per month. At lower FARs the gap widens but the individual sensitivities carry large uncertainties due to low number statistics. For higher FARs this gap narrows to a separation of roughly 4% at a FAR of 1000 per month. We suspect that the difference between the two approaches is on the order that could be explained by different initializations of the training procedure.
On dataset 4 the submission from the Virgo-AUTh group manages to maintain a stable sensitivity for the full range of tested FARs. The submission from TPI FSU Jena, on the other hand, is dominated by background triggers and seemingly struggles to adjust to the non-Gaussian noise characteristics. For high FARs the sensitivity is on a similar scale as the submission from the Virgo-AUTh group and as was observed on previous datasets, backing up the hypothesis that rejecting background triggers is the main problem. This is surprising, as both algorithms were optimized on dataset 4 but performed similarly only on datasets 1 to 3. One reason for this result may be the neural network architectures used by the different groups. The Virgo-AUTh group uses a very deep ResNet that may be better suited to represent non-Gaussian noise artifacts. The architecture from the TPI FSU Jena group is a more straightforward convolutional architecture that may be limited in its ability to learn appropriate parameters.
The algorithms from the MFCNN group presented in section III A and the CNN-Coinc group presented in section III C also show similarities in sensitivity. Both are significantly less sensitive than the leading machine learning submission on all datasets. For datasets 1, 2, and 3, the MFCNN contribution achieves 32.5%, 30.8%, and 23.5% of the sensitive distances of the leading machine learning contribution, respectively. The CNN-Coinc submission reaches 42%, 25.5%, and 27% of the sensitivity of the leading machine learning contribution at the point of farthest separation. For dataset 4 the submission from the MFCNN and CNN-Coinc groups do comparatively better. They retain ≥ 68% and ≥ 50% of the sensitive distance of the leading machine learning submission down to a FAR of 10 per month, respectively. At a FAR of 1 per month the CNN-Coinc submission does not detect any signals, whereas the MFCNN still retains 60% of the sensitivity of the leading machine learning contribution.
On the first three datasets one can observe a steep gradient of the sensitivity curves at varying FARs for the MFCNN and CNN-Coinc submissions. At even higher FARs the curves level off again and return to a similar A summary of the analysis results for all submissions and all datasets. The columns labeled "Sensitivity" give the values for the sensitive distance at the three FARs 10 2 per month, 10 1 per month, and 10 0 per month rounded to the second decimal place. The values lie on the lines in Figure 2. The columns labeled "Runtime" list the time for evaluation of the foreground and background set in seconds, respectively. The runtime column labeled "average" lists the mean time obtained from evaluating the foreground and background data. Entries labeled "N/A" are not available, because they were not measured. The PyCBC times labeled with * are only approximations. The analysis did not run on the challenge hardware but made use of a compute cluster. Shown times are the result of scaling the computational costs to 16 CPU cores. The PyCBC times labeled with * * are approximations obtained in the same manner as the approximations labeled with * , but make use of a larger filter bank. The times of the cWB group marked with * * * are approximations derived from dividing the CPU core-seconds reported by the search by 16 to normalize it to the challenge hardware.
slope observed at low FARs. The sudden increase leads to the MFCNN submission being more sensitive than the modeled PyCBC search by up to 15% on dataset 3 for FARs > 200 per month. This behavior is not present in any of the other submissions and we were not able to find a clear explanation. However, we observe that both algorithms have different trigger rates on the foreground and background set. If the background is estimated from the foreground data only, the sensitivity of both algorithms drops sharply. All other algorithms are robust to this change. We show these sensitivity curves in Figure 3 of the appendix.
For all datasets we compare the leading machine learning submission to the submission from PyCBC presented in section III B. We also compare it to the submission from cWB presented in section III F for datasets 3 and 4. These two are traditional, state-of-the-art search algorithms that have already been used successfully in past observation runs [1,13,124].
For dataset 1 we find that the machine learning search is able to achieve between 94% and 99% of the sensitivity obtained with PyCBC. These results are remark-ably close and improve significantly on the findings from [74], which targeted a very similar dataset. However, the gap between the machine learning detection algorithm and the PyCBC search widens for lower FARs. Therefore, we expect that the PyCBC contribution will be able to attribute a substantially higher significance to many events. This is amplified by the ability of PyCBC to trivially increase the amount of data that can be used for background estimation by introducing time-slides between detectors [74,90].
For dataset 2 the leading machine learning contribution gets even closer to the traditional algorithm from the PyCBC group. At low FARs ≤ 20 per month it retains ≥ 93.5% of the sensitivity achieved by the PyCBC submission. For high FARs ≥ 200 per month it even manages to outperform the PyCBC submission and is up to 1.5% more sensitive.
From dataset 2 to dataset 3 all submissions experience a slight increase of the measured sensitive distance. This may be surprising at first but can be explained by the distribution of the effective spin. For dataset 3 the spin orientations are distributed isotropically, which causes the average effective spin to be smaller than in dataset 2. This leads to few systems with large effective spin. The PyCBC search gains up to 3% in sensitivity at low FARs, although it loses about 1% in sensitivity at high FARs. A similar change can be observed in the submission from TPI FSU Jena. Since both the leading machine learning contribution and the PyCBC search gain similar amounts of sensitivity from dataset 2 to dataset 3 the comparison between the two does not change substantially. The submission from the TPI FSU Jena group is now up to 2.5% more sensitive at high FARs and still about 6% less sensitive at low FARs. The Virgo-AUTh, the MFCNN, and the CNN-Coinc submissions increase their sensitive distance by a larger fraction, suggesting that they benefit more from the signal population being closer to the distribution of signals in their training set. Dataset 3 is also the first dataset for which results from the cWB search are available. We find that cWB retains ≥ 80% of the sensitive distance obtained by PyCBC over all tested FARs. Subsequently the leading machine learning submission achieves a sensitive distance greater by 15% to 23% over the range of tested FARs.
For dataset 4 the leading machine learning contribution now comes from the Virgo-AUTh group. Compared to PyCBC their algorithm retains ≥ 87% of the sensitivity down to a FAR of 10 per month. For smaller FARs the sensitivity gap widens quickly. At a FAR of 1 per month the machine learning search achieves 70% of the sensitivity of PyCBC. The cWB submission evolves similarly to PyCBC and retains ≥ 79% of the sensitive distance. At high FARs the leading machine learning search manages a sensitive distance up to 27% larger than that of cWB. For low FARs the sensitive distance falls off quicker than that of cWB. At a FAR of 1 per month the cWB search is 12.5% more sensitive than the Virgo-AUTh submission. For lower FARs we expect this difference to be-come larger, as the production level search algorithms are tuned for lower FARs than tested in this work. In comparison to the sensitivity difference on dataset 3 the machine learning submission from Virgo-AUTh does not retain as much sensitivity on real noise as the PyCBC or cWB submissions.
The results on dataset 1 demonstrate that machine learning detection algorithms are already capable of rivaling traditional search algorithms for simulated data at FARs ≥ 1 per month. A previous study [74] had identified the capability of machine learning searches to build an internal representation of the signal morphology as the main problem to achieve comparable sensitivities to traditional algorithms. Such a signal representation would allow the algorithms to compare detections in multiple detectors and require them to be consistent. The two leading machine learning algorithms in this challenge seem to have overcome this limitation, at least for high FAR detections.
For dataset 2 we expected machine learning searches to decline in sensitivity more strongly than traditional searches. This expectation was provoked by the short duration of data that is processed by most machine learning searches at each step. As the signals injected into dataset 2 are of longer duration than those used in dataset 1, the machine learning algorithms inherently lose some amount of sensitivity due to considering only small parts of the signal. We estimate this loss to account for at most a 1% difference in sensitivity. However, we observe the opposite effect for the two leading machine learning algorithms, which get even closer in sensitivity to the PyCBC submission compared to dataset 1. This may be caused by the distribution of signals in the training data used for the machine learning algorithms. Since both algorithms optimized for dataset 4, most signals in the training data will have non-zero spin. Therefore, the challenge set for dataset 2 is closer in nature to the training data, which may have introduced a bias that leads to higher sensitivities for spinning systems or in other words a slightly reduced sensitivity to non-spinning systems.
Dataset 3 was intended to test if machine learning searches are capable of outperforming traditional algorithms for precessing systems and signals carrying higher order mode information. We do not find substantial evidence in support of this hypothesis from the sensitivity curves. However, the challenge set 3 contains only very few signals with strong evidence for precession and higher order modes, as most signals are still relatively short. The impact on the overall sensitivity from these signals is, therefore, minor. Surprisingly, the leading machine learning search is still on par with PyCBC and manages to be significantly more sensitive even at the lowest tested FARs than the unmodeled cWB search. It must be noted that the cWB submission was not optimized for the parameter space used in this challenge. We, thus, expect this gap to narrow if more effort were to be used to tune the cWB pipeline.
The change in the relative difference in sensitivity be-tween the PyCBC submission and the leading machine learning contribution, as well as the change in difference to the cWB submission, from dataset 3 to dataset 4 suggests that many machine learning algorithms currently used by the community are not yet capable of treating real noise as well as sophisticated traditional algorithms. We suspect that one major factor may be non-Gaussian noise artifacts that are misclassified as signals by machine learning algorithms, while the traditional searches excise them from the data or reject them on other bases. Another reason may be the non-stationary character of the noise that may lead to different sensitivities at different times. However, this would have also been a factor in dataset 3, where the PSDs used to simulate the noise change over the duration of the challenge set. However, since the leading machine learning search does retain sensitivity at all FARs it must have learned to reject most non-Gaussian noise artifacts, which is in line with expectations from studies carried out at higher FARs [62,64,66,69].
B. Found and missed injections
We generate found-missed plots for all submissions and show a few selected ones. The ones not included in this paper can be found in the associated data release [84]. These plots highlight specific areas in parameter space where the machine learning searches are already competitive and those where more work is required. Specifically, we provide plots for chirp-mass M c versus decisive effective chirp-distance D c,eff , τ 0 versus mass-ratio q, and the effective precession spin χ p [125] versus inclination with respect to the line of sight θ jn . To first order τ 0 is the time to merger from the lower frequency cutoff of the waveform [126,127]. The decisive effective chirpdistance is a measure for how strong the signal can be observed in the detector that has the worse sensitivity due to source location and orientation. The effective chirpdistance is the chirp-distance at which a source with the same intrinsic parameters and sky location but an optimal orientation would have been observed from at the same amplitude as the injected signal. The decisive effective chirp-distance is then the larger of the two effective chirp-distances from the two detectors. Therefore, the M c /D c,eff plot informs about the ability to detect signals as a function of the SNR in the detector that is less sensitive to the signal. We also include information on the ranking statistic like quantity returned for each detected event, to highlight how strongly it is correlated to the SNR. The τ 0 versus q plot highlights how well long and short duration signals are recovered. It also gives information on the mass ratio, which is an important parameter for the strength of precession effects. The main plot used to determine the impact precession effects and higher order modes have on the detectability of signals is the χ p versus θ jn plot.
In Figure 4 we show the found injections from dataset In Figure 5 we compare the found injections from dataset 2 in the τ 0 -q-plane for the PyCBC and TPI FSU Jena submissions. The plots show that the two searches are competitive in the comparable mass region and identify similar signals. The main difference between the two searches can be observed in the τ 0 distribution of found signals. Most of the signals with large values for τ 0 , i.e. long duration signals, are missed by the TPI FSU Jena submission. These crucially include many signals that the PyCBC submission identifies with relatively high confidence. Therefore, the short duration of the input windows used by the TPI FSU Jena submission still seem to be a limiting factor for the sensitivity. This limitation will likely be more severe if longer duration signals from sources like BNS or NSBH systems were considered.
In Figure 6 we compare the θ jn and χ p values of the injections from dataset 3 that are found by one algorithm but missed by the other. The two algorithms come from the PyCBC group and the TPI FSU Jena group. If either algorithm adapted better to signals with strong precession or higher order modes content, we would expect to see a clustering from that search in the scatter plot. However, we do not observe this clustering, which backs up our observation from the sensitivity curves that the machine learning algorithm from the TPI FSU Jena group has not learned a better representation of precessing systems or signals with higher order mode content than the modeled PyCBC search, which only includes non-precessing signals in its template bank. However, the amount of impact precession or higher order modes have on the detectability of short duration signals used in this study are small. A real test of this hypothesis would require the analysis of long duration signals.
C. Runtimes
All runtimes in this section are given in terms of wallclock times obtained on equivalent hardware, which is listed in Table II. The runtimes are largely independent of the dataset for all submissions. We, therefore, discuss them only in summary. An overview of the times can be found in Table IV. They were measured by applying each algorithm to the foreground and background of each challenge set. We report the time between the algorithm call and it returning. To avoid bottlenecks, all files were transferred to the local storage of the individual compute nodes before calling the algorithm. The output was also written to said local storage and transferred back only after the algorithm returned. It should be noted that the runtimes are heavily dependent on the amount of optimization of the algorithms. The main objective for this challenge was the sensitivity and not the runtime.
The PyCBC and cWB submissions are exceptions as their runtimes were not measured on the same hardware. Instead they were run on compute clusters making heavy use of parallelized work over multiple CPUs. The times reported here are approximations by normalizing the compute time to 16 CPU cores available in the compute nodes used for this challenge. Furthermore, for the evaluation of dataset 1 PyCBC used a different template bank than those for dataset 2 to 4 was used. This bank was substantially smaller, resulting in faster evaluation. cWB times were reported to us only on the foreground data in CPU core seconds.
We find that of the machine learning algorithms the submission from the TPI FSU Jena group is the fastest, evaluating an entire month of archival data in about 1 h. It utilizes a single GPU when evaluating the network. The second fastest algorithm is the submission from the Virgo-AUTh group. It evaluates a month of data in 1.5 h on a single GPU and is thus about 50% slower than the fastest algorithm. Notably, the two fastest algorithms are also the two most sensitive machine learning searches. The algorithm from the CNN-Coinc group requires almost 4 h on a single GPU to evaluate the same amount of data but is significantly less sensitive. However, none of these algorithms are limited by the GPU performance. The differences in execution time can be mainly attributed to the difference in optimization of the pre-processing steps. The submission from the MFCNN group on the other hand does not apply any pre-processing directly. They instead use a neural network to carry out this computation. They operate on all 8 available GPUs and manage to evaluate the month of data in ≈ 11.5 h.
For dataset 1 the PyCBC submission has a runtime comparable to that of the submission from the Virgo-AUTh group. On all other datasets it requires roughly 43 h to evaluate the month of data. The large difference in runtime between the datasets is caused by the smaller template bank that is used only for dataset 1. Contrary to the machine learning algorithms, the PyCBC submission did not utilize GPUs and ran on CPUs only. However, PyCBC is a production level search pipeline and as such has been optimized to run on CPUs. It is not limited by the pre-processing but rather by the matched fil-ter operation. It should be noted that PyCBC is still the most sensitive search presented here and gains in computational efficiency could be obtained by reducing the number of templates. This would effectively trade off search sensitivity for lower computational cost.
The PyCBC submission is implemented on the CPU as a GPU implementation is inherently more difficult to optimize. GPUs, on the other hand, are usually far more efficient from a cost to performance and energy to performance standpoint [128]. One advantage of machine learning algorithms is that they make use of well optimized libraries such as PyTorch [95] or TensorFlow [129] that utilize GPUs for their computations. This makes the implementation of search algorithms on GPUs relatively straightforward and allows researchers to focus on optimizing the sensitivity of their algorithm rather than having to spend time on optimizing the algorithmic implementation.
The runtimes in this challenge are measured under the assumption that the lowest required FAR is 1 per month. In a real search lower FARs are beneficial especially for rare signals. Therefore, most traditional searches are tuned to be most sensitive at FARs well below the level tested in this challenge. PyCBC for instance can extend its background by introducing time-slides [90], thereby potentially lowering the FARs of detected events. This process is a trivial operation that requires a fraction of the computational cost of the actual filtering stage. If machine learning algorithms are not specifically designed to allow for a similar approach, lowering the FARs of detections requires multiple complete re-evaluations of the time-shifted data. This would in turn lead to a linear increase in the computational cost, i.e. lowering the potential FAR of an event by an order of magnitude would lead to an order of magnitude increase in the computational cost.
VI. CONCLUSIONS
In this paper we have presented the results of the first Machine Learning Gravitational Wave Search Mock Data Challenge (MLGWSC-1). The study compiled curves showing the sensitive distances from 4 different machine learning submissions and compared them to 2 state-of-the-art traditional search algorithms; the modeled PyCBC [90] pipeline and the unmodeled coherent wave burst search [22,115]. We established a common dataset and means for evaluation. We hope that other researchers will continue to make use of the resources presented in this work to allow for quantitative comparisons between different machine learning approaches and to traditional filtering techniques. As research continues and machine learning search algorithms become more sensitive, we want to motivate other groups to host new challenges, focusing on other parts of parameter space or targeting different observing strategies.
The key observations of this challenge are:
1. Machine learning search algorithms are competitive in sensitivity compared to state-of-the-art searches on simulated data and the limited parameter space explored in this challenge.
2. Most of the tested machine learning algorithms struggle to effectively handle real noise, which is contaminated with non-Gaussian noise artifacts.
3. Traditional search algorithms are capable of detecting signals at lower FARs, thus making detections more confident.
4. The tested machine learning searches struggle to identify long duration signals.
Therefore, the main challenges for current machine learning searches are the operation on real noise, the confidence in detections due to comparatively high FARs, and the detection of long duration signals. The last of those three is a major hurdle to confidently detect signals from BNS and NSBH systems. Improvements in any of these fields would be beneficial. Specifically, we identify the following key research areas:
1. Improve the ability to compare signal parameters, or representations thereof, between detectors to check for consistency and reject noise artifacts.
2. Improve the ability to calculate large amounts of background, for instance by designing algorithms that can trivially evaluate time-slides of the input data.
3. Increase the duration of data that is processed by machine learning algorithms to enable the detection of long duration signals.
This challenge shows the potential of machine learning algorithms to act as GW detection pipelines. We have shown that these algorithms are competitive in a realistic scenario to state-of-the-art searches today. They operate at low computational cost and allow for a trivial implementation of the algorithms on highly efficient GPUs, rather than relying on CPUs. We believe that this work justifies more research on this topic, especially in areas where machine learning may have a tangible impact on the rapid identification of GWs.
However, we do acknowledge that the research carried out here operates on a limited parameter space. Moreover, the targeted parameter space is not the computationally expensive part of the search space of traditional searches. About 1% of the total size of the template bank used in [14] is dedicated to the area this study searches.
To have the greatest impact on real searches machine learning algorithms need to be extended to target either the low mass region, where signals are long and the computational cost of matched filtering rises rapidly, or the high mass region where signals and noise artifacts are difficult to distinguish.
We also want to mention that we did not receive a submission utilizing one of the most promising neural network architectures for GW detection of the recent past. A WaveNet based architecture, that uses dilated convolutions, has been reported to do well for this kind of task [64,67,130]. We also did not receive submission based on many other neural network architectures that have been used in the past, such as autoencoders [72,79,80,131], inception networks [46,68], or two dimensional convolutions that analyze time-frequency decompositions [69]. We hope that some of these approaches will be adopted to the requirements of this challenge and evaluated on the datasets presented here, to allow for a quantitative comparison.
Future mock data challenges could target longer duration signals, concentrating on BNS and NSBH systems. These are potentially EM-bright and would, therefore, be of particular interest. Furthermore, these signals stem from regions of parameter space where traditional searches are computationally expensive to run. For even longer signals, sub-solar mass black holes could be targeted. Existing modeled searches in those regions make use of several million templates and are computationally limited [132]. Another avenue may be very massive systems, which can be difficult to distinguish from noise artifacts. Finally, we recommend that future mock data challenges drop the notion of a foreground and background set and only provide data files containing injections. This would eliminate further sources of error and be more true to a realistic application, where no true GW-free background exists.
The cWB team gratefully acknowledges the computational resources provided by LIGO-Virgo. This material is based upon work supported by NSF's LIGO Laboratory, which is a major facility fully funded by the National Science Foundation. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center, a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. The work by S. K. was supported by NSF Grants No. PHY 1806165 and PHY 2110060.
This publication is based upon work from COST Action CA17137, supported by COST (European Cooperation in Science and Technology).
EAH is supported by Laboratory Directed Research and Development (LDRD) funding from Argonne Na-
FIG. 1 .
1An illustration of the range for the intrinsic parameters covered by this challenge. The left panel (a) shows a typical range for the component masses used by state of the art searches [14]. The color indicates the duration of the waveform from 20 Hz. The triangles show the parameter regions covered by this challenge. The right panel (b) shows the component-spin χi distribution of the different datasets in this challenge.
. The four individual panels of Figure 2 show the sensitive distances as a function of the FAR for all submissions. The panels contain the results for dataset 1 to 4 from left to right and top to bottom. In
FIG. 2 .
2The sensitive distances of all submissions and all four datasets as functions of the FAR. Submissions that made use of a machine learning algorithm at their core are shown with solid lines, others with dashed lines. The FAR was calculated on a background set that does not contain any injections.
FIG. 3 .
3tional Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357, and the U.S. National Science Foundation Awards OAC-2209892 and OAC-1931561. CM is supported by the Science and Technology Research Council (grant no. ST/V005634/1) and the European Cooperation in Science and Technology (COST) action CA17137. F.O. was supported by the Max Planck Society'The sensitive distances of all submissions and all four datasets as functions of the FAR. The sensitive distances are calculated using only the data from the foreground file. The FAR is determined from the false positives on that data. Submissions that made use of a machine learning algorithm at their core are shown with solid lines, others with dashed lines. This figure differs from Figure 2 as the algorithms from MFCNN and CNN-Coinc behave differently on the foreground and the background. FIG. 5. The injections from dataset 2 identified by the PyCBC and TPI FSU Jena submissions with a FAR ≤ 10 3 per month in the signal duration τ0 versus mass ratio q plane. The scatter plot shows injections that are found only by one of the two algorithms. Injections that are missed or found by both are only shown in the 1D marginal distributions. FIG. 6. The injections from dataset 3 identified by the PyCBC and TPI FSU Jena submissions with a FAR ≤ 10 3 per month in the inclination to spin axis θjn to χp plane. The scatter plot shows injections that are found only by one of the two algorithms. Injections that are missed or found by both are only shown in the 1D marginal distributions.
TABLE I .
IA summary of the distributions shared between all datasets from which parameters are drawn.
TABLE II. Main hardware specifications available to each search algorithm during final testing.Hardware type
Specification
CPU
2× Intel Xeon Silver 4215,
8(16)
cores(threads)
at
2.5 GHz
GPU
8× NVIDIA RTX 2070 Super
(8 GB VRAM)
RAM
192 GB
TABLE III. A selective list of the core Python packages available to algorithms during evaluation. A complete list is given at[84].Python package
Version
bilby
1.1.3
pycbc
efeaeb6
tensorflow-gpu
2.6.0
tensorflow-probability
0.14.0
torch
1.9.1+cu11
TABLE IV .
IV
we find that the chirp-mass distribution from the TPI FSU Jena submission favors chirp-masses in the region M c ∈[20,35], which is not true for the PyCBC submission. We attribute this bias to the training set, which contained signals drawn from the distributions used for dataset 3 and 4. The probability distribution of the chirpmass for these sets is shaped such that about 51% of signals are being drawn from the mass range 20 M to 35 M . A similar bias is not so evident for the other machine learning submissions but may be masked by other effects. The PyCBC submission uses a uniform prior on the chirp mass and thus avoids this bias.1 in the M c -D c,eff -plane for the PyCBC and TPI FSU
Jena submissions, respectively. Both plots clearly show
that closer injections are generally attributed a higher
confidence to be a real signal. This indicates that the
ranking statistic like quantities for both algorithms are
actually correlated with the signal strength. Similar cor-
relations can be observed for all submissions. From Fig-
ure 4
There has previously been a public Kaggle challenge[83]. First in the sense of this paper refers to our setup of providing continuous data.
This excludes submissions by the organization group. However, no member of the organization group accessed the challenge-data before the submission deadline or altered their algorithm after the submission deadline.
The corresponding authors for the MFCNN submission are He Wang, Shichao Wu, Zong-Kuan Guo, Zhoujian Cao, and Zhixiang Ren
The corresponding author for the PyCBC submission is Alexander H. Nitz.
The corresponding author for the CNN-Coinc submission is Marlin B. Schäfer.
The corresponding authors for the Virgo-AUTh submission are Paraskevi Nousi, Nikolaos Stergioulas, Panagiotis Iosif, Alexandra E. Koloniari, Anastasios Tefas, and Nikolaos Passalis.
The corresponding authors for the cWB submission are Francesco Salemi, Gabriele Vedovato, Sergey Klimenko, Tanmaya Mishra.
VII. ACKNOWLEDGEMENTSWe want to thank Narenraju Nagarajan and Pascal Müller for their valuable scientific input and contributions to the code of this challenge.We acknowledge the Max Planck Gesellschaft and the Atlas cluster computing team at Albert-Einstein Institut (AEI) Hannover for support.O.Z. thanks the Carl Zeiss Foundation for the financial support within the scope of the program line "Breakthroughs".The MFCNN team members would like to acknowledge that the submission was supported by the Peng Cheng Laboratory Cloud Brain (No. PCL2021A13).The Virgo-AUTh team members would like to acknowledge the support provided by the IT Center of the Aristotle University of Thessaloniki (AUTh) throughout the progress of this work, as results presented in this work have been produced, in part, using the AUTh High Performance Computing Infrastructure and Resources, and thank the COST network CA17137 "G2Net" for support. P.I. acknowledges support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No. 759253. This research has made use of data or software ob-tained from the Gravitational Wave Open Science Center (gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre Na-
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3 per month in the chirp-mass Mc versus decisive effective chirp-distance D c,eff plane. The blue bars in the histograms show the one dimensional marginal distributions of the found injections. The gray bars show the distribution of injected signals, including those missed by the search. The color shows the. The injections from dataset 1 identified by the PyCBC and TPI FSU Jena submissions with a FAR ≤ 10. stat" value attributed to the injection by the algorithm. The red lines in the colorbar highlight the thresholds on the "stat" to achieve different FARsFIG. 4. The injections from dataset 1 identified by the PyCBC and TPI FSU Jena submissions with a FAR ≤ 10 3 per month in the chirp-mass Mc versus decisive effective chirp-distance D c,eff plane. The blue bars in the histograms show the one dimensional marginal distributions of the found injections. The gray bars show the distribution of injected signals, including those missed by the search. The color shows the "stat" value attributed to the injection by the algorithm. The red lines in the colorbar highlight the thresholds on the "stat" to achieve different FARs.
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"Evaluation of a Search Interface for Preference-Based Ranking -Measuring User Satisfaction and System Performance"
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"Wilko Van Hoek [email protected] \nGESIS -Leibniz-Institute for the Social Science\nGESIS -Leibniz-Institute for the Social Science\nCologne, Bonn, CologneGermany, Germany, Germany\n",
"Daniel Hienert [email protected] \nGESIS -Leibniz-Institute for the Social Science\nGESIS -Leibniz-Institute for the Social Science\nCologne, Bonn, CologneGermany, Germany, Germany\n"
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"GESIS -Leibniz-Institute for the Social Science\nGESIS -Leibniz-Institute for the Social Science\nCologne, Bonn, CologneGermany, Germany, Germany",
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] | [] | Finding a product online can be a challenging task for users. Faceted search interfaces, often in combination with recommenders, can support users in finding a product that fits their preferences. However, those preferences are not always equally weighted: some might be more important to a user than others (e.g. red is the favorite color, but blue is also fine) and sometimes preferences are even contradictory (e.g. the lowest price vs. the highest performance). Often, there is even no product that meets all preferences. In those cases, faceted search interfaces reach their limits. In our research, we investigate the potential of a search interface, which allows a preference-based ranking based on weighted search and facet terms. We performed a user study with 24 participants and measured user satisfaction and system performance. The results show that with the preference-based search interface the users were given more alternatives that best meet their preferences and that they are more satisfied with the selected product than with a search interface using standard facets. Furthermore, in this work we study the relationship between user satisfaction and search precision within the whole search session and found first indications that there might be a relation between them. | 10.1145/3240167.3240170 | [
"https://export.arxiv.org/pdf/2302.06440v1.pdf"
] | 52,815,077 | 2302.06440 | 5a5fb7724308187556a264d2fa5da243f3d2e38d |
Evaluation of a Search Interface for Preference-Based Ranking -Measuring User Satisfaction and System Performance
Dagmar Kern [email protected]
GESIS -Leibniz-Institute for the Social Science
GESIS -Leibniz-Institute for the Social Science
Cologne, Bonn, CologneGermany, Germany, Germany
Wilko Van Hoek [email protected]
GESIS -Leibniz-Institute for the Social Science
GESIS -Leibniz-Institute for the Social Science
Cologne, Bonn, CologneGermany, Germany, Germany
Daniel Hienert [email protected]
GESIS -Leibniz-Institute for the Social Science
GESIS -Leibniz-Institute for the Social Science
Cologne, Bonn, CologneGermany, Germany, Germany
Evaluation of a Search Interface for Preference-Based Ranking -Measuring User Satisfaction and System Performance
Author Keywords Search interfaceInformation filteringPreference-based rankingWeighted facetsEvaluation of whole sessions ACM Classification Keywords H52 Information Interfaces and Presentation (eg HCI): User Interfaces -Evaluation/methodologyGraphical user interfaces (GUI); H33 Information Storage and Retrieval : Information Search and Retrieval -Query formulation
Finding a product online can be a challenging task for users. Faceted search interfaces, often in combination with recommenders, can support users in finding a product that fits their preferences. However, those preferences are not always equally weighted: some might be more important to a user than others (e.g. red is the favorite color, but blue is also fine) and sometimes preferences are even contradictory (e.g. the lowest price vs. the highest performance). Often, there is even no product that meets all preferences. In those cases, faceted search interfaces reach their limits. In our research, we investigate the potential of a search interface, which allows a preference-based ranking based on weighted search and facet terms. We performed a user study with 24 participants and measured user satisfaction and system performance. The results show that with the preference-based search interface the users were given more alternatives that best meet their preferences and that they are more satisfied with the selected product than with a search interface using standard facets. Furthermore, in this work we study the relationship between user satisfaction and search precision within the whole search session and found first indications that there might be a relation between them.
INTRODUCTION
The number of consumers who browse or buy products online grows further [10], and they are all facing the challenge to choose the best option out of a huge set of alternatives. Web shops and service providers (e.g. hotel booking services or apartment finders) try to support users in finding the right product. One well-established method for that is to provide search facets in the user interface. [8]. Facets allow users to filter products by predefined categories or features. In this way, users can exclude products they are not interested in and obtain a smaller and more manageable number of alternatives. An overview about faceted search in general is given by [21,23]. Studies have shown that search interfaces providing facets are considered as intuitive and easy to use, see e.g. [8]. Furthermore, they provide a high level of control and transparency [24]. However, user preferences can not always be encoded into a boolean logic. Sometimes, some features are not mandatory but nice to have and some of them are considered more important than others. But, using facets means that all selected facet terms are mandatory and equally weighted. Selecting less important facet terms may remove potentially interesting products unintentionally, while specifying only a few criteria may lead to a too large result set where interesting alternatives are not obvious [22].
To counterbalance disadvantages of facet search, recommender systems [17] are often additionally applied to suggest possible alternatives that might fit users' preferences. In current systems, the recommendations rely on user profiles and the use of automated recommendation techniques [17]. In general, common commercial recommender systems offer no or little options for users to explicitly influence the recommendations, leaving them no opportunity to express their preferences. However, in the research literature, one can find some evidence that, regarding the user satisfaction, it is beneficial to put users in control of their recommendations (e.g. [1,7]) and of the ranking process of search results in general (e.g. [14]). One possibility to give users control over recommendation and ranking is to let them weight terms according to their preferences (e.g. [11,22]). From a user perspective, these systems have been evaluated as very helpful, and they are able to increase user satisfaction (e.g. [4,7]). However, little attention has been given so far considering both sides to evaluate a search system -user feedback as well as system performance and their relationship to one another. We want to close this gap by providing new insights about a search interface that uses preference-based ranking in the form of weighted facet terms. We want to know if the user's perceived satisfaction and system support can be backed up by analyzing system performance measures. For that purpose, we tracked the changes in recall and precision over the course of whole search sessions. arXiv:2302.06440v1 [cs.IR] 13 Feb 2023
RELATED WORK
Most of the existing commercial search interfaces using facets still offer few opportunities for the user to adapt the search query explicitly to her preferences. However, in research, several attempts have been made to include user preferences in product search. For example, Stolze [20] proposed a soft navigation approach for finding products in an electronic product catalog. He distinguished between hard and soft constraints and allows weighting the importance of product features. The proposed system requires from the user to learn a rule based syntax. Therefore, it is stated to be more suitable for frequent users. However, in the research field of search interfaces using facets, there are surprisingly few approaches focusing on allowing the user to express their preferences through weighting facet terms. Han et al. [6] focus in their people search system on using slider-based facets to specify the importance of three predefined categories. Their results show that the users consistently interact with the sliders to fine-tune the result ranking to achieve a better ranking. An approach very similar to ours is presented by Voigt et al. [22]. They distinguish in their VisBoard application between must-have and optional facet terms that can be weighted by the user with drag-and-drop in a configuration area. They utilized this approach for the specific task of selecting a visualization component based on its characteristics. Unfortunately, they only performed a preliminary user study with five participants showing the general potential of this approach to support the user expressing her preferences.
In the field of recommender systems, a lot of research approaches show the benefit of including user's preference to increase recommendation accuracy, user satisfaction and user experience. Critiquing-based recommender systems (e.g. see [4,2]) allow users to interactively criticize recommendations and thus put the users in control of finding a product that fits their preferences. With the possibility to weight critiques (e.g. "less expensive" or "compromise distance"), a user can perform trade-offs while searching for products, a concept which can still be very rarely found in commercial systems. Studies have shown that the critiquing-based approach leads to a higher decision accuracy compared to non critiquing-based systems such as a ranked list with one ranking criteria at a time [15,16]. In TasteWeight [1] users can adjust their taste interactively during a recommendation session via slider-weights components. Thus, they weight suggested recommendation terms and influence directly the recommendations. Results of a user study showed a positive effect on user satisfaction. Harper et al. [7] also put the user in control of her recommendations by providing means for tuning system generated recommendations according to her preferences (e.g., "show more popular items"). Results of their user study show that if users have control over a recommender system, they evaluate suggested recommendations more positively than automatically generated recommendations. A study with SetFusion [13], an interactive system that allows weighting the influence of three different recommender algorithms, also shows that due to the interaction and visualization, the users have a greater sense of perceived control and transparency. The approaches above focus primarily on ranking or criticizing a generated set of recommendations and recommended terms. Loepp et al. [11] in contrast, integrate different recommender algorithms with several interactive filter techniques in one hybrid recommender system. This system allows the selection of hard and soft filter criteria from different facets by the user. The selected facet values can be weighted by the user and serve as input for collaborative and content-based recommender techniques. Results of a user study showed that users feel more in control with the hybrid recommender system than with a standard faceted filtering system. They find the interaction to be more appropriate for generating recommendations.
In the field of information retrieval different concepts have been proposed to involve the user's preferences more interactively in the ranking of search results. A core concept is relevance feedback [18], in which the user implicitly or explicitly influences the ranking by marking some result items more relevant than others. Another concept that is strongly related to relevance feedback, is query expansion [3]. Here, user search terms are expanded with additional terms originating from knowledge sources, search results, the document corpus or the user's history. These expanded terms are often assigned a different weight in the overall query, to balance the effect of query expansion. On the user interface side, Frei and Qui [14] allow the user to weight query terms in the context of document retrieval. They showed that weighted queries perform significantly better than Boolean retrieval regarding usefulness and precision. In the context of a digital library, an approach for conditional weighting like preference A is more important than B was formally introduced by [19].
In this paper, we build up on different concepts of the works above: allowing users to enter query terms, a recommender for facets, the possibility to weight free and facet terms, a certain fuzziness for specific facets and a highly interactive search interface. We especially concentrate on a thorough evaluation measuring user feedback and system performance based on the whole search session.
HOTEL SEARCH -AN EXAMPLE APPLICATION FOR A
PREFERENCE-BASED RANKING SYSTEM
Searching for a hotel includes a wide variety of facets (hotel features) on the system side as well as some different preferences on the user side. We chose this as a use case for studying the potential of a preference-based ranking system. Our search interface allows optional search terms to enhance the result list with alternatives which do not necessarily fulfill all of the user's preferences. The user can specify which of her preferences have to be matched exactly ("must-have terms") and which preferences are nice to have ("optional terms"). Furthermore, the user can explicitly exclude hotels containing features which are not wanted ("must-not have"). "Must-have" and "must-not have" terms cause a trimming of the result list, while the weighting of "optional terms" influences the ranking so that the best-matched hotels are at the top.
User Interface
The search interface consists of three main components: input field, weighting area, and search result list. Figure 1 shows the implemented interface. In the input field, the user can enter her preferences one after another ( Figure 1a). While typing, a recommender suggests a list of facet terms or facet categories that match the current character string. If the search term does not match a facet term, it is marked as a free text term. In the weighting area ( Figure 1b) all search terms which formulate the search query are listed with the possibility to weight each of them. Initially, all search terms have the same almost highest weight. This means that all products are still in the result list. With the provided slider the user can weight the impact of each term. By setting the slider to the minimum of the scale, the search term is considered as a "must-not have" and by setting the slider to the maximum of the scale the search term is considered as a "must-have". With criteria marked as "must-haves" and "must-not haves" the number of results can be decreased. With a click on the recycle bin icon, the search term can be easily removed from the search query. Any interaction in the weighting area leads directly to a new calculation of the search results ( Figure 1c) and provides immediate feedback to the user. On the left side of each result item, information on all important hotel features are shown (Figure 1d). On the right side matched and mismatched search terms are visualized (Figure 1e). This gives the user the opportunity to judge the agreement of search results with her preferences and provides transparency over the ranking process.
Determining the Result Set
The weights for each search criterion are mapped to floating point numbers between zero and one. This weighting factor is used to determine which hotel will be included in the result set. If the weight is set to one, a hotel will be retrieved if it satisfies the required criterion. In contrary, if the weight is set to 0, the weighting acts as a NOT operator, in which hotels that satisfy the criterion will be eliminated from the result list. This allows users to exclude unwanted criteria. In both cases, hotels will not be included if they lack the given criterion. These cases can be considered as filtering the result set.
Ranking Model
Each hotel in the result lists gets a summed relevance score (srs) according to that srs the result list is ordered. The hotel with the highest srs is on top of the list followed by the others in descending order. The summed relevance score for each hotel is computed as srs = ∑ n i=1 w Ci * rs Ci , whereas (C1-n) are the hotel's criteria the search terms are mapped to. w is the user defined weight for each criterion and rs a relevance score that depends on the criterion class the search term is mapped to (see Table 1). Search terms that cannot be mapped to the given facets are assigned to the class Text, for which a tf-idf scoring is applied to calculate a relevance score taking all textual description of the hotel into consideration. A hotel criterion that fulfills a nominal facet (like "breakfast" or "single room") gets a relevance score of 1. For criteria that fulfill numerical facet terms, we suggest using a fuzzy relevance scoring approach whereby hotels that fully match the stated criterion get the highest rs and hotels whose feature fall in a range around the stated value get a lower rs according to the applied function. We propose the usage of a "Gaussian", "linear" or a "tri-linear" scoring function. Figure 2a shows a customizable Gaussian function that can be applied to a criterion for which a single value is given. An optional offset parameter can widen the range where the score will be the maximum. A linear function can be applied where a maximum or minimum is stated (shown in Figure 2b), for example, for user ratings . In case a scoring is applied to a criterion given by a range r (e.g. "price from $100-$120") a tri-linear function can be used which primarily favors the selected range and secondly the border ranges in order of direction ( Figure 2c). Here, each range is valued differently, so that one range border gets a higher value than the other. In our price example, this means that the lowest price of the selected range gets the highest relevance score (rs). The rs decreases linearly until the highest price of the selected range is reached. Prices that do not fell into the price range get lower rs whereby lower prices get higher rs than higher prices, in each case the rs is linearly descending from the lowest to the highest price in
Tiergarten Neighbor- hood - - - Breakfast Meals 1 EQUAL 1 Restaurant Meals 0.8 EQUAL 1
srs (Hotel Hackerscher Markt) = 0.9 * f Tri (95) + 0.9 * f Gau (0.6) + 1 + 1 = 3.26 Table 2. Example mapping from the search terms to the hotel criteria of the first hotel in the results of Figure 1 including provided user weights, applied functions and calculated relevance scores.
each range border. For the first hotel in Figure 1, Table 2 shows the mapping of the search terms to the hotel criteria, the weights provided by the user, the relevance functions that are used and the calculated relevance score for each criterion. The hotel gets, therefore, a summed relevance score of 3.26.
Implementation
For implementing our search interface we utilized Elastic Search 1 as a search engine. The software architecture was designed to support various types of data and data sources. The prototype is implemented in Java-EE and provides a generic library and the actual hotel search application. The user interface is a generic JSF web fragment which provides a custom component and java controller (bean) 2 .
The hotel search application contains specific facets with different classes and functions. For the facet "neighborhood", a Gauss scoring (Figure 2a) is applied to consider the distance of a hotel to a specified neighborhood. That means hotels that are nearer to the neighborhood indicated in the search query are weighted higher than those farther away. For the "customer rating values" and "hotel stars" linear functions ( Figure 2b) are used. For the price range, a tri-linear function (shown in Figure 2c) is used with a range extended by 20% to each side of the specified price maximum and minimum. For all nominal facets, an equal value comparison is applied.
EVALUATION
We performed a lab study to evaluate the presented hotel search interface against a standard search interface using facets (see Figure 3). Additionally to the explicit user feedback and user task performance, we are interested in the system performance during a whole search session and how this relates to the user's feeling about system support.
Apparatus Prototypes
For the user study, we used the hotel search prototype introduced above -in the following called weighted prototype.
For the comparison with a conventional search interface using 1 https://www.elastic.co/de/products/elasticsearch 2 The prototype's source code is available here: https://git.gesis.org/iir/preferenced-based-search. facets we developed a second prototype as a representative of a state-of-the-art hotel search interface (see Figure 3) -in the following called facet prototype. This prototype provides filter functionalities in the form of facets, which are separated into different categories (see Figure 3a). For each facet term, the number of remaining results in the result list after choosing this facet is shown after the term in brackets. Selected facets are represented on the right-top side of the interface where the user can also delete selected facets by clicking on the recycle bin icon (see Figure 3b). The results shown in the result list (see Figure 3c) match all selected facet terms. The representation of the result list is the same as in the weighted prototype. However, an additional sorting function is available to sort the results by relevance, price, hotel stars and customer ratings (see Figure 3d). Both prototypes include a logging component to record user actions as well as the status of each presented result list.
Scenario
In order to compare the two prototypes and to analyze whole search sessions under the same conditions, we created a use case all participants had to perform with each prototype. The scenario includes "must-haves", "must-not haves" and "optional" preferences for a hotel in Berlin. The participants were asked to book a hotel for Paul who wanted to attend a conference in Berlin. His intended price range is between e60 and e120. Participants were provided with Paul's preferences: "Paul would prefer to stay in district 'Mitte' (neighborhood) or in another district with good access to public transport (transport). By no means he wants to stay in the district 'Tiergarten' (neighborhood). In any case, he would like to have breakfast (meal). Furthermore, he would appreciate if the hotel has a restaurant (meal) or a bar (entertainment). Additionally, he would welcome, if the hotel has a fitness center (sport) and that he can pay on invoice (payment type). He could do without these two last items, whereas paying on invoice would be more important for him than the fitness center." The categories of the conditions were explicitly mentioned in the scenario because we are not interested in comparing how fast the par-ticipants could find the desired category compared to how fast they could type in the search terms. There is no hotel in the data set that matches all criteria. Otherwise participants would find this hotel with both prototypes very quickly -it would be the best hit after selecting all facets. In our scenario, Paul's preferences have to be weighted against each other and compromises have to be made finding a hotel that matches Paul's preferences properly.
Data sets and hotel relevance scores
We created a data set of 150 hotels in Berlin. The initial set of accommodations is based on information gained through the public Yelp-API. We received a list of hotels in Berlin with addresses, user comments, user ratings and categories. Each hotel was enriched by further features assigned to 18 different categories (such as price, neighborhood, type of room, type of catering, customer rating, etc.). This information was taken either from the hotel's website or information provided by Booking.com. To be able to have two comparable sets of hotels that can be used in the evaluation and to avoid a learning effect, we copied the hotel data set and changed only the names of the hotels. All information in the data set were provided in German.
For evaluating how well a selected hotel fits Paul's preferences, we generated a graded relevance score (grs) for all hotels in our dataset. The algorithm is illustrated in Figure 4. First, we checked if the hotel match the "must have" / "must-not have" criteria. When these mandatory criteria are fulfilled the hotel gets a first grs of one. In our case, that means, the hotel has a price between e60 and e120, breakfast is included and the hotel is in the district "Mitte" and if it is not in "Mitte" it has access to public transport, and it is not in the district "Tiergarten". All other hotels that do not meet these criteria are considered to be not relevant and are not considered further. They get a grs of zero. Then, we checked the additional optional criteria and awarded additional relevance scores depending on Paul's preferences. That means, a hotel with a fitness center gets two additional scores, as this was Paul's least preferred criterion. Hotels that provide the opportunity to pay by invoice gets three additional scores, as this was more preferred by Paul than the fitness center and hotels with a restaurant or a bar get four additional scores, as Paul preferred these features most. A hotel that meets all criteria can gain a grs of 10 (= 1 (mandatory criteria) +2 (fitness studio) +3 (paying on invoice) +4 (restaurant or bar)). In our data set only 15 hotels meet Paul's must-have preferences. No hotel meets all requirements, but five hotels have a graded relevance score of eight.
Setup
For the user study, we used a laptop with internet access. Through the Firefox browser 46.0.1 both prototypes were accessed on our server. To make sure that all participants see exactly the same part of the user interface a 21 monitor with the same resolution was used in all sessions. Furthermore, an external keyboard and a mouse were provided to the participants as input modalities. The activities on the screen were recorded with the screen capture software Camtasia for further analysis. The used questionnaires were on paper.
Methodology
Data collection took place in a laboratory setting at a university and our institute in single sessions. We used a within-subject design approach with the two prototypes and the two data sets being the independent variables. The four resulting experimental conditions were randomly but equally assigned to the participants. The study took about 45 minutes per participant. As dependent variables, we collected time-on-task, clicks-ontask, graded relevance scores of selected hotels, subjective ratings, free-form text comments. Furthermore, we calculated the normalized discounted cumulative gain (NDCG) [9] of each result list.
Procedure
The study was performed in single sessions and followed a detailed trail protocol with a counter-balanced order of the four experimental conditions. First, the experimenter explained the purpose of the study and that all activities on the screen are recorded. The participant agreed to the procedure by signing a consent form. Before the actual experiment started, the participant filled out a questionnaire in which we asked a few questions regarding demographic information and the experience with hotel search systems. Afterwards, the experimenter showed and explained the first prototype and asked the participant to familiarize herself with the interface for about four minutes. Then, the participant was given the assignment with the scenario description. The task ends with selecting a hotel for Paul. There was no time limit for the task. In a questionnaire, the participant was asked why she selected the hotel, how satisfied she is with her choice, how well the system supported her while searching for a hotel and if the ranking of the result list was comprehensible. The same procedure was followed with the second prototype. At the end of the study, the participant filled out a final questionnaire. In this, we asked for advantages, disadvantages of each prototype as well as for suggestions for improvements with open questions. Each participant received e10 as a compensation for expenses. Participants 24 native speaking German participants took part in the user study. Half of them were university students from different fields of study. They were recruited through an announcement on notice boards or mailing lists. The other half were recruited by word-of-mouth recommendation and they all had either a university degree or a completed apprenticeship. 12 participants were female. Participants' age ranged from 18 to 51 years (M=28.04, SD=8.3). Three of them have never used a hotel search system. 16 of the participants are familiar with more than one hotel search system like Booking.com or Trivago.com and 17 used such a system at least once in the last six months. 81% of these participants are satisfied or very satisfied with the functionalities such systems provide and 76% are also satisfied or very satisfied with the search results of such systems.
EVALUATION RESULTS
In the following section, we describe the results of the user study by first focusing on the selected hotels, then on the provided user feedback and finally on system performance. Given the fact that all participants had to perform the task following a given scenario we are able to analyze changes in recall and precision over the whole search sessions and combine them with the provided user feedback on system support. We want to know if there are relations between user feedback and system performance measures. If not stated others, we use a Wilcoxon signed-rank test to compute differences between two paired groups with α ≤ 0.05.
Analysis of selected hotels
In almost all sessions, hotels with the highest possible graded relevance score were selected with both prototypes (see Table 3). Having a closer look at the selected hotels, it is striking that with the facet prototype 17 participants chose a hotel in the neighborhood "Mitte", while with the weighted prototype only nine participants did the same. This might be an indication of a different level of elaborateness during the task performance. While the neighborhood was an equally weighted condition by Paul we found evidence in the log files that ten of the participants using the facet prototype did not consider that alternative at all. Furthermore, the log files showed that in total 16 participants had not examined all of Paul's preferences in the facet prototype. Beside of the preference "outside Mitte with access to public transport", the preference "fitness center" was often remained unconsidered. In the following, we will also have a closer look at those two user groups -those who examined all of Paul's preferences in the facet prototype (full examiners, n=8) and those who did not (incomplete examiners, n=16).
One interesting finding is the significant difference in price of the selected hotels (see Table 3). With the weighted prototype, the accommodation price is significantly lower than with the facet prototype (facet M=99.63, weighted M=80.25, p=0.011). Whereby, in the group of full examiners there is no significant difference in the hotel price (facet M=85.82, weighted M=74). Nine participants also stated price as an important factor while searching for a hotel.
In the weighted prototype condition, two participants failed in solving the task correctly based on the provided information about the hotel. They chose a hotel outside the neighborhood Mitte without access to public transport. However, one of them answered to our question why she chose this specific hotel with "There is good access to public transport." which indicates real knowledge about the neighborhood. With the facet prototype, also one participant chose a hotel outside Mitte with no access to public transport. We do not know if these two participants have the same reason (knowledge about the neighborhood) without stating it in the questionnaire. 18 participants provided further comments to the reasons why they chose a hotel. 15 stated that they included the user rating in their relevance decision process. For nine participants the price played an important role and comments from four participants allow the conclusion that they had further knowledge about Berlin, especially about the neighborhoods and the distances.
Quantitative User Feedback
We asked the participants on five-point likert scales how satisfied they are with the selected hotel ("very satisfied"=1 to "not at all satisfied"=5), how well they felt supported in their search by the system ("very well"=1 to "not well at all"=5) and if the result sorting was comprehensible ("very comprehensible"=1 to "not at all comprehensible"=5). The results show that there is no significant difference regarding support and comprehension of both presented search systems. However, participants are significantly more satisfied with the selected hotel when they use the weighted prototype (M=1.67) than the facet prototype (M=2.13, Z=-2.082, p<.05). A closer look at our to user groups showed that this is true for the incomplete examiner (weighted M=1.63, facet M=2.19, p=0.039) but there was no significant difference in the group of full examiner. They were similar satisfied with the hotel selected in the weighted prototype (M=1.75) and in the facet prototype (M=1.88).
For analyzing time-on-task and clicks-on-task needed to perform the task, we consulted the recorded screencast videos as well as the log file data. In both cases, we could find significant differences between the two prototypes. Participants are significantly faster and significantly fewer clicks are needed in the facet prototype than in the weighted prototype. In the facet prototype it took 5.3 minutes (σ = 2.9) and 16.5 clicks (σ = 10) on average to select a hotel compared to the weighted prototype with 7.36 minutes (σ = 2.96) and 25.83 clicks (σ = 9.76) on average (time-on-task: Z=-2.714, p<.05, clicks: Z=-3.244, p<.05). These results seems not surprisingly given the fact that the number of incomplete examiner is with 2/3 relatively high.
In the group of full examiner, we could not find a significant difference in time-on-task (weighted prototype M=6.6 minutes, facet prototype M=7.52 minutes) and clicks (weighted prototype M=18.13, facet prototype M=27.13). While in the group of incomplete examiner, the differences in time-on-task (weighted prototype M=7.64 minutes, facet prototype M=4.16 minutes) and clicks (weighted prototype M=24.69, facet prototype M=11.06) are significant (time: p<.0001; clicks: p=.000).
Qualitative User Feedback
In the final questionnaire, we collected qualitative user feedback in open questions on advantages, disadvantages and suggestions for both prototypes. In the following, we only provide feedback comments that were stated by more than one participant.
Weighted prototype's advantages 22 participants answered the question about advantages for the weighted prototype. Altogether we collected 34 different statements. The benefit most often stated, by eleven participants, was the opportunity to weight optional criteria. Five persons liked the "must-not have"-opportunity and also five found the interface well structured. Three participants appreciated the free-text search function and two other liked that they got more potential results that they can compare.
Weighted prototype's disadvantages 18 participants provided 28 disadvantage statements for the weighted prototype. Six persons disliked that there was no category list. Five missed an explicit sort-by function for price or ratings. Four were annoyed by typing in the search criteria.
Suggestions for the weighted prototype We collected 18 suggestions for the weighted prototype, provided by 16 participants. Six persons would like to have an explicit sort function by price or ratings. Showing all available criteria was suggested by four participants.
Facet prototype's advantages 24 participants provided feedback to the question about advantages of the facet prototype, wherewith we collected altogether 30 statements. 15 persons liked to select the facet terms from lists shown on the left side of the interface, as they know it from online shops and travel portals. Four participants appreciated the opportunity to sort the results by a predefined sorting criterion. Two persons mentioned positively that they do not have to type in a search term and another two liked the clear arrangement.
Facet prototype's disadvantages We received 24 statements from 19 participants to the question about disadvantages of the facet prototype. Six persons missed that there was no possibility to weight the criteria. For four participants the interface was too overloaded, and three criticized the cutting of hotels that did not fulfill all criteria. Two participants stated that it was cumbersome to find out which criteria led to a noresult list and two others missed the opportunity to compare alternatives that did not match all criteria.
Suggestions for facet prototype 18 participants provided suggestions for the facet prototype. Altogether 21 comments could be collected of which eight would like to have a weighting function. Three would add a search field and two the possibility to exclude results matching a "must-not have" criterion.
System Performance Measure
When analyzing the facet and the weighted prototype regarding system performance, we have to keep in mind that their individual functionalities influence recall and precision differently. In a faceted search system, recall is massively influenced by the filtering functionality of the facets. The weighting of Figure 5. Boxplots of the ratio between relevant hotels that were visible during all participants' sessions and the total number of relevant hotels, grouped by systems.
facets, in contrast, mainly influences the precision. Concerning individual result lists, the two systems are hardly comparable. Therefore, we will evaluate them on their own on the level of the complete search session and combine results with user feedback.
Relevant hotels seen
In our context, recall would be defined as the ratio of the number of relevant hotels that the system retrieved for a user query and the total number of relevant hotels in the data set. However, in our study, the data set contained only 150 hotels.
Stating queries with all relevant hotels within the result set is not a complex task. Instead of looking at the complete result list that a system retrieved, we will only consider the visible part of it, namely, all the results that have been displayed on the result pages the users browsed through. Also, we will not only focus on single queries, but we will incorporate all queries within a session.
Let us consider a participant's session in which multiple searches are conducted to find a suitable hotel. These queries can be triggered by entering search terms, filtering or reordering the result list, changing the weight of a criterion, and including or excluding criteria. Each query action will lead to a new result list of which the first 15 hotels are displayed on the first result page. These hotels form the visible part of the result list. If the participant proceeds to the next page of a result list, the number of visible results is increased. If we collect all visible hotels from each participant's search session, we can assess how many of the relevant hotels have been visible to the participants using a specific system. Figure 5 shows boxplots of the ratio between relevant hotels that were visible during all participants' sessions and the total number of relevant hotels, grouped by systems. We calculated the ratio with a decreasing relevance threshold and grouped the results. The first two boxplots on the left show the ratio of hotels with a graded relevance score (grs) higher or equal 8 for the facet and the weighted prototype. The third and fourth boxplots show the ratio of visible hotels with a grs higher or equal 7 and so forth. It can be observed that when using the weighted system until a relevance threshold of 4 a high mean of 1.0 and a first quartile of around 0.8 was achieved whereas the mean and first quartile using the facet prototype was lower. Overall, a non-parametric Mann-Withney test (α ≤ 0.05) found the number of relevant hotels seen with the weighted system to be significantly higher. The results of the test are as follows: grs≥8 (u = 2.668, p = .008), grs≥7 (u = 2.645, p = .008), grs≥5 (u = 2.544, p = .011), grs≥4 (u = 2.900, p = .004), grs≥3 (u = 3.099, p = .002), and grs≥1 (u = 3.478, p = .001). Having a closer look at the two groups, it is not surprisingly that the group of incomplete examiner is again responsible for that result. They have seen significantly more hotels in the weighted prototype than in the facet prototype (for example grs≥8 facet prototype M=3.50 vs. weighted prototype M=4.50, p=.002; and for grs≥1 facet prototype M=7.94 vs. weighted prototype M=12.13, p<.0001). In the group of full examiner there are no significant differences.
Result list precision
For analyzing the quality of the result lists, we used the normalized discounted cumulative gain (NDCG) [9]. Plotting the NDCG values of all searches conducted during a search session against time gives an overview of the development of the participant's session from the perspective of precision. As both systems' modes of operation have different impacts on the NDCG, the NDCG plots of the two systems should not be compared. Therefore, we will not compare the two systems concerning precision values, but we will investigate the relationship between precision and the perceived satisfaction with the systems' which was asked by the question "How well did you feel supported by the system?" ("very well"=1 to "not well at all"=5). Figure 6 shows the NDCG plots for the weighted prototype and Figure 7 for the facet prototype. Each point represents the NDCG value of a search at a specific point of time during the session. In both figures, the plots are grouped along the participants' answers to the question how well they felt supported by the system during their search. As the answers "not well" and "not well at all" were only given by one or none participant we only show the plots for "very well supported", "well supported" and "neither well nor not well supported". For example, in the upper plot of Figure 6 ("very well"), each point represents the NDCG value of a search result list of a search session, where the participant felt very well (n=10) supported by the system. In addition to the data points, we generated LOESS smoothing curves 3 which helped to identify common characteristics. The dashed blue line is a LOESS curve created for the complete data set regardless of the participant's answer to the question of system support satisfaction. The solid red lines are a LOESS curve generated for the specific group. 4 In addition to the LOESS curves, for each plot, there exists a vertical dot dashed green line, which indicates the point of 3 LOESS was first introduced by [5]. In this paper we use the implementation which is part of the R-Project: https://stat.ethz.ch/ R-manual/R-devel/library/stats/html/loess.html 4 All LOESS curves are generated with a degree of 2 and a span of 0.75. The LOESS curves generated for the whole data set in Figure 6 and Figure 7 show one common characteristic. The search process seems to be divided into two phases. First a take-off, where the precision of search requests increases until roughly 180 seconds into the session, where a break of slope indicates the beginning of the second phase. During the take-off phase, the steepness of the curve indicates a fast improvement of search precision. For the weighted prototype ( Figure 6), this first phase ends at an NDCG value of around 0.4. After this, the NDCG still increases, but with a lower gradient. For the facet prototype (Figure 7), the break of slope builds the maximum of an NDCG value of around 0.7. After the maximum, the curve slowly declines until a minimum of around 0.4 after roughly 600 seconds. Notably, the break of slope lies before the median task processing time. This means that at least half of the participants (in most cases even 75%) were still working on the task at the time point of the break of slope. This supports the assumption that the search sessions are indeed divided into two phases, as it means that the break of slope does not coincide with the completion of the task.
Comparing the LOESS curve of the complete data sets and the curves of each individual subgroup one can observe a relationship between the satisfaction with the system's support and the precision of the search requests of the participants. In Figure 7. NDCG plots for the facet prototype, grouped along the answer to the question 'how well did you feel supported by the system?'. Figure 6, the LOESS curve of the group of participants that felt very well supported by the system primarily lies above the curve of the complete data set. Even if it is also divided into two phases, the take-off phase is steeper, and the break of slope lies on a higher level. Participants in this group were faster and better in formulating more precise search requests than the users in the other groups and felt better supported by the system. Similar results can be observed in the facet prototype condition.
When comparing the individual groups' LOESS curves with the overall curves, we can identify similar relationships between those two for both systems. The lower the perceived level of support is, the lower the curve. The curve for the group "very well" is the highest, the curve for "well" is close to the overall curve, and the curve for "neither well nor not well" lies mostly underneath the overall curve. Overall, we can conclude that there seems to be a connection between perceived system support and the development of the search precision.
DISCUSSION
In this paper, we compare two different concepts for searching products: (1) a pure facet concept, well established in all kinds of (product) search engines and (2) a preference-based approach using weighted facets which allows users to express preferences of certain product, in our case hotel features. We took user feedback, system performance and a combination of both into consideration to evaluate both approaches.
Our evaluation results show that the participants are significantly more satisfied with the selected hotels found in the weighted prototype than in the facet prototype. One possible explanation for that is given by the recall analysis which showed that there were significantly more relevant hotels visible during the whole search session. Therefore, participants were able to compare more hotels, even those that only partly fulfill their requirements, and might get a better feeling for their buying decision. This is inline with McSherry's findings that a decision maker wants to be informed of all items that are likely to be of interest [12]. The high number of relevant hotels shown is explainable by the different interaction techniques to compare alternative hotels. In the weighted prototype, based on the task, users themselves push more relevant hotels higher in the list with the interactive sliders, and therefore these hotels are better visible. In the facet prototype, the effort to see more relevant hotels is higher as the user has to select and deselect each facet and facet combinations explicitly to see its influence on the results. It seems that the incomplete examiner were not willing to take the extra effort. Apparently they are willing to select a hotel more quickly that matches at least parts of the preferences without knowing other, probably better, alternatives. Therefore, it is not surprising that these participants were faster and needed fewer clicks in the facet prototype to select a hotel. The time the full examiner took to find a hotel in the weighted and in the facet prototype did not differ significantly. Half of them were even faster in the weighted prototype than in the facet prototype. Furthermore, no participant complained in the questionnaire that finding a hotel with the weighted prototype, in general, took too much time or effort. The fact that with the weighted prototype the hotel price of the selected hotel is significantly lower than with the facet prototype could provide an incentive for some users to spend more effort for the search process.
Besides our comparison of the two prototypes regarding user satisfaction and recall, we were able to find similar characteristics within the search process of our participants by analyzing the search precision. Overall, the search sessions are divided into two phases. During the first phase, the take-off phase, the requirements defined in the scenario are transferred into the system, which leads to an increase in precision. During the second phase, the precision decreases and increases alternatively leading to a change of the precision curves slope. The reason here might be, that there is no perfect hotel for the task given and the participants had to change their queries to find alternatives that are close to the criteria given.
Regarding result list quality we observed a relationship between the perceived system support and the precision of the participants. When plotting the NDCG values grouped by the participants answer to the question how well they felt supported by the system, one could observe, that a certain groups' precision curve lies above the average if the user felt very well supported and below if she felt neither well nor not well supported. This indicates that there might be a relation between the perceived system support and the precision of the participants' searches. However, this method is new, and we
have not yet understood enough to draw solid conclusions, but we believe that the analysis of the search precision can aid in the task of measuring user satisfaction. In our case, the two prototypes do not allow for a comparison of the result list precision, as they generate those list differently. When comparing two similar systems, this method could produce comparable results, which would allow to evaluating two different versions of a system.
One important lesson learned from our study have to be mentioned. Knowledge about Berlin, the city we chose as an example in our study, might have influenced the results, as participants selected hotels knowing that the neighborhood has a good connection to public transport. In further user studies, we will use a fictitious city. Furthermore, there are some other limitations in our approach which should not go unmentioned. The number of hotels in our data set with 150 hotels is rather low. We could not find an existing hotel data set with a sufficient number of facets for each hotel. So, we created our own by manually enhancing the dataset with a lot of different facets. However, we cannot preclude that the dataset size might have an effect on our evaluation results. In future user studies, we will use a data set with a higher number of hotels. In order to perform a combined analysis of user feedback with system performance it was necessary that all users performed the same task in both conditions. Also in this case, we cannot preclude that with a different task the results might be different. This is also an aspect we have to address in future research to examine further the relation between system performance and user satisfaction measures.
CONCLUSIONS
In this paper, we present an evaluation of a search interface using a preference-based ranking approach. Users can select, exclude and weight (optional) search criteria by their preferences and thus influence the ranking of the result list. In a user study, we compared this search interface to an interface using standards facets. 24 participants had the task to find a hotel according to predefined preferences with both search interfaces. We evaluated the interfaces from a user and a system performance perspective and found out that:
• Users are significantly more satisfied with the selected hotel found with the weighted prototype.
• Users were given more relevant hotels in the result lists with the weighted prototype.
• There is no significant difference regarding time-on-task and clicks when users examine all preferences in both prototypes.
• Users, who do not consider all preferences in their search queries in the facet prototype were significantly faster and needed fewer clicks to select a hotel than with the weighted prototype.
• Users chose a significantly cheaper hotel with the weighted prototype.
• Both user interfaces show characteristic differences in the evolvement of precision during a search session.
• Users that were able to generate result lists with a higher precision seem to felt better supported by a system.
The last two results based on observations on the analysis of the relation between the participants' answers to the question how well they felt supported by the system and the precisions (measured by NDCG) of the result lists. In future work, we want to research the potential of analyzing the evolvement of search precision over whole search sessions as an indication for user satisfaction in more detail.
Figure 1 .
1The preference-based hotel search interface consists of a) input field, b) weighting area, c) result list, d) list of product features, e) visual feedback on matched or mismatched search terms.
Figure 2 .
2a) Customizable Gaussian function for relevance scoring, b) linear function applied on directed Likert-scales, c) Tri-linear descendant directed scoring function.
Figure 3 .
3Comparative prototype used in the user study as a representative of a faceted search interface.
Figure 4 .
4Algorithm for calculating Hotel's graded relevance score (grs) based on Paul's preferences provided in the scenario.
Figure 6 .
6NDCG plots for the weighted prototype, grouped along the answer to the question 'How well did you feel supported by the system?'. time, where the median of the group's participants has finished their task.
Table 3 .
3Number of hotels selected by the participants for each prototype divided according to the graded relevance score, neighborhood "Mitte" and average price.
ACKNOWLEDGEMENTWe would like to thank Marco Janc for implementing the prototype. Additionally, we thank him and Maria Lusky for supporting us with the execution of the user study. We are also very grateful to those who participated in our study.
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"PARAMETERIZATION-BASED NEURAL NETWORK: Predicting Non-linear Stress-Strain Response of Composites",
"PARAMETERIZATION-BASED NEURAL NETWORK: Predicting Non-linear Stress-Strain Response of Composites"
] | [
"Haotian Feng \nDept. of Mechanical Engineering\nDept. of Mechanical Engineering Dept. of Civil & Env. Engineering\nUniversity of Wisconsin-Madison Madison\nUniversity of Wisconsin-Madison Madison\n53706, 53706WI, WI\n",
"Pavana Prabhakar *[email protected] \nDept. of Mechanical Engineering\nDept. of Mechanical Engineering Dept. of Civil & Env. Engineering\nUniversity of Wisconsin-Madison Madison\nUniversity of Wisconsin-Madison Madison\n53706, 53706WI, WI\n"
] | [
"Dept. of Mechanical Engineering\nDept. of Mechanical Engineering Dept. of Civil & Env. Engineering\nUniversity of Wisconsin-Madison Madison\nUniversity of Wisconsin-Madison Madison\n53706, 53706WI, WI",
"Dept. of Mechanical Engineering\nDept. of Mechanical Engineering Dept. of Civil & Env. Engineering\nUniversity of Wisconsin-Madison Madison\nUniversity of Wisconsin-Madison Madison\n53706, 53706WI, WI"
] | [] | Composite materials like syntactic foams have complex internal microstructures that manifest highstress concentrations due to material discontinuities occurring from hollow regions and thin walls of hollow particles or microballoons embedded in a continuous medium. Predicting the mechanical response as non-linear stress-strain curves of such heterogeneous materials from their microstructure is a challenging problem. This is true since various parameters, including the distribution and geometric properties of microballoons, dictate their response to mechanical loading. To that end, this paper presents a novel Neural Network (NN) framework called Parameterization-based Neural Network (PBNN), where we relate the composite microstructure to the non-linear response through this trained NN model. PBNN represents the stress-strain curve as a parameterized function to reduce the prediction size and predicts the function parameters for different syntactic foam microstructures. We show that compared to several common baseline models considered in this paper, the PBNN can accurately predict non-linear stress-strain responses and the corresponding parameterized functions using smaller datasets. This is enabled by extracting high-level features from the geometry data and tuning the predicted response through an auxiliary term prediction. Although built in the context of the compressive response prediction of syntactic foam composites, our NN framework applies to predict generic non-linear responses for heterogeneous materials with internal microstructures. Hence, our novel PBNN is anticipated to inspire more parameterization-related studies in different Machine Learning methods. This paper presents a novel Neural Network (NN) for predicting the non-linear stress-strain behavior of heterogeneous materials through a parameterization-based approach. Specifically, we relate the composite microstructure to the non-linear response through this trained NN model. We build this framework in the context of the compressive response prediction of polymeric foams called 'syntactic foams' with their microstructure as the inputs. Syntactic foams have complex internal microstructures with high-stress concentration regions and macroscale non-linear compressive stress-strain responses, making it more challenging to predict their stress-strain responses. To that end, the NN presented in this paper applies to predicting generic non-linear responses for composites with internal microstructures.Syntactic foams, specifically polymer ones, are closed-cell composite foams with hollow spheres or particles called 'microballoons' embedded in a polymer matrix. The presence of hollow spheres results in several excellent mechanical properties, including lower density, higher specific strength, lower thermal expansion coefficient, and lower moisture absorption[1,2]. Due to these properties, syntactic foam materials are widely used as buoyancy materials for marine applications as a component of sea-related products and offshore products[3,4]. Moreover, syntactic foams have been arXiv:2212.12840v2 [physics.app-ph] 13 May 2023 | null | [
"https://export.arxiv.org/pdf/2212.12840v2.pdf"
] | 255,125,281 | 2212.12840 | 2ce6a803495990a83c38f876310a97a6249dffd6 |
PARAMETERIZATION-BASED NEURAL NETWORK: Predicting Non-linear Stress-Strain Response of Composites
Haotian Feng
Dept. of Mechanical Engineering
Dept. of Mechanical Engineering Dept. of Civil & Env. Engineering
University of Wisconsin-Madison Madison
University of Wisconsin-Madison Madison
53706, 53706WI, WI
Pavana Prabhakar *[email protected]
Dept. of Mechanical Engineering
Dept. of Mechanical Engineering Dept. of Civil & Env. Engineering
University of Wisconsin-Madison Madison
University of Wisconsin-Madison Madison
53706, 53706WI, WI
PARAMETERIZATION-BASED NEURAL NETWORK: Predicting Non-linear Stress-Strain Response of Composites
Parameterized Function · Neural Network · Stress-Strain Curve · Reinforced Composites · Finite Element Analysis
Composite materials like syntactic foams have complex internal microstructures that manifest highstress concentrations due to material discontinuities occurring from hollow regions and thin walls of hollow particles or microballoons embedded in a continuous medium. Predicting the mechanical response as non-linear stress-strain curves of such heterogeneous materials from their microstructure is a challenging problem. This is true since various parameters, including the distribution and geometric properties of microballoons, dictate their response to mechanical loading. To that end, this paper presents a novel Neural Network (NN) framework called Parameterization-based Neural Network (PBNN), where we relate the composite microstructure to the non-linear response through this trained NN model. PBNN represents the stress-strain curve as a parameterized function to reduce the prediction size and predicts the function parameters for different syntactic foam microstructures. We show that compared to several common baseline models considered in this paper, the PBNN can accurately predict non-linear stress-strain responses and the corresponding parameterized functions using smaller datasets. This is enabled by extracting high-level features from the geometry data and tuning the predicted response through an auxiliary term prediction. Although built in the context of the compressive response prediction of syntactic foam composites, our NN framework applies to predict generic non-linear responses for heterogeneous materials with internal microstructures. Hence, our novel PBNN is anticipated to inspire more parameterization-related studies in different Machine Learning methods. This paper presents a novel Neural Network (NN) for predicting the non-linear stress-strain behavior of heterogeneous materials through a parameterization-based approach. Specifically, we relate the composite microstructure to the non-linear response through this trained NN model. We build this framework in the context of the compressive response prediction of polymeric foams called 'syntactic foams' with their microstructure as the inputs. Syntactic foams have complex internal microstructures with high-stress concentration regions and macroscale non-linear compressive stress-strain responses, making it more challenging to predict their stress-strain responses. To that end, the NN presented in this paper applies to predicting generic non-linear responses for composites with internal microstructures.Syntactic foams, specifically polymer ones, are closed-cell composite foams with hollow spheres or particles called 'microballoons' embedded in a polymer matrix. The presence of hollow spheres results in several excellent mechanical properties, including lower density, higher specific strength, lower thermal expansion coefficient, and lower moisture absorption[1,2]. Due to these properties, syntactic foam materials are widely used as buoyancy materials for marine applications as a component of sea-related products and offshore products[3,4]. Moreover, syntactic foams have been arXiv:2212.12840v2 [physics.app-ph] 13 May 2023
extended to other applications like the aerospace and automotive industry. For effective design and optimization, such extended applications require predicting their mechanical properties, especially the stress-strain responses.
Past researchers have focused on establishing the mechanical properties of syntactic foams through experimental testing and numerical studies. Gupta et al. [5] performed compression tests and showed that compressive strength and modulus would increase when the microballoon radius decreases. Shahapurkar et al. [6] experimentally showed that compressive strength decreased with increasing cenosphere volume fraction for both modified and unmodified surfaces. Moreover, the authors showed a reduction in the compressive modulus but an increase in the compressive strength for arctic-conditioned samples. Jayavardhan et al. [7] conducted quasi-static compressive testing of glass microballoon reinforced high-density polyethylene syntactic foams with different densities. The authors showed that the compressive modulus increased as the microballoon volume fraction increased, but the yield strength, densification stress, and overall energy absorption were reduced. Exploring the influence of many parameters of syntactic foams experimentally is challenging and time-consuming. Thus later, Prabhakar et al. [8] developed the computational modeling to establish a fundamental understanding of densification mechanics of polymeric syntactic foams under compressive loading accounting for microballoon volume fraction, microballoon wall thickness, bonding between the microballoons and the matrix, and the crushing strength of microballoons. The authors further utilized multiple linear regression to understand the influence of structural and material parameters on its densification properties. Wang et al. [9] investigated how the strength distribution of a batch of hollow glass microspheres (HGM) influences the compression strength of syntactic foams. The authors discovered that the compressive strength of syntactic foams improved with an increase in HGM's strength.
The above research explored the syntactic foam composite's time-independent (quasi-static) mechanical properties. However, analyzing how mechanical properties of syntactic foam composite change under different strains or loading rates is also essential, i.e., the strain-dependent (dynamic) mechanical properties. Woldesenbet et al. [10] analyzed the effect of density and strain rate on the properties of syntactic foam and showed a considerable increase in peak strength of syntactic foams for higher strain rates and higher density. Song et al. [11] investigated the dynamic compressive properties of an epoxy syntactic foam at various strain rates under lateral confinement with a split Hopkinson pressure bar. The authors discovered that the quasi-static and dynamic stress-strain behavior has an elastic-plastic-like shape, whereas an elastic-brittle behavior was observed under uniaxial loading. Li et al. [12] experimentally analyzed the compressive responses of glass microballoon epoxy syntactic foams within a range of strain rates. The authors combined testing results with finite element stress analysis to determine the foam's localized damage and failure modes. Shunmugasamy et al. [13] utilized microCT-scanning and scanning electron microscopy to understand the effect of high strain rate loading on the deformation and fracture characteristics of syntactic foams and further understand the strain rate dependence of failure mechanisms. Zhang et al. [14] investigated the compressive response of epoxy syntactic foam with strain rate and temperature dependency. The authors also developed a non-linear phenomenological model to describe the responses of syntactic foam and its temperature-strain rate equivalence. These works have shown the different approaches (experimental and analytical) to obtaining the strain-dependent mechanical properties of syntactic foam composites. However, these analyses only focus on limited microballoon distributions by considering only a few samples. To explore the full range of parameters of syntactic foams and to design these foams effectively, an approach to relate the stress-strain responses to microballoon properties and distributions is needed. To achieve this, we will utilize Deep Learning methods to determine the complete stress-strain responses of syntactic foams given the volume fraction and wall thickness of randomly distributed microballoons.
The emergence of Machine Learning methods facilitates research to predict the mechanical properties of composite materials with the graph neural network as an essential tool. Graph Neural Network is developed based on Deep Convolutional Neural Network [15] (DCNN) and Generative Adversarial Network [16] (GAN). Researchers have been focusing on solving engineering design and analysis problems with Graph Neural Networks, like [17,18]. Regarding composite materials, Chen et al. [19] compared how different Machine Learning techniques, like regression model, DCNN, and Gaussian process, can accelerate the composite material design. Feng et al. [20] proposed a Difference-based Neural Network to enhance the stress distribution prediction within different composite micromechanical models, especially for models with stress concentrations in the stress distribution contours. Sepasdar et al. [21] proposed a modified U-Net framework to predict the damage and failure within microstructure-dependent composite materials. Feng et al. [22] further proposed a Physics-Constraint Neural Network to understand the forward and inverse predictions of woven composite models in the mesoscale.
Besides predicting the linear elastic mechanical properties with Machine Learning, researchers also utilized Machine Learning to predict the non-linear constitutive behaviors and mechanical responses of composite materials, like the entire stress-strain curve. Hashash et al. [23] developed a Neural Network constitutive model to replace the commonly used integration procedures in Finite Element Analysis. Then a consistent material stiffness matrix is derived based on the Neural Network constitutive model instead of conventional plasticity-based models. This new model leads to efficient convergence of the Finite Element Newton iterations. Bos et al. [24] developed a Neural Network-based constitutive model to capture the elastic-plastic stress-strain curve. The authors proposed to sub-sample the stressstrain curve at several discrete points and then predicted the stress values at different discrete strain values. The final curve was obtained by interpolating the discrete points. However, this approach still needs several discrete points for prediction, and there are errors during interpolation. Yang et al. [25] combined principal component analysis (PCA) and convolutional neural networks to predict the entire stress-strain behavior of binary composite materials. The authors showed that the PCA could effectively transform the stress-strain curve into a latent space and the prediction error is less than 10%. Kosmerl et al. [26] proposed a Neural Network by combining a convolutional neural network and residual neural network to predict the stress-strain curve for single-walled carbon nanotube configurations. These methods have shown the promising aspect of Neural Networks in obtaining the non-linear stress-strain behaviors of a targeting model. However, one key drawback of the above methods is that we need a large training dataset to better predict and represent the latent space features. Such massive training datasets are usually extremely challenging to generate as they come from experiments or numerical simulations.
In this paper, we propose a Parameterization-based Neural Network (PBNN) where we represent the non-linear stress-strain response of the chosen composite (here hollow particle reinforced geometries) with a parameterized function space. The parameters in the function can be different for different stress-strain curves. Then, we utilize the concepts of self-supervised learning [27] and transfer learning [28] to effectively extract the latent features from the composite geometric model using an Encoder-Decoder Neural Network. The key benefits of our approach are:
1. We use an Encoder-Decoder Neural Network for latent feature extraction from composite geometries (named Feature Extraction module). This Feature Extraction module reduces the input dimension from a 256-by-256 image to a 128-by-1 vector, simplifying the following Neural Network prediction task. It is easier to generate different matrices to represent the composite geometries, while it is time-consuming to solve each model numerically.
2. We use a parameterized representation of the stress-strain curve to capture the shape of the true stress-strain response. Otherwise, we usually need a 'physically meaningful' or higher-order smooth function to represent the stress-strain curve. This parameterized representation vastly reduces the prediction size as it requires fewer data during training to achieve a relatively good prediction.
3. We propose a module named 'Modification module' that predicts an auxiliary term to increase the stress-strain curve prediction accuracy. This Modification module serves as a constraint by predicting an extra data point on the stress-strain curve. Then it modifies the stress-strain curve predicted through Neural Network by adding a carefully constructed polynomial equation.
4.
Our proposed PBNN is not limited to the stress-strain curve prediction with a known function but works for other response prediction problems. For example, we can fit any arbitrary curve with a polynomial function, which will be the targeting parameterized function for our proposed PBNN.
Overview of the Machine Learning Framework
The PBNN framework consists of two key modules: 1) the Feature Extraction module and 2) the Curve Prediction module, as shown in Figure 1. The Feature Extraction module extracts a high-level feature vector from the syntactic foam geometry. Then the feature vector is brought into the Curve Prediction module, consisting mainly of a Dense module and a Modification module, to predict the final stress-strain curve. We perform the Machine Learning training on NVIDIA GeForce RTX 2080 SUPER with 3072 CUDA cores.
Feature Extraction Module
The framework for the Feature Extraction module is shown in Figure 2, which is constructed based on the Encoder-Decoder structure. The Encoder will extract the high-level features from the input model, and the Decoder will expand the extracted high-level features back to the input model. Here the Decoder is needed to train the Feature Extraction Module, as the loss function of the Feature Extraction Module is defined based on the input model. The module simplifies the Neural Network training problem by reducing the complex 256-by-256 image input into a 128-by-1 vector. This feature vector can be treated as a high-level equivalence of the corresponding input geometry. Then the prediction is simplified into a problem as predicting the stress-strain curve from a 128-by-1 feature vector instead of a 256-by-256 image matrix. Figure 2: Feature Extraction module framework: the yellow color blocks belong to the Encoder module, and the blue color blocks belong to the Decoder module, as shown in Figure 1. Light yellow and blue blocks are convolutional and deconvolutional layers, respectively. The dark yellow and dark blue colors are the ReLU activation layer. Block 'conv5' outputs a 'Latent Vector,' the high-level feature vector of the syntactic foam geometry. It will be further used in the Curve Prediction module.
Curve Prediction module and Modification module
The Curve Prediction module predicts the stress-strain curve from the extracted feature vector. This module consists of two main sub-modules: the Dense and Modification modules. The Dense module consists of three dense layers. The first two dense layers have 64 neurons at each layer, with a tanh activation function. A linear activation function follows the last dense layer and has a neuron size equal to the prediction size, which equals the size of function parameters plus one. For example, if the size of targeting parameters is three, the last dense layer will predict four neurons; three are function parameters, and the last is the 'end stress'. The 'end stress' is an auxiliary term that refers to the stress value at the maximum strain considered (15% strain in this paper). Predicted function parameters will determine the initial expression of the stress-strain curve. The Modification module will utilize the predicted 'end stress' to form a 'Modification function', which will be further added to the initial expression to reconstruct the initial curve by shifting it closer to the true stress-strain curve.
The Modification module is a critical component of this prediction framework. By predicting the function parameters, the value of each point on the curve will be a non-linear function of all coefficients, making it extremely challenging to add constraints to the values of these points. Moreover, the shape of the function can be sensitive to given parameters, so adjusting the function parameters might significantly change the curve shape. Thus, we need to add additional constraints to the predicted curve but simultaneously keep the prediction physically meaningful. So we incorporated the Modification function to serve as the constraint. The Modification function can be expressed as Equation 1.
g(λ) = (σ end − P (λ = 1.15))(λ − 1) 2 /0.15 2(1)
Where λ is the principle stretch defined as the ratio of the deformed length to the undeformed length along the principal axes. When a uniaxial stress is applied, λ is related to uniaxial strain as λ = + 1. σ end is the auxiliary prediction term, denoting the predicted 'end stress' from the dense module corresponding to stretch λ=1.15. P (λ) denotes the fitting function we choose to represent the stress-strain curve. Detailed expression of function P (λ) considered in this paper can be referred to in Section 4.2. P (λ = 1.15) denotes the true value of 'end stress' obtained from the fitting function. The term P (λ = 1.15) ensures the 'end stress' of the final stress-strain curve matches the predicted 'end stress'. '0.15' is the normalization term, representing the maximum strain for the stress-strain response considered. This normalization term ensures that the g(λ = 1.15) represents the gap between true 'end stress' and predicted 'end stress'. This g(λ = 1.15) is effectively pulling the 'end stress' of the predicted curve to the 'end stress' of the true curve.
Since this normalization term depends on the maximum strain considered, it needs to be updated accordingly for a different dataset used. We designed the Modification function as a quadratic function because the stress prediction error generally increases as the strain value increases and is easier to construct. The base term λ − 1 ensures the initial point of the curve is not shifted after adding the Modification function. When λ=1, there is no change to the final stress-strain curve. At the 'end stress', when λ=1.15, we only end up with σ end − P (λ = 1.15). Then the final stress-strain curve expression will be F (λ) = P (λ) + g(λ).
Syntactic Foam Computational Modelling
The syntactic foam micromechanical models are modeled using the Finite Element Method. A description of computational modeling, including boundary conditions and materials, is presented in Section 3.1 and 3.2. The parametric space of syntactic foam models and corresponding stress-strain curves are shown in Section 3.3.
Modelling and Boundary Value Problem
This paper considers 2D micromechanical syntactic foam models with randomly distributed microballoons in a matrix region. All micromechanical model dimensions are maintained at 0.295 mm x 0.295 mm, and the microballoon outer radius is maintained at 0.0225 mm. These micromechanical models are implemented within Finite Element software ABAQUS [29] with a mesh consisting of linear plane stress elements (CPS3 and CPS4). The mesh size is determined through mesh convergence analysis such that results are within 10% of the converged solution to reach a balance between convergence and computational cost. Since we have not considered strain-softening material behavior in our models, there is no pathological mesh dependency.
A schematic representation of the syntactic foam micromechanical model is shown in Figure 3. Here, Ω m represents the matrix region, Ω p represents the microballoon wall region, and Ω v represents the hollow region or void inside the microballoons. Γ 1−4 are the external boundaries of the micromechanics domain, and Γ i are the interfaces between each microballoon and the matrix region. The volume fraction of matrix is given by V m = Ωm Ω . The volume fraction of microballoons, including that of the particle wall and void, in a syntactic foam composite, is
V mb = 1−V m = Ωp+Ωv Ω .
Compressive displacement ∆ applied is applied on Γ 4 . Γ 1 is fixed from deforming only along the y-axis and Γ 2 only along the x-axis. A flat boundary condition is considered on Γ 3 to achieve uniform deformation along the y-axis. These
x = −∆ applied on Γ 4 u x = 0 on Γ 2 u y = 0 on Γ 1 u y = constant on Γ 3(2)
Constitutive Materials Description
This paper considers High-Density Polyethylene (HDPE) as the material for the matrix. HDPE manifests a non-linear behavior under compression. The compressive stress-strain material data for HDPE is obtained from [7,8], and is used as the input to fit an Ogden hyperelastic model. Prabhakar et al. [8] has shown that the Ogden function effectively represents the stress-strain relationship for pure HDPE material. The glass microballoons (GMBs) are modeled as a linear elastic material. HDPE and GMB material properties can be found respectively in [8]. This study considers the interfacial behavior between the GMBs and HDPE matrix as perfectly bonded.
Parametric Space and Stress-strain Curves
The syntactic foam micromechanical model has several significant geometric parameters which govern the parametric space: (1) Number of microballoons: We consider the microballoon volume fraction to range from 10% to 50%. The volume fraction represents the ratio between the volume occupied by the microballoon and the whole model. A higher volume fraction implies more microballoons within a fixed matrix volume. (2) Position of microballoons: Each microballoon can be randomly located within the matrix. (3) Size of microballoons: We consider uniform microballoon outer diameters, but two different microballoon wall thicknesses are considered: 1.08 and 2.16 micrometers. The 1.08-micrometer thickness is denoted as 'thin-wall', and the 2.16-micrometer thickness is denoted as 'thick-wall'. The wall-thickness significantly affects the mechanical properties of syntactic foam composites, including stress-strain response, modulus, and failure modes [1]. Examples of different syntactic foam micromechanical models are shown in Figure 4. This paper solves 6825 thin-wall and thick-wall syntactic foam models to obtain their stress-strain curves. Figure 5 shows sample stress-strain curves of thick-wall and thin-wall syntactic foam models with different microballoons volume fractions.
Machine Learning Model Inputs
Two inputs are used to set up the Machine Learning training process: the syntactic foam geometry and the corresponding stress-strain curve. We use 18,000 syntactic foam geometries, where 6825 models are solved to obtain their stress-strain responses. That is, 18,000 syntactic foam geometries were used to train the Feature Extraction module, and the 6825 syntactic foam models (thick-wall and thin-wall) were used to obtain stress-strain curves for further training the Curve Prediction module.
Syntactic Foam Geometry Representation
The 2D geometric representation of a syntactic foam includes three parts: the matrix labeled as '1', the microballoon wall labeled as '2', and the hollow region labeled as '0' shown in Figure 4 in blue, red, and white, respectively. Since the wall of the microballoon is very thin compared to the whole model, we construct a 256-by-256 Cartesian Map to ensure at least three layers of pixels along the microballoon wall thickness. Since boundary conditions are consistent for all models to calculate the stress-strain curve, boundary conditions are not considered part of the inputs.
Stress-Strain Curve Representation
The stress-strain curves for the syntactic foams are non-linear responses relating the compressive stresses with strains on the micromechanical RUC models. To predict the stress-strain curves using Machine Learning algorithms, we can treat them as a piecewise function by connecting several discrete points or a smooth function with a particular expression. In this paper, we consider three different representations: (1) linear piecewise function, (2) cubic polynomial function, and (3) Ogden function. We utilize the Mean Squared Error (MSE), defined as Equation 3, to quantify the fitting errors of different representations.
M SE = 1 n n i=1 (Y i −Ŷ i ) 2(3)
where n is the number of data points, Y i is the i-th observed value andŶ i is the i-th predicted value.
Linear piecewise function representation
To balance the need between capturing the shape of the curve with a linear piecewise function and not making the Machine Learning prediction too difficult, we consider 21 uniformly distributed discrete strain values from 0 to 15%, corresponding to 21 discrete points along the stress-strain curve. Then the PBNN will attempt to directly predict these 21 discrete points to represent the stress-strain response. After obtaining the values for all 21 discrete points, the overall curve will be the union of several discrete linear piecewise functions in different subdomains. Each j-th subdomain consists of two points X j and X j+1 , where X j is the j-th discrete strain point having the corresponding stress value of Y j . The linear piecewise function in the j-th subdomain can be expressed as in Equation 4.
Y = F j (X) = Y j + Y j+1 − Y j X j+1 − X j (X − X j )(4)
where X ∈ [X j , X j+1 ]. The ultimate linear piecewise function P (λ) can be expressed as P (λ) = j F j (λ) f or ∀λ ∈ [X j , X j+1 ], λ = + 1 is the principle stretch value in the defined domain of interest, as mentioned in Section 2.2 and refers to the strain value. j is a collection of all subdomains.
Cubic polynomial function representation
A polynomial function can also be used to represent the stress-strain curve. By testing the fitting errors of polynomials with different order using polynomial regression [30], we pick the cubic polynomial function expression and calculate the fitting coefficients. Since we know the stress value is zero when strain is zero or λ is 1, we carefully constructed the cubic polynomial function as Equation 5.
P (λ) = a 3 (λ − 1) 3 + a 2 (λ − 1) 2 + a 1 (λ − 1)(5)
Here a 1 , a 2 , a 3 are the coefficients of the cubic polynomial function. The obtained coefficient values are shown in Figure 6. We notice that the cubic polynomial function representations have fitting mean squared errors between 0.41 to 0.92 for all models considered. As the stress value ranges from 0 to around 20MPa, we believe such a fitting error is acceptable. Further, the cubic polynomial function can give a good representation of the stress-strain relationship.
Ogden function representation
As the HDPE manifests a non-linear behavior under compression, the stress-strain curve can also be fitted using the Ogden function through the non-linear fitting [8]. We consider the incompressible case as shown in Equation 6. Detailed derivations of the expression can also be found in our prior work [8].
P (λ) = n i=1 2µ i α i (λ αi−1 − λ − 1 2 αi−1 )(6)
This paper considers the 3rd-order Ogden model, which has n = 3 in Equation 6. Through non-linear regression, we can obtain the values of coefficients in the Ogden function as shown in Figure 7. From non-linear regression, we notice that the Ogden function fitting errors are between 0.02 to 0.54. Since α i appears on the denominator, we want to keep the sign of α i consistent for all models in the non-linear regression such that the Neural Network is less likely to predict a denominator value close to zero. This issue is avoided by adding constraints to the parameter values during non-linear regression, such that the sign of the regression coefficients will always be positive or negative. From Figure 7, the signs are consistent for all α i . Consequently, the Ogden function can also represent the stress-strain curve.
(a) (b) Figure 7: Values of parameters in the Ogden functions for all stress-strain curves considered: (a) µ 1 , µ 2 and µ 3 ; (b) α 1 , α 2 and α 3 . Figure 8 shows how each representation compares with the true stress-strain data. Here we notice that the linear piecewise function (black curve) can accurately capture the trend of the stress-strain curve in a non-smooth manner. The cubic polynomial function (pink curve) gives a smooth representation of the curve but is less accurate than the linear piecewise function. The Ogden function (red curve) gives the best approximation of true stress-strain curve, also a visually smoother solution than the linear piecewise function. This is because we use the Ogden hyperelastic model to fit the matrix material description for the FEA simulations that are used to generate the input composite stress-strain responses for this paper.
Loss Function and Training Process
We propose different training processes for different curve representations. We utilize the Mean Squared Error (MSE) as the loss function for linear piecewise and cubic polynomial function representations. For both representations, The Neural Network uses MSE as a loss function for both representations and is trained for 500 epochs, each with 30 steps. We propose a modified MSE loss function 'MMSE' for training the Ogden function representation. We know from the Ogden function that P (λ = 1) = dP dλ at (λ = 1) = µ 1 + µ 2 + µ 3 . The term P (λ = 1) represents the slope at the initial point on the stress-strain curve and is independent of the choice of α i . From our non-linear regression results in Figure 7, the value of µ i is more spread out than α i . Thus adding this constraint to the loss function is important to Figure 8: An example of stress-strain curve representation: blue dots are the true data, the black curve is fitted using linear piecewise function, the pink curve is fitted using the cubic polynomial function, and the red curve is fitted using the Ogden function.
constrain the prediction of different µ i and improve the prediction accuracy. Moreover, as the Modification function g(λ) is a quadratic function of λ − 1, adding the Modification function will not change the value of P (λ = 1).
The MMSE can be written as Equation 7.
M M SE = M SE(y,ỹ) + α · M SE[(P (λ = 1) | y) − (P (λ = 1) |ỹ)](7)
Where MSE is defined in Equation 3, α is a constant weight added to the loss term generated by initial slope P (λ = 1).
In this paper, we use α = 0.01. However, during the training process, we notice that directly training the Network with MMSE will lead to a blown-up prediction since the additional loss term creates more difficulty in finding the global minimum. So instead, we propose a 'hybrid' training process: the Neural Network is trained for 500 epochs. For the first 300 epochs, we utilize MSE as the loss function. Then we utilize MMSE for the subsequent 100 epochs and then use MSE again for the last 100. During this training process, MMSE tries to change the gradient descent direction from MSE and help avoid falling into saddle points or local minima.
Results and Discussion
In this section, we evaluate the performance of our Feature Extraction and Curve Prediction modules on different stress-strain curve representations. The Feature Extraction module is validated by checking if the 'Latent Vector' can effectively predict back to the original geometry. The Curve Prediction module is validated by checking how close the predicted curve is to the true curve. We randomly split the data into 60% training, 20% cross-validation, and 20% testing for all datasets used.
Feature Extraction Module Results
We first train the Feature Extraction module using 18000 syntactic foam geometries, with 60% training, 20% testing, and 20% cross-validation. Figure 9 shows the predicted syntactic foam geometries from the Feature Extraction module.
We can see that the predicted syntactic foam geometries can capture most of the critical features of the syntactic foams, even for complex geometries with high volume fractions, like Figure 9(a) and (b).
Moreover, to quantitatively measure how the Feature Extraction Module performs, we define an error rate (ER) to measure the prediction accuracy, as: where N x = N y = 256 is the size of the geometry Cartesian Map, mentioned in Section 4.1. G o represents the original geometry Cartesian Map, and G p represents the predicted geometry Cartesian Map. The error rate calculates the ratio between incorrectly predicted node values in the Cartesian Map and the total Cartesian Map nodes (N x * N y ). To evaluate how the training dataset affects the performance of the Feature Extraction Module, we first pick 2000 samples as the test set, and train the module with different training data sizes. We use three random split methods (we call random split seed) and calculate the error rate with different random splits to account for randomness. shows how the training loss decreases as we increase the training epochs (we run a total of 1000 epochs and each epoch has 40 steps) for training sample sizes of 7000, 10000, and 12500 each. We observe that the Feature Extraction Module optimization is converged within 1000 epochs. Further, Figure 11 shows how the prediction error changes with increasing training data size. Despite different random split methods, the prediction error rate decreases as we increase the training sample size. When the training data size is larger than 15000, we have a consistent prediction error rate below 3.5%. Thus in this paper, we choose a training data size of 18000 to train the Feature Extraction Module for extracting the corresponding high-level latent features of the syntactic foam geometries.
ER = Nx i=1 Ny j=1 1 [Go(i,j) =Gp(i,j)] N x * N y (8) (a) (b) (c) (d)
Curve Prediction Module Results
After fully training the Feature Extraction module, we can use the extracted feature vectors to predict the stress-strain curves using the Curve Prediction module. The prediction error is evaluated based on the errors at 21 uniformly distributed points along the curve (same as the discrete point locations when defining linear piecewise function). The errors are calculated based on Norm-2 Error (NE), defined in Equation 9
.
N E = n i=1 (Y (i) −Ŷ (i)) 2(9)
n = 21 represents the total size of the data points considered in one sample. The overall prediction errors are evaluated based on Mean Norm-2 Error and Max Norm-2 Error, which calculates the average and maximum Norm-2 Error in the testing sample set. Table 1 shows the prediction errors of different stress-strain curve representation methods using PBNN. Besides, Figure 12 gives two examples of the Stress-Strain curve prediction using different representations. Comparing the predictions for different representations, we can conclude that:
1. The Ogden function and cubic polynomial function representations give similar prediction accuracy by providing a smooth curve close to the true curve. The polynomial function representation has a lower Mean Norm-2 Error, but Ogden function representation has a lower Max Norm-2 Error. This is because the cubic polynomial function has fewer parameters to train, making obtaining a better prediction during the training process easier. However, as shown in Figure 8, the cubic polynomial function has fitting errors to the true curve.
2. The linear piecewise function representation gives a relatively worst prediction by having the most significant prediction error. Moreover, the linear piecewise function gives a non-smooth prediction, and the relative relationship between adjacent points might not obey the physical truth (like the horizontal line at the beginning of the blue curve in Figure 12(a), which is not true). Furthermore, we potentially introduce an additional error if we attempt to smooth the prediction obtained from the linear piecewise function.
(a) (b) Figure 12: Predicted stress-strain curves from the Curve Prediction module: (a) Example 1 (b) Example 2. The blue dots are the true data, the black curve is fitted using the linear piecewise function, the pink curve is fitted using the cubic polynomial function, and the red curve is fitted using the Ogden function.
Importance of Feature Extraction module, Modification module, and Hybrid Training Process
We introduce the ideas of different modules and training methods for PBNN. To understand the importance of the different aspects in PBNN, like the Feature Extraction module, Modification module, and the hybrid training process (only for Ogden function representation), this section compares PBNN's prediction error with several baseline models. We use the Ogden function representation for illustration purposes.
Baseline model 0 We first choose a baseline model that always outputs the mean stress-strain curve from the training dataset. The norm-2 error for the Baseline model 0 represents the variance of the testing dataset. To show the improved efficacy of other baseline models proposed, their norm-2 error should be lower than that of the Baseline model 0. Figure 13 shows a comparison between the mean curve and the training (a) and testing (b) datasets.
(a) (b) Figure 13: Comparison between the mean curve of the Baseline model 0 (black asterisk) and (a) training dataset (b) testing dataset
Baseline model 1 As discussed in Section 2.1, we utilize the Feature Extraction module to extract latent space features using easily obtained syntactic foam geometries. To validate the effectiveness of the Feature Extraction module, we consider a baseline model in which we directly use an Encoder structure (as discussed in Figure 2) to predict the Ogden parameters (named Baseline-1). To validate how the Modification module performs, we choose another baseline model by removing the Modification module from PBNN (named Baseline-2). We compare the prediction between PBNN with and without the Modification module. The detailed frameworks for the two models are shown in Figure 14, with and without the blue dashed line modules. Figure 14: Stress-Strain curve prediction using PBNN framework without modification module (Baseline-2) is shown using red color. PBNN with modification module is shown by adding blue dashed line modules to Baseline-2. Baseline-2 directly predicts the Ogden parameters from extracted latent features and constructs the Stress-Strain curve with the Ogden function. While PBNN also uses the Modification module to generate the Modification function P 2 and constructs the Stress-Strain curve by adding the Ogden function and Modification function.
Baseline model 2
Baseline model 3 (only for Ogden function representation) To understand how our proposed hybrid training improves the prediction accuracy, we consider another baseline model (named Baseline-3), which has the same structure as PBNN but only uses MSE as the loss function instead of MMSE.
Prediction results Here we show the prediction accuracy using our training data with 6825 syntactic foam models. Three different data-splitting methods are used, and the average error is calculated. The results are shown in Table 2.
From the results, we notice that:
1. The Baseline-0 has the largest mean Norm-2 Error compared to all the other models, while the max norm-2 error is lower than Baseline-1 and Baseline-2. This means Baseline-1 and Baseline-2 could make predictions beyond the range of the test set curves. On the other hand, Baseline-3 and PBNN have better predictions than Baseline-0 for both mean norm-2 error and max norm-2 error.
2. Baseline-2 has slightly better prediction than Baseline-1, meaning that the Feature Extraction module can extract the high-level features better and improve the curve prediction accuracy.
3. Baseline-3 significantly improves the prediction accuracy compared to Baseline-1 and Baseline-2, proving that the modification module is essential to achieving better prediction accuracy.
4. PBNN can give a better prediction than Baseline-3, proving that the hybrid training procedure improves the prediction accuracy.
Moreover, we can visualize the effect of the modification module in Figure 15, which validates the predictions of PBNN and Baseline-2 for two randomly picked test datasets (test set 1 and test set 2) from the total 20% test data set. The graph shows that the modification module could effectively push the initially predicted curve closer to the true curve and enhance the stress-strain curve prediction accuracy. Figure 15: Importance of the modification module: stress-strain curve prediction between Baseline-2 and PBNN on test set samples: (a) Curve prediction on test sample 1 (b) Curve prediction on test sample 2. The blue curve is the true Stress-Strain curve, the green curve is the predicted curve from Baseline-2, and the red curve is the predicted curve from PBNN.
Conclusions
This paper proposes a Parameterization-base Neural Network (PBNN) framework to predict the non-linear stress-strain responses of composites. We choose syntactic foam composites to develop these frameworks due to their complex internal architecture and corresponding mechanical responses. Instead of predicting discrete points along the stressstrain curve, we propose the stress-strain curve representations using the cubic polynomial function and Ogden function and utilize PBNN to predict corresponding function parameters. This vastly reduces the computational cost and data size needed for training. By comparing different baseline models, we further show that PBNN achieves a better prediction accuracy of the stress-strain curve than other baseline models.
The main conclusions and contributions of this paper are:
1. This is the first attempt to our knowledge to predict non-linear stress-strain responses by treating them as a parameterized function, especially for the complex composite material analysis.
Figure 1 :
1Overall framework for predicting the non-linear mechanical responses with PBNN: (1) Orange block is the Feature Extraction module, which extracts the high-level features (Latent Vector) of the syntactic foam geometry using an Encoder-Decoder structure. (2) Green block is the Curve Prediction module, which predicts the stress-strain curve from the extracted high-level features using a Dense module and a Modification module.
Figure 3 :
3An example of 2D syntactic foam micromechanical model boundary conditions are commonly used in micromechanics modeling of composites, and the corresponding boundary conditions are:û
Figure 4 :Figure 5 :
45Example of 2D RUC models of thick-wall syntactic foam geometries with different volume fractions of (a) 10% (b) 30% (c) 50%: Blue color refers to the matrix, red color refers to the wall of microballoon, and white color refers to the hollow region. Example of thick-wall and thin-wall syntactic foam stress-strain curves with different volume fractions: (a) thick-wall syntactic foam stress-strain curves; (b) thin-wall syntactic foam stress-strain curves. 'VF=10%' denotes 10% microballoon volume fraction
Figure 6 :
6Values of parameters in the cubic polynomial functions for all stress-strain curves considered for: (a) a 1 ; (b) a 2 ; (c) a 3 .
Figure 9 :
9True syntactic foam geometry versus Predicted syntactic foam geometry obtained from the Feature Extraction module. Thick-wall syntactic foam -(a) real and (b) predicted geometry. Thin-wall syntactic foam -(c) real and (d) predicted geometry
Figure 10 :
10Training loss profile of Feature Extraction Module (a) using 7000 samples (b) using 10000 samples (
Figure 11 :
11Prediction error rate of Feature Extraction Module, on 2000 testing samples (a) random split seed 1 (b) random split seed 2 (c) random split seed 3
Table 1 :
1Curve Prediction Error for different curve representations with PBNNMean Norm-2 Error Max Norm-2 Error
Linear Piecewise Function
7.31
21.64
Cubic Polynomial Function
5.03
18.73
Ogden Function
5.89
14.89
Table 2 :
2Curve Prediction Error for different frameworksMean Norm-2 Error Max Norm-2 Error
Baseline-0
8.14
18.26
Baseline-1
7.51
30.12
Baseline-2
6.69
30.08
Baseline-3
6.14
18.18
PBNN
5.89
14.89
(a)
(b)
. We have shown that our method could simplify the Machine Learning problem and generate a 'physically meaningful' prediction by utilizing the Feature Extraction module and the Modification module. The Feature Extraction module extracts the high-level features from the microstructure geometry into a latent vector, serving as reduced-order input to the ML framework. The Modification module improves the prediction accuracy by referring to an auxiliary prediction and reconstructing the predicted stress-strain curve expression.3. We have demonstrated that PBNN can predict general polynomial functions (like a cubic polynomial function) or complex highly non-linear functions (like an Ogden function) from internal material microstructures.4. Our method is not limited to syntactic foam or composite material stress-strain prediction, and we can use a similar approach for all curve-related predictions. Our method can also be extended to develop knowledge/physics-guided Machine Learning algorithms with the proposed Feature Extraction module and Modification module.
AcknowledgementsThe authors would also like to acknowledge the support from the University of Wisconsin Graduate Fellowship for partially supporting Haotian Feng's doctoral studies.AcknowledgementsData AvailabilityData will be made available by the authors upon reasonable request.• Code and Data availability The entire PBNN framework and baseline models can be found on our GitHub page: https: //github.com/Isaac0047/Parameterization-based-Neural-Network.git. This includes the entire implementation code with model generation, data processing, PBNN framework setup and baseline models setup. A detailed description the steps involved with running the PBNN framework is included in the readme file on this GitHub page.• Authors' contributions P.P. and H.F. conceptualized and developed this study. H.F. implemented the research presented in this paper. H.F. and P.P. evaluated the outcomes of the work and drafted the manuscript.
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| [] |
[
"High Chern number van der Waals magnetic topological multilayers MnBi 2 Te 4 /hBN",
"High Chern number van der Waals magnetic topological multilayers MnBi 2 Te 4 /hBN"
] | [
"Mihovil Bosnar \nDonostia International Physics Center\n20018Donostia-San SebastiánSpain\n\nDepartamento de Polímeros y Materiales Avanzados: Física\nFacultad de Ciencias Químicas\nQuímica y Tecnología\nUniversidad del País Vasco UPV/EHU\n20018Donostia-San SebastiánSpain\n",
"Alexandra Yu Vyazovskaya \nTomsk State University\n634050TomskRussia\n\nSaint Petersburg State University\n199034Saint PetersburgRussia\n",
"Evgeniy K Petrov \nTomsk State University\n634050TomskRussia\n\nSaint Petersburg State University\n199034Saint PetersburgRussia\n",
"Evgueni V Chulkov \nDonostia International Physics Center\n20018Donostia-San SebastiánSpain\n\nDepartamento de Polímeros y Materiales Avanzados: Física\nFacultad de Ciencias Químicas\nQuímica y Tecnología\nUniversidad del País Vasco UPV/EHU\n20018Donostia-San SebastiánSpain\n\nTomsk State University\n634050TomskRussia\n\nSaint Petersburg State University\n199034Saint PetersburgRussia\n",
"Mikhail M Otrokov \nCentro de Física de Materiales (CFM-MPC)\nCentro Mixto (CSIC-UPV/EHU)\n20018Donostia-San SebastiánSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n"
] | [
"Donostia International Physics Center\n20018Donostia-San SebastiánSpain",
"Departamento de Polímeros y Materiales Avanzados: Física\nFacultad de Ciencias Químicas\nQuímica y Tecnología\nUniversidad del País Vasco UPV/EHU\n20018Donostia-San SebastiánSpain",
"Tomsk State University\n634050TomskRussia",
"Saint Petersburg State University\n199034Saint PetersburgRussia",
"Tomsk State University\n634050TomskRussia",
"Saint Petersburg State University\n199034Saint PetersburgRussia",
"Donostia International Physics Center\n20018Donostia-San SebastiánSpain",
"Departamento de Polímeros y Materiales Avanzados: Física\nFacultad de Ciencias Químicas\nQuímica y Tecnología\nUniversidad del País Vasco UPV/EHU\n20018Donostia-San SebastiánSpain",
"Tomsk State University\n634050TomskRussia",
"Saint Petersburg State University\n199034Saint PetersburgRussia",
"Centro de Física de Materiales (CFM-MPC)\nCentro Mixto (CSIC-UPV/EHU)\n20018Donostia-San SebastiánSpain",
"Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain"
] | [] | Chern insulators are two-dimensional magnetic topological materials that conduct electricity along their edges via the one-dimensional chiral modes. The number of these modes is a topological invariant called the first Chern number C, that defines the quantized Hall conductance as Sxy = Ce 2 /h. Increasing C is pivotal for the realization of low-power-consumption topological electronics, but there has been no clear-cut solution of this problem so far, with the majority of existing Chern insulators showing C = 1. Here, by using state-of-the-art theoretical methods, we propose an efficient approach for the realization of the high-C Chern insulator state in MnBi2Te4/hBN van der Waals multilayer heterostructures. We show that a stack of n MnBi2Te4 films with C = 1 intercalated by hBN monolayers gives rise to a high Chern number state with C = n, characterized by n chiral edge modes. This state can be achieved both under the external magnetic field and without it, both cases leading to the quantized Hall conductance Sxy = Ce 2 /h. Our results therefore pave way to practical high-C quantized Hall systems. arXiv:2212.13457v1 [cond-mat.mes-hall] | 10.1038/s41699-023-00396-y | [
"https://export.arxiv.org/pdf/2212.13457v1.pdf"
] | 255,186,390 | 2212.13457 | c91d9deda5cfbafcaed0fd07ccf096fd5e98565f |
High Chern number van der Waals magnetic topological multilayers MnBi 2 Te 4 /hBN
Mihovil Bosnar
Donostia International Physics Center
20018Donostia-San SebastiánSpain
Departamento de Polímeros y Materiales Avanzados: Física
Facultad de Ciencias Químicas
Química y Tecnología
Universidad del País Vasco UPV/EHU
20018Donostia-San SebastiánSpain
Alexandra Yu Vyazovskaya
Tomsk State University
634050TomskRussia
Saint Petersburg State University
199034Saint PetersburgRussia
Evgeniy K Petrov
Tomsk State University
634050TomskRussia
Saint Petersburg State University
199034Saint PetersburgRussia
Evgueni V Chulkov
Donostia International Physics Center
20018Donostia-San SebastiánSpain
Departamento de Polímeros y Materiales Avanzados: Física
Facultad de Ciencias Químicas
Química y Tecnología
Universidad del País Vasco UPV/EHU
20018Donostia-San SebastiánSpain
Tomsk State University
634050TomskRussia
Saint Petersburg State University
199034Saint PetersburgRussia
Mikhail M Otrokov
Centro de Física de Materiales (CFM-MPC)
Centro Mixto (CSIC-UPV/EHU)
20018Donostia-San SebastiánSpain
Basque Foundation for Science
IKERBASQUE
48009BilbaoSpain
High Chern number van der Waals magnetic topological multilayers MnBi 2 Te 4 /hBN
Chern insulators are two-dimensional magnetic topological materials that conduct electricity along their edges via the one-dimensional chiral modes. The number of these modes is a topological invariant called the first Chern number C, that defines the quantized Hall conductance as Sxy = Ce 2 /h. Increasing C is pivotal for the realization of low-power-consumption topological electronics, but there has been no clear-cut solution of this problem so far, with the majority of existing Chern insulators showing C = 1. Here, by using state-of-the-art theoretical methods, we propose an efficient approach for the realization of the high-C Chern insulator state in MnBi2Te4/hBN van der Waals multilayer heterostructures. We show that a stack of n MnBi2Te4 films with C = 1 intercalated by hBN monolayers gives rise to a high Chern number state with C = n, characterized by n chiral edge modes. This state can be achieved both under the external magnetic field and without it, both cases leading to the quantized Hall conductance Sxy = Ce 2 /h. Our results therefore pave way to practical high-C quantized Hall systems. arXiv:2212.13457v1 [cond-mat.mes-hall]
I. INTRODUCTION
The Chern insulator (CI) state is a quantum phase of two-dimensional (2D) gapped materials with broken time-reversal invariance and nontrivial electronic band topology [1][2][3][4]. It is most straightforwardly probed via Hall measurements, the hallmark being a vanishing longitudinal conductance S xx along with a transversal conductance S xy quantized to integer multiples of the conductance quantum, Ce 2 /h [5,6]. Here, e is the electron charge, h is the Planck's constant, and C is a dimensionless integer called the first Chern number, corresponding to the number of the 1D gapless chiral modes residing at the CI film's edge. The existence of these modes is guaranteed by the nontrivial band topology of CI.
The edge modes of a CI conduct electricity without dissipation, which could be useful for the construction of novel highly efficient chiral interconnects for low-power-consumption electronics [7,8]. However, the contact resistance between a metal electrode and CI in the envisioned interconnect devices is a bottleneck limiting their performance. To reduce this resistance as much as possible, the number of chiral edge modes, i.e., the Chern number C, should be as large as possible [7,8]. Therefore, it is of great interest and importance to engineer CIs with high Chern number.
Historically, the CI state was first observed in 2D electron gases in 1980 in a transport phenomenon that * [email protected] † [email protected] is now known as quantum Hall effect (QHE) [5]. The QHE in this system stems from the formation of Landau levels under the external magnetic field, which drives the system into a topologically-nontrivial state. However, well-defined Landau levels are only possible in systems with high carrier mobility under strong external magnetic fields, which prevents this QHE from a wide applied use.
Notwithstanding, the developments in the research field of magnetic topological insulators (TIs) in the last decade have allowed a qualitative leap towards QHE without Landau levels. In particular, the quantum anomalous Hall effect (QAHE), a special kind of the QHE that occurs without the external magnetic field, has been observed [9]. It is mainly realized in the thin films of TIs of the (Bi,Sb) 2 Te 3 family doped by Cr or/and V atoms [9][10][11][12][13][14]. In these systems C = 1, and although it is theoretically possible to increase C by increasing the dopant concentration and the film thickness [15], this has not been experimentally realized to date.
Instead, complex materials engineering has been resorted to in order to achieve C = n > 1 state based on the magnetically doped TIs using the following idea. Rather than seeking a high-C state in a particular system, it can be realized by stacking of n CI layers with C = 1 each.
In this case, it is necessary, however, that the adjacent CI layers are efficiently decoupled from each other by a normal, i.e. topologically trivial, insulator layer. In this way, C up to 4 and 5 have been achieved in (Cr,V) x (Bi,Sb) 2−x Te 3 /CdSe [16] and heavily Cr-doped (Bi,Sb) 2 Te 3 /Cr x (Bi,Sb) 2−x Te 3 multilayers [17], respectively. Although these studies represent a proof-of-concept of the C enhancement approach, it is well known that the potential of magnetically doped TIs for the QAH-based applications is quite limited. Namely, due to a strong disorder in the pnictogen sublattice, which is randomly occupied by Bi, Sb and magnetic dopants, both electronic [18,19] and magnetic [20] properties of such materials are strongly inhomogeneous. Therefore, the observation of the QAHE in these systems appears to be limited to about 2 K at best [12,21], with no further improvements achieved over the last several years [12].
Recently, new systems showing the C = 1 QAHE have emerged, such as the intrinsic magnetic topological insulators of the MnBi 2 Te 4 (MBT) family [22][23][24], the twisted bilayer graphene [25] and the transition metal dichalcogenide moiré superlattices [26], opening new opportunities for C engineering. MBT, shown in Fig. 1(a), appears as particularly promising due to its van der Waals (vdW) nature, intrinsic combination of nontrivial band topology and long-range antiferromagnetic order (T Néel = 25 K), as well as large predicted surface band gap [22,23,[27][28][29][30][31][32][33][34][35][36][37][38][39]. The QAHE in thin MBT flakes has been achieved up to about 1.4 K, leaving a large room for the observation temperature enhancement. Indeed, a recent study [33] demonstrates the actual potential of this material by registering the C = 1 (C = 2) QHE up to 30 K (13 K) in its thin flakes, ferromagnetically polarized by external magnetic field. Remarkably, the quantization in this case appears without the Landau levels, in contrast to the conventional QHE observed in 2D electron gas [5].
Here, inspired by the recent progress on the Q(A)HE in MBT, we propose a novel MBT-based high Chern number material. Namely, we design a multilayer vdW heterostructure, in which thin MBT CI films are stacked on top of each other, interlayed by hexagonal boron nitride (hBN) monolayers that decouple and insulate them from one another ( Fig. 1(b)). As an inert wide band gap insulator, hBN is an ideal material for such a decoupling, widely used in vdW heterostructure devices as an encapsulating layer or substrate for the stacked 2D materials [40][41][42][43]. Using the state-of-the-art density functional theory and tight-binding calculations, we show that the weak vdW bonding between MBT and hBN essentially preserves the C = 1 CI state in the individual MBT layers. This state can correspond to either (i) the QH insulator phase achieved in thin MBT films under the external magnetic field, but without the Landau levels or (ii) the QAH insulator phase at zero field, if the MBT films are made of the odd number of septuple layer blocks. In either case, stacking n MBT films with C = 1 interlayed by (n − 1) hBN monolayers gives rise to a C = n CI state, with n as large as allowed by the vdW heterostructures growth technology. Our results provide excellent platform for realization of the high-C Chern insulators.
II. RESULTS
A. Crystal and electronic structure of MBT/hBN interface MnBi 2 Te 4 crystallizes in the trigonal R3m-group structure [44,45], made up of septuple layer (SL) blocks, in which hexagonal atomic layers are stacked in the Te-Bi-Te-Mn-Te-Bi-Te sequence, as shown in Fig. 1(a). Neighboring SLs are bound by vdW forces. Below T Néel = 25 K, MnBi 2 Te 4 orders antiferromagnetically due to the antiparallel alignment between the alternating ferromagnetically-ordered Mn layers [22,31], with the local moments pointing out-of-plane (Fig. 1a).
We start our study from the consideration of structural, magnetic and electronic properties of the MBT/hBN bilayer made of the 2 SL thick MBT film (MBT 2SL ) and hBN monolayer. This can be considered a minimal system because it contains all of the essential characteristics of MBT, such as intra-and interlayer exchange couplings as well as the non-trivial topology in the forced FM state [27], so it can be used to test whether they are affected by hBN.
The MBT and hBN basal planes are symmetry compatible and show a good lattice parameter matching in the MBT
(1 × 1)/hBN( √ 3 × √ 3) configuration, with a mismatch of only about 0.6 %.
The optimal hBN adsorption geometry was then determined by the comparison of total energies of structurally optimized MBT 2SL /hBN in four high-symmetry registries of such configuration, shown in Supplementary Fig. S3 Fig. 2(a)), E AFM and E FM (in meV) are the energies of the respective interlayer magnetic states relative to that of the AFM-hollow case (whose energy is set to zero), interlayer FM and AFM spin configurations of MBT 2SL , assuming the out-of-plane magnetic moment direction, to determine a possible influence of hBN on the interlayer exchange coupling, which is in MBT significantly weaker than the intralayer one (the latter is addressed below, as well). For more details, see Supplementary Note SII A 1.
∆E A/F = E AFM − E FM (in meV per
The relevant numerical results are listed in Table. I. It can be seen that for all adsorption registries the interlayer distance d between MBT and hBN is about 3.5 − 3.6 A, and the AFM spin configuration is lower than the FM configuration by at least ∆E A/F = E AFM − E FM 1 meV per Mn pair. The lowest energy is obtained for the hollow site geometry, shown in Fig. 2(a) and Supplementary Fig. S3(a). The large d on one hand and the overall similarity of the energy differences ∆E A/F in MBT 2SL /hBN and MBT 2SL [27] on the other suggest the vdW bonding of MBT and hBN.
Having determined the MBT/hBN interface geometry, we can explore the effect of hBN on the MBT 2SL electronic structure and topology. Since we are interested in the CI state, the FM interlayer spin alignment is considered here, for which MBT 2SL has been predicted to have C = 1 [27]. Noteworthy, in experiments the FM alignment in MBT is achieved by the external magnetic field application [23,32,33,46].
Comparison of the band structure along the K − Γ − M path for MBT 2SL /hBN in the hollow site registry and pure MBT 2SL is shown in Fig. 2(b). Their similarity near the Fermi level is immediately obvious and the band gaps, 63.5 and 62.7 meV, respectively, are very close. The hBN states can be found about 1.5 eV below the Fermi level (and deeper) as well as over 3.2 eV above it. Furthermore, Fig. 2(c) shows that the Fermi level dependencies of the anomalous Hall conductance, S xy (E F ), of MBT 2SL /hBN and MBT 2SL are very well matched, too. In particular, in both cases S xy is constant inside the band gap, where the actual Fermi level is, and equal to one conductance quantum e 2 /h, indicating the CI state with C = 1. Accordingly, the edge spectral function of MBT 2SL /hBN in Fig. 2(d) features a single chiral edge mode traversing the band gap, similar to MBT 2SL . Finally, a Wilson loop method calculation for MBT 2SL /hBN yields |C| = 1 as well, with the sign depending on the magnetization direction, as expected for a CI [4].
The calculations of the band structures, S xy (E F ) and C for the other three high-symmetry registries show that the same results hold for all of them (see Table I We note that recently it has been reported elsewhere [47] that while hBN and MBT are bound by vdW interaction, the interlayer FM configuration in MBT 2SL /hBN becomes significantly lower in energy than the AFM one (by up to 45 meV) for any adsorption registry. We have attempted to reproduce those results by retracing the steps outlined in Ref. [47] (see Supplementary Note SII A 2), but arrived to the same results that we present here. We believe that our result is correct, on the physical basis that the vdW interaction along with the insulating character of hBN should not produce such a drastic effect on magnetism as it was found in Ref. [47].
Finally, we confirm the implicit assumptions of preference for the FM intralayer spin configuration and the out-of-plane easy axis direction by total energy calculations on MBT 1SL /hBN (see Supplementary Notes SII A 3 and SII A 4, respectively). The former calculation reveals that the FM configuration is by 16.9 meV (per Mn pair) lower than the AFM configuration while the latter yields a positive magnetic anisotropy energy of 0.07 meV per Mn atom (vs. 0.074 meV in pure MBT 1SL ), meaning that the easy axis indeed stays out-of-plane. Thus, neither the intralayer magnetic order of MBT nor its magnetic anisotropy are not changed by interfacing with hBN. The above results concerning the insensitivity of magnetic, electronic and topological properties of MBT to hBN should hold for thicker MBT films as well because of the vdW nature of the bond. Thus, we conclude that hBN can be efficiently used to decouple MBT CI layers from each other, without altering their properties.
B. High-C state in the forced FM phase
We can now proceed with the study of the topological properties of the MBT/hBN multilayer heterostructures. We first note that in experiments the CI state in the forced FM phase, achieved through the application of the external magnetic field, is observed in thin MBT flakes made of both even and odd number of SLs [23, 32-39, 48, 49]. According to the previous density functional theory calculations [27], the minimal MBT film thickness required for the realization of such a C = 1 state is two SL blocks. Let us therefore consider nMBT 2SL /hBN, n > 1, heterostructures, in which n films of MBT 2SL are interlayed by (n − 1) hBN monolayers, as schematically depicted in Fig. 1(b). We will assume that all MBT 2SL films are FM-polarized in the +z direction by the external magnetic field. These heterostructures were constructed based on the structure of the MBT/hBN/MBT system that was determined after a series of calculations outlined in the Supplementary Note SII B 1. Fig. 3 shows the calculated low-energy band structures along the K − Γ − M path for nMBT 2SL /hBN with n = 2, 3, 4 and 5. The band structures basically correspond to that of a free-standing MBT 2SL repeated n times, slightly shifted in energy due to a slight variation of the electrostatic potential across the multilayer. The bands stemming from the spatial inversion equivalent MBT 2SL layers come in pairs, as it is seen in insets to Fig. 3.
In the corresponding S xy (E F ) dependencies, shown in Fig. 4(a), there are plateaus in the band gap that are equal to an integer number of conductance quanta, the integer being equal to n, suggesting C = n state in the respective multilayers. Accordingly, in the plots of the calculated edge spectral functions, shown in Fig. 4(b), two (three) edge modes can be seen for the n = 2 (n = 3) system.
From these results we can conclude that C = n in the examined heterostructures and infer by induction that the same will hold for heterostructures with greater n.
C. High-C QAH state: odd number of SLs
In the thin films made of an odd number of SLs, MBT has been predicted [27,29] and subsequently experimentally confirmed [23] to show the intrinsic C = 1 QAHE. We therefore now turn to the topological properties of the MBT/hBN heterostructures based on the thinnest possible odd-SL MBT film, i.e. MBT 3SL [27]. A comparison of S xy (E F ) calculated for MBT 3SL /hBN and MBT 3SL is shown in Fig. 5. A good matching between the two can be observed, especially for the flat portion in the band gap where S xy = e 2 /h for both, confirming insensitivity of the C = 1 QAH state of MBT 3SL to interfacing with hBN.
As we are seeking the high-C QAH state realized at zero external field, the question arises whether it is supported by the magnetic ground state of nMBT 3SL /hBN, n > 1. While we have shown above that the magnetic ordering inside MBT film is not affected by hBN, the interlayer exchange coupling between the local Mn moments through hBN should now be studied. Our total-energy calculations show that the AFM configuration is ∼ 0.03 meV (per 2 Mn atoms) lower in energy than the FM one (see Supplementary Note SIII). Such a small energy difference is due to the large separation between neighboring Mn planes (about 17.8Å) and vdW coupling between MBT and hBN. Although this number is at the limit of our calculation accuracy, assuming that its sign is correct yields a zero (non-zero) net magnetization in nMBT 3SL /hBN with even (odd) n. As the net C is a sum of C's of individual MBT 3SL films, with the sign of C changing with the magnetization direction, our result predicts that an nMBT 3SL /hBN heterostructure should have C equal to 0 (1) in its ground state.
However, the weakness of the interlayer exchange coupling through hBN makes a reliable prediction of its sign hardly possible using the density functional theory, so it should be defined in future experiments. If the FM coupling through hBN occurs experimentally, the Chern numbers of the individual MBT films will sum up to create the high-C state. However, even if the experiment would show the worst-case scenario of AFM coupling through hBN, a metamagnetic state that is appropriate for achieving the zero-field high-C state in the nMBT 3SL /hBN (n > 1) multilayers can nevertheless be "prepared" by the external magnetic field, in a manner we will describe below. In this state, the alignment of the MBT SLs across hBN would be FM, while the interlayer alignment inside MBT films stays AFM, thus guaranteeing an overall non-zero net magnetization increasing with n, in turn yielding the desired C = n state. Crucial for the realization of such a state is a fact that the magnetic anisotropy energy of MBT 3SL is ∼ 0.22 meV [27], several times larger than the energy difference of AFM and FM configurations across hBN. Note that the magnetic anisotropy energy of MBT 3SL should not change appreciably upon its interfacing with hBN, as shown in Sec. II A. We therefore propose the following procedure to induce the above described metamagnetic state. First, the external magnetic field of about 10 [23,32,33] should be applied to overcome the AFM coupling inside individual MBT 3SL films and polarize all SLs along the same direction. Next, the external magnetic field should be gradually reduced to zero leading to recovery of the uncompensated interlayer AFM state with non-zero magnetization in each individual MBT 3SL film [23,27]. However, their net magnetizations will remain parallel to each other even if the exchange coupling across hBN is AFM because the energy barrier due to anisotropy prevents the magnetization relaxation into the only slightly more favorable AFM configuration across hBN. Indeed, a similar metamagnetic state has been recently experimentally observed in bulk MnBi 4 Te 7 and MnBi 6 Te 10 [50][51][52][53][54], where the uniaxial anisotropy dominates over the AFM interlayer exchange coupling [54].
The magnetic anisotropy energy of MBT 3SL is slightly larger, while the exchange coupling across hBN is significantly weaker than those in MnBi 4 Te 7 and MnBi 6 Te 10 (c.f. Ref. [53]), making the proposed realization of the metamagnetic state in nMBT 3SL /hBN feasible.
We now have a firm basis to claim a possibility of the high-C QAH state realization in nMBT 3SL /hBN, n > 1. As in the preceding section, the high-C state can be demonstrated by the S xy (E F ) and the edge electronic structure calculations, results of which is shown in Fig. 5. Analogously to the nMBT 2SL /hBN multilayers, the S xy (E F ) for nMBT 3SL /hBN with n = 2 shows a clear plateau within the band gap, where the conductance is equal to two conductance quanta, 2e 2 /h, i.e. C = n = 2. Moreover, the edge electronic structure shows two edge modes traversing the band gap.
From these results it can be inferred that the nMBT 3SL /hBN, n > 1, heterostructures have C = n either in the metamagnetic state or even intrinsically, depending on the actual interlayer coupling over hBN. Similar results are expected for MBT/hBN multilayers based on MBT 5SL and MBT 7SL films.
III. DISCUSSION
Using ab initio and tight-binding calculations, we studied novel MnBi 2 Te 4 /hBN multilayer heterostructures in which thin MnBi 2 Te 4 films are interlayed by hexagonal boron nitride monolayers. The van der Waals bonding between hBN and MnBi 2 Te 4 preserves the magnetic and electronic properties of the the latter, in particular, the C = 1 Chern insulator state. Taking advantage of this, we showed that a stack of n MnBi 2 Te 4 films with C = 1 in the MnBi 2 Te 4 /hBN multilayer gives rise to a high Chern number state, C = n, characterized by n chiral edge modes.
There are two ways to achieve this state in the proposed heterostructures. The first way is to use the external magnetic field to drive MnBi 2 Te 4 films into a forced ferromagnetic state, which is nowadays widely used to observe a new kind of the quantum Hall effect that does not require formation of Landau levels [23,[32][33][34][35][36][37][38][39]48]. In this case, both even and odd septuple layer MnBi 2 Te 4 films can be used. One may well expect the high Chern number state to persist up to the temperatures as high as 20 − 30 K, as previously observed for the C = 1 state in MnBi 2 Te 4 thin flakes [33,37]. The second way relies on use of the odd septuple layer films as they realize the quantum anomalous Hall state intrinsically [23]. Although a prior application of the external field might be needed to align the net magnetizations of the individual MnBi 2 Te 4 slabs (as it is done in the Cr-doped (Bi,Sb) 2 Te 3 [9,55]), the high Chern number quantum anomalous Hall state can later be observed in remanence, i.e. at zero field. Currently, the observation temperature of the C = 1 quantum anomalous Hall effect in MnBi 2 Te 4 is about 1.4 K. Improving the structural quality of MnBi 2 Te 4 should allow the resolution of the intensely debated issue of its Dirac point gap size [22,[56][57][58][59][60] and push the effect observation temperature towards MnBi 2 Te 4 's Néel point. The steps in this direction are currently underway [35,48,61].
MnBi 2 Te 4 thin films currently used in the quantized transport measurements are mostly obtained by exfoliation [23,[32][33][34][35][36][37][38][39]. Using exfoliation and transfer, which is a standard technique for construction of the van der Waals heterostructures [42,43], should be suitable to realize the here proposed MnBi 2 Te 4 /hBN multilayers as well. However, this approach will not necessarily favor formation of the interface structure with the lowest energy, but it could rather result in an arbitrary non-symmetric alignment between MnBi 2 Te 4 and hBN. Fortunately, our results show that the Chern insulator state of individual MnBi 2 Te 4 layers is not sensitive to the MnBi 2 Te 4 /hBN interface registry. Therefore the exfoliation and transfer strategy should be appropriate to synthesize the here proposed MnBi 2 Te 4 /hBN multilayers. The accumulated world-wide experience in the synthesis of two-dimensional van der Waals heterostructures is expected to greatly facilitate a prompt realization of the high Chern number state in the MnBi 2 Te 4 /hBN system.
In conclusion, we presented a novel concept of realization of a high Chern number magnetic topological insulator state, which relies on the use of fundamentally different, but highly compatible van der Waals materials, assembled in a multilayer heterostructure. While there is hardly a good alternative to hBN as an inert wide band gap monolayer insulator, other choices of the Chern insulators are in principle possible, such as transition metal dichalcogenide moiré superlattices
IV. METHODS
The first-principles calculations were carried out on the level of density functional theory (DFT) as implemented in the Vienna Ab-initio Simulation Package (VASP) [65][66][67][68][69]. All calculations shared several common parameters: the VASP's PAW datasets [69] were used, the plane wave cutoff was set to 500 eV, the spin-orbit coupling (SOC) was enabled, the exchange-correlation functional was that of Perdew, Burke and Ernzerhof (PBE) [70], while the vdW interaction was taken into the account through the Grimme D3 model with the Becke-Jonson cutoff [71,72]. The Mn 3d-states were treated employing the GGA+U approach [73] within the Dudarev scheme [74]. The U eff = U − J value for the Mn 3d-states was chosen to be equal to 5.34 eV, as in previous works [22,56].
The DFT calculations were carried out with 11×11×1 Monkhorst-Pack grid used to sample the Brillouin zone (BZ) for all but n = 5 system, where 9 × 9 × 1 Monkhorst-Pack grid was used instead, and the electronic convergence threshold was 10 −6 eV. For structural relaxations, VASP's conjugate gradient algorithm was employed until the force on each atom decreased under 0.01 eV/Å, and the electronic occupations were smeared by the Gaussian function of width of 0.01 eV. For the static calculations the tetrahedron method with Blöchl corrections was used to treat the electronic occupations instead. The noncollinear intralayer AFM configuration was calculated using hexagonal cells containing three atoms per layer [( √ 3× √ 3)R30 • in-plane periodicity] (see Supplementary Note SII A 3) and a 7×7×1 BZ sampling. For purpose of electronic structure visualization the band energies for selected systems were calculated along the K − Γ − M path by a non-self consistent DFT calculation, again with Gaussian smearing of 0.01 meV.
The magnetic anisotropy energy was calculated as explained in Ref. [22], using the 25 × 25 × 1 Monkhorst-Pack grid and electronic convergence threshold of 10 −8 eV.
The obtained Kohn-Sham functions were used to construct the maximally localized Wannier functions by the Wannier90 code [75,76]. We refer reader to the Supplementary Note. SI for further details regarding the wannierization. The Wannier functions were in turn used to calculate anomalous Hall conductances of the proposed heterostructures through the Kubo formula as implemented in the WannierBerri code [77]. In WannierBerri calculations the broadening of 10 K was used. For hBN-covered MBT 2SL film, as well as MBT 2SL /hBN/MBT 2SL heterostructure, the Chern number was also calculated by the Wilson loop method as implemented in Z2Pack [78,79].
The edge spectral functions were calculated by the Green function method [80] from the tight-binding Wannier Hamiltonian as implemented in WannierTools [81]. The number of principal layers in these calculations was set to four, while the details of the cell used can be found in the Supplementary Note SIV.
DATA AVAILABILITY STATEMENT
Inputs and results are available from the corresponding authors upon a reasonable request.
FIG. 1 :
1(a) Side view of the bulk MnBi 2 Te 4 (MBT) crystal structure. Red arrows denote Mn local moments. (b) Schematic depiction of the proposed system: MBT films are separated by hBN monolayers to make a vdW multilayer heterostructure with Chern number equal to the number of MBT films, C = n. Black arrows depict the direction of the edge currents.
(a)-(d). These energies were calculated starting from both FIG. 2: (a) Top and side views of the hollow site MBT/hBN adsorption registry. (b) The band structures of MBT 2SL and MBT 2SL /hBN with hBN contribution shown in green. (c) Fermi energy dependence of the anomalous Hall conductance S xy (E F ) in the units of conductance quantum e 2 /h for MBT 2SL and MBT 2SL /hBN. E F = 0 corresponds to the center of the band gap. (d) The edge electronic band structures for MBT 2SL (left) and MBT 2SL /hBN (right). The regions with a continuous spectrum correspond to the 2D bulk states projected onto the 1D Brillouin zone. The edge crystal structure is shown in Supplementary Figure S7. The data in (b-d) were calculated for the FM interlayer alignment in MBT 2SL (see text).
and the Supplementary Fig. S3(e)-(k) for visualization). These findings clearly confirm the vdW nature of bond between hBN and MBT, which preserves the CI state in the forced FM phase of the MBT 2SL film.
FIG. 3 :
3Electronic band structures calculated along the K − Γ − M path in the 2D Brillouin zone of the nMBT 2SL /hBN multilayers for (a) n = 2, (b) n = 3, (c) n = 4 and (d) n = 5. In the insets, the colored blocks depict the equivalent MBT 2SL layers from which the bands of corresponding color dominantly stem. The black lines between the blocks represent hBN layers that separate them. FIG. 4: The Fermi energy dependence of anomalous Hall conductance S xy (E F ) in the units of conductance quantum e 2 /h for MBT 2SL /hBN and nMBT 2SL /hBN, n = 2, . . . , 5. E F = 0 corresponds to the center of the band gap in each case. (b) The edge electronic structure of nMBT 2SL /hBN for n = 2 and n = 3.
FIG. 5 :
5(a) The Fermi energy dependence of the anomalous Hall conductance S xy (E F ) in the units of conductance quantum e 2 /h for MBT 3SL , MBT 3SL /hBN and 2MBT 3SL /hBN. E F = 0 corresponds to the center of the band gap in each case. (b) The edge electronic structure of 2MBT 3SL /hBN.
TABLE I :
IStructural, magnetic, transport and
topological characteristics of the MBT 2SL /hBN bilayer
for the four adsorption registries: Hollow, Bridge,
Top-B, Top-N. d (inÅ) is the MBT-hBN interlayer
distance (see
Mn pair) is the total energy difference of the interlayer AFM and FM states, S xy is the anomalous Hall conductance for the Fermi level lying within the fundamental band gap, and C is the Chern number obtained for the interlayer FM state in MBT 2SL .Hollow
Bridge
Top-B
Top-N
d
3.482
3.601
3.573
3.600
EAFM
0.0
5.3
1.6
10.2
EFM
1.2
6.4
2.9
11.3
∆E A/F
−1.2
−1.1
−1.3
−1.1
Sxy
e 2 /h
e 2 /h
e 2 /h
e 2 /h
C
1
1
1
1
[ 26 ]
26or compounds of the MnBi 2 Te 4 family[24, 62- 64]. The here proposed design allows a wide-range tuning of the Chern number, its upper limit being restricted only by the van der Waals heterostructures growth technology, making such multilayers interesting for future fundamental research and efficient interconnect technologies.
AUTHOR CONTRIBUTIONSThe problem was conceptualized byMMOCOMPETING INTERESTSAuthors declare no competing interests.ADDITIONAL INFORMATIONSupplementary informationSupporting Information acompanies the manuscript.
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The authors thank J. Ibañez-Azpiroz, S. S. Tsirkin, I. Souza, and M. Garnica for stimulating discussions. The authors thank J. Ibañez-Azpiroz, S. S. Tsirkin, I. Souza, and M. Garnica for stimulating discussions.
M B , M M , acknowledge support of the Spanish Ministerio de Ciencia e Innovacion (Grant no. PID2019-103910GB-I00) and the University of the Basque Country. Grant no. IT1527-22M.B. and M.M.O. acknowledge support of the Spanish Ministerio de Ciencia e Innovacion (Grant no. PID2019-103910GB-I00) and the University of the Basque Country (Grant no. IT1527-22).
acknowledge support of the Russian Science Foundation (grant №18-12-00169-p). A V Yu, E K , A.Yu.V. and E.K.P. acknowledge support of the Russian Science Foundation (grant №18-12-00169-p).
acknowledges support from Saint Petersburg State University (Grant ID 90383050). The calculations were performed using computational resources of. E V C , Donostia International Physics Center. Research Park of Saint Petersburg State UniversityComputing CenterE.V.C. acknowledges support from Saint Petersburg State University (Grant ID 90383050). The calculations were performed using computational resources of Donostia International Physics Center (http://dipc.ehu.es/cc/) and Research Park of Saint Petersburg State University "Computing Center" (http://www.cc.spbu.ru/).
| [] |
[
"ICRC 2021 ONLINE ICRC 2021 A web application for monitoring cosmic rays and solar activity",
"ICRC 2021 ONLINE ICRC 2021 A web application for monitoring cosmic rays and solar activity"
] | [
"David Pelosi [email protected] \nTHE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy\n",
"Nicola Tomassetti [email protected] \nTHE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy\n",
"Matteo Duranti \nTHE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy\n"
] | [
"THE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy",
"THE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy",
"THE ASTROPARTICLE PHYSICS CONFERENCE\nTHE ASTROPARTICLE PHYSICS CONFERENCE\nDepartment of Physics and Earth's Science\nINFN -Sezione di Perugia -Perugia\nUniversità degli Studi di Perugia\nBerlin, Berlin |Germany, Germany, Italy, Italy"
] | [
"th International Cosmic Ray Conference (ICRC 2021)"
] | The flux of cosmic rays in the heliosphere is subjected to variations that are related to the Sun's magnetic activity. To study this effect, updated time series of multichannel observations are needed. Here we present a web application that collects real-time data on solar activity proxies, interplanetary plasma parameters, and charged cosmic-ray data. The data are automatically retrieved on daily basis from several space missions or observatories. With this application, the data can be visualized and download into a common format. Along with observational data, the application aims to provide real-time calculations for the solar modulation of cosmic rays in the heliosphere. | 10.22323/1.395.1259 | [
"https://export.arxiv.org/pdf/2210.05696v1.pdf"
] | 237,523,399 | 2210.05696 | 66ee26f8dadf3afd3663785c597e1e7357ad40ce |
ICRC 2021 ONLINE ICRC 2021 A web application for monitoring cosmic rays and solar activity
July 2021 37. July 12th -23rd, 2021
David Pelosi [email protected]
THE ASTROPARTICLE PHYSICS CONFERENCE
THE ASTROPARTICLE PHYSICS CONFERENCE
Department of Physics and Earth's Science
INFN -Sezione di Perugia -Perugia
Università degli Studi di Perugia
Berlin, Berlin |Germany, Germany, Italy, Italy
Nicola Tomassetti [email protected]
THE ASTROPARTICLE PHYSICS CONFERENCE
THE ASTROPARTICLE PHYSICS CONFERENCE
Department of Physics and Earth's Science
INFN -Sezione di Perugia -Perugia
Università degli Studi di Perugia
Berlin, Berlin |Germany, Germany, Italy, Italy
Matteo Duranti
THE ASTROPARTICLE PHYSICS CONFERENCE
THE ASTROPARTICLE PHYSICS CONFERENCE
Department of Physics and Earth's Science
INFN -Sezione di Perugia -Perugia
Università degli Studi di Perugia
Berlin, Berlin |Germany, Germany, Italy, Italy
ICRC 2021 ONLINE ICRC 2021 A web application for monitoring cosmic rays and solar activity
th International Cosmic Ray Conference (ICRC 2021)
July 2021 37. July 12th -23rd, 2021Online -Berlin, Germany Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ arXiv:2210.05696v1 [physics.space-ph] 11 Oct 2022 HVO
The flux of cosmic rays in the heliosphere is subjected to variations that are related to the Sun's magnetic activity. To study this effect, updated time series of multichannel observations are needed. Here we present a web application that collects real-time data on solar activity proxies, interplanetary plasma parameters, and charged cosmic-ray data. The data are automatically retrieved on daily basis from several space missions or observatories. With this application, the data can be visualized and download into a common format. Along with observational data, the application aims to provide real-time calculations for the solar modulation of cosmic rays in the heliosphere.
The flux of cosmic rays in the heliosphere is subjected to variations that are related to the Sun's magnetic activity. To study this effect, updated time series of multichannel observations are needed. Here we present a web application that collects real-time data on solar activity proxies, interplanetary plasma parameters, and charged cosmic-ray data. The data are automatically retrieved on daily basis from several space missions or observatories. With this application, the data can be visualized and download into a common format. Along with observational data, the application aims to provide real-time calculations for the solar modulation of cosmic rays in the heliosphere.
th International Cosmic Ray Conference (ICRC 2021) July 12th -23rd, 2021
Online -Berlin, Germany
Introduction
During their motion inside the heliosphere, cosmic rays (CRs) interact with the heliospheric plasma. Specifically, the solar wind and its embedded magnetic field reshape their energy spectra. The resulting effect, known as solar modulation, is caused by basic transport processes such as diffusion, drift, reacceleration, convection, and adiabatic cooling [1]. Due to solar modulation, the energy spectrum of CRs observed near-Earth is considerably different from that in the surrounding interstellar medium, the so-called Local Interstellar Spectrum (LIS). Solar modulation is energyand time-dependent. The modification is larger for CRs at low energies (e.g. with kinetic energies below a few GeV) and shows a quasi-peridical time variation which reflects a clear correlation with the 11-year cycle of solar activity. To monitor solar activity, a widely used proxy is the SunSpot Number (SSN), i.e. the number of the observed dark spots in the Sun's photosphere. The SSN is widely used as a good proxy for solar activity, varies with a period of 11 years, known as solar cycle. Investigating the solar modulation phenomenon is of crucial importance to achieve a deep understanding of the dynamics of charged particles in the heliosphere, as well as to reliably predict the radiation dose received by electronics and astronauts in space missions. Given the ever-growing number of satellites orbiting around Earth and human space missions to Moon and Mars planned in the next decades, forecasting the level of CR radiation in low-Earth or deep-space orbits is of great importance. For this purpose, several predicting models for the solar modulation of CRs have been developed [1][2][3][4][5]. Recent CR data collected in space include the time-resolved flux measurements of the experiments EPHIN/SOHO (since 1995 to 2018) [6], PAMELA (2006-2016) [7], and AMS-02 (since 2011 and still operative for all ISS lifetime) [8], along with the direct LIS data from the Voyager probes in the interstellar space [9]. The temporal variations of CRs fluxes are also measured indirectly by Neutron Monitors (NMs) [10]. The counting rate of a NM detector associated with -type CR particles is defined as an integration of the total flux ( , ), above the local geomagnetic cutoff rigidity , convoluted with the yield function of the detector = ( , ). The total NM rate is then obtained by the sum of the contributing CR species. Models also need heliospheric and solar data such as, in particular, the SSN as function of time, which is provided by the SILSO/SIDC database of the Royal Observatory of Belgium [11]; the strength of the polar magnetic field and the tilt angle of the heliospheric current sheet, provided on 10-day basis by the Wilcox Solar Observatory [12]. Also important are interplanetary data about the solar wind, such as its radial speed and the density of its proton component, that are distributed by NASA missions WIND and ACE [13].
The Heliophysics Virtual Observatory (HVO) has been developed to make data an easy and quick access to all these data involving charged radiation, solar activity, and interplanetary plasma. HVO is a web application that collects all the observational data mentioned above into a unique platform which is updated automatically on daily basis. The HVO functionalities include the possibility of visualizing, manipulating and downloading updated data. We also present a simplified real-time model of near-Earth proton flux integrated into a specific section of HVO.
HVO
Real-time model
The transport of CRs in the heliosphere is governed by the Parker equation:
= −( ì + ì ) · ∇ + ∇ · (K · ∇ ) + 1 3 (∇ · ì ) ln +(1)
The equation describes the temporal evolution of CR phase space density = ( , ), where = / is the magnetic rigidity of CRs, ì is their averaged drift velocity, ì is the solar wind velocity, K is the symmetric part of CR diffusion tensor, and is any local source of CRs [1]. The Parker equation is often resolved in under simplifying approximation, for example in the so-called Force-Field (FF) model [2]. The FF model assumes steady-state conditions (i.e. negligible short-term modulation effects), radially expanding wind ( ), an isotropic and separable diffusion coefficient ≡ 1 ( )· 2 ( ), negligible losses and no drift. In spite of its questionable assumptions, the FF model is widely used as it provides a simple and practical way to describe the near-Earth CR flux and its long-term evolution. From Eq. 1, the CR flux ( , ) is given by = 2 . In terms of kinetic energy per nucleon , for a CR nucleus with charge number and mass number , the FF equation for the near-Earth ( = 1 AU) flux at the epoch is given by:
( , ) = ( + ) 2 − 2 ( + + ( )) 2 − 2 LIS ( + ( )),(2)
where the parameter is the so-called modulation potential. It has the units of an electric potential and lies in the typical range 100-1000 MV. The parameter can be interpreted as the averaged energy loss per charge units of CR particles in the heliosphere. The implementation of a CR modulation model under the FF approximation depends on the knowledge of two key elements: the time-series of the parameters, and the input LIS model = ( ) describing the energy spectrum of CRs outside the heliosphere. In this work, we have used new LIS models based on the latest results from Voyager 1 and AMS-02 [14] and the values of the modulation potential reconstructed by Usoskin et al. 2011 [15], from NM data on monthly basis, since 1964 to 2011. To set up the reconstruction after 2011 to the present epoch, instead of repeating the Usoskin methodology (based on the calibration of NMs and the evaluation of their yield function), we have adopted a simplified method. For a given NM detector, the NM counting rate ( ) and the parameter ( ) are well anti-correlated and we can establish a quadratic relation between them:
( ( )) = + · ( ) + · ( ) 2(3)
We determined the coefficient , , and as best-fit values using NM counts of many stations, analyzed against the -parameter from Usoskin et al. 2011 [15]. This approach enabled us to obtain a prediction of the parameter for any epoch for which the NM rate is known. Inserting the parametric model of LIS and the value at epoch in Eq. 2, we obtain a real-time evaluation of the near-Earth CR flux, based on FF calculations calibrated against NM data. This simple model has been integrated into HVO.
The Heliophysics Virtual Observatory
The investigation of the solar modulation phenomenon requires a large variety of heliospheric and radiation data. HVO [16] is a project developed under the CRISP scientific program of experi- mental study and phenomenological modeling of space weather, within the framework agreement between Università degli Studi di Perugia and Agenzia Spaziale Italiana (ASI). HVO is a web application that daily extracts data with Python scripts from several databases (listed in Resources section). It visualizes and makes them available in a standardized format. HVO is implemented with the JavaScript ROOT package JSROOT. The application allows users to manipulate data graphs directly from the web, to download data as machine readable text, graphic objects in ROOT format or PNG images. HVO has is organized in three main sections. The first one is dedicated to solar data such as SSN in daily, monthly, yearly, and smoothed formats extracted from the SILSO/SIDC data center [11], observations of the Sun's polar magnetic field strength and tilt angle of the HCS reconstructed with the classic and radial model from the Wilcox Solar observatory [12]. The second section contains heliospheric data such as the radial speed of the solar wind and the density of its proton component, updated on monthly bases from WIND and ACE missions [13]. The third section contains cosmic radiation data from several NM stations and a real-time model for the flux of CR protons, as presented in Sect. 2. An interactive user interface provides the possibility to select one or more NM stations, to choose the time resolution of the rates (daily, monthly, yearly and by Carrington rotation), set the proton CR energy and time range. For each selected NM station, HVO provides the graph of the counting rate ( ), the corresponding calculation of the modulation potential ( ) (from Eq. 3), and the estimated near-Earth flux of CR protons ( , ) (from Eq. 2). An example of the HVO functionalities is shown in Fig. 1 and Fig. 2.
We have presented a web application aimed at monitoring solar activity and cosmic radiation, as well as providing real-time calculation of the energy spectra of CRs in proximity of the Earth. HVO is a useful tool for the CR astrophysics and space physics community. Future development may include an improvement of the real-time CR flux model, its extension the to other charged species and other locations in the interplanetary space. In this respect, HVO can also integrated with state of the art numerical models of CRs transport in the heliosphere [3,4], that will enable us to forecast the CR radiation at an interplanetary level. Other extension of HVO may include space weather data, e.g. on the occurrence of solar flares and coronal mass ejections, geomagnetic storms, or other interplanetary disturbance phenomena.
Averaged tilt angle of the HCS measured with the classic and radial model for any Carrington rotation between 1/1/1996 and 23/6/2021. (c) Sun's Polar Field strength (SFS) for northern and southern hemisphere in the time interval 1Monthly rate ( ) of the NM stations: OULU, NEWK, APTY, JUNG. Time interval 1
Figure 1 :
1Data available on HVO.
Calculated time-series of (1/1/1996 -23/6/2021) using the rates of NM stations: OULU, NEWK, APTY, JUNG. (b) Estimated near-Earth proton flux at = 1 GeV using the time-series of reconstructed from the rates of NM stations: OULU, NEWK, APTY, JUNG.
Figure 2 :
2Real-time flux model in HVO
AcknowledgmentsWe acknowledge the support of ASI under agreement ASI-UniPG 2019-2-HH.0.
. M S Potgieter, 10.12942/lrsp-2013-3Liv. Rev. Sol. Phys. 103M.S. Potgieter, Liv. Rev. Sol. Phys.,10, 3 (2013)
. H , 10.1007/s11214-011-9819-3Space Sci. Ref. 176299H. Moraal, Space Sci. Ref.,176, 299 (2013)
. N Tomassetti, 10.1103/PhysRevD.96.103005Phys. Rev. D. 96103005N. Tomassetti, Phys. Rev. D,96, 103005 (2017)
. N Tomassetti, 10.3847/2041-8213/aa9373ApJ Lett. 84932N. Tomassetti, et al., ApJ Lett.,849, L32 (2017)
. N Tomassetti, Phys. Rev. Lett. 121251104N. Tomassetti, et al., Phys. Rev. Lett.,121, 251104 (2018)
. P Kühl, R Gómez-Herrero, B Heber, 10.1007/s11207-016-0879-0Sol. Phys. 291P. Kühl, R. Gómez-Herrero, B. Heber, Sol. Phys,291, 2016 (965-974)
. M Martucci, 10.3847/2041-8213/aaa9b2ApJ Lett. 8542M. Martucci, et al., ApJ Lett.,854, L2 (2018)
. M Aguilar, 10.1103/PhysRevLett.121.051101PRL. 12051101M. Aguilar, et al., PRL,120, 051101 (2018)
. A C Cummings, 10.3847/0004-637X/831/1/18ApJ. 83118A. C. Cummings, et al., ApJ,831, 18 (2016)
. Silso/Sidc -Solar, SILSO/SIDC -Solar Influences Data analysis Center [http://www.sidc.be/silso]
. Wso -Wilcox Solar, Observatory, WSO -Wilcox Solar Observatory [http://wso.stanford.edu]
. Nasa/Spdf -Space, Physics Data Facility. NASA/SPDF -Space Physics Data Facility [https://spdf.gsfc.nasa.gov]
. C Corti, 10.3847/0004-637X/829/1/8ApJ. 829C. Corti, et al., ApJ,829, 8 (2016);
. N Tomassetti, 10.1103/PhysRevLett.121.251104PRL. 121251104N. Tomassetti, et al., PRL,121, 251104 (2018)
. I G Usoskin, 10.1029/2010JA016105J. Geophys. Res. -Space Physics. 116I. G. Usoskin, et al., J. Geophys. Res. -Space Physics,116, A2 (2011)
HVO -Heliophysics Virtual Observatory. HVO -Heliophysics Virtual Observatory [https://crisp.unipg.it/hvo]
| [] |
[
"SBFT Tool Competition 2023 -Fuzzing Track",
"SBFT Tool Competition 2023 -Fuzzing Track"
] | [
"Dongge Liu *[email protected] \nMPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA\n",
"Jonathan Metzman [email protected] \nMPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA\n",
"Marcel Böhme †[email protected] \nMPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA\n",
"Oliver Chang [email protected] \nMPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA\n",
"Abhishek Arya [email protected] \nMPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA\n"
] | [
"MPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA",
"MPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA",
"MPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA",
"MPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA",
"MPI-SP\nGoogle, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA"
] | [] | This report outlines the objectives, methodology, challenges, and results of the first Fuzzing Competition held at SBFT 2023. The competition utilized FUZZBENCH to assess the code-coverage performance and bug-finding efficacy of eight participating fuzzers over 23 hours. The competition was organized in three phases. In the first phase, participants were asked to integrate their fuzzers into FUZZBENCH and allowed them to privately run local experiments against the publicly available benchmarks. In the second phase, we publicly ran all submitted fuzzers on the publicly available benchmarks and allowed participants to fix any remaining bugs in their fuzzers. In the third phase, we publicly ran all submitted fuzzers plus three widely-used baseline fuzzers on a hidden set and the publicly available set of benchmark programs to establish the final results. | 10.48550/arxiv.2304.10070 | [
"https://export.arxiv.org/pdf/2304.10070v2.pdf"
] | 258,236,080 | 2304.10070 | 81ff1931259ec884fd0306341e73281294dbae7f |
SBFT Tool Competition 2023 -Fuzzing Track
16 May 2023
Dongge Liu *[email protected]
MPI-SP
Google, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA
Jonathan Metzman [email protected]
MPI-SP
Google, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA
Marcel Böhme †[email protected]
MPI-SP
Google, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA
Oliver Chang [email protected]
MPI-SP
Google, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA
Abhishek Arya [email protected]
MPI-SP
Google, Google, Germany Monash, Google, GoogleUSA, USA, Australia, USA, USA
SBFT Tool Competition 2023 -Fuzzing Track
16 May 2023Index Terms-fuzzingevaluationopen-source
This report outlines the objectives, methodology, challenges, and results of the first Fuzzing Competition held at SBFT 2023. The competition utilized FUZZBENCH to assess the code-coverage performance and bug-finding efficacy of eight participating fuzzers over 23 hours. The competition was organized in three phases. In the first phase, participants were asked to integrate their fuzzers into FUZZBENCH and allowed them to privately run local experiments against the publicly available benchmarks. In the second phase, we publicly ran all submitted fuzzers on the publicly available benchmarks and allowed participants to fix any remaining bugs in their fuzzers. In the third phase, we publicly ran all submitted fuzzers plus three widely-used baseline fuzzers on a hidden set and the publicly available set of benchmark programs to establish the final results.
I. INTRODUCTION
We report on the organization of the first fuzzing competition at the 16th International Workshop on Search-Based and Fuzz Testing (SBFT) held on the 14th of May 2023 in Melbourne, Australia. The objectives of this competition were (i) to evaluate the performance of the fuzzers submitted to this competition in terms of coverage and bug finding ability, (ii) to gather experience and feedback on the sound benchmarking of fuzzing tools, and (iii) to stress test the FUZZBENCH benchmarking platform which has been built particularly for this purpose.
Throughout the competition we paid particular attention to the mitigation of different forms of bias. For instance, in order to avoid overfitting to a particular set of benchmarks (confirmation bias), we allowed participants to develop, integrate, and evaluate their fuzzers privately on a publically available set of benchmarks while conducting the actual competition on a set of benchmarks that included a large number of hidden benchmarks. In order to avoid survivorship bias, we do not evaluate their bug finding ability on a given set of bugs that we already know how to find. Instead, we evaluate their bug finding ability in terms bugs found by any fuzzer. We make sure to use the same AddressSanitizer (ASAN) instrumented binaries across all fuzzers.
In summary, we found that the AFLRUSTRUST fuzzer performed well in terms of both, the coverage achieved and bugs found. The fuzzers LIBAFL LIBFUZZER, HASTEFUZZ, and AFL+++ excelled on coverage-based benchmarks, while PASTIS and AFLSMART++ found more bugs than the average fuzzer. We present the final ranking and more concrete results live at the tool competition. FUZZBENCH can conduct bug-based or code coveragebased experiments [2]. Throughout out the course of an experiment, and upon its completion, FUZZBENCH generates a report detailing the performance of each fuzzer. The report compares fuzzers based on their performance across all benchmarks as well as on individual benchmarks and shows effect size (Vargha Delaney 12 ) and statistical significance (Mann Whitney U test). The comparison across all benchmarks contains two rankings, one based on their average rank on each individual benchmark and one based on their performance relative to the best performing fuzzer on each individual benchmark. FUZZBENCH reports include a critical difference diagram so that users can see if differences between fuzzers based on average rank is statistically significant. The report's comparison on individual benchmarks consists of graphs and data showing, the number of crashes found and the growth of code coverage throughout the experiment.
To request an experiment, the interested researcher submits a pull request to the Github repository where the fuzzer is integrated or privately emails [email protected]. A typical experiment in FUZZBENCH involves about 20 trials of 10 fuzzers running on 20 benchmarks for 23 hours. This is about 10-CPU years, which is cost prohibitive for most researchers. Researchers can use FUZZBENCH by integrating with a simple Python and Docker based API. This integration usually is less than 100 lines of code.
FUZZBENCH has had an enormous impact on fuzzer development and research. Over 900 experiments have been conducted using the FUZZBENCH service. FUZZBENCH has been discussed in over 100 academic papers. And FUZZBENCH has been used to guide the development of popular fuzzers such as AFL++, HONGGFUZZ and LIBFUZZER. FUZZBENCH experiments have most desirable qualities that Klee et al. [3] described most evaluations as lacking, including: statistically sound comparisons and statistical tests, long timeouts and realworld programs.
III. COMPETITION SETUP
Phases. The competition was organized in three phases. In the first phase, participants were asked to integrate their fuzzers into FUZZBENCH and allowed them to privately run local experiments against the publicly available benchmarks. In the second phase, we publicly ran all submitted fuzzers on the publicly available benchmarks and allowed participants to fix any remaining bugs in their fuzzers. In the third phase, we publicly ran all submitted fuzzers plus three widely-used baseline fuzzers on a hidden set and the publicly available set of benchmark programs to establish the final results.
Performance metrics. In our competition, we measure both the code coverage achieved and the bug-finding capacity to compare the performance of the submitted fuzzers [3], [4]. As benchmarking platform, we use FUZZBENCH which measures line coverage across all coverage-based benchmarks and the time it takes to generate the first crashing input across all bugbased benchmarks. To facilitate a more intuitive comparison of fuzzer performance in both categories, we present a relative median score for each fuzzer.
We compute the coverage-based score for each fuzzer as follows. As it is impractical to determine the total number of reachable lines in each coverage-based benchmark bc [5], we compute the relative coverage score score(bc, f ) for a fuzzer f by dividing the median value of its line coverage over 20 trials (i.e., cov(bc, f, n) where n = 1..20) by the maximum line coverage attained by all fuzzers F on that specific benchmark:
score(bc, f ) = cov(bc, f ) max i∈F max n=1..20 cov(bc, i, n) (1) cov(bc, f ) = Med n=1..20 (cov(bc, f, n))(2)
We compute the bug-based score for each fuzzer as follows. Many fuzzer-generated crashing inputs may expose the same bug, and the same bug may yield different stack traces [6], [7]. In order to circumvent challenges of bug deduplication, we include only one reproducible bug in each benchmark and measure the time it takes to generate the first input that causes the benchmark binary to crash. Therefore, considering that each bug-based benchmark bb comprises only one bug, we calculate the relative score score(bb, f ) of a fuzzer f using the following method:
score(bb, f ) = Med n=1..20 (bug(bb, f, n)) (3) bug(bb, f, n) = 1 if f finds a bug in bb in trial n 0 otherwise(4)
In instances where multiple fuzzers detect an equal number of bugs across all benchmarks, we additionally provide their average time required for bug discovery as an auxiliary metric.
Benchmarks. The 53 benchmarks employed in this study were selected from a diverse range of real-world open-source projects integrated into OSS-FUZZ. This approach ensures that researchers can evaluate their fuzzers on the latest, popular, and actively maintained real-world open-source programs. Meanwhile, project maintainers can benefit from state-of-theart fuzzers.
To guarantee the reproducibility of fuzzer performance, each benchmark is anchored to a specific commit. In particular, the commit for each bug-based benchmark are carefully chosen such that the bug present have been fixed or published within one year. This approach prevents security vulnerability leakage while maintaining benchmarks up-to-date for research evaluation purposes.
Benchmarks are divided into public and private sets. The public benchmark set, consisting of 5 bug-based and 24 coverage-based benchmarks, is made available to participants for build and runtime errors identification upon joining the competition. In contrast, the private benchmark set, comprising of 10 bug-based and 14 coverage-based benchmarks, is withheld until the final evaluation to mitigate overfitting.
Preventing overfitting in fuzzing competitions is typically challenging since participants usually require access to the benchmark source code to identify and resolve compatibility issues. However, FUZZBENCH's design effectively addresses this issue by separating the benchmarks and fuzzers. This allows fuzzers to be built and run on private benchmarks using the same code that was tested on the public ones, contributing to a fair and impartial evaluation of fuzzer performance.
Fuzzers. The competition evaluates a total of 12 fuzzers, including 8 fuzzers submitted by participants and 4 fuzzers used as baseline. The participant-submitted fuzzers are AFL+++ 6 , AFLRUSTRUST 7 , AFLSMART++ 8 , HASTEFUZZ 9 , LEARN-PERFFUZZ 10 , LIBAFL LIBFUZZER 11 , PASTIS 12 , and SYM-SAN 13 . The four baseline fuzzers encompass AFL 14 , AFL++ 15 , HONGGFUZZ 16 , and LIBFUZZER 17 . We selected AFL and AFL++ as baselines, as most participants extended them to construct their own. The fuzzers HONGGFUZZ and LIBFUZZER were chosen due to their contribution to the discovery of bugs in the bug-based benchmarks under OSS-FUZZ production environment.
Platform and Configuration. The competition is conducted on Google Cloud virtual machines. We concurrently measure 20 trials per fuzzer on each benchmark, with each trial executing one fuzzer instance on one benchmark. Each trial was run on a dedicated clean Ubuntu20.04 virtual machine instance equipped with 1 vCPU and 3.75 GB memory. For some benchmarks, seed corpora were available, mirroring the production environment in OSS-FUZZ.
IV. EVALUATION RESULTS
We present and discuss the results of coverage-based and bug-based benchmarking separately. From previous experiments [4], we do not expect a strong agreement between rankings established by coverage-based versus bug-based benchmarking, but they each provide important and interesting insights about the capabilities of the fuzzers.
A. Coverage-based Benchmarking
We first focus on the fuzzers' ability to cover the most code possible. Bugs cannot be found in code that is not covered.
We find that LIBAFL LIBFUZZER leads in 23 out of 38 coverage-based benchmarks, significantly more than any other fuzzer. However, its overall performance is negatively impacted by the near-zero coverage exhibited on three benchmarks: draco, dropbear, and proj4. In particular, LIBAFL LIBFUZZER generated merely two input cases for draco and crashed immediately after initiating dropbear. To facilitate debugging, FUZZBENCH has provided researchers with the input corpora and fuzzer logs.
HASTEFUZZ consistently performs well on all coveragebased benchmarks, securing its position as one of the best fuzzers. Although its relative median scores ranked first on only 16 benchmarks, it remained within the 90% relative median range on 31 benchmarks and secured a position within the top three on 35 benchmarks. Notably, it exhibited the lowest standard deviation across all benchmarks (approximately 8.15), which is less than half of the second-lowest (AFL+++, 17.61).
Both AFL+++ and AFLRUSTRUST display competitive performance across the majority of benchmarks. Their relative scores ranked first on 16 and 12 benchmarks, respectively, achieved within the 90% range on 31 and 29 benchmarks, and secured top three positions on 33 and 22 benchmarks.
As for baseline fuzzers, AFL++ emerged as the bestperforming and outperformed most other fuzzers on the majority of benchmarks. Its average relative score is 92.67, whereas 15 https://github.com/google/fuzzbench/tree/SBFT'23/fuzzers/aflplusplus 16 https://github.com/google/fuzzbench/tree/SBFT'23/fuzzers/honggfuzz 17 https://github.com/google/fuzzbench/tree/SBFT'23/fuzzers/libfuzzer the highest average score among the remaining participant fuzzers is below 90.
A notable observation is that many top-performing fuzzers exhibit a high degree of similarity in their coverage performance, primarily due to their shared underlying fuzzer architecture. To measure "coverage similarity", we consider the coverage achieved by two fuzzers across different benchmarks and compute the cosine similarity. We find that the cosine similarity between AFLRUSTRUST and AFL+++ surpasses 0.99, signifying their nearly identical relative median scores across all benchmarks. Likewise, the cosine similarities among AFLRUSTRUST and HASTEFUZZ, HASTEFUZZ and AFL++, AFL++ and AFL+++ are all above 0.98. In contrast, the cosine similarities between LIBFUZZER and AFL++, LIB-FUZZER and AFLRUSTRUST, LIBFUZZER and AFL+++ are approximately 0.93.
Our analysis reveals that certain benchmarks are adept at distinguish the coverage performance of fuzzers. For instance, after excluding outliers, the openthread benchmark exhibits the highest interquartile range of 22.25, along with a standard deviation of 18.91. The range of fuzzer scores on this benchmark spans from 98 to 49, indicating that the topperforming fuzzer achieves approximately double the relative coverage of the lowest-performing one.
Similarly, the scores on the lcms benchmark range from 95 to 19, yielding a standard deviation of 22.79 and an interquartile range of 18.50. For the freetype2 benchmark, the standard deviation is 19.98, with an interquartile range of 21.75 and fuzzer scores ranging from 22 to 95. Furthermore, no fuzzer's relative median score exceeds 68 on the botan benchmark, suggesting that the maximum of median scores of all fuzzers is approximately two-thirds of the highest line coverage across all trials.
Conversely, some benchmarks display a high degree of similarity in performance across fuzzers, thereby offering limited utility in differentiating and ranking them. For example, all fuzzers are within 98% of the top-performing fuzzer's score on the libjpeg benchmark, and almost all of them achieve the same line coverage on the firestore benchmark.
B. Bug-based Benchmarking
In terms of bug finding, many fuzzers display similar performance on bug-based benchmarks. For instance, AFLRUST-RUST and PASTIS both have the highest relative median score (53.33), indicating that their median-performing fuzzer trials covered 8 out of 15 bugs across all benchmarks. Likewise, participant-submitted fuzzers AFL+++ and HASTEFUZZ covered 6 bugs, equal to the performance of baseline fuzzers AFL++ and LIBFUZZER.
Seven benchmarks were found to be particularly useful in differentiating fuzzers in this competition, as they exhibited diverse performance among fuzzers: aspell, assimp, file, bloaty, ffmpeg, libaom, and libxml2. Both AFLRUSTRUST and PASTIS discovered 6 out of the 7 bugs in these benchmarks, outperforming other fuzzers. However, AFLRUSTRUST and PASTIS had slightly different bug-finding patterns; AFLRUSTRUST covered a comparatively rare bug in file but missed a more commonly found bug in ffmpeg.
While half of the fuzzers found more than 4 bugs overall, the symbolic-based fuzzer SYMSAN discovered only 1 bug in assimp. Interestingly, LIBAFL LIBFUZZER, which performed well across coverage-based benchmarks and found bugs in 5 benchmarks, was the only fuzzer that missed the bug in assimp. This result could be attributed to its relatively low coverage on this specific benchmark.
We also examined the average time required for fuzzers to discover a bug. PASTIS proved to be the fastest in detecting bugs on average, with AFLRUSTRUST and AFLSMART++ following closely behind. Notably, the cosine similarity between AFL++ and AFL+++ exceeds 0.98, suggesting that they frequently identify bugs at approximately the same time. Likewise, the cosine similarity between HONGGFUZZ and PASTIS surpasses 0.9, indicating a comparable speed in causing crashes within the benchmark. LIBAFL LIBFUZZER appears to possess a distinct design, resulting in the lowest similarity score when compared to any other fuzzers.
The bug-based benchmarks in this competition also underscore the "asymmetry" between coverage-based and bugbased rankings, as highlighted by Böhme et al. [4]. For instance, HASTEFUZZ excelled in coverage-based benchmarks yet discovered fewer bugs. Conversely, AFL identified more bugs than AFL++, despite covering less code. Although code coverage is a well-established and easily measurable benchmarking metric, these findings stress the significance of taking bug-finding capabilities into consideration when optimizing for higher coverage and evaluating fuzzers. Essentially, fuzzers are intended to detect bugs, with coverage serving as a heuristic to estimate their bug-finding potential.
Bug-based benchmarking presents several challenges that we tackled in different ways. Firstly, acquiring the source code of real-world bugs is arduous, and the performance measured by artificial bugs might not accurately reflect reality. FUZZBENCH addresses this issue by using bugs filed by OSS-FUZZ when fuzzing actual open-source projects, providing a ground truth for bugs that had been and need to be discovered in production.
Secondly, a systematic approach for selecting appropriate bug benchmarks for evaluation remains absent. For instance, if all fuzzers exhibit similar performance on certain benchmarks, those bugs offer limited value into fuzzer assessment. To mitigate this concern, we incorporated benchmarks that were hidden during development and only revealing during final evaluation, culminating in nine benchmarks that demonstrate varying bug-discovery performances among fuzzers in this competition.
Thirdly, determining the superior fuzzer performance becomes difficult when multiple fuzzers can discover the same bug. To address this, we employ an auxiliary metric, i.e., measuing the average time required by each fuzzer to discover a bug. While FUZZBENCH evaluates this metric at 15-minute intervals, which may occasionally compromise accuracy, we highlight that this potential risk does not unfairly benefit any specific fuzzer.
Finally, ascertaining whether multiple crashes correspond to the same bug by grouping backtraces poses a considerable challenge. To tackle this issue, the competition restricts each benchmark to include only one known bug. Each associated open-source project is subjected to rigorous testing using multiple fuzzers over an extended period to minimize the likelihood of multiple reproducible bugs coexisting within a single benchmark.
V. CONCLUSION AND FUTURE WORK
In this competition, FUZZBENCH evaluates participant fuzzers and common baselines, comparing them using a variety of statistical tools. The assessment encompasses two key metrics: code coverage and bug-finding. Benchmarks for both metrics are derived from real-world open-source projects, and all fuzzers are tested under uniform production-like environment.
Moving forward, FUZZBENCH aims to enhance the statistical analysis by providing more detailed information, particularly concerning lines or bugs that fuzzers failed to cover. Additionally, FUZZBENCH plans to incorporate a larger collection of bug-based benchmarks to facilitate more comprehensive statistical reasoning.
II. FUZZBENCH: FUZZER BENCHMARKING PLATFORM FUZZBENCH[1] is a free, open source fuzzer benchmarking service built to make fuzzer benchmarking easy and rigorous. It allows researchers, who are interested in evaluating their fuzzers against other state-of-the-art fuzzers, to launch largescale experiments in a free and reproducible manner.The FUZZBENCH infrastructure consists of a large number of publicly available benchmark programs taken from OSS-FUZZ 1 . The benchmark programs are open source C/C++ programs carefully integrated by their maintainers, and include programs like Curl 2 , OpenSSL 3 , PHP 4 , and systemd 5 . Because the source code for most FUZZBENCH experiments is made public and the specific FUZZBENCH version can be pinned, reproducing FUZZBENCH experiments is often much easier than reproducing bespoke experiments used in other research.
https://google.github.io/oss-fuzz/ 2 https://github.com/curl/curl 3 https://github.com/openssl/openssl 4 https://github.com/php/php-src 5 https://github.com/systemd/systemd
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"A BERT-based Dual Embedding Model for Chinese Idiom Prediction",
"A BERT-based Dual Embedding Model for Chinese Idiom Prediction"
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"Minghuan Tan \nSchool of Information Systems\nSchool of Information Systems\nSingapore Management University\nSingapore\n",
"Jing Jiang [email protected] \nManagement University\n\n"
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"School of Information Systems\nSchool of Information Systems\nSingapore Management University\nSingapore",
"Management University\n"
] | [
"Proceedings of the 28th International Conference on Computational Linguistics"
] | Chinese idioms are special fixed phrases usually derived from ancient stories, whose meanings are oftentimes highly idiomatic and non-compositional. The Chinese idiom prediction task is to select the correct idiom from a set of candidate idioms given a context with a blank. We propose a BERT-based dual embedding model to encode the contextual words as well as to learn dual embeddings of the idioms. Specifically, we first match the embedding of each candidate idiom with the hidden representation corresponding to the blank in the context. We then match the embedding of each candidate idiom with the hidden representations of all the tokens in the context thorough context pooling. We further propose to use two separate idiom embeddings for the two kinds of matching. Experiments on a recently released Chinese idiom cloze test dataset show that our proposed method performs better than the existing state of the art. Ablation experiments also show that both context pooling and dual embedding contribute to the improvement of performance.This work is licensed under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/. | 10.18653/v1/2020.coling-main.113 | [
"https://www.aclweb.org/anthology/2020.coling-main.113.pdf"
] | 226,246,075 | 2011.02378 | c0603bafe053f3706d85e7442d290ec2da592251 |
A BERT-based Dual Embedding Model for Chinese Idiom Prediction
OnlineCopyright OnlineDecember 8-13, 2020
Minghuan Tan
School of Information Systems
School of Information Systems
Singapore Management University
Singapore
Jing Jiang [email protected]
Management University
A BERT-based Dual Embedding Model for Chinese Idiom Prediction
Proceedings of the 28th International Conference on Computational Linguistics
the 28th International Conference on Computational LinguisticsBarcelona, SpainOnlineDecember 8-13, 20201312
Chinese idioms are special fixed phrases usually derived from ancient stories, whose meanings are oftentimes highly idiomatic and non-compositional. The Chinese idiom prediction task is to select the correct idiom from a set of candidate idioms given a context with a blank. We propose a BERT-based dual embedding model to encode the contextual words as well as to learn dual embeddings of the idioms. Specifically, we first match the embedding of each candidate idiom with the hidden representation corresponding to the blank in the context. We then match the embedding of each candidate idiom with the hidden representations of all the tokens in the context thorough context pooling. We further propose to use two separate idiom embeddings for the two kinds of matching. Experiments on a recently released Chinese idiom cloze test dataset show that our proposed method performs better than the existing state of the art. Ablation experiments also show that both context pooling and dual embedding contribute to the improvement of performance.This work is licensed under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/.
Introduction
In this paper, we study Chinese idiom prediction, a language understanding problem that has not been extensively explored before in computational linguistics. Chinese idioms, mainly Chengyu (成语) (set phrases) (Wang and Yu, 2010;Wang, 2019), have fixed forms in structure; the component characters (mostly four) cannot be changed. Chinese idioms are characterized by rich contents, concise forms and frequent use (Wang, 2019) with properties of structural regularity, semantic fusion, and functional integrity (Shao, 2018;Wang, 2019). Chinese idioms are commonly used in both written and spoken Chinese, and understanding Chinese idioms is important for learning Chinese as a second language.
The meaning of each Chinese idiom may not be literally understood through the composition of its characters, especially for those which are derived from historical stories or formulated using ancient Chinese grammars. For example, "一定不易" is literally interpreted as "it must be not easy" in modern Chinese. However, the idiom is constructed from grammars and word senses of ancient Chinese. Its idiomatic meaning is "once decided, never change", which is not even close to the literal meaning. As a result, the usage of Chinese idioms poses a challenge on language understanding not only for humans but also for artificial intelligence. Due to their pervasive usage, Chinese idiom prediction is an important task in Chinese language understanding.
There have been several studies focusing on representing Chinese idioms using neural network models (Jiang et al., 2018;Liu et al., 2019b), but they were limited by the amount of data available for training. Recently, Zheng et al. (2019) released a large-scale Chinese IDiom Dataset (ChID) to facilitate machine comprehension of Chinese idioms. The ChID dataset contains more than 500K passages and 600K blanks, making it possible for researchers to train deep neural network models. The dataset is in cloze test style that target Chinese idioms in passages are replaced by blanks. For each blank, a set of candidate Chinese idioms is provided and the task is to pick the correct one based on the context. Table 1 Passage: 戴尔克·施特略夫把自己的工作全部撂下，整天服侍病人，又体贴，又关切。他的手脚 非常利索，把病人弄得舒舒服服。大夫开了药，他总是连哄带骗地劝病人按时服用，我从来没 想到他的手段这么巧妙。无论做什么事他都不嫌麻烦。尽避他的收入一向只够维持夫妻两人的 生活，从来就不宽裕，现在他却 ，购买时令已过、价钱昂贵的美味，想方设法叫思特里 克兰德多吃一点东西(他的胃口时好时坏，叫人无法捉摸)。 Dirk Stroeve, giving up his work entirely, nursed Strickland with tenderness and sympathy. He was dexterous to make him comfortable, and he exercised a cunning of which I should never have thought him capable to induce him to take the medicines prescribed by the doctor. Nothing was too much trouble for him. Though his means were adequate to the needs of himself and his wife, he certainly had no money to waste; but now he was in the purchase of delicacies, out of season and dear, which might tempt Strickland's capricious appetite.
Candidates:
• 月明星稀 The moon is bright and stars are few; with a clear moon and few stars • 苦尽甘来 bitterness ends and happiness begins • 坐吃山空 even a great fortune can be depleted by idleness 大手大脚 extravagant or wasteful • 斤斤计较 haggle over every ounce • 不见天日 a world of darkness; total absence of justice • 好吃懒做 be fond of eating and averse to work; be gluttonous and lazy Table 1: An example showing a passage with a blank and seven candidate idioms. The idiom with the solid circle is the ground truth idiom. The passage is from a Chinese translation of The Moon and Sixpence. Translations of idioms are extracted from online dictionary http://dict.cn.
shows an example from the testing set of ChID. We can see that among the seven candidates, most can fit into the local context "现在他却 " ("but now he was ") well grammatically, but to select the best answer we need to understand the entire passage.
In this paper, we propose a BERT-based dual embedding model for the Chinese idiom prediction task. We first present two baseline models that use BERT to process and match passages and candidate answers in order to rank the candidates. Observing that these baselines do not explicitly model the global, long-range contextual information in the given passage for Chinese idiom prediction, we propose a context-aware pooling operation to force the model to explicitly consider all contextual words when matching a candidate idiom with the passage. Furthermore, we propose to split the embedding vector of each Chinese idiom into two separate vectors, one modeling its local properties and the other modeling its global properties. We expect the embedding for local properties to capture the syntactic properties of an idiom, while the embedding for global properties to capture its topical meaning. In addition, using idiom embeddings makes it possible for us to consider the entire Chinese idiom vocabulary as the candidate set, which is computationally intractable compared to pretrained BERT models with multiple-sequence classification. we apply this enlarged candidates heuristic to all the models with idiom embeddings to further strengthen the performance.
To evaluate the effectiveness of the BERT-based dual embedding model, we conduct experiments on the ChID dataset. Our experiments show that our method can outperform several existing methods tested by Zheng et al. (2019) as well as our baseline methods. We also find that both context-aware pooling and dual embedding contribute to the performance improvement. To prove the effectiveness of our model, we also evaluate it against a public leaderboard of ChID Competition. The results show that our model is competitive compared to the top-ranked systems. We can also achieve better performance with a large margin compared with several methods using pretrained language models. We also conduct further analysis using a gradient-based attribution method to check if our model can indeed capture global information to make correct predictions. Some case studies show that indeed our method makes use of more global contextual information to make predictions.
Related Work
Cloze-style Reading Comprehension
Cloze-style reading comprehension is an important form in assessing machine reading abilities. Researchers created many large-scale cloze-style reading comprehension datasets like CNN/Daily Mail (Hermann et al., 2015), Children's Book Test (CBT) (Hill et al., 2015) and RACE (Lai et al., 2017). These datasets have inspired the design of various neural-based models (Hermann et al., 2015;Chen et al., 2016) and some become benchmarks for machine reading comprehension. The dataset ChID used in this paper is also a large scale cloze-style dataset but focuses on Chinese idiom prediction.
Pre-trained Language Models
Language model pre-training has been proven to be effective over a list of natural language tasks at both sentence level (Bowman et al., 2015) and token level (Tjong Kim Sang and De Meulder, 2003;Rajpurkar et al., 2016). Existing strategies of using pre-trained language models include feature-based methods like ELMO (Peters et al., 2018) and fine-tuning methods such as OpenAI GPT (Radford et al., 2018) and BERT (Devlin et al., 2019). BERT-based fine-tuning strategy and its extensions (Cui et al., 2019;Liu et al., 2019a) are pushing performance of neural models to near-human or super-human level. In this paper, we use pre-trained Chinese BERT with Whole Word Masking (Cui et al., 2019) as text sequence processor.
Modelling Figurative Language
Figurative (or non-literal) language is different from literal language where words or characters in literal language act in accordance with conventionally accepted meanings or denotation. In figurative language, meaning can be detached from the words or characters while a more complicated meaning or heightened effect is reattached. As a special type of figurative language, idioms have been actively researched in tasks like Idiom Identification (Muzny and Zettlemoyer, 2013), Idiom Recommendation (Liu et al., 2019b) and Idiom Representation (Gutiérrez et al., 2016;Jiang et al., 2018;Zheng et al., 2019). In this paper, we will focus on the representations of Chinese idioms using a BERT-based approach.
Method
Task Definition and Dataset
We formally define the Chinese idiom prediction task as follows. Given a passage P , represented as a sequence of tokens (p 1 , p 2 , . . . , p n ), where each token is either a Chinese character or the special "blank" token [MASK], and a set of K candidate Chinese idioms denoted as A = {a 1 , a 2 , . . . , a K }, our goal is to select an idiom a * ∈ A that best fits the blank in P . See the example in Table 1.
We assume that a set of training examples in the form of triplets, each containing a passage, a candidate set and the ground truth answer, is given. We denote the training data as
{(P i , A i , a * i )} N i=1 .
We use V to denote the vocabulary of all Chinese idioms observed in the training data, i.e., V = ∪ N i=1 A i . To facilitate the study of Chinese idiom comprehension using deep learning models, Zheng et al. (2019) released the ChID dataset. The dataset was created in the "cloze" style. The authors collected diverse passages from novels and essays on the Internet and news articles from THUCTC (Guo et al., 2016). The authors then replaced target Chinese idioms found in these passages with the blank token. To construct the candidate answer set for each blank, the authors considered synonyms, near-synonyms and other idioms either irrelevant or opposite in meaning to the ground truth idiom (Zheng et al., 2019).
BERT Baselines
Previous methods applied to the ChID dataset are not based on BERT (Devlin et al., 2019) or Transformer (Vaswani et al., 2017) architecture. Because of the success of BERT for many NLP tasks, here we first present two BERT baselines. The first one treats a Chinese idiom as a sequence of characters.
It combines the passage with each candidate idiom into a single sequence and processes multiple sequences, one for each candidate, using BERT. The second one treats a Chinese idiom as a single token that has its own embedding vector. The method uses BERT to process the passage and then matches the encoded passage with each candidate idiom's embedding. These baselines can be regarded as standard ways to solve the Chinese idiom prediction problem using BERT.
For the second baseline that uses idiom embeddings, we also present a heuristic that uses an enlarged candidate set to improve learning. This heuristic is only applicable to the second baseline because it would be computationally too expensive for the first baseline.
BERT Baseline with Idioms as Character Sequences: A straightforward way to apply BERT for Chinese idiom prediction is as follows. Given a passage P = (p 1 , p 2 , . . . , [MASK], . . . , p n ) and a candidate answer a k ∈ A, we first concatenate them into a single sequence ([CLS], p 1 , p 2 , . . . , p n , [SEP], a k,1 , a k,2 , a k,3 , a k,4 , [SEP]), where a k,1 to a k,4 are the four Chinese characters that idiom a k is composed of. We can then directly use BERT to process this sequence and obtain the hidden representation for [CLS] on the last (L-th) layer, denoted by h L k,0 ∈ R d . To select the best answer idiom, we first use a linear layer to process h L k,0 for k = 1, 2, . . . , K and then use standard softmax to obtain the probabilities of each candidate. To train the model, we use standard negative log likelihood as the loss function.
BERT Baseline with Idiom Embeddings: Many Chinese idioms are non-compositional and therefore their meanings should not be directly derived from the embeddings of its four individual characters, as the baseline above does. E.g., "狐假虎威" literally means a fox assuming the majesty of a tiger, but it is usually used to describe someone flaunting his powerful connections. Therefore, we hypothesize that learning a single embedding vector for the entire idiom can help the understanding of idioms.
In this second BERT baseline, instead of concatenating the passage and a candidate answer into a single sequence for BERT to process, we keep them separated. We only use BERT to process the passage sequence ([CLS], p 1 , p 2 , . . . , [MASK], . . . , p n , [SEP]). Afterwards, we use the hidden representation of [MASK] at the last (L-th) layer, denoted as h L b , to match each candidate answer. In this way, no matter how many candidate answers there are, BERT is used to process the passage only once. On the other hand, each Chinese idiom has a hidden embedding vector, which is to be learned.
We use a k to denote the embedding vector for candidate a k ∈ A. The hidden representation h L b is fused with each candidate idiom via element-wise multiplication. Then the probability of selecting a k among all the candidates A is defined as follows:
p k = exp(w · (a k h L b ) + b) K k =1 exp(w · (a k h L b ) + b) ,(1)
where w ∈ R d and b ∈ R are model parameters, and is element-wise multiplication. To train the model, we again use negative log likelihood as the loss function.
Heuristic with Enlarged Candidate Set: The ChID dataset uses only a small set of negative answers in each candidate set and these negatives are fixed for each example during training. It is reasonable to expect that most of the remaining Chinese idioms not in the candidate set are also negative answers and including them in the training data may help. We therefore use a heuristic that considers an enlarged candidate set to further boost the performance.
To apply this heuristic, we define a candidate set A to be the same as V (i.e., the vocabulary containing all Chinese idioms observed in the training data), and then define a second term in the loss function that is the negative log likelihood of selecting the correct answer from this enlarged candidate set.
Note that because A is large, this heuristic is not feasible to be applied to the character sequencebased BERT baseline, because it would require inserting each candidate into the passage for BERT to process, which would be computationally too expensive. Therefore, this enlarged candidate set heuristic is only applied to the idiom embedding-based BERT baseline. Specifically, we can define the probability of selecting answer a ∈ A as follows:
q a = exp(a · h L b ) c∈A exp(c · h L b )
.
(2)
Let q * i denote the probability of selecting the ground truth idiom among all candidates in A for the i-th training example, and p * i denote the probability of selecting the correct answer among the original candidate set A for the i-th training example. Our training loss function is then defined as follows:
L = − N i=1 (log(p * i ) + log(q * i )).(3)
Our Dual Embedding Model
The BERT baselines presented above are reasonable baselines, but they have a potential problem. We observe that in order for an idiom to fit into a passage well, it has to not only grammatically (i.e., syntactically) fit into the local context surrounding the [MASK] token but also show semantic relevance to the whole passage. In the example shown in Table 1, a correct answer has to first be an adjective rather than, say, a noun or a verb. In addition, given the global context of the entire passage, it is understood that the correct answer should convey the meaning of "extravagant." Based on the observation above, we introduce the following two changes to the second BERT baseline, i.e., the idiom embedding-based BERT baseline, introduced in Section 3.2.
Context-aware Pooling
As we have pointed out earlier, oftentimes Chinese idioms have non-compositional meanings, and to evaluate whether a Chinese idiom is suitable in a passage, we need to understand the semantic meaning of the entire passage. Therefore, it is important for us to not only try to match an idiom with the local context it is to be placed in (which can roughly be modeled by h L b ) but also to match it with the entire passage. Let us use a k to denote the embedding for idiom a k . Recall that H L = (h L 0 , h L 1 , . . . , h L n ) represents the hidden states of the last layer of BERT after it processes the passage sequence. Our method with context-aware pooling can be represented as follows:
p k = exp(a k · h L b + max n i=0 (a k · h L i )) K k =1 exp(a k · h L b + max n i=0 (a k · h L i )) .(4)
Dual Embeddings
Because we need to match an idiom with both h L b and the entire passage, the second idea we propose is to split the embedding of an idiom into two "sub-embedding" vectors, which we refer to as "dual embeddings." Let us use a u k and a v k to denote the two embeddings for idiom a k . We then calculate the probability of selecting candidate a k as follows:
p k = exp(a u k · h L b + max n i=0 (a v k · h L i )) K k =1 exp(a u k · h L b + max n i=0 (a v k · h L i )) .(5)
We also adopt the heuristic of enlarged candidate set from Section 3.2. With the candidate set A to be the same as V, we still use dual embeddings to represent each idiom, but when we match the dual embeddings with the passage, we use both a u and a v to match h L b only. This is because it would be too expensive to match a v of each candidate with the entire sequence of hidden states H L as we now have many candidates. So we define the probability of selecting answer a ∈ A , i.e., selecting the ground truth answer from the entire vocabulary of Chinese idioms, as follows:
q a = exp(a u · h L b + a v · h L b ) c∈A exp(c u · h L b + c v · h L b ) .(6)
Experiments
In this section, we evaluate our proposed dual embedding method using the ChID dataset. We also use an attribution method to visualize how each proposed method works on some selected cases. Out: This is an out-of-domain test dataset. The passages come from essays (whereas the training and development data comes from news and novels). Statistics of the data can be found in Table 2.
Evaluation on ChID-Official
Methods Compared: We compare the following different methods. Performance of the first three baselines are directly taken from (Zheng et al., 2019). It is worth noting that the three baselines use BiLSTM as their backbones while our methods use BERT (Transformer) as our backbones. Although BiLSTM with attention can also capture the global contextual information in the passages, our experiments below will show that empirically our BERT-based methods are more effective. Language Model (LM): This method is based on standard bidirectional LSTM (BiLSTM) (Hochreiter and Schmidhuber, 1997;Zhou et al., 2016). It uses BiLSTM to encode the given passage and obtain the hidden state of the blank. Then it compares the blank state with the embedding vector of each candidate idiom to choose the best idiom.
Attentive Reader (AR): This method also uses BiLSTM but augments it with attention mechanism. It is based on the Attentive Reader model by (Hermann et al., 2015).
Standard Attentive Reader (SAR): This is an altered version of Attentive Reader, where attention weights are computed using a bilinear matrix (Chen et al., 2016).
BL-CharSeq: This is the first BERT baseline treating idioms as character sequences.
BL-IdmEmb (w/o EC):
This is the second BERT baseline using idiom embeddings. In this version, we do not use enlarged candidate set.
BL-IdmEmb: This baseline is the same as BL-IdmEmb (w/o EC) but incorporates the heuristic of enlarged candidate set.
Ours-CP: This is our method with contextual pooling (CP) as presented in Section 3.3.1. This method also incorporates the enlarged candidate set heuristic.
Ours-Full (CP+DE): This is our method with both context pooling (CP) and dual embedding (DE), as presented in Section 3.3.2. This method also uses the enlarged candidate set heuristic.
Evaluation Metrics: A standard metric for the task of Chinese idiom prediction is accuracy, which is the percentage of test examples where our predicted idiom is the same as the ground truth idiom. Here besides accuracy, we also consider another setting where we do not have a pre-defined set of candidate idioms, or in other words, we consider all Chinese idioms in our vocabulary as candidates. For this (Voorhees, 1999;Radev et al., 2002), a well-established metric for ranking problems, as the evaluation metric. We use 4 Nvidia 1080Ti GPU cards and a batch size of 10 per card with a total 5 training epochs. The initial learning rate is set to 5e −5 with 1000 warm-up steps. We use the optimizer AdamW in accordance with a learning rate scheduler WarmupLinearSchedule. Our code has been made available online 3 .
Results:
We show the comparison of the performance of the various methods together with the human performance in Table 3. For Human, LM, AR and SAR, the performance shown in the table is taken directly from ChID (Zheng et al., 2019).
We can observe the following from the table.
(1) In general, methods using BERT (including both the baselines and our methods) perform substantially better than previous methods based on BiLSTMs. This is not surprising and confirms the general observation that pre-trained BERT is generally very effective for many NLP tasks. (2) Our two methods that use context pooling to explicitly incorporate more contextual information consistently work better than the BERT-based baselines that do not perform context pooling. This shows the importance of using context pooling to encode long-range contextual information for the task of Chinese idiom prediction. (3) Comparing Ours-Full (CP+DE) with Ours-CP, we can see that Ours-Full (CP+DE) consistently outperforms Ours-CP, for all evaluation splits in terms of both accuracy and MRR. This shows that our full model using dual embeddings coupled with context-aware pooling makes the model more expressive and captures the underlying meanings of Chinese idioms better. It is also worth noting that on the Out split, Ours-Full (CP+DE) achieves significant improvement over Ours-CP, showing better generalization ability of the dual embeddings.
It is interesting to observe that although we hypothesize that the meanings of Chinese idioms are oftentimes not compositional, BL-CharSeq performs better than BL-IdmEmb (w/o EC). We suspect that this is because the BL-CharSeq method allows cross attention between the passage and the characters in each candidate idiom, whereas BL-IdmEmb (w/o EC) encodes both the passage and a candidate as a vectors without allowing any cross attention between them. However, the design of BL-IdmEmb (w/o EC) allows a large number of candidates to be considered, and when we use the enlarged candidate set, we see that BL-IdmEmb performs similarly to BL-CharSeq. When we subsequently incorporate context pooling and dual embedding, we are able to achieve better performance than BL-CharSeq. Overall, we can see that the experiment results demonstrate that both context-aware pooling and dual embeddings are effective, and our proposed full method generally can outperform all the other methods we consider that represent the state of the art.
Evaluation on ChID-Competition
In the second set of experiments, we use ChID-Competition 4 , which is the data for an online competition 5 on Chinese idiom comprehension. Different from ChID, for each entry in ChID-Competition, a list of passages are provided with the same candidate set, and therefore some heuristic strategies can be used (for instance, the exclusion method). The challenge is that ground truth answers will be similar in semantic meanings, and prediction models need to focus on their differences while comparing similar contexts to make the correct predictions. ChID-Competition is divided into Train, Dev, Test and Out splits for different evaluation stages.
To further test the competency of our model, we evaluate the full model Ours-Full on ChID-Competition. Considering the differences between ChID-Official and ChID-Competition, we use some heuristic methods to postprocess the predictions in order to optimize the results globally for a candidate set. Without changing the training paradigm, we treat this problem an assignment problem during postprocessing and use Linear Sum Optimization to optimize the assignment. The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. The method we used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm. Suppose for each blank, we get a probability distribution over the candidate set C. Then define a cost matrix Z where Z i,j represents the log probability of the i-th blank choosing c j . Formally, let X be a boolean matrix where X i,j is 1 if the i-th blank chooses the candidate j. Our optimization problem can be written as
min i j Z i,j X i,j ,(7)
so that each candidate is assigned to at most one blank, and each blank to at most one candidate. The comparison between our method and previous methods is listed in Table 4. In the first section of the table, we list the top-ranked competitors from the competition leaderboard. It is worth noting that these systems are used for competition purposes and may not be publicly available. We then show the results using several pre-trained language models, where the results are found on the CLUE leaderboard 6 . Finally, we list our own full model Ours-Full, which used a larger pre-trained RoBERTa for Chinese 7 . The experiment results show that our full model achieves competitive results compared with the top ranked systems of the competition.
Further Analysis Through Attribution Method
To better understand how our models achieve consistent improvement, we adopt the gradient based attribution method, Integrated Gradients (IG) (Sundararajan et al., 2017), to visualize how each character contributes to the final prediction. To make the visualization more readable, we first perform Chinese word segmentation to merge characters into words. The attribution value of a word is the highest absolute value of all merged characters.
We show some cases in Figure 1, where red color represents positive correlation with the prediction and blue color represents negative correlation with the prediction. For the example on the left, both "供不应求" (in great demand) and "大名鼎鼎" (famous) are positive idioms with a sense of "being abundant in", but the correct answer is "大名鼎鼎" based on the context, because the context suggests that this idiom serves as an adjective to modify a person, and only "大名鼎鼎" is used to describe a person. On the one hand, we hypothesize that BL-IdmEmb may have learned the correlation between "多年" (for many years) and "供不应求," and thus makes a wrong prediction solely based on this signal. On the other hand, Ours-CP chooses "大名鼎鼎", likely because it is consistent with the word "顾 问" (consultant), which is a person, together with the conjunction word "以及" (and), suggesting that context-aware pooling may have helped the understanding of the context.
For the example on the right hand side of the figure, the two candidates "斤斤计较" (to haggle over every ounce) and "大手大脚" (extravagant) are antonyms and represent different attitudes towards spending money. Both idioms suit the context well syntactically. However, the context has the word "却" (but) and the word "价钱昂贵" (expensive), suggesting the person is extravagant with money, making "大 手大脚" the correct candidate. This example shows that for more complex contextual understanding, Ours-Full has advantages over Ours-CP.
Conclusion
In this paper, we proposed a BERT-based dual embedding method to study Chinese idiom prediction. We used a dual-embedding to not only capture local context information but also match the whole context passage. Our experiments showed that our dual-embedding design can improve the performance of the base model, and both the idea of context-aware pooling and the idea of dual embedding can help improve the idiom prediction performance compared to the baseline methods on the ChID dataset.
Figure 1 :
1Example cases with attribution values of words shown in red and blue. Red indicates positive correlation with the prediction while blue indicates negative correlation with the prediction.
Table 2 :
2Some statistics of the ChID dataset. Split: In the first set of experiments, We use the official release of ChID 1 , denoted as ChID-Official. The data has a training set, a development set and a few different test sets. Besides the standard test set Test, the authors also constructed the following test sets: Ran: In this test set, the candidate idioms are randomly sampled from the vocabulary V. No synonyms or near-synonyms were intentionally added as candidates. Sim: In this test set, the candidates are sampled from the top-10 similar idioms and are more challenging than the Ran test dataset. The only difference of Test, Ran and Sim is the candidate sets.Data
Table 3 :
3The experiment results on ChID. We only compute MRR for methods that have idiom embed-
dings.
setting, we use Mean Reciprocal Rank (MRR)
Similarly, to train the model, we use negative log likelihood as shown before.
https://github.com/zhengcj1/ChID-Dataset
https://github.com/ymcui/Chinese-BERT-wwm 3 https://github.com/VisualJoyce/ChengyuBERT
https://github.com/zhengcj1/ChID-Dataset/tree/master/Competition 5 https://biendata.com/competition/idiom/ 6 We show representative systems on the leaderboard as of the submission date of this paper. https://github.com/ CLUEbenchmark/CLUE. 7 https://github.com/brightmart/roberta_zh
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| [
"https://github.com/zhengcj1/ChID-Dataset",
"https://github.com/ymcui/Chinese-BERT-wwm",
"https://github.com/VisualJoyce/ChengyuBERT",
"https://github.com/zhengcj1/ChID-Dataset/tree/master/Competition",
"https://github.com/brightmart/roberta_zh"
] |
[
"Unsupervised Latent Tree Induction with Deep Inside-Outside Recursive Autoencoders",
"Unsupervised Latent Tree Induction with Deep Inside-Outside Recursive Autoencoders"
] | [
"Andrew Drozdov [email protected] \nCollege of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n\n",
"Pat Verga \nCollege of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n\n",
"Mohit Yadav \nCollege of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n\n",
"Mohit Iyyer [email protected] \nCollege of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n\n",
"Andrew Mccallum [email protected] \nCollege of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n\n"
] | [
"College of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n",
"College of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n",
"College of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n",
"College of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n",
"College of Information and Computer Sciences\nUniversity of Massachusetts Amherst\n"
] | [] | We introduce deep inside-outside recursive autoencoders (DIORA), a fully-unsupervised method for discovering syntax that simultaneously learns representations for constituents within the induced tree. Our approach predicts each word in an input sentence conditioned on the rest of the sentence and uses inside-outside dynamic programming to consider all possible binary trees over the sentence. At test time the CKY algorithm extracts the highest scoring parse. DIORA achieves a new state-of-the-art F1 in unsupervised binary constituency parsing (unlabeled) in two benchmark datasets, WSJ and MultiNLI. | 10.18653/v1/n19-1116 | [
"https://arxiv.org/pdf/1904.02142v1.pdf"
] | 102,350,747 | 1904.02142 | 01083f6a848a9f2bdfbf195bbdac5cf1f9465d23 |
Unsupervised Latent Tree Induction with Deep Inside-Outside Recursive Autoencoders
Andrew Drozdov [email protected]
College of Information and Computer Sciences
University of Massachusetts Amherst
Pat Verga
College of Information and Computer Sciences
University of Massachusetts Amherst
Mohit Yadav
College of Information and Computer Sciences
University of Massachusetts Amherst
Mohit Iyyer [email protected]
College of Information and Computer Sciences
University of Massachusetts Amherst
Andrew Mccallum [email protected]
College of Information and Computer Sciences
University of Massachusetts Amherst
Unsupervised Latent Tree Induction with Deep Inside-Outside Recursive Autoencoders
We introduce deep inside-outside recursive autoencoders (DIORA), a fully-unsupervised method for discovering syntax that simultaneously learns representations for constituents within the induced tree. Our approach predicts each word in an input sentence conditioned on the rest of the sentence and uses inside-outside dynamic programming to consider all possible binary trees over the sentence. At test time the CKY algorithm extracts the highest scoring parse. DIORA achieves a new state-of-the-art F1 in unsupervised binary constituency parsing (unlabeled) in two benchmark datasets, WSJ and MultiNLI.
Introduction
Syntactic parse trees are useful for downstream tasks such as relation extraction (Gamallo et al., 2012), semantic role labeling (Sutton and Mc-Callum, 2005;He et al., 2018), machine translation (Aharoni and Goldberg, 2017;Eriguchi et al., 2017;Zaremoodi and Haffari, 2018), and text classification (Li and Roth, 2006;Tai et al., 2015). Traditionally, supervised parsers trained on datasets such as the Penn TreeBank (Marcus et al., 1994) are used to obtain syntactic trees. However, the treebanks used to train these supervised parsers are typically small and restricted to the newswire domain. Unfortunately, models trained on newswire treebanks tend to perform considerably worse when applied to new types of data, and creating new domain specific treebanks with syntactic annotations is expensive and timeconsuming.
Motivated by the desire to address the limitations of supervised parsing and by the success of large-scale unsupervised modeling such as ELMo and BERT (Peters et al., 2018a;Devlin et al., * Equal contribution, randomly ordered.
Under the current circumstances he says their scenario no longer seems unrealistic 2018), we propose a new deep learning method of unsupervised parser training that can extract both shallow parses (i.e., noun phrases or entities) and full syntactic trees from any domain or language automatically without requiring any labeled training data. In addition to producing parses, our model simultaneously builds representations for internal constituents that reflect syntactic and semantic regularities which can be leveraged by downstream tasks.
Our model builds on existing work developing latent tree chart parsers (Socher et al., 2011b;Le and Zuidema, 2015;Yogatama et al., 2017;Maillard et al., 2017;Choi et al., 2018). These methods produce representations for all internal nodes in the tree (cells in the chart), each generated as a soft weighting over all possible sub-trees ( §2). Unfortunately, they still require sentence-level annotations during training, as they are all trained to optimize a downstream task, typically natural language inference.
To address these limitations, we present deep inside-outside recursive autoencoders (DIORA) which enable unsupervised discovery and representation of constituents without requiring any supervised training data. DIORA incorporates the inside-outside algorithm (Baker, 1979;Lari and Young, 1990) into a latent tree chart parser. The bottom-up inside step calculates a representation for all possible constituents within a binary tree over the input sentence. This step is equivalent to the forward-pass of previous latent tree chart parsers (Maillard et al., 2017). These inside representations only encode the current subtree, ignor- (i, j) a (i, j) b (i, j) Figure 2: The inside and outside pass of DIORA for the input 'the cat drank'. a) The inside pass: The blue inside vectorā(k) for the phrase 'the cat drank' is a weighted average of the compositions for the two possible segmentations -((the cat), drank) and (the, (cat drank)). The scalar weights come from a learned compatibility function. b) The outside pass: The red outside vectorb(k) for the phrase 'the cat' is a function of the outside vector of its parent 'the cat drank' and the inside vector of its sibling 'drank'.
i 0 j 0 i 1 j 1 i 0 j 0 a(k)b (k)
ing all outside context. Thus, we perform an additional top-down outside calculation for each node in the tree, providing external context into the subtree representations in each chart cell. The model is then trained with the objective that the outside representations of the leaf cells should reconstruct the corresponding leaf input word, analogous to masked language model (Devlin et al., 2018) pretraining, except by using dynamic programming we predict every word from a completely unmasked context. The single most likely tree can be recovered using the CKY algorithm and compatibility scores between constituents. Previous work either predict trees that are not well aligned with known treebanks (Yogatama et al., 2017;Choi et al., 2018), or has no mechanism for explicitly modeling phrases, requiring a complex procedure to extract syntactic structures (Shen et al., 2018a).
To probe different properties of our model, we run experiments on unsupervised parsing, segment recall, and phrase representations. DIORA achieves multiple new state-of-the-art results for unsupervised constituency parsing (absolute improvements of 13.7%, 11.5%, and 7.8% on WSJ, WSJ-40, and MultiNLI), has a greater recall on more constituent types than a strong baseline, and produces meaningful phrase representations.
DIORA: Deep Inside-Outside Recursive Autoencoders
Our goal is to design a model and unsupervised training procedure that learns structure from raw text. The design of DIORA is based on our hypothesis is that the most effective compression of a sentence will be derived from following the true syntactic structure of the underlying input. Our approach builds on previous latent tree chart parsers which are augmented with the insideoutside algorithm (Baker, 1979;Lari and Young, 1990) and trained to reproduce each input word from its outside context. Based on our hypothesis, loosely inspired by the linguistic "substitution principle" (Frege, 1960), the model will best reconstruct the input by discovering and exploiting syntactic regularities of the text. The inside pass of our method recursively compresses the input sequence, at each step inputting the vector representations of the two children into a composition function ( §2.1.1) that outputs an inside vector representation of the parent. This process continues up to the root of the tree, eventually yielding a single vector representing the entire sentence (Figure 2a). This is loosely analogous to the compression step of an autoencoder and equivalent to existing latent tree chart parsers forward pass (Maillard et al., 2017). Following this, we initiate the outside pass of our algorithm with a generic (root) representation that is learned as a separate parameter. As the outside step of the inside-outside algorithm (Figure 2b), we unfold until finally producing representations of the leaf nodes. These leaves are then optimized to reconstruct the input sentence as done in an autoencoder-based deep neural network.
Filling the Chart with Inside-Outside
Each inside representation is the root of a particularly sub-tree, and that representation is generated by considering only the descendant constituents within that sub-tree, ignoring any outside context. After the inside representations are calculated, we perform a top-down outside pass to compute outside representations. The outside representations are encoded by looking at only the context of a given sub-tree. Once the chart is filled, each constituent k (cell in the chart) is associated with an inside vectorā(k), an outside vectorb(k), inside compatibility scoreē(k) and outside compatibility scoref (k).
The input to our model is a sentence x made up of T tokens, x 0 , x 1 , ..., x T −1 . Each token x i has a corresponding pre-trained embedded vector v i .
Inside Pass
For each pair of neighboring constituents i and j 1 , we compute a compatibility score and a composition vector. The score and vector that represent a particular span k are computed using a soft weighting over all possible pairs of constituents, that together fully cover the span (we refer to this set of constituent pairs as {k}).
Vectors for spans of length 1 are initialized as a non-linear transformation 2 of the embedded input v i , and the scores associated with these spans are set to 0:
x o u = σ σ tanh (U ψ v k + b) a(k) = o + tanh(x u) e(k) = 0
Higher levels of the chart are computed as a weighted summation of constituent pairs:
a(k) = i,j∈{k} e(i, j) a(i, j) e(k) = i,j∈{k} e(i, j)ê(i, j)
The compatibility functionê is meant to produce a score for how likely a pair of neighboring cells are to be merged. We implement this as a bilinear function of the vectors from neighboring spans, using a learned parameter matrix S. We additionally add the individual scores from each two merging cells. Intuitively, these individual scores correspond to how likely each of the cells would exist in the final binary tree independently. The formula for the compatibility function (and its normalized form e) is defined as follows:
e(i, j) = exp(ê(i, j)) î ,ĵ∈{k} exp(ê(î,ĵ)) e(i, j) = φ(ā(i),ā(j); S α ) +ē(i) +ē(j)
Where the bilinear projection φ is defined as:
φ(u, v; W ) = u W v
For the composition function a we used either a TreeLSTM (Tai et al., 2015) or a 2-layer MLP (see Appendix A.1 for more precise definitons on both methods). In order for the remainder of equations to remain agnostic to the choice of composition function, we refer to the function as Compose, which produces a hidden state vector h and, in the case of TreeLSTM, a cell state vector c, resulting in:
a(i, j) = Compose α (ā(i),ā(j))
Outside Pass
The outside computation is similar to the inside pass (depicted in Figure 2b).
The root node of the outside chart is learned as a bias. Descendant cells are predicted using a disambiguation over the possible outside contexts. Each component of the context consists of a sibling cell from the inside chart and a parent cell from the outside chart.
The function f is analogous to the function e. It is normalized over constituent pairs i, j for the span k, and is used to disambiguate among the many outside contexts. The function b generates a phrase representation for the missing sibling cell. Equations for the outside computation follow:
b(k) = i,j∈{k} f (i, j) b(i, j) f (k) = i,j∈{k} f (i, j)f (i, j) b(i, j) = Compose β (ā(i),b(j)) f (i, j) = φ(ā(i),b(j); S β ) +ē(i) +f (j)
In the majority of our experiments, the Compose used in b shares parameters with a used in the inside pass, as do the compatibility functionsê andf (see §3.4 for results on the effects of parameter sharing).
Training Objective
To train our model we use an autoencoder-like language modeling objective. In a standard autoencoder, the entire input x is compressed into a single lower dimensional representation. This representation, z, is then decompressed and trained to reconstruct x. In our model, we never condition the reconstruction of x on a single z because the root's outside representation is initialized with a bias rather than the root's own inside vector. Instead, we reconstruct x conditioned on the many sub-tree roots, each of which is only a compression of a subset of the input.
To approximate this reconstruction we use a max-margin loss considering a set {x * } of N negative examples that are sampled according to their frequency from the vocabulary (further details in Appendix A.2). The terminal outside vectorb(i) is trained to predict its original input v i .
The per-instance loss function is described in Equation 1:
L x = T −1 i=0 N −1 i * =0 max(0, 1 −b(i) ·ā(i) +b(i) ·ā(i * ))(1)
The max-margin loss does not provide a gradient if the predicted vector is closer to its ground truth than the negative example by a margin greater than 1. For that reason, we also experimented with an objective based on cross-entropy, described in Equation 2:
Z * = N −1 i * =0 exp(b(i) ·ā(i * )) L x = − T −1 i=0 log exp(b(i) ·ā(i)) exp(b(i) ·ā(i)) + Z *(2)
DIORA CKY Parsing
To obtain a parse with DIORA, we populate an inside and outside chart using the input sentence. We can extract the maximum scoring parse based on our single grammar rule using the CKY procedure (Kasami, 1966;Younger, 1967). The steps for this procedure are described in Algorithm 1 and its runtime complexity in Appendix A.3.
Experiments
To evaluate the effectiveness of DIORA, we run experiments on unsupervised parsing, unsuper- for each k ∈ chart | SIZE(k) = 1 do 3:
x k ← 0 Calculate a maximum score for each span, and record a backpointer.
4:
for each k ∈ chart do 5:
x k ← max i,j∈{k} [x i + x j + e(i, j)] 6: π i k , π j k ← arg max i,j∈{k} [x i + x j + e(i, j)]
Backtrack to get the maximal tree. j ← BACKTRACK(π j k ) 12:
return (i, j) 13:
return BACKTRACK(k ← root)
vised segment recall, and phrase similarity. The model has been implemented in PyTorch (Team, 2018) and the code is published online. 3 For training details, see Appendix A.2.
Unsupervised Parsing
We first evaluate how well our model predicts a full unlabeled constituency parse. We look at two data sets used in prior work (Htut et al., 2018), The Wall Street Journal (WSJ) section of Penn Tree Bank (Marcus et al., 1994), and the automatic parses from MultiNLI (Williams et al., 2018b). WSJ has gold human-annotated parses and MultiNLI contains automatic parses derived from a supervised parser (Manning et al., 2014). In addition to PRPN (Shen et al., 2018a), 4 we compare our model to deterministically constructed left branching, right branching, balanced, and random trees. We also compare to ON-LSTM (Shen et al., 2018b), an extension of the PRPN model, RL-SPINN (Yogatama et al., 2017), an unsupervised shift-reduce parser, and ST-Gumbel (Choi et al., 2018), an unsupervised chart parser. The latter two of these models are trained to predict the downstream task of natural language inference (NLI).
Binarized WSJ and MultiNLI results
For the full WSJ test set and MultiNLI datasets we follow the experimental setup of previous work (Williams et al., 2018a). We binarize target trees using Stanford CoreNLP (Manning et al., 2014) and do not remove punctuation (experiments in §3.1.2 do remove punctuation). Latent tree models have been shown to perform particularly poorly on attachments at the beginning and end of the sequence (Williams et al., 2018a). To address this, we incorporate a postprocessing heuristic (denoted as +PP in result tables) 5 . This heuristic simply attaches trailing punctuation to the root of the tree, regardless of its predicted attachment.
In Table 1, we see that DIORA +PP achieves the highest average and maximum F1 from five random restarts. This model achieves a mean F1 7 points higher than ON-LSTM and an increase of over 6.5 max F1 points. We also see that DIORA exhibits much less variance between random seeds than ON-LSTM. Additionally, we find that PRPN-UP and DIORA benefit much more from the +PP heuristic than PRPN-LM. This is consistent with qualitative analysis showing that DIORA and PRPN-UP incorrectly attach trailing punctuation much more often than PRPN-LM.
On the MultiNLI dataset, PRPN-LM is the top performing model without using the +PP heuristic while DIORA matches PRPN-UP (Table 2. Using the heuristic, DIORA greatly surpasses both variants of PRPN. However, it is worth noting that this is not a gold standard evaluation and instead evaluates a model's ability to replicate the output of a trained parser (Manning et al., 2014). A second caveat is that SNLI (Bowman et al., 2015) and MultiNLI contain several non-newswire domains. Syntactic parsers often suffer significant performance drops when predicting outside of the newswire domain that the models were trained on.
WSJ-10 and WSJ-40 results
We also compare our models to two subsets of the WSJ dataset that were used in previous unsupervised parsing evaluations. WSJ-10 and WSJ-40 contain sentences up to length 10 and 40 respectively after punctuation removal. We do not binarize either of these two splits in order to compare to previous work (see Appendix A.4 details on WSJ split differences). Not binarizing the target trees sets an upper-bound on the performance of our models, denoted as UB in Table 3. We compare against previous notable models for this task: CCM (Klein and Manning, 2002) uses the EM algorithm to learn probable nested bracketings over a sentence using gold or induced part-of-speech tags, and PRLG (Ponvert et al., 2011) performs constituent parsing through consecutive rounds of sentence chunking.
In Table 3, we see that DIORA outperforms the previous state of the art for WSJ-40, PRLG, in max F1. The WSJ-10 split has been difficult for latent tree parsers such as DIORA, PRPN, and ON-LSTM, none of which (including our model) are able to improve upon previous non-neural methods. However, when we compare trends between WSJ-10 and WSJ-40, we see that DIORA does a better job at extending to longer sequences.
Unsupervised Phrase Segmentation
In many scenarios, one is only concerned with extracting particular constituent phrases rather than a full parse. Common use cases would be identifying entities, noun phrases, or verb phrases for downstream analysis. To get an idea of how well our model can perform on phrase segmentation, we consider the maximum recall of spans in our predicted parse tree. We leave methods for cutting the tree to future work and instead consider the maximum recall of our model which serves as an upper bound on its performance. Recall here is the percentage of labeled constituents that appear in our predicted tree relative to the total number of constituents in the gold tree. These scores are separated by type and presented in Table 4.
In Table 4 we see the breakdown of constituent recall across the 10 most common types. DIORA achieves the highest recall across the most types and is the only model to perform effectively on verb-phrases. Interestingly, DIORA performs worse than PRPN-LM at prepositional phrases.
Phrase Similarity
One of the goals of DIORA is to learn meaningful representations for spans of text. Most language modeling methods focus only on explicitly modeling token representations and rely on ad-hoc postprocessing to generate representations for longer spans, typically relying on simple arithmetic functions of the individual tokens.
To evaluate our model's learned phrase representations, we look at the similarity between spans of the same type within labeled phrase datasets. We look at two datasets. is a named entity dataset containing 19 different entity types. For each of the labeled spans with length greater than one, we first generate its phrase representation. We then calculate its cosine similarity to all other labeled spans. We then calculate if the label for that query span matches the labels for each of the K most similar other spans in the dataset. In Table 5 we report precision@K for both datasets and various values of K.
The first baseline we compare against produces phrase representations from averaging contextinsensitive (CI) ELMo vectors of individual tokens with the span. The second uses sentenceinsensitive (SI) ELMo vectors, running the full ELMo over only the relevant tokens and ignoring the rest of the sentence. We also look at ELMo's output when given the entire sentence. When analyzing our baselines that run the full ELMo, we follow the procedure described in (Peters et al., 2018b) and represent phrases as a function of its first and last hidden state. We extract these states from the final ELMo layer (3rd BiL-STM) as these consistently gave the best performance among other options. For DIORA, we use the concatenation of the inside and outside representations ([ā;b] For CoNLL 2000, we find that our model outperforms all baselines for all values of K. This demonstrates DIORA's ability to capture and represent syntactic information within phrases. For CoNLL 2012, we find that DIORA outperforms both ELMo CI and ELMo SI while ELMo performs best overall. ELMo CI is surprisingly effective on this dataset even though it performed more poorly on CoNLL 2000. These results indicate that DIORA is capturing syntax quite well, but still has room to improve on more fine-grained semantic representations.
Impact of Modeling Choices
To test the impact of our modeling choices, we compared the performance of two different losses and four different composition functions on the full WSJ validation set. The losses were covered in Equations 1 (Margin) and 2 (Softmax). The two primary methods of composition we considered were TreeLSTM (Tai et al., 2015) and MLP (a 2-hidden layer neural network). In addition, we experimented with a simple kernel of the MLP input [x; y; x y; x − y] and with a setting where both the inside and outside parameters are shared.
The results are shown in Table 6. We see that MLP composition consistently performs better than with TreeLSTM, that MLP benefits from the Softmax loss, and that the best performance comes from sharing parameters. All other experimental results use this highly performant setting unless otherwise specified.
The convoy of about 100 vehicles was the first to make deliveries to the capital in about 10 days
The court ruled that the news media did n't reveal Twiggy 's problems at the time The following month the company put itself up for sale
The following month the company put itself up for sale He added that the U.S. has cut off aid to some rebel units when it was determined that those units broke the cease-fire He added that the U.S. has cut off aid to some rebel units when it was determined that those units broke the cease-fire
We simply do n't agree with that or the findings of their investigation We simply do n't agree with that or the findings of their investigation
Qualitative Results
Looking at our model's output, we see that some trees are an exact replication of the binarized ground truth (Fig. 3), or very close (Fig. 4). For future work we intend to explore common patterns in DIORA's learned structure, although some patterns are already recognizable, such as the affinity to group particles and verbs (Fig. 5).
Related Work
Latent Tree Learning A brief survey of neural latent tree learning models was covered in ( Softmax 50.8 56.7 Table 6: F1 for different model variants on the binary WSJ validation set with included punctuation.
The binary trees are as-is (∅) or modified according to the post-processing heuristic (+P P ). The mean F1 is shown across three random seeds.
Neural Inside-Outside Parsers The Inside-Outside Recursive Neural Network (IORNN) (Le and Zuidema, 2014) is closest to ours. It is a graph-based dependency parser that uses beam search and can reliably find accurate parses when retaining a k-best list. In contrast, our model produces the most likely parse given the learned compatibility of the constituents. The Neural CRF Parser (Durrett and Klein, 2015), similar to DIORA, performs exact inference on the structure of a sentence, although requires a set of grammar rules and labeled parse trees during training. DIORA, like Liue et al. (2018), has a single grammar rule that applies to any pair of constituents and does not use structural supervision.
Learning from Raw Text Unsupervised learning of syntactic structure has been an active research area (Brill et al., 1990), including for unsupervised segmentation (Ando and Lee, 2000;Goldwater et al., 2009;Ponvert et al., 2011) and unsupervised dependency parsing (Spitkovsky et al., 2013). Some models exploit the availability of parallel corpora in multiple languages (Das and Petrov, 2011;Cohen et al., 2011). Others have shown that dependency parsing can be used for unsupervised constituency parsing (Spitkovsky et al., 2013;Klein and Manning, 2004), or that it's effective to prune a random subset of possible trees (Bod, 2006). These approaches aren't necessarily orthogonal to DIORA. For instance, our model may benefit when combined with an unsupervised dependency parser.
Conclusion
In this work we presented DIORA, an unsupervised method for inducing syntactic trees and representations of constituent spans. We showed inside-outside representations constructed with a latent tree chart parser and trained with an autoencoder language modeling objective learns syntactic structure of language effectively. In experiments on unsupervised parsing, chunking, and phrase representations we show our model is comparable to or outperforms previous methods, achieving the state-of-the-art performance on unsupervised unlabeled constituency parsing for the full WSJ (with punctuation), WSJ-40, and NLI datasets. We also show our model obtains higher segment recall than a comparable model and outperforms strong baselines on phrase representations on a chunking dataset. While the current model seems to focus primarily on syntax, future work can improve the model's ability to capture fine-grained semantics. Potential avenues include training larger models over much larger corpora, extra unsupervised or weakly-supervised phrase classification objectives, and other modeling enhancements. We are also eager to apply DIORA to other domains and languages which do not have rich linguistically annotated training sets.
x f i f j o u = σ σ σ σ tanh U h i h j + b + 0 ω ω 0 0 c = c i f i + c j f j + x u h = o + tanh(c)
The constant ω is set to 1 for the inside, 0 for the outside. U and b are learned.
MLP. MLP (Multi-Layer Perceptron) is a deep non-linear composition with the following form:
h = W 1 (W 0 h i , h j + b) + b 1 The operator h i , h j is a concatenation [h i ; h j ].
For the MLP Kernel h i , h j is more involved to support further interaction between the two input vectors [h i ; h j ; h i h j ; h i − h j ]. The variables W 0 , W 1 , b, b 1 are learned and c is unused.
A.2 Training Details
Training Data. Sentences of length ≤ 20 from the SNLI and MultiNLI training sets. Optimization. We train our model using stochastic gradient descent with the Adam optimization algorithm (Kingma and Ba, 2014). Cells were normalized to have magnitude of 1, following Socher et al. (2011a). For instance,ā(k) := a(k)/ ā(k) 2 . Gradients are clipped to a maximum L2-norm of 5. Hyperparameters. Chosen using grid search over cell-dimension {400D, 800D} and learning rate {2, 4, 8, 10, 20} · 10 −4 . Early Stopping. Using unlabeled parsing F1 against the binarized WSJ validation set. Vocabulary. The model is trained in an openvocabulary setting using pre-trained contextinsensitive character embeddings from ELMo (Peters et al., 2018a). Batching. Batches were constructed such that they contained sentences of uniform length. Using batch size 128 for 400D and 64 for 800D.
Sampling. N negatives are sampled for each batch. All experiments use N = 100. Training Steps. 1M parameter updates, taking 3 days using 4x Nvidia 1080ti.
A.3 Runtime Complexity
The runtime complexities for DIORA's methods are shown in Table 7. The parallel column represents the complexity when the values for all constituent pairs are computed simultaneously, assuming that these computations are independent and do not depend on values that have yet to be computed. Linear complexity is theoretically feasible depending on batch size, input length, and number of computational cores. In practice, one might experience super-linear performance.
Although both the inside pass and outside pass have an upper bound of n 3 operations, the outside pass will have more operations than the inside pass for sentences of length > 1.
As a point of reference, our implementation computes the loss over the entire WSJ corpus in 5 minutes 30 seconds at a rate of 3,500 words per second using a single GPU.
Method
Serial Parallel
Inside Pass O(n 3 ) O(n) Outside Pass O(n 3 ) O(n) Training Objective O(n · N ) O(n) CKY O(n 3 ) O(n)
A.4 Reproducing Parsing Results
In
A.5 Parse Trees
Examples of parse trees derived from the compatibility scores are shown in Figures 6, 7, and 8. Some punctuation has been removed for easier readability.
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The wholesaler of cash and carry merchandise reported fiscal fourthquarter earnings that were better than analysts had expected Figure 6: Examples where DIORA achieves 100% accuracy compared with the binarized ground truth.
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In addition a big loan that First Boston made to Ohio Mattress Co was n't repaid on time when its 450 million junk financing for a buy-out of the bedding company was withdrawn
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Within a year Kao Corp. a major cosmetics company plans to eliminate 1,000 clerical jobs by putting on a central computer network some work such as credit reports currently performed in 22 separate offices
Within a year Kao Corp. a major cosmetics company plans to eliminate 1,000 clerical jobs by putting on a central computer network some work such as credit reports currently performed in 22 separate offices Authorities at London 's Heathrow Airport are investigating the disappearance of a Paul Gauguin watercolor Young Tahitian Woman in a Red Pareo that has two sketches on its verso -LRB-opposite -RRB-side
Authorities at London 's Heathrow Airport are investigating the disappearance of a Paul Gauguin watercolor Young Tahitian Woman in a Red Pareo that has two sketches on its verso -LRB-opposite -RRB-side But in its ruling last April the New York court said that all producers of the anti-miscarriage drug should share liability when the manufacturer of a specific dose ca n't be determined But in its ruling last April the New York court said that all producers of the anti-miscarriage drug should share liability when the manufacturer of a specific dose ca n't be determined
The Fed said the Comptroller of the Currency is expected to begin a Community Reinvestment Act examination of First Union 's Florida and North Carolina banking units in the next two weeks
The Fed said the Comptroller of the Currency is expected to begin a Community Reinvestment Act examination of First Union 's Florida and North Carolina banking units in the next two weeks
The appeals-court decision last year was particularly surprising because the same court had dismissed a similar case in 1970 involving singer Nancy Sinatra and a tire ad also a Young & Rubicam product
The appeals-court decision last year was particularly surprising because the same court had dismissed a similar case in 1970 involving singer Nancy Sinatra and a tire ad also a Young & Rubicam product
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The resulting # 1.9 billion merchandise trade deficit was partly offset by an assumed surplus of # 300 million in so-called invisible items which include income from investments services and official transfers
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For the third quarter net premiums were 742 million up 9.6 % from 677 million in last year 's quarter because of the expiration of the National Indemnity quota share reinsurance agreement The three units are a nationwide pharmaceutical and health-products distributor a small sporting-goods chain and a combination catalog showroom and toy-store chain
The three units are a nationwide pharmaceutical and health-products distributor a small sporting-goods chain and a combination catalog showroom and toy-store chain Dreyfus alone has seen its money market funds grow from 1 billion in 1975 to closes to 15 billion today Dreyfus alone has seen its money market funds grow from 1 billion in 1975 to closes to 15 billion today : Examples where DIORA achieves 100% recall compared with the raw (n-ary) ground truth, but less than 100% accuracy on the binarized ground truth. DIORA is shown above the ground truth. DIORA's output is shown above the ground truth.
On Tuesday the House approved a labor-backed amendment that would require the Transportation Department to reject airline acquisitions if the person seeking to purchase a carrier had run two or more airlines previously that have filed for protection from creditors under Chapter 11 of the federal Bankruptcy Code
On Tuesday the House approved a labor-backed amendment that would require the Transportation Department to reject airline acquisitions if the person seeking to purchase a carrier had run two or more airlines previously that have filed for protection from creditors under Chapter 11 of the federal Bankruptcy Code
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There is also speculation that Mr. Newhouse could bring in a powerhouse businessman or another Newhouse family member to run the business side in combination with a publishing executive like Robert Gottlieb who left Random House 's Alfred A. Knopf to run the New Yorker also owned by the Newhouse family
The Warner Bros. studio and Sony signaled they are close to a settlement yesterday asking a Los Angeles Superior Court to postpone a hearing scheduled for tomorrow on Warner 's request for a preliminary injunction blocking Mr. Guber and Mr. Peters from taking the top posts at Columbia Pictures Entertainment Inc
The Warner Bros. studio and Sony signaled they are close to a settlement yesterday asking a Los Angeles Superior Court to postpone a hearing scheduled for tomorrow on Warner 's request for a preliminary injunction blocking Mr. Guber and Mr. Peters from taking the top posts at Columbia Pictures Entertainment Inc
Rep. Edwards the California Democrat is one who pledges that he would immediately challenge Mr. Bush in the courts arguing a line-item veto would expand a president 's powers far beyond anything the framers of the Constitution had in mind
Rep. Edwards the California Democrat is one who pledges that he would immediately challenge Mr. Bush in the courts arguing a line-item veto would expand a president 's powers far beyond anything the framers of the Constitution had in mind
The Merc received considerable criticism in 1987 when it was discovered that its compliance director Kevin P. Conway who then was responsible for policing the exchange 's busy oil and metal pits was engaged in other personal business activities on Exchange time including out-of-state trips according to a New York Merc report prepared last year
The Merc received considerable criticism in 1987 when it was discovered that its compliance director Kevin P. Conway who then was responsible for policing the exchange 's busy oil and metal pits was engaged in other personal business activities on Exchange time including out-of-state trips according to a New York Merc report prepared last year
For example one of my favorite movies is the 1949 British comedy Kind Hearts and Coronets in which the entire comedy is based on actor Dennis Price 's murdering eight titled relatives -LRB-all played by Alec Guinness -RRB-because they snubbed his mother and stand in the way of his acquiring the family title For example one of my favorite movies is the 1949 British comedy Kind Hearts and Coronets in which the entire comedy is based on actor Dennis Price 's murdering eight titled relatives -LRB-all played by Alec Guinness -RRB-because they snubbed his mother and stand in the way of his acquiring the family title
The leveraged buy-out firm of Kohlberg Kravis Roberts & Co. which owns 46 % of the common equity of SCI TV indicated in the debt plan that it would reduce its equity stake to 15 % giving the rest of its stake to bondholders in the restructuring
The leveraged buy-out firm of Kohlberg Kravis Roberts & Co. which owns 46 % of the common equity of SCI TV indicated in the debt plan that it would reduce its equity stake to 15 % giving the rest of its stake to bondholders in the restructuring Recognition also said it obtained a commitment from Chemical Bank and Bank of Boston to convert an estimated 18 million in bank debt to a new 24-month secured term loan to be repaid through the sale of certain assets Recognition also said it obtained a commitment from Chemical Bank and Bank of Boston to convert an estimated 18 million in bank debt to a new 24-month secured term loan to be repaid through the sale of certain assets
The prices of cattle and hog futures contracts dropped sharply because traders speculated that the stock market plunge Friday will linger in the minds of U.S. consumers long enough to prompt them to rein in their spending at the supermarket which would hurt demand for beef and pork
The prices of cattle and hog futures contracts dropped sharply because traders speculated that the stock market plunge Friday will linger in the minds of U.S. consumers long enough to prompt them to rein in their spending at the supermarket which would hurt demand for beef and pork Figure 8: DIORA can perform close to the ground truth even on long sentences. In this figure, n-ary trees are shown for the ground truth. DIORA's output is shown above the ground truth.
Figure 1 :
1An unlabeled binary constituency parse from DIORA matching the ground truth.
Figure 3 :
3DIORA can match the ground truth exactly.Ferro also said it would cancel the unused portion of a 1987 buy-back plan for administrative reasons Ferro also said it would cancel the unused portion of a 1987 buy-back plan for administrative reasonsIn the stands people waved ANC flags wore ANC T-shirts sang ANC songs and chanted ANC slogansIn the stands people waved ANC flags wore ANC T-shirts sang ANC songs and chanted ANC slogans
Figure 4 :
4At times, DIORA exhibits contrary behavior to the ground truth inevitably leading to some error. DIORA's output is shown above the ground truth. 6
Figure 5 :
5DIORA often groups verbs and particles (top), sometimes exactly as the ground truth (middle). Occasionally, errors are particle-like (bottom). DIORA's output is shown above the ground truth. 6
Williams et al., 2018a). The first positive result for neural latent tree parsing was shown in(Htut et al., 2018), which used a language modeling objective. The model in(Liue et al., 2018) uses an inside chart and an outside procedure to calculate marginal probabilities in order to align spans between sentences in entailment.
F1
At the 932 million T. Rowe Price High Yield Fund investors yanked out about 182 million in the past two months At the 932 million T. Rowe Price High Yield Fund investors yanked out about 182 million in the past two months Import values are calculated on a cost insurance and freight -LRB-c.i.f -RRB-basis while exports are accounted for on a free-on-board -LRB-f.o.b -RRB-basis Import values are calculated on a cost insurance and freight -LRB-c.i.f -RRB-basis while exports are accounted for on a free-on-board -LRB-f.o.b -RRB-basis
Also it was not a funny time over here what with the Vietnam War the '68 Democratic convention assassinations and riots Also it was not a funny time over here what with the Vietnam War the '68 Democratic convention assassinations and riots The Tennessee Valley Authority issued 4 billion in bonds in the federal utility 's first public debt offering in 15 years The Tennessee Valley Authority issued 4 billion in bonds in the federal utility 's first public debt offering in 15 years
Figure 7
7Figure 7: Examples where DIORA achieves 100% recall compared with the raw (n-ary) ground truth, but less than 100% accuracy on the binarized ground truth. DIORA is shown above the ground truth. DIORA's output is shown above the ground truth.
BUSINESSLAND INC. San Jose computer retail company annual sales of 1.1 billion NYSE said all 16 corporate office and stores in the area were open with the exception of a retail center in San Francisco 's business district BUSINESSLAND INC. San Jose computer retail company annual sales of 1.1 billion NYSE said all 16 corporate office and stores in the area were open with the exception of a retail center in San Francisco 's business district
arXiv:1904.02142v1 [cs.CL] 3 Apr 20190.7 ⋅ + 0.3 ⋅
The
cat
drank
The
cat drank
(a) Inside Pass
(b) Outside Pass
The
cat drank
The
cat drank
e(i, j)
e(i, j)
a
+PP refers to post-processing heuristic that attaches trailing punctuation to the root of the tree. The top F1 value in each column is bolded.for more
Model
F1 µ
F1 max
δ
LB
13.1
13.1
12.4
RB
16.5
16.5
12.4
Random
21.4
21.4
5.3
Balanced
21.3
21.3
4.6
RL-SPINN †
13.2
13.2
-
ST-Gumbel -GRU † 22.8 ±1.6
25.0
-
PRPN-UP
38.3 ±0.5
39.8
5.9
PRPN-LM
35.0 ±5.4
42.8
6.2
ON-LSTM
47.7 ±1.5
49.4
5.6
DIORA
48.9 ±0.5
49.6
8.0
PRPN-UP +PP
-
45.2
6.7
PRPN-LM +PP
-
42.4
6.3
DIORA +PP
55.7 ±0.4
56.2
8.5
Table 1: Full WSJ (test set) unsupervised unlabeled
binary constituency parsing including punctuation.
† indicates trained to optimize NLI task. Mean and max
are calculated over five random restarts. PRPN F1 was
calculated using the parse trees and results provided by
Htut et al. (2018). The depth (δ) is the average tree
height. Model
F1 median F1 max
δ
Random
27.0
27.0
4.4
Balanced
21.3
21.3
3.9
PRPN-UP
48.6
-
4.9
PRPN-LM
50.4
-
5.1
DIORA
51.2
53.3
6.4
PRPN-UP +PP -
54.8
5.2
PRPN-LM +PP -
50.4
5.1
DIORA +PP
59.0
59.1
6.7
Table 2 :
2NLI unsupervised unlabeled binary con-
stituency parsing comparing to CoreNLP predicted
parses. PRPN F1 was calculated using the parse trees
and results provided by Htut et al. (2018). F1 median
and max are calculated over five random seeds and the
top F1 value in each column is bolded. Note that we
use median rather than mean in order to compare with
previous work.
CoNLL 2000 (Tjong Kim Sang and Buchholz, 2000) is a shallow parsing dataset containing spans of noun phrases, verb phrases, etc. CoNLL 2012 (Pradhan et al., 2012)WSJ-10
WSJ-40
Model
F1 µ
F1 max F1 µ
F1 max
UB
87.8
87.8 85.7
85.7
LB
28.7
28.7 12.0
12.0
RB
61.7
61.7 40.7
40.7
CCM †
-
63.2 -
-
CCM gold † -
71.9 -
33.7
PRLG †
-
72.1 -
54.6
PRPN N LI
66.3 ±0.8 68.5 -
-
PRPN ‡
70.5 ±0.4 71.3 -
52.4
ON-LSTM ‡ 65.1 ±1.7 66.8 -
-
DIORA
67.7 ±0.7 68.5 60.6 ±0.2 60.9
Table 3 :
3WSJ-10 and WSJ-40 unsupervised nonbinary unlabeled constituency parsing with punctuation removed. † indicates that the model predicts a full, non-binary parse with additional resources. ‡ indicates model was trained on WSJ data and PRPN N LI was trained on MultiNLI data. CCM uses predicted POS tags while CCM gold uses gold POS tags. PRPN F1 was calculated using the parse trees and results provided byHtut et al. (2018). LB and RB are the left and right-branching baselines. UB is the upper bound attainable by a model that produces binary trees.
).Label
Count DIORA P-UP P-LM
NP
297,872
0.767
0.687 0.598
VP
168,605
0.628
0.393 0.316
PP
116,338
0.595
0.497 0.602
S
87,714
0.798
0.639 0.657
SBAR
24,743
0.613
0.403 0.554
ADJP
12,263
0.604
0.342 0.360
QP
11,441
0.801
0.336 0.545
ADVP
5,817
0.693
0.392 0.500
PRN
2,971
0.546
0.127 0.144
SINV
2,563
0.926
0.904 0.932
Table 4 :
4Segment recall from WSJ separated by phrase type. The 10 most frequent phrase types are shown above, and the highest value in each row is bolded. P-UP=PRNP-UP, P-LM=PRPN-LM
Table 5 :
5P@1, P@10, and P@100 for labeled chunks from CoNLL-2000 and CoNLL 2012 datasets. For all metrics, higher is better. The top value in each column is bolded. Diora uses the concatenation of the inside and outside vector at each cell which performed better than either in isolation.
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A Appendices
A.1 Composition and Input Transform
TreeLSTM. The TreeLSTM (Tai et al., 2015)
function produces a hidden state vector h and cell
state vector c given two input vectors h i and h j .
Table 7 :
7Runtime complexity for methods associated with DIORA in terms of sentence length n and number of negative examples per token N . Each column represents the complexity when the values for each constituent are computed serially or in parallel.
Table 8 ,
8we've organized a reference for cre-
ating various splits of the WSJ for the purpose
of evaluating unsupervised parsing. Some splits
use only the test set (section 23), others use all of
the training, validation, and test data. Optionally,
punctuation is stripped and sentences greater than
a specified length are ignored. Predictions can be
compared to the full parse trees in the annotated
data, or to a binarized version. The PARSEVAL
specification calculated bracketing F1 considering
all spans, although some previous work diverts
from PARSEVAL and ignores spans that are triv-
ially correct (ones over the entire sentence).
Table 8 :
8Settings for unlabeled binary bracketing evaluation for different splits of the WSJ corpus.
BANKERS ACCEPTANCES : 8.45 % 30 days ; 8.33 % 60 days ; 8.32 % 90 days ; 8.15 % 120 days ; 8.06 % 150 days ; 7.96 % 180 days BANKERS ACCEPTANCES : 8.45 % 30 days ; 8.33 % 60 days ; 8.32 % 90 days ; 8.15 % 120 days ; 8.06 % 150 days ; 7.96 % 180 days
The symbols i, j, and k are identifiers of spans from the input x. The symbol i * identifies a token from the set of negative examples {x * }.2 This function shares its bias term b with Compose α , although U ψ is not tied to any other weights.
https://github.com/iesl/diora 4 We consider the PRPN models using LM stopping criteria, which outperformed UP.
We did not have access to predictions or an implementation of the concurrent ON-LSTM model and therefore could not apply the +PP heuristic.
Ground truth parses are binarized unless otherwise specified. All examples of DIORA parses are already binary. Some punctuation has been removed for easier readability.
AcknowledgementsWe are grateful to Carolyn Anderson, Adina Williams, Phu Mon Htut, and our colleagues at UMass for help and advice, and to the UMass NLP reading group and the anonymous reviewers for feedback on drafts of this work. This work was supported in part by the Center for Intelligent Information Retrieval, in part by the National Science Foundation (NSF) grant numbers DMR-1534431, IIS-1514053 and CNS-0958392. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsor.
Towards string-to-tree neural machine translation. Roee Aharoni, Yoav Goldberg, Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics. the 55th Annual Meeting of the Association for Computational LinguisticsRoee Aharoni and Yoav Goldberg. 2017. Towards string-to-tree neural machine translation. In Pro- ceedings of the 55th Annual Meeting of the Asso- ciation for Computational Linguistics.
Mostlyunsupervised statistical segmentation of japanese: Applications to kanji. Rie Kubota Ando, Lillian Lee, Proceedings of the 1st North American chapter of the Association for Computational Linguistics conference. the 1st North American chapter of the Association for Computational Linguistics conferenceAssociation for Computational LinguisticsRie Kubota Ando and Lillian Lee. 2000. Mostly- unsupervised statistical segmentation of japanese: Applications to kanji. In Proceedings of the 1st North American chapter of the Association for Com- putational Linguistics conference, pages 241-248. Association for Computational Linguistics.
Trainable grammars for speech recognition. K James, Baker, The Journal of the Acoustical Society of America. 65S1James K Baker. 1979. Trainable grammars for speech recognition. The Journal of the Acoustical Society of America, 65(S1):S132-S132.
| [
"https://github.com/iesl/diora"
] |
[
"Self-dual gravity and color/kinematics duality in AdS 4",
"Self-dual gravity and color/kinematics duality in AdS 4"
] | [
"Arthur Lipstein \nDepartment of Mathematical Sciences\nDurham University\nDH1 3LEDurhamUK\n",
"Silvia Nagy \nDepartment of Mathematical Sciences\nDurham University\nDH1 3LEDurhamUK\n"
] | [
"Department of Mathematical Sciences\nDurham University\nDH1 3LEDurhamUK",
"Department of Mathematical Sciences\nDurham University\nDH1 3LEDurhamUK"
] | [] | We show that self-dual gravity in Euclidean four-dimensional Anti-de Sitter space (AdS4) can be described by a scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalisation of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS4 version of the so-called kinematic algebra. We also obtain the 3-point interaction vertex of self-dual gravity in AdS4 from that of self-dual Yang-Mills by replacing the structure constants of the Lie group with the structure constants of the new kinematic algebra, implying that self-dual gravity in AdS4 can be derived from self-dual Yang-Mills in this background via a double copy. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS4 by expanding around the self-dual sector. Moreover, we show that the new kinematic Lie algebra can be lifted to a deformed version of the w1+∞ algebra, which plays a prominent role in celestial holography. | null | [
"https://export.arxiv.org/pdf/2304.07141v2.pdf"
] | 258,170,072 | 2304.07141 | ca80172e443982819d0e3576e5691f946a6dea35 |
Self-dual gravity and color/kinematics duality in AdS 4
15 May 2023
Arthur Lipstein
Department of Mathematical Sciences
Durham University
DH1 3LEDurhamUK
Silvia Nagy
Department of Mathematical Sciences
Durham University
DH1 3LEDurhamUK
Self-dual gravity and color/kinematics duality in AdS 4
15 May 2023
We show that self-dual gravity in Euclidean four-dimensional Anti-de Sitter space (AdS4) can be described by a scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalisation of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS4 version of the so-called kinematic algebra. We also obtain the 3-point interaction vertex of self-dual gravity in AdS4 from that of self-dual Yang-Mills by replacing the structure constants of the Lie group with the structure constants of the new kinematic algebra, implying that self-dual gravity in AdS4 can be derived from self-dual Yang-Mills in this background via a double copy. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS4 by expanding around the self-dual sector. Moreover, we show that the new kinematic Lie algebra can be lifted to a deformed version of the w1+∞ algebra, which plays a prominent role in celestial holography.
I. INTRODUCTION
Self-dual Yang-Mills (SDYM) and gravity (SDG) have provided a very fruitful setting for studying the mathematical structure of perturbative quantum gravity in asympotically flat background. For example, in lightcone gauge they can be described by very simple scalar theories [1][2][3][4][5][6][7][8][9] which make various properties such as color/kinematics duality and the double copy manifest, as shown in [10] and further explored in [11][12][13][14][15][16][17][18][19][20][21][22][23]. Color/kinematics duality is a relation between the color structures and kinematic numerators appearing in Feynman diagrams [24] which lies at the heart of the double copy relating gravity to the square of gauge theory, allowing one to reduce complicated calculations in the former to simpler calculations in the latter [25][26][27].
Another notable feature of SDYM and SDG is their integrability [4][5][6][7][28][29][30][31], which in the case of SDG may be linked to an infinite dimensional symmetry known as the w 1+∞ algebra. This algebra is closely related to the kinematic algebra in SDG [32] and may play a fundamental role in describing 4d quantum gravity in asympotically flat background via a two-dimensional conformal field theory (CFT) living in the sphere at null infinity, known as the celestial CFT [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Celestial CFT provides a framework to recast soft theorems of scattering amplitudes and their underlying asympotic symmetries in the language of 2d CFT.
The holographic description of quantum gravity is best understood in Anti-de Sitter space (AdS), where the dual description is provided by a CFT on the boundary [50]. Furthermore, the study of boundary correlators in four dimensional Anti-de Sitter space (AdS 4 ) is relevant for cosmology after Wick rotating to four dimensional de Sitter space (dS 4 ) [51][52][53]. There has recently been a great deal of progress formulating color/kinematics duality and the double copy in (A)dS [54][55][56][57][58][59][60][61][62][63][64][65][66][67] (and there are also the beginnings of a larger programme to extend the double copy to curved backgrounds [68][69][70][71][72][73][74]) although a systematic understanding is still lacking.
In this paper we set out to find a simple description of SDG in AdS 4 in order to gain a deeper understanding of how color/kinematics duality and the double copy work in this background. After generalising the self-duality equation to nonzero cosmological constant, we show that the solution for the metric can be elegantly written in terms of a scalar field obeying a simple generalisation of the one found long ago by Plebanski for SDG in flat space [1]. In particular it describes a scalar field with in-teractions encoded by a deformed Poisson bracket. From this we deduce that SDG can be derived from SDYM in this background by replacing the color algebra with a deformed kinematic algebra which reduces to the flat space one as the AdS radius goes to infinity. Even more surprisingly, we find that that this kinematic algebra can be lifted to a deformed version of the w 1+∞ algebra suggesting exciting new connections between AdS/CFT and flat space holography.
This paper is organised as follows. In section II, we consider SDYM in AdS 4 , which obeys the same equation of motion as flat space (the fact that we are working in AdS is only encoded by boundary conditions). In section III we then look at SDG in AdS 4 . We introduce an appropriate generalisation of the self-duality condition, and use it to extract a simple Plebanski-like scalar equation. This exhibits a modified Poisson bracket and double copy structure. In section IV we show that SDG in this background encodes a new kinematic algebra and can be obtained by combining this with the flat space kinematic algebra via an asymmetric double copy. In section V, we then lift the two distinct kinematic algebras to two w 1+∞ algebras, one of which is deformed. We present our conclusions in VI. There is also an appendix providing more details on the derivation of our SDG solution.
II. SELF-DUAL YANG-MILLS
We will consider four dimensional Euclidean AdS 4 with unit radius in the Poincaré patch:
ds 2 AdS = dt 2 + dx 2 + dy 2 + dz 2 z 2 ,(1)
where 0 < z < ∞ is the radial coordinate. In a general background, the self-duality constraint for Yang-Mills theory (YM) reads
F µν = √ g 2 ǫ µνρλ F ρλ ,(2)
where g µν is the background metric and g is its determinant. In conformally flat spaces, such as AdS 4 , this just reduces to the self-duality constraint in R 4 , since four dimensional YM is classically scale invariant. Indeed, for the metric in (1), the √ g yields a factor of z −4 while the inverse metrics used to raise indices of the field strength give z 4 , so these factors just cancel out.
We work in the so-called light-cone coordinates:
u = it + z, v = it − z, w = x + iy,w = x − iy,(3)
in which the metric is given by
ds 2 AdS = 4 (dw dw − du dv) (u − v) 2 ,(4)
and ǫ uvww = −1. The non-trivial self-duality contraints can then be written as
F uw = F vw = 0, F uv = F ww .(5)
Following [2], we will also impose lightcone gauge A u = 0. We then find that the self-duality constraints are solved by
A w = 0, Aw = ∂ u Φ, A v = ∂ w Φ,(6)
where Φ is a scalar field in the adjoint representation which satisfies the following equation of motion:
R 4 Φ + i [∂ u Φ, ∂ w Φ] = 0,(7)
where R 4 = −∂ u ∂ v + ∂ w ∂w. This can in turn be derived from the following Lagrangian by introducing a Lagrange multiplier fieldΦ:
L SDYM = Tr Φ ( R 4 Φ + i [∂ u Φ, ∂ w Φ]) .(8)
This is the same action that was previously derived for SDYM in flat space [2][3][4][5][7][8][9] since AdS 4 can be conformally mapped to half of R 4 . On the other hand, since there is a boundary at z = 0, momentum along the z direction will not be conserved, which will become visible when computing boundary correlators in this background. We save a detailed analysis for future work.
With a view to the gravity formulation, we find it useful to split the spacetime coordinates as
x i = (u, w), y α = (v,w),(9)
and introduce the operators
Π α = (Π v , Πw) = (∂ w , ∂ u ).(10)
Finally, we define the Poisson bracket [10] {f, g} :
= ∂ w f ∂ u g − ∂ u f ∂ w g = ε αβ Π α f Π β g,(11)
and notice that it appears naturally in the scalar equation of motion:
R 4 Φ − i 2 [{Φ, Φ}] = 0,(12)
where we introduced the notation
[{f, g}] = ε αβ [Π α f, Π β g] .(13)
Thus SDYM manifestly exhibits color-kinematics duality, as it posesses both a commutator and a Poisson bracket structure.
III. SELF-DUAL GRAVITY
In this section, we will first review SDG in flat background and then describe the generalisation to AdS 4 . A. Self-duality in asymptotically flat gravity
In asymptotically flat gravity, the self-duality condition is given by
R µνρσ = 1 2 √ gǫ ηλ µν R ηλρσ .(14)
The above is the appropriate form of the condition in Euclidean signature. One can go to Lorenzian signature by rescaling the right-hand-side by a factor of i, and the coordinates apropriately. Crucially, the self-duality condition encodes both the equations of motion and the algebraic Bianchi identity for the Riemann tensor, which can be seen by cotracting two of the indices in (14) to get
R µρ = 1 2 ǫ σηλ µ R ηλρσ = 0.(15)
Writing the metric as
ds 2 = dw dw − du dv + h µν dx µ dx ν ,(16)
we find that (14), together with the light-cone gauge choice h uµ = 0 leads to
h iµ = 0, h αβ = Π α Π β φ,(17)
with Π α as defined in the YM sector (10) and the scalar φ satisfying
R 4 φ − {{φ, φ}} = 0,(18)
where we introduced the notation
{{f, g}} = 1 2 ε αβ {Π α f, Π β g},(19)
and {, } is the Poisson bracket introduced previously in (11). This then alows us to give elegant double copy rules in the self-dual sector via [10]:
Φ → φ, i 2 [{ , }] → {{ , }}.(20)
B. Self-duality in AdS4 gravity We wish to generalise the self-duality condition to AdS 4 . To this end, we introduce the tensor:
T µνρσ = R µνρσ − 1 3 Λ(g µρ g νσ − g νρ g µσ ),(21)
where Λ is the cosmological constant. We now define our duality relation as
T µνρσ = 1 2 √ gǫ ηλ µν T ηλρσ .(22)
Upon contracting with g νσ we find
R µρ − Λg µρ = 1 2 √ gǫ σηλ µ R ηλρσ = 0,(23)
where the left-hand-side is equivalent to the Einstein equation with a cosmological constant in the absence of matter sources
R µν − 1 2 Rg µν + Λg µν = 0,(24)
and the right-hand-side is again the algebraic Bianchi identity for the Riemann tensor.
To see more concretely how T µνρσ is the natural generalisation of the Riamann tensor appearing in (14) to spaces with a non-zero cosmological constant, it helps to consider the Weyl tensor in 4 dimensions:
C ρσ µν = R ρσ µν − 2R [ρ [µ g σ] ν] + 1 3 Rg [ρ [µ g σ] ν](25)
In asymptotically flat spaces, upon application of the vacuum equation of motion R µν = R = 0, we get the wellknown result that the Weyl tensor becomes equal to the Riemann tensor. However, in the presence of a cosmological constant, the relevant equations become
R µν = Λg µν , R = 4Λ .(26)
Upon plugging these into the Weyl tensor, we recover exactly the form of T µνρσ from (21) [98]. This result is also natural from the spinorial formulation of tensors in General Relativity, where the so-called Weyl spinors, arising from the Weyl tensor, encode the self-dual and anti-selfdual degrees of freedom.
We will now specialise to a background with cosmological constant Λ = −3, corresponding to AdS 4 background with unit radius.
C. Solution
In this section, we will show that the solution to the self-duality constraint in AdS 4 is a remarkably simple generalisation of the flat space one in (18) when written in terms of a deformed Poisson bracket. Let us begin by introducing the modified Poisson bracket:
{f, g} * = {f, g} + 2 u−v (f ∂ w g − g∂ w f ) .(27)
Using a deformation of the operators (10)
Π = (Π v ,Πw) = ∂ w , ∂ u − 4 u−v ,(28)
we can write (27) as
{f, g} * = 1 2 ε αβ (Π α fΠ β g − Π α gΠ β f ).(29)
In this form, the Poisson bracket which previously appeared in flat space can be recovered simply by replacing Π with its undeformed version Π:
{f, g} = {f, g} * |Π →Π .(30)
We also observe the following relation between the brackets:
{f, g} * = (u − v) 4 f (u − v) 2 , g (u − v) 2 .(31)
Let us proceed to solve the self-duality equation in (22). First, working again in light-cone gauge, we make the following ansatz motivated by the solution in flat space:
ds 2 = 4 dw dw − du dv + h αβ dy α dy β (u − v) 2 ,(32)
where h αβ are unfixed functions. We provide details of how to solve the resulting self-duality equations in Appendix A. At the end of the day, we find the following simple solution:
h αβ = Π (αΠβ) φ,(33)
where φ is a scalar field satisfying the following equation of motion: (34) where the modified double Poisson is defined as follows:
1 u − v R 4 φ u − v − φ u − v , φ u − v * = 0,{{f, g}} * = 1 2 ε αβ {Π α f, Π β g} * ,(35)
with {, } * defined in (29). Setting f = g, this becomes
{{f, f }} * = ∂ 2 w f ∂ 2 u f − (∂ u ∂ w f ) 2 + 2 u − v ∂ w f ∂ u ∂ w f − ∂ u f ∂ 2 w f .(36)
Hence, the equation of motion in (34) provides a natural generalisation of the equation of motion for SDG in flat space given in (18). In particular, it exhibits an asymetric double copy structure
Φ → φ u − v , i 2 [{ , }] → {{ , }} * ,(37)
up to a rescaling of the kinetic term, which will be explored further in the next section. Alternatively, the equation of motion in (34) can be written as follows:
√ g AdS − m 2 φ + 4 φ u − v , φ u − v * = 0,(38)
where AdS φ = g −1/2 ∂ µ √ gg µν ∂ ν φ with g µν the background AdS 4 metric, and m 2 = −2 (corresponding to a conformally coupled scalar in AdS 4 ). This equation of motion is in turn encoded by the following Larangian:
L SDG = √ gφ AdS − m 2 φ + 4φ φ u − v , φ u − v * ,(39)
whereφ is a Lagrange multiplier field.
Finally, (34) admits the following solutions, which are related to planewave solutions by a Weyl rescaling:
φ = (u − v)e ik·x ,(40)
where k · x ≡ uk u + vk v + wk w +wkw is the flat space inner product and k u k v − k w kw = 0 (we refer to this as the on-shell condition). Note that the momenta are complex since we are working in Euclidean signature. Since there is a boundary at z = 0, momentum along the z direction will not be conserved and the natural observables are boundary correlators Fourier transformed to momentum space [75][76][77][78].
IV. COLOR/KINEMATICS DUALITY
It is straightforward to read off Feynman rules from the Lagrangians in (8) and (39). First we expand the scalar fields in the SDYM action as Φ = Φ a T a , where T a generators of the gauge group satisfying Tr T a T b = δ ab and T a , T b = if abc T c . Using on-shell plane wave external states for SDYM and external states of the form (40) for SDG, we obtain the following 3-point vertices (which would be relevant when computing three-point boundary correlators):
V SDYM = 1 2 X (k 1 , k 2 ) f a1a2a3 , V SDG = 1 2 X (k 1 , k 2 )X (k 1 , k 2 ) ,(41)
where
X (k 1 , k 2 ) = k 1u k 2w − k 1w k 2u , X (k 1 , k 2 ) = X (k 1 , k 2 ) − 2i u − v (k 1 − k 2 ) w .(42)
The objects X andX obey Jacobi identities analogous to f a1a2a3 and can therefore be thought of as structure constants of kinematic Lie algebras:
0 = X (k 1 , k 2 ) X (k 3 , k 1 + k 2 ) + cyclic =X (k 1 , k 2 )X (k 3 , k 1 + k 2 ) + cyclic.(43)
These relations do not rely on momentum conservation, and encode color/kinematics duality. Moreover, we find that the SDG vertex can be obtained from the SDYM one by replacing the color structure constant with the deformed kinematic structure contant:
f a1a2a3 →X (k 1 , k 2 ) ,(44)
which encodes the double copy. Whereas in flat background there is only one kinematic algebra and the SDG vertex is obtained by simply squaring X [10], in AdS 4 there are two distinct kinematic algebras and SDG arises from an asymmetrical double copy. The kinematic structure constants naturally arise from Poisson brackets on plane waves:
e ik1·x , e ik2·x = X (k 1 , k 2 ) e i(k1+k2)·x , e ik1·x , e ik2·x * =X (k 1 , k 2 ) e i(k1+k2)·x ,(45)
where k · x is defined below (40). Note that when we plug the solutions in (40) into the deformed Poisson bracket in (39), this is indeed equivalent to acting on plane waves since we divide by the conformal factor (u − v). The kinematic Jacobi identity in (43) is a consequence of the following general property of the deformed Poisson bracket:
{f, {g, h} * } * + {g, {h, f } * } * + {h, {f, g} * } * = 0. (46)
for arbitrary functions f, g and h. Note that the deformed Poisson bracket satisfies a deformed Leibniz rule:
f g (u−v) 2 , h * = 1 (u−v) 2 f {g, h} * + 1 (u−v) 2 g {f, h} * ,(47)
or alternatively
{f g, h} * = f {g, h} * + g {f, h} * − 2f g∂ w h u − v ,(48)
although does not play an important role in our analysis.
V. w1+∞ ALGEBRAS
As shown in [32], the kinematic algebra which appears in SDG can be lifted to a w 1+∞ algebra, which plays an important role in the study of scattering amplitudes in the context of celestial CFT [33]. In particular, the w 1+∞ algebra contains the extended BMS algebra underlying soft graviton theorems of scattering amplitudes [45,46]. In this section, we will follow similar steps to those in [32] to show that the two kinematic algebras derived in the previous section can be lifted to two w 1+∞ algebras, one of which is deformed.
For an on-shell state, the momentum satisfies kw/k u = k v /k w = ρ, where ρ is some number. It is then possible to expand an on-shell plane wave as follows:
e ik·x = ∞ a,b=0 (ik u ) a (ik w ) b a!b! e ab ,(49)
where e ab = (u + ρw) a (w + ρv) b . This is naturally interpreted as an expansion in soft momenta. Letting w p m = 1 2 e p−1+m,p−1−m and plugging this into the Poisson brackets in (11) and (27) then gives
{w p m , w q n } = (n(p − 1) − m(q − 1)) w p+q−2 m+n , {w p m , w q n } * = {w p m , w q n } + (m + q − p − n) u − v w p+q−3/2 m+n+1/2 .(50)
We recognize the first line as the w 1+∞ algebra [33,79], and the second line appears to be a deformed version of this algebra. In the limit where z = (u − v) → ∞ (which corresponds to the flat space limit), the deformation vanishes and the two algebras coincide. This suggests that self-dual gravity in AdS 4 is integrable, and it contains two distinct infinite dimensional symmetry algebras.
VI. CONCLUSION
We have shown that SDG in AdS 4 can be described by a scalar field whose interactions are encoded by a deformed Poisson bracket, providing a surprisingly simple generalisation of the Plebanski action for SDG in flat space. Our action implies a new kinematic algebra dual to the color algebra appearing in SDYM which is a deformation of the flat space kinematic algebra. Moreover, the new kinematic algebra can be lifted to a deformation of the w 1+∞ algebra, implying a new relation between AdS/CFT and flat space holography which extends beyond the flat space limit. Indeed, to our knowledge w 1+∞ symmetry was not previously identified in the context of AdS/CFT. It would be interesting to see how our SDG action compares to previous proposals in [80][81][82][83][84][85][86][87], as well as how it can be realised in twistor space [35,41,88]. In particular, scalar equations were proposed long ago in [87] and more recently in [82], although they appear to be nontrivially related to ours and the deformed Poisson structure is not manifest.
There are a number of other directions for future study. Perhaps the most immediate task is to compute tree-level boundary correlators of SDYM and SDG in AdS 4 and investigate how they encode color/kinematics duality and w 1+∞ symmetry. In doing so, we must take into account the fact that momentum along the radial direction is not conserved and that the bulk-to-bulk propagators must satisfy nontrivial boundary conditions as a result of the boundary at z = 0. Note that the classical solutions in (40) correspond to bulk-to-boundary propagators and can be mapped to plane waves via a Weyl transformation. One slightly nonstandard aspect of these calculations will be the need to work in lightcone gauge, since previous treatments usually worked in axial gauge [75,76]. We can then investigate if the correlators exhibit universal behaviour in the soft or collinear limit, analogous to those in flat space, and explore how this is encoded by the w 1+∞ symmetry. Recent work relating soft theorems to Ward identities in 3d CFT may be of use in this regard [89][90][91][92][93]. In flat space, the scattering amplitudes of SDYM and SDG are 1-loop exact rational functions [94,95] which also exhibit color/kinematics duality [96]. It would be very interesting to see if any of these properties extends to loop-level boundary correlators in AdS 4 .
As mentioned above, we can obtain SDG in AdS 4 from an asymmetrical double copy by combining the flat space kinematic algebra (which appears in SDYM) with a deformed kinematic algebra. It would be interesting to see what gravitational theory arises from squaring the deformed kinematic algebra, or alternatively what gauge theory arises from combining the deformed kinematic algebra with the color algebra. Our approach may also provide a framework for defining color/kinematics duality and the double copy in Einstein gravity via an expansion around the self-dual sector. Indeed, the 4-point tree-level wavefunction coefficient for gravitons in dS 4 (which can be obtained by analytic continuation from AdS 4 ) can be deduced from an ansatz resembling an asymmetric double copy with deformed kinematic numerators [54].
Finally, and perhaps most ambitiously, it would be interesing to identify the CFT dual to SDG in AdS 4 . Given that the bulk theory has an infinite dimensional symmetry it seems very likely that it is integrable, and it should be possible to prove this by generalising the arguments in [4-7, 12, 30, 31]. SDG in AdS 4 may therefore provide an exactly solvable toy model of AdS/CFT. Moreover, introducing a Moyal deformation analogous to the one recently implemented for SDG in flat space [42] may describe a chiral higher spin theory in AdS 4 [97]. There are many exciting avenues which we hope to explore in the future.
ACKNOWLEDGMENTS
We thank Roberto Bonezzi, Gregory Korchemsky, Kirill Krasnov, Lionel Mason, Ricardo Monteiro, Malcolm Perry, Andrea Puhm, and David Skinner for useful discussions. AL is supported by the Royal Society via a University Research Fellowship. SN is supported in part by STFC consolidated grant T000708.
Appendix A: Details of the gravity solution
In this Appendix, we will explain how to derive (34). First we write the self-duality constraint in (22) as
∆ µνρσ = −T µνρσ + 2 (u − v) 2 ǫ µνηλ T ηλ ρσ ,(A1)
where the left-hand-side must vanish and we have plugged in the ansatz in (32). We then make an ansatz for h αβ in terms of derivatives of a scalar field and inverse powers of (u − v) which reduces to (17) as (u − v) → ∞. We then find that the following choice of coefficients in this ansatz causes most of the components of ∆ to trivially vanish:
h vv = ∂ 2 w φ, h vw = ∂ u ∂ w φ − 2 u − v ∂ w φ, hww = ∂ 2 u φ − 4 u − v ∂ u φ + 4 (u − v) 2 φ. (A2)
This is indeed equivalent to (33).
Furthermore, we find that the nonzero components of ∆ can be written as follows:
∆ vuwv = −∆w wwv = 4 (u − v) 2 ∂ w eom, ∆w vwv = − 4 (u − v) 2 ∂ 2 w eom, ∆w vvu = 4 (u − v) 3 ((u − v)∂ u ∂ w − ∂ w ) eom, ∆w wwu = 4 (u − v) 4 (u − v) 2 ∂ 2 u − 4(u − v)∂ u + 6 eom, ∆w vww = 4 (u − v) 3 ((u − v)∂ u ∂ w − 3∂ w ) eom, ∆w vwv = − 4 (u − v) 4 (u − v) 2 (∂ w ∂w − ∂ u ∂ v ) +(u − v) (∂ u + 3∂ v ) − 2∂ w φ∂ w + 2] eom,(A3)
where
eom = (∂ u ∂ v − ∂ w ∂w) φ + 1 u − v (∂ u − ∂ v ) φ − 2φ (u − v) 2 + h vv hww − h 2 vw + 1 (u − v) 2 (∂ w φ) 2 .(A4)
Hence, all the components of ∆ will vanish if we impose eom = 0, which can be interpreted as the equation of motion for the scalar field. Note that the quantity eom can be written as eom = kin + pot,
where
kin = (u − v) 2 4 √ g − AdS + m 2 φ, pot = (u − v) 2 φ u − v , φ u − v * ,(A6)
where AdS φ = g −1/2 ∂ µ √ gg µν ∂ ν φ with g µν the background AdS 4 metric, and m 2 = −2 (corresponding to a conformally coupled scalar in AdS 4 ). The kinetic and potential terms correspond to the first and second lines of (A4), respectively. Noting that a conformally coupled scalar can be mapped to a massless scalar via a Weyl rescaling, we see that eom = 0 is equivalent to the equation of motion in (34).
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We thank Malcolm Perry for this observation. We thank Malcolm Perry for this observation.
| [] |
[
"A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences *",
"A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences *"
] | [
"Joshua Brody [email protected] \nDepartment of Computer Science\nDepartment of Computer Science\nDartmouth College Hanover\n03755NHUSA\n",
"Amit Chakrabarti \nDartmouth College Hanover\n03755NHUSA\n"
] | [
"Department of Computer Science\nDepartment of Computer Science\nDartmouth College Hanover\n03755NHUSA",
"Dartmouth College Hanover\n03755NHUSA"
] | [] | The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their n-bit input strings is large (i.e., at least n/2 + √ n) or small (i.e., at most n/2 − √ n); they do not care if it is neither large nor small. This ( √ n) gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm.Thus far, for randomized communication, an (n) lower bound on this problem was known only in the one-way setting. We prove an (n) lower bound for randomized protocols that use any constant number of rounds.As a consequence we conclude, for instance, that ε-approximately counting the number of distinct elements in a data stream requires (1/ε 2 ) space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream.In the process, we also obtain tight n − ( √ n log n) lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an (n) lower bound on the one-way randomized communication complexity.IntroductionThis paper concerns communication complexity, which is a heavily-studied basic computational model, and is a powerful abstraction useful for obtaining results in a variety of settings not necessarily involving communication. To cite but two examples, communication complexity has been applied to prove lower bounds on circuit depth (see, e.g.,[KW90]) and on query times for static data structures (see, e.g., [MNSW98, Pǎt08]). The basic setup involves two players, Alice and Bob, each of whom receives an input string. Their goal is to compute some function of the two strings, using a protocol that involves exchanging a small number of bits. When communication complexity is applied as a lower bound technique -as it often is -one seeks to * Work supported in part by an NSF CAREER Award CCF-0448277 and NSF grant EIA-98-02068.1 prove that there does not exist a nontrivial protocol, i.e., one that communicates only a sublinear number of bits, for computing the function of interest. Naturally, such a proof is more challenging when the protocol is allowed to be randomized and err with some small probability on each input. The textbook by Kushilevitz and Nisan [KN97] provides detailed coverage of the basics of communication complexity, and of a number of applications, including the two mentioned above. In this paper, we only recap the most basic notions, in Section 2.Our focus here is on a specific communication problem -the Gap-Hamming-Distance problemthat, to the best of our knowledge, was first formally studied by Indyk and Woodruff [IW03] in FOCS 2003. They studied the problem in the context of proving space lower bounds for the Distinct Elements problem in the data stream model. We shall discuss their application shortly, but let us first define our communication problem precisely.The Problem. In the Gap-Hamming-Distance problem, Alice receives a Boolean string x ∈ {0, 1} n and Bob receives y ∈ {0, 1} n . They wish to decide whether x and y are "close" or "far" in the Hamming sense. That is, they wish to output 0 if (x, y) ≤ n/2 − √ n and 1 if (x, y) ≥ n/2 + √ n. They do not care about the output if neither of these conditions holds. Here, denotes Hamming distance. In the sequel, we shall be interested in a parametrized version of the problem, where the thresholds are set at n/2 ± c √ n, for some parameter c ∈ R + .Our Results. While we prove a number of results about the Gap-Hamming-Distance problem here, there is a clear "main theorem" that we wish to highlight. Technical terms appearing below are defined precisely in Section 2.Theorem 1 (Main Theorem, Informal). Suppose a randomized 13 -error protocol solves the Gap-Hamming-Distance problem using k rounds of communication. Then, at least one message must be n/2 O(k 2 ) bits long. In particular, any protocol using a constant number of rounds must communicate (n) bits in some round. In fact, these bounds apply to deterministic protocols with low distributional error under the uniform distribution.Notice that our lower bound applies to the maximum message length, not just the total length. At the heart of our proof is a round elimination lemma that lets us "eliminate" the first round of communication, in a protocol for the Gap-Hamming-Distance problem, and thus derive a shorter protocol for an "easier" instance of the same problem. By repeatedly applying this lemma, we eventually eliminate all of the communication. We also make the problem instances progressively easier, but, if the original protocol was short enough, at the end we are still left with a nontrivial problem. The resulting contradiction lower bounds the length of the original protocol. We note that this underlying "round elimination philosophy" is behind a number of key results in communication complexity [MNSW98, Sen03, CR04, ADHP06, Cha07, VW07, CJP08].Besides the above theorem, we also prove tight lower and upper bounds of n − ( √ n log n) on the oneway deterministic communication complexity of Gap-Hamming-Distance. Only (n) lower bounds were known before. We also prove an (n) one-way randomized communication lower bound. This matches earlier results, but our proof has the advantage of being purely combinatorial. (We recently learned that Woodruff [Woo09] had independently discovered a similar combinatorial proof. We present our proof nevertheless, for pedagogical value, as it can be seen as a generalization of our deterministic lower bound proof.)Motivation and Relation to Prior Work. We now describe the original motivation for studying the Gap-Hamming-Distance problem. Later, we discuss the consequences of our Theorem 1. In the data stream | 10.1109/ccc.2009.31 | [
"https://arxiv.org/pdf/0902.2399v2.pdf"
] | 11,277,470 | 0902.2399 | c16bbebe9840996b10281905e096dcce61e49367 |
A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences *
17 Feb 2009 February 17, 2009
Joshua Brody [email protected]
Department of Computer Science
Department of Computer Science
Dartmouth College Hanover
03755NHUSA
Amit Chakrabarti
Dartmouth College Hanover
03755NHUSA
A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences *
17 Feb 2009 February 17, 2009
The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their n-bit input strings is large (i.e., at least n/2 + √ n) or small (i.e., at most n/2 − √ n); they do not care if it is neither large nor small. This ( √ n) gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm.Thus far, for randomized communication, an (n) lower bound on this problem was known only in the one-way setting. We prove an (n) lower bound for randomized protocols that use any constant number of rounds.As a consequence we conclude, for instance, that ε-approximately counting the number of distinct elements in a data stream requires (1/ε 2 ) space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream.In the process, we also obtain tight n − ( √ n log n) lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an (n) lower bound on the one-way randomized communication complexity.IntroductionThis paper concerns communication complexity, which is a heavily-studied basic computational model, and is a powerful abstraction useful for obtaining results in a variety of settings not necessarily involving communication. To cite but two examples, communication complexity has been applied to prove lower bounds on circuit depth (see, e.g.,[KW90]) and on query times for static data structures (see, e.g., [MNSW98, Pǎt08]). The basic setup involves two players, Alice and Bob, each of whom receives an input string. Their goal is to compute some function of the two strings, using a protocol that involves exchanging a small number of bits. When communication complexity is applied as a lower bound technique -as it often is -one seeks to * Work supported in part by an NSF CAREER Award CCF-0448277 and NSF grant EIA-98-02068.1 prove that there does not exist a nontrivial protocol, i.e., one that communicates only a sublinear number of bits, for computing the function of interest. Naturally, such a proof is more challenging when the protocol is allowed to be randomized and err with some small probability on each input. The textbook by Kushilevitz and Nisan [KN97] provides detailed coverage of the basics of communication complexity, and of a number of applications, including the two mentioned above. In this paper, we only recap the most basic notions, in Section 2.Our focus here is on a specific communication problem -the Gap-Hamming-Distance problemthat, to the best of our knowledge, was first formally studied by Indyk and Woodruff [IW03] in FOCS 2003. They studied the problem in the context of proving space lower bounds for the Distinct Elements problem in the data stream model. We shall discuss their application shortly, but let us first define our communication problem precisely.The Problem. In the Gap-Hamming-Distance problem, Alice receives a Boolean string x ∈ {0, 1} n and Bob receives y ∈ {0, 1} n . They wish to decide whether x and y are "close" or "far" in the Hamming sense. That is, they wish to output 0 if (x, y) ≤ n/2 − √ n and 1 if (x, y) ≥ n/2 + √ n. They do not care about the output if neither of these conditions holds. Here, denotes Hamming distance. In the sequel, we shall be interested in a parametrized version of the problem, where the thresholds are set at n/2 ± c √ n, for some parameter c ∈ R + .Our Results. While we prove a number of results about the Gap-Hamming-Distance problem here, there is a clear "main theorem" that we wish to highlight. Technical terms appearing below are defined precisely in Section 2.Theorem 1 (Main Theorem, Informal). Suppose a randomized 13 -error protocol solves the Gap-Hamming-Distance problem using k rounds of communication. Then, at least one message must be n/2 O(k 2 ) bits long. In particular, any protocol using a constant number of rounds must communicate (n) bits in some round. In fact, these bounds apply to deterministic protocols with low distributional error under the uniform distribution.Notice that our lower bound applies to the maximum message length, not just the total length. At the heart of our proof is a round elimination lemma that lets us "eliminate" the first round of communication, in a protocol for the Gap-Hamming-Distance problem, and thus derive a shorter protocol for an "easier" instance of the same problem. By repeatedly applying this lemma, we eventually eliminate all of the communication. We also make the problem instances progressively easier, but, if the original protocol was short enough, at the end we are still left with a nontrivial problem. The resulting contradiction lower bounds the length of the original protocol. We note that this underlying "round elimination philosophy" is behind a number of key results in communication complexity [MNSW98, Sen03, CR04, ADHP06, Cha07, VW07, CJP08].Besides the above theorem, we also prove tight lower and upper bounds of n − ( √ n log n) on the oneway deterministic communication complexity of Gap-Hamming-Distance. Only (n) lower bounds were known before. We also prove an (n) one-way randomized communication lower bound. This matches earlier results, but our proof has the advantage of being purely combinatorial. (We recently learned that Woodruff [Woo09] had independently discovered a similar combinatorial proof. We present our proof nevertheless, for pedagogical value, as it can be seen as a generalization of our deterministic lower bound proof.)Motivation and Relation to Prior Work. We now describe the original motivation for studying the Gap-Hamming-Distance problem. Later, we discuss the consequences of our Theorem 1. In the data stream
model, one wishes to compute a real-valued function of a massively long input sequence (the data stream) using very limited space, hopefully sublinear in the input length. To get interesting results, one almost always needs to allow randomized approximate algorithms. A key problem in this model, that has seen much research [FM85, AMS99, BJK + 04, IW03,Woo09], is the Distinct Elements problem: the goal is to estimate the number of distinct elements in a stream of m elements (for simplicity, assume that the elements are drawn from the universe [m] := {1, 2, . . . , m}).
An interesting solution to this problem would give an nontrivial tradeoff between the quality of approximation desired as the space required to achieve it. The best such result [BJK + 04] achieved a multiplicative (1 + ε)-approximation using space O(1/ε 2 ), where the O-notation suppresses log m and log(1/ε) factors. It also processed the input stream in a single pass, a very desirable property. Soon afterwards, Indyk and Woodruff [IW03] gave a matching (1/ε 2 ) space lower bound for one-pass algorithms for this problem, by a reduction from the Gap-Hamming-Distance communication problem. In SODA 2004, Woodruff [Woo04] improved the bound, extending it to the full possible range of subconstant ε, and also applied it to the more general problem of estimating frequency moments F p := n i=1 f p i , where f i is the frequency of element i in the input stream. A number of other natural data stream problems have similar space lower bounds via reductions from Gap-Hamming, a more recent example being the computation of the empirical entropy of a stream [CCM07].
The idea behind the reduction is quite simple: Alice and Bob can convert their Gap-Hamming inputs into suitable streams of integers, and then simulate a one-pass streaming algorithm using a single round of communication in which Alice sends Bob the memory contents of the algorithm after processing her stream. In this way, an (n) one-way communication lower bound translates into an (1/ε 2 ) one-pass space lower bound. Much less simple was the proof of the communication lower bound itself. Woodruff's proof [Woo04] required intricate combinatorial arguments and a fair amount of complex calculations. Jayram et al. [JKS07] later provided a rather different proof, based on a simple geometric argument, coupled with a clever reduction from the INDEX problem. A version of this proof is given in Woodruff's Ph.D. thesis [Woo07]. In Section 5, we provide a still simpler direct combinatorial proof, essentially from first principles.
All of this left open the tantalizing possibility that a second pass over the input stream could drastically reduce the space required to approximate the number of distinct elements -or, more generally, the frequency moments F p . Perhaps O(1/ε) space was possible? This was a long-standing open problem [Kum06] in data streams. Yet, some thought about the underlying Gap-Hamming communication problem suggested that the linear lower bound ought to hold for general communication protocols, not just for one-way communication. This prompted the following natural conjecture.
Conjecture 2. A 1
3 -error randomized communication protocol for the Gap-Hamming-Distance problem must communicate (n) bits in total, irrespective of the number of rounds of communication.
An immediate consequence of the above conjecture is that a second pass does not help beat the (1/ε 2 ) space lower bound for the aforementioned streaming problems; in fact, no constant number of passes helps. Our Theorem 1 does not resolve Conjecture 2. However, it does imply the (1/ε 2 ) space lower bound with a constant number of passes. This is because we do obtain a linear communication lower bound with a constant number of rounds.
Finer Points.
To better understand our contribution here, it is worth considering some finer points of previously known lower bounds on Gap-Hamming-Distance, including some "folklore" results. The earlier one-way (n) bounds were inherently one-way, because the INDEX problem has a trivial two-round protocol. Also, the nature of the reduction implied a distributional error lower bound for Gap-Hamming only under a somewhat artificial input distribution. Our bounds here, including our one-way randomized bound, overcome this problem, as does the recent one-way bound of Woodruff [Woo09]: they apply to the uniform distribution. As noted by Woodruff [Woo09], this has the desirable consequence of implying space lower bounds for the Distinct Elements problem under weaker assumptions about the input stream: it could be random, rather than adversarial.
Intuitively, the uniform distribution is the hard case for the Gap-Hamming problem. The Hamming distance between two uniformly distributed n-bit strings is likely to be just around the n/2± ( √ n) thresholds, which means that a protocol will have to work hard to determine which threshold the input is at. Indeed, this line of thinking suggests an (n) lower bound for distributional complexity -under the uniform distribution -on the gapless version of the problem. Our proofs here confirm this intuition, at least for a constant number of rounds.
It is relatively easy to obtain an (n) lower bound on the deterministic multi-round communication complexity of the problem. One can directly demonstrate that the communication matrix contains no large monochromatic rectangles (see, e.g. [Woo07]). Indeed, the argument goes through even with gaps of the form n/2 ± (n), rather than n/2 ± ( √ n). It is also easy to obtain an (n) bound on the randomized complexity of the gapless problem, via a reduction from DISJOINTNESS. Unfortunately, the known hard distributions for DISJOINTNESS are far from uniform, and DISJOINTNESS is actually very easy under a uniform input distribution. So, this reduction does not give us the results we want.
Furthermore, straightforward rectangle-based methods (discrepancy/corruption) fail to effectively lower bound the randomized communication complexity of our problem. This is because there do exist very large near-monochromatic rectangles in its communication matrix. This can be seen, e.g., by considering all inputs (x, y) with x i = y i = 0 for i ∈ [n/100].
Connection to Decision Trees and Quantum Communication.
We would like to bring up two other illuminating observations. Consider the following query complexity problem: the input is a string x ∈ {0, 1} n and the desired output is 1 if |x| ≥ n/2 + √ n and 0 if |x| ≤ n/2 − √ n. Here, |x| denotes the Hamming weight of x. The model is a randomized decision tree whose nodes query individual bits of x, and whose leaves give outputs in {0, 1}. It is not hard to show that (n) queries are needed to solve this problem with 1 3 error. Essentially, one can do no better than sampling bits of x at random, and then (1/ε 2 ) samples are necessary to distinguish a biased coin that shows heads with probability 1 2 + ε from one that shows heads with probability 1 2 − ε. The Gap-Hamming-Distance problem can be seen as a generalization of this problem to the communication setting. Certainly, any efficient decision tree for the query problem implies a correspondingly efficient communication protocol, with Alice acting as the querier and Bob acting as the responder (say). Conjecture 2 says that no better communication protocols are possible for this problem.
This query complexity connection brings up another crucial point. The quantum query complexity of the above problem can be shown to be O( √ n), by the results of Nayak and Wu [NW99]. This in turn implies an O( √ n log n) quantum communication protocol for Gap-Hamming, essentially by carefully "implementing" the quantum query algorithm, as in Razborov [Raz02]. Therefore, any technique that seeks to prove an (n) lower bound for Gap-Hamming (under classical communication) must necessarily fail for quantum protocols. This rules out several recently-developed methods, such as the factorization norms method of Linial and Shraibman [LS07] and the pattern matrix method of Sherstov [She08].
Connections to Recent Work. Our multi-round (n) bound turns out to also have applications [ABC09] to the communication complexity of several distributed "functional monitoring" problems, studied recently by Cormode et al. [CMY08] in SODA 2008. Also, our lower bound approach here uses and extends a subspace-finding technique recently developed by Brody [Bro09] to prove lower bounds on multiparty pointer jumping.
Basic Definitions, Notation and Preliminaries
We begin with definitions of our central problem of interest, and quickly recall some standard definitions from communication complexity. Along the way, we also introduce some notation that we use in the rest of the paper.
Definition 1.
For strings x, y ∈ {0, 1} n , the Hamming distance between x and y, denoted (x, y), is defined as the number of coordinates i ∈ [n] such that x i = y i .
Definition 2 (Gap-Hamming-Distance problem).
Suppose n ∈ N and c ∈ R + . The c-Gap-Hamming-Distance partial function, on n-bit inputs, is denoted GHD c,n and is defined as follows.
GHD c,n (x, y) = 1 , if (x, y) ≥ n/2 + c √ n , 0 , if (x, y) ≤ n/2 − c √ n , ⋆ , otherwise.
We also use GHD c,n to denote the corresponding communication problem where Alice holds x ∈ {0, 1} n , Bob holds y ∈ {0, 1} n , and the goal is for them to communicate and agree on an output bit that matches GHD c,n (x, y). By convention, ⋆ matches both 0 and 1.
Protocols. Consider a communication problem
f : {0, 1} n × {0, 1} n → {0
, 1, ⋆} n and a protocol P that attempts to solve f . We write P(x, y) to denote the output of P on input (x, y): note that this may be a random variable, dependent on the internal coin tosses of P, if P is a randomized protocol. A deterministic protocol P is said to be correct for f if ∀ (x, y) : P(x, y) = f (x, y) (the "=" is to be read as "matches"). It is said to have distributional error ε under an input distribution ρ if Pr (x,y)∼ρ [P(x, y) = f (x, y)] ≤ ε. A randomized protocol P, using a public random string r, is said to be have error ε if ∀ (x, y) : Pr r [P(x, y) = f (x, y)] ≤ ε. A protocol P is said to be a k-round protocol if it involves exactly k messages, with Alice and Bob taking turns to send the messages; by convention, we usually assume that Alice sends the first message and the recipient of the last message announces the output. A 1-round protocol is also called a one-way protocol, since the entire communication happens in the Alice → Bob direction.
Communication Complexity. The deterministic communication complexity D( f ) of a communication problem f is defined to be the minimum, over deterministic protocols P for f , of the number of bits exchanged by P for a worst-case input (x, y). By suitably varying the class of protocols over which the minimum is taken, we obtain, e.g., the ε-error randomized, one-way deterministic, ε-error one-way randomized, and ε-error
ρ-distributional deterministic communication complexities of f , denoted R ε ( f ), D → ( f ), R → ε ( f ), and D ρ,ε ( f ),
respectively. When the error parameter ε is dropped, it is tacitly assumed to be 1 3 ; as is well-known, the precise value of this constant is immaterial for asymptotic bounds.
Definition 3 (Near-Orthogonality). We say that strings x, y ∈ {0, 1} n are c-near-orthogonal, and write
x ⊥ c y, if | (x, y) − n/2| < c √ n.
Here, c is a positive real quantity, possibly dependent on n. Notice that GHD c,n (x, y) = ⋆ ⇔ x ⊥ c y.
The distribution of the Hamming distance between two uniform random n-bit strings -equivalently, the distribution of the Hamming weight of a uniform random n-bit string -is just an unbiased binomial distribution Binom(n, 1 2 ). We shall use the following (fairly loose) bounds on the tail of this distribution (see, e.g., Feller [Fel68]).
Fact 3. Let T n (c) = Pr x [x ⊥ c 0 n ], where x is distributed uniformly at random in {0, 1} n . Let T (c) = lim n→∞ T n (c). Then 2 −3c 2 −2 ≤ T (c) ≈ e −2c 2 c √ 2π ≤ 2 −c 2 .
There are two very natural input distributions for GHD c,n : the uniform distribution on {0, 1} n × {0, 1} n , and the (non-product) distribution that is uniform over all inputs for which the output is precisely defined. We call this latter distribution µ c,n .
Definition 4 (Distributions).
For n ∈ N, c ∈ R + , let µ c,n denote the uniform distribution on the set
{(x, y) ∈ {0, 1} n × {0, 1} n : x ⊥ c y}. Also, let U n denote the uniform distribution on {0, 1} n .
Using Fact 3, we can show that for a constant c and suitably small ε, the distributional complexities D U n ×U n ,ε (GHD c,n ) and D µ c,n ,ε (GHD c,n ) are within constant factors of each other. This lets us work with the latter and draw conclusions about the former. The latter has the advantage that it is meaningful for any ε < 1 2 , whereas the former is only meaningful if ε < 1 2 T (c). Let B(x, r) denote the Hamming ball of radius r centered at x. We need use the following bounds on the volume (i.e., size) of a Hamming ball. Here, H : [0, 1] → [0, 1] is the binary entropy function.
Fact 4. If r = c √ n, then ( √ n/c) r < |B(x, r)| < n r . Fact 5. If r = αn for some constant 0 < α < 1, then |B(x, r)| ≤ 2 n H (α) .
Main Theorem: Multi-Round Lower Bound
Some Basics
In order to prove our multi-round lower bound, we need a simple -yet, powerful -combinatorial lemma, known as Sauer's Lemma [Sau72]. For this, we recall the concept of Vapnik-Chervonenkis dimension. Let S ⊆ {0, 1} n and I ⊆ [n]. We say that S shatters I if the set obtained by restricting the vectors in S to the coordinates in I has the maximum possible size, 2 |I | . We define VC-dim(S) to be the maximum |I | such that S shatters I . When d = αn for some constant α, then the above sum can be upper bounded by 2 n H (α) . This yields the following corollary.
Corollary 7. If |S| ≥ 2 n H (α) , for a constant α, then VC-dim(S) ≥ αn.
We now turn to the proof proper. It is based on a round elimination lemma that serves to eliminate the first round of communication of a GHD protocol, yielding a shorter protocol, but for GHD instances with weakened parameters. To keep track of all relevant parameters, we introduce the following notation.
Definition 5. A [k, n, s, c, ε]-protocol is a deterministic k-round protocol for GHD c,n that errs on at most an ε fraction of inputs, under the input distribution µ c,n , and in which each message is s bits long.
The next lemma gives us the "end point" of our round elimination argument.
Lemma 8. There exists no [0, n, s, c, ε]-protocol with n > 1, c = o( √ n), and ε < 1 2 . Proof. With these parameters, µ c,n has nonempty support. This implies Pr µ c,n [GHD c,n (x, y) = 0] = Pr µ c,n [GHD c,n (x, y) = 1] = 1 2 . Thus, a 0-round deterministic protocol, which must have constant output, cannot achieve error less than 1 2 .
The Round Elimination Lemma
The next lemma is the heart of our proof. To set up its parameters, we set t 0 = (48 ln 2) · 2 11k , t = 2 15k , and b = T −1 (1/8), and we define a sequence (n i , s i , c i , ε i ) k i=0 as follows:
n 0 = n , n i+1 = n i /3 , s 0 = t 0 s , s i+1 = ts i , c 0 = 10 , c i+1 = 2c i , ε 0 = 2 −2 11k , ε i+1 = ε i /T (c i+1 ) . for 0 ≤ i < k .
(1)
Lemma 9 (Round Elimination for GHD). Suppose
0 ≤ i < k and s i ≤ n i /20. Suppose there exists a [k − i, n i , s i , c i , ε i ]-protocol. Then there exists a [k − i − 1, n i+1 , s i+1 , c i+1 , ε i+1 ]-protocol. Proof. Let (n, s, c, ε) = (n i , s i , c i , ε i ) and (n ′ , s ′ , c ′ , ε ′ ) = (n i+1 , s i+1 , c i+1 , ε i+1 ). Also, let µ = µ c,n , µ ′ = µ c ′ ,n ′ , GHD = GHD c,n and GHD ′ = GHD c ′ ,n ′ . Let P be a [k − i,[P(x, y) = GHD(x , y) | x = x 0 ] ≤ 2ε .(2)
By the error guarantee of P and Markov's inequality, the number of good strings is at least 2 n−1 . There are 2 s ≤ 2 n/20 different choices for Alice's first message. Therefore, there is a set M ⊆ {0, 1} n of good strings such that Alice sends the same first message m on every input x ∈ M, with |M| ≥ 2 n−1−n/20 ≥ 2 n H (1/3) . By Corollary 7, VC-dim(M) ≥ n/3. Therefore, there exists a set I ⊆ [n], with |I | = n/3 = n ′ , that is shattered by M. For strings x ′ ∈ {0, 1} n ′ and x ′′ ∈ {0, 1} n−n ′ , we write x ′ • x ′′ to denote the string in {0, 1} n formed by plugging in the bits of x ′ and x ′′ (in order) into the coordinates in I and [n] \ I , respectively. We now give a suitable (k − i − 1)-round protocol Q for GHD ′ , in which Bob sends the first message. Consider an input (x ′ , y ′ ) ∈ {0, 1} n ′ × {0, 1} n ′ , with Alice holding x ′ and Bob holding y ′ . By definition of shattering, there exists an x ′′ ∈ {0, 1} n−n ′ such that x := x ′ • x ′′ ∈ M. Alice and Bob agree beforehand on a suitable x for each possible x ′ . Suppose Bob were to pick a uniform random y ′′ ∈ {0, 1} n−n ′ and form the string y := y ′ • y ′′ . Then, Alice and Bob could simulate P on input (x, y) using only k − i − 1 rounds of communication, with Bob starting, because Alice's first message in P would always be m. Call this randomized protocol Q 1 . We define Q to be the protocol obtained by running t instances of Q 1 in parallel, using independent random choices of y ′′ , and outputting the majority answer. Note that the length of each message in Q is ts = s ′ . We shall now analyze the error.
Suppose
x ′′ ⊥ b y ′′ . Let d 1 = (x, y) − n/2, d 2 = (x ′ , y ′ ) − n ′ /2 and d 3 = (x ′′ , y ′′ ) − (n − n ′ )/2. Clearly, d 1 = d 2 + d 3 . Also, |d 1 | ≥ |d 3 | − |d 2 | ≥ c ′ √ n ′ − b √ n − n ′ ≥ (c ′ − b √ 2) √ n √ 3 ≥ c √ n ,
where we used (1) and our choice of b. Thus, x ⊥ c y. The same calculation also shows that d 1 and d 3 have the same sign, as |d 3 | > |d 2 |. Therefore GHD(x , y) = GHD ′ (x ′ , y ′ ).
For the rest of the calculations in this proof, fix an input x ′ for Alice, and hence, x ′′ and x as well. For a fixed y ′ , let E(y ′ ) denote the event that P(x, y) = GHD(x , y): note that y ′′ remains random. Using the above observation (at step (3) below), we can bound the probability that Q 1 errs on input (x ′ , y ′ ) as follows.
Pr y Q 1 (x ′ , y ′ ) = GHD ′ (x ′ , y ′ ) | y ′ ≤ Pr y P(x, y) = GHD(x , y) ∨ GHD(x , y) = GHD ′ (x ′ , y ′ ) | y ′ ≤ Pr y ′′ E(y ′ ) + Pr y GHD(x , y) = GHD ′ (x ′ , y ′ ) | y ′ ≤ Pr y ′′ E(y ′ ) + Pr y ′′ x ′′ ⊥ b y ′′ (3) ≤ Pr y ′′ E(y ′ ) + T (b) = Pr y ′′ E(y ′ ) + 1/8 ,(4)
where step (4) follows from our choice of b. To analyze Q, notice that during the t-fold parallel repetition of Q 1 , y ′ remains fixed while y ′′ varies. Thus, it suffices to understand how the repetition drives down the sum on the right side of (4). Unfortunately, for some values of y ′ , the sum may exceed 1 2 , in which case it will be driven up, not down, by the repetition. To account for this, we shall bound the expectation of the first term of that sum, for a random y ′ .
To do so, let z ∼ µ | x be a random string independent of y. Notice that z is uniformly distributed on a subset of {0, 1} n of size 2 n T (c), whereas y is uniformly distributed on a subset of {0, 1} n of size 2 n T (c ′ ).
(We are now thinking of x as being fixed and both y ′ and y ′′ as being random.) Therefore,
E y ′ Pr y ′′ E(y ′ ) = Pr y E(y ′ ) = Pr y [P(x, y) = GHD(x , y)] ≤ Pr z [P(x, z) = GHD(x , z)] · T (c)/T (c ′ ) ≤ 2εT (c)/T (c ′ ) ,(5)
where (5) holds because x, being good, satisfies (2). Thus, by Markov's inequality,
Pr y ′ Pr y ′′ E(y ′ ) ≥ 1 8 ≤ 16εT (c)/T (c ′ ) .(6)
If, for a particular y ′ , the bad event Pr y ′′ [E(y ′ )] ≥ 1 8 does not occur, then the right side of (4) is at most 1/8 + 1/8 = 1/4. In other words, Q 1 errs with probability at most 1/4 for this y ′ . By standard Chernoff bounds, the t-fold repetition in Q drives this error down to (e/4) t /4 ≤ 2 −t /10 ≤ ε 0 ≤ ε. Combining this with (6), which bounds the probability of the bad event, we get
Pr y ′ ,r Q(x ′ , y ′ ) = GHD ′ (x ′ , y ′ ) ≤ 16εT (c)/T (c ′ ) + ε ≤ ε/T (c ′ ) = ε ′ ,
where r denotes the internal random string of Q (i.e., the collection of y ′′ s used).
Note that this error bound holds for every fixed x ′ , and thus, when (x ′ , y ′ ) ∼ µ ′ . Therefore, we can fix Bob's random coin tosses in Q to get the desired [k − i − 1, n ′ , s ′ , c ′ , ε ′ ]-protocol.
The Lower Bound
Having established our round elimination lemma, we obtain our lower bound in a straightforward fashion. Remark. This is a formal restatement of Theorem 1.
Theorem 10 (Multi-round Lower Bound). Let P be a k-round
Proof. For simplicity, assume c ≤ c 0 = 10. Our proof easily applies to a general c = O(1) by a suitable modification of the parameters in (1). Also, assume n ≥ 2 4k 2 , for otherwise there is nothing to prove.
By repeating P (48 ln 2) · 2 11k = t 0 times, in parallel, and outputting the majority of the answers, we can reduce the error to 2 −2 11k = ε 0 . The size of each message is now t 0 s = s 0 . Fixing the random coins of the resulting protocol gives us a [k, n 0 , s 0 , c 0 , ε 0 ]-protocol P 0 .
Suppose s i ≤ n i /20 for all i, with 0 ≤ i < k. We then repeatedly apply Lemma 9 k times, starting with P 0 . Eventually, we end up with a [0, n k , s k , c k , ε k ]-protocol. Examining (1), we see that n k = n/3 k , s k = 2 15k 2 s 0 = (48 ln 2)2 15k 2 +11k s, and c k = 10 · 2 k . Notice that n k ≥ 2 4k 2 /3 k > 1 and c k = o( √ n k ). We also see that c i k i=1 is an increasing sequence, whence ε i+1 /ε i = 1/T (c i+1 ) ≤ 1/T (c k ) ≤ 2 3c k 2 +2 , where the final step uses Fact 3. Thus,
ε k ≤ ε 0 2 3c 2 k +2 k = 2 −2 11k · 2 (3(10·2 k ) 2 +2)·k = 2 −2 11k +300k·2 2k +2k < 1 2 .
In other words, we have a [0, n k , s k , c k , ε k ]-protocol with n k > 1, c k = o( √ n k ) and ε k < 1 2 . This contradicts Lemma 8. Therefore, there must exist an i such that s i ≥ n i /20. Since s i k i=1 is increasing and n i k i=1 is decreasing, s k ≥ n k /20. By the above calculations, (48 ln 2)2 15k 2 +11k s ≥ n/(20 · 3 k ), which implies s ≥ n/2 O(k 2 ) , as claimed.
Notice that, for constant k, the argument in the above proof in fact implies a lower bound for deterministic protocols with small enough constant distributional error under µ c,n . This, in turn, extends to distributional error under the uniform distribution, as remarked earlier.
Tight Deterministic One-Way Bounds
The main result of this section is the following.
Theorem 11. D → (GHD c,n ) = n − ( √ n log n) for all constant c.
Definition 6. Let x 1 , x 2 , y ∈ {0, 1} n . We say that y witnesses x 1 and x 2 or that y is a witness for (x 1 , x 2 ) if x 1 ⊥ c y, x 2 ⊥ c y, and GHD c,n (x 1 , y) = GHD c,n (x 2 , y).
Intuitively, if (x 1 , x 2 ) have a witness, then they cannot be in the same message set. For if Alice sent the same message on x 1 and x 2 and Bob's input y was a witness for (x 1 , x 2 ) then whatever Bob were to output, the protocol would err on either (x 1 , y) or (x 2 , y). The next lemma characterizes which (x 1 , x 2 ) pairs have witnesses.
Lemma 12.
For all x 1 , x 2 ∈ {0, 1} n , there exists y that witnesses (x 1 , x 2 ) if and only if (x 1 , x 2 ) ≥ 2c √ n.
Proof. On the one hand, suppose y witnesses (x 1 , x 2 ). Then assume WLOG that (x 1 , y) ≤ n/2−c √ n and (x 2 , y) ≥ n/2 + c √ n. By the triangle inequality, (x 1 ,
x 2 ) ≥ (x 2 , y) − (x 1 , y) = 2c √ n. Conversely, suppose (x 1 , x 2 ) ≥ 2c √ n. Let L = {i : x 1 [i] = x 2 [i]}, and let R = {i : x 1 [i] = x 2 [i]}.
Suppose y agrees with x 1 on all coordinates from R and half the coordinates from L. Then, (x 1 , y) = |L|/2 = (n − (x 1 , x 2 ))/2 ≤ n/2 − c √ n. Furthermore, y agrees with x 2 on no coordinates from R and half the coordinates from L, so (x 1 , y) = |L|/2 + |R| ≥ n/2 + c √ n.
We show that it is both necessary and sufficient for Alice to send different messages on x 1 and x 2 whenever (x 1 , x 2 ) is "large". To prove this, we need the following theorem, due to Bezrukov [Bez87] and a claim that is easily proved using the probabilistic method (a full proof of the claim appears in the appendix).
One Round Randomized Lower Bound
Next, we develop a one-way lower bound for randomized protocols. Note that our lower bound applies to the uniform distribution, which, as mentioned in Section 1, implies space lower bounds for the Distinct Elements problem under weaker assumptions about the input stream. Woodruff [Woo09] recently proved similar results, also for the uniform distribution. We include our lower bound as a natural extension of the deterministic bound.
Theorem 15. R → ε (GHD c,n ) = (n).
Proof. For the sake of clarity, fix c = 2 and ε = 1/10, and suppose P is a one-round, ε-error, o(n)-bit protocol for GHD c,n . By Markov's inequality, at most a 1/2-fraction of x are bad. Next, fix Alice's message m to maximize the number of good x, and let M = {x ∈ {0, 1} n : x is good and Alice sends m on input x}. It follows that
|M| ≥ 2 n−1 /2 o(n) > 2 n(1−o(1)) .
Our goal is to show that since |M| is large, we must err on a > 2ε-fraction of y ∈ Y x for some x ∈ M, contradicting the goodness of x. Note that it suffices to show that a 4ε fraction of y ∈ Y x 1 witness x 1 and x 2 .
|M| ≥ 2 n(1−o(1)) , so by Fact 5 and Theorem 13, There exist x 1 , x 2 with (x 1 , x 2 ) ≥ 1 − o(1). Next, we'd like to determine the probability that a random y ∈ Y x 1 witnesses (x 1 , x 2 ). Without loss of generality, let x 1 = 0 n . Let w(x) := Pr y∈Y x 1 [GHD(x, y) = GHD (x 1 , y)]. The following lemma shows that w(x) is an increasing function of |x|. We leave the proof until the appendix.
Lemma 16. For all x, x ′ ∈ {0, 1} n , w(x) ≥ w(x ′ ) ⇔ |x| ≥ |x ′ |, with equality if and only if |x| = |x ′ |.
We compute w(x) by conditioning on |y|:
w(x) = n 1 ≤n/2−c √ n
Pr (x, y) ≥ n/2 + c √ n| |y| = n 1 · Pr[|y| = n 1 ] .
Fix |x| =: m, pick a random y with |y| = n 1 , and suppose there are k coordinates i such that
x i = y i . Then, (x, y) = (m − k) + (n 1 − k) = m + n 1 − 2k. Hence, (x, y) ≥ n/2 + c √ n ⇐⇒ k ≤ m + n 1 2 − n 4 − c 2 √ n .
Note that given a random y with weight |y| = n 1 , the probability that exactly k of m coordinates have x i = y i = 1 follows the hypergeometric distribution Hyp(k; n, m, n 1 ). Therefore, we can express the probability Pr |y|=n 1 [ (x, y) ≥ n/2 + c √ n] as
Pr |y|=n 1 (x, y) ≥ n/2 + c √ n = k≤ m+n 1 2 − n 4 − c 2 √ n
Hyp(k; n, m, n 1 ) .
Finally, we show that w(x) > 4ε for a suitably large constant |x| with the following claims, whose proofs are left to the appendix. Its easy to see from the previous two claims that w(x) > 0.95 · (2/3) > 4ε.
Concluding Remarks
Our most important contribution here was to prove a multi-round lower bound on a fundamental problem in communication complexity, the Gap-Hamming Distance problem. As a consequence, we extended several known (1/ε 2 )-type space bounds for various data stream problems, such as the Distinct Elements problem, to multi-pass algorithms. These resolve long-standing open questions.
The most immediate open problem suggested by our work is to resolve Conjecture 2. It appears that proving the conjecture true is going to require a technique other than round elimination, or else, an extremely powerful round elimination lemma that does not lose a constant fraction of the input length at each step. On the other hand, proving the conjecture false is also of great interest, and such a proof might extend to nontrivial data stream algorithms, albeit with a super-constant number of passes.
Note the one-to-one correspondence between strings in B and strings in C obtained by flipping the nth bit. Now, consider any y ∈ B such that y witnesses ( 0, x ′ ) but not ( 0, x). Flipping the nth bit of y yields a string y ′ ∈ C such that Y witnesses ( 0, x) but not ( 0, x ′ ). Hence among y ∈ B ∪ C there is an equal number of witnesses for x and x ′ . For any y ∈ A, y n = 0, whence |y − x ′ | = |y − x| + 1. Therefore, any y that witnesses ( 0, x) must also witness ( 0, x ′ ), whence w(x) ≤ w(x ′ ).
Many claims in this paper require tight upper and lower tail bounds for binomial and hypergeometric distributions. We use Chernoff bounds where they apply. For other bounds, we approximate using normal distributions. We use Feller [Fel68] as a reference. N (x) is the cumulative distribution function of the normal distribution. We use it in Fact 3 to approximate T (x). Here, we'll also use it to approximate tails of the binomial and hypergeometric distributions.
Lemma 21 (Feller, Chapter VII, Lemma 2.).
For all x > 0,
φ(x) 1 x − 1 x 3 < N (x) < φ(x) 1
x .
Theorem 22 (Feller, CHapter VII, Theorem 2.). For fixed z 1 , z 2 , Pr[n/2 + (z 1 /2) √ n ≤ |y| ≤ n/2 + (z 2 /2) √ n] ∼ N (z 1 ) − N (z 2 ).
Theorem 23.
For any γ such that γ = ω(1) and γ = o(n 1/6 ), we have k>n/2+γ √ n/2 n k ∼ N (γ ).
Claim 28.
For any x L ∈ {0, 1} n L , GHD(x L , y L ) is defined for at least a ≥ e −2(c ′ ) 2 /5c ′ -fraction of y L ∈ {0, 1} n L .
Proof. Without loss of generality, assume x L = 0. Then, GHD(x L , y L ) is defined for all y such that |y| ≤ n L /2 − c ′ √ n L or |y| ≥ n L /2 + c ′ √ n L . Note that for any constant x > c ′ ,
Pr y [|y| ≤ n L 2 − c ′ √ n L ] ≥ Pr[ n L 2 − x √ n L ≤ |y| ≤ n L 2 − c ′ √ n L ] ≥ N (2c ′ ) − N (2x) ≥ φ(2c ′ ) 1 2c ′ − 1 (2c ′ ) 3 − φ(2x) 2x = e −(2c ′ ) 2 /2 √ 2π ( 1 2c ′ − 1 (2c ′ ) 3 − e −2x 2 2x √ 2π ≥ e −2(c ′ ) 2 10c ′
Pr[|y| ≥ n L /2 + c ′ √ n L ] is bounded in the same fashion.
Lemma 6 (
6Sauer's Lemma). Suppose S ⊆ {0, 1} n has VC-dim(S) < d. Then
1 3
3-error randomized communication protocol for GHD c,n , with c = O(1), in which each message is s bits long. Then s ≥ n 2 O(k 2 ) .
Theorem 13 .
13Call a subset A ⊆ {0, 1} n d-maximal if it is largest, subject to the constraint that (x, y) ≤ d for all x, y ∈ A.1. If d = 2t then B(x, t) is d-maximal for any x ∈ {0, 1} n . 2. If d = 2t + 1 then B(x, t) ∪ B(y, t) is d-maximal for any x, y ∈ {0, 1} n such that (x, y) = 1.Claim 14. It is possible to cover {0, 1} n with at most 2 n−O( √ n log n) Hamming balls, each of radius c √ n. Proof of Theorem 11. For the lower bound, suppose for the sake of contradiction that there is a protocol where Alice sends only n − c √ n log n bits. By the pigeonhole principle, there exists a set M ⊆ {0, 1} n of inputs of size |M| ≥ 2 n /2 n−c √ n log n = 2 c √ n log n = n c √ n upon which Alice sends the same message. By Theorem 13, the Hamming ball B(x, c √ n) is 2c √ n-maximal, and by Fact 4, |B(x, c √ n)| < |M|. Therefore, there must be x 1 , x 2 ∈ M with (x 1 , x 2 ) > 2c √ n. By Lemma 12, there exists a y that witnesses (x 1 , x 2 ). No matter what Bob outputs, the protocol errs on either (x 1 , y) or on (x 2 , y). For a matching upper bound, Alice and Bob fix a covering C = {B(x 0 , r)} of {0, 1} n by Hamming balls of radius r = c √ n.On input x, Alice sends Bob the Hamming ball B(x 0 , r) containing x. Bob selects some x ′ ∈ B(x 0 , r) such that x ′ ⊥ c y and outputs GHD(x ′ , y). The correctness of this protocol follows from Lemma 12, as (x, x ′ ) ≤ 2c √ n since they are both in B(x 0 , c √ n). The cost of the protocol is given by Claim 14, which shows that it suffices for Alice to send log 2 n−O( √ n log n) = n − O( √ n log n) bits to describe each Hamming ball.
Definition 8 .
8For x ∈ R, let φ(x) := e −x 2
n, s, c, ε]-protocol. Assume, WLOG, that Alice sends the first message in P.Call a string x 0 ∈ {0, 1} n "good" ifPr
(x,y)∼µ
Definition 7 .
7For x ∈ {0, 1} n , let Y x := {y : x ⊥ 2 y}. Say that x is good if Pr y∈Y x [P(x, y) = GHD(x , y)] ≤ 2ε. Otherwise, call x bad.
Claim 17. Conditioned on |y| ≤ n/2 − 2 √ n, we have Pr[|y| ≥ n/2 − 2.1 √ n] ≤ 1 3 . Claim 18. For all d < n/2 − 2.1 √ n, we have Pr[ (x 2 , y) ≥ n/2 + d √ n] ≥ 0.95.
Claim 24 (Restatement of Claim 17). Conditioned on |y| ≤ n/2 − 2 Pr[|y| ≥ n/2 − 2.1 √ n] ≤ 1/3. Proof. By Theorem 22 and Lemma 21, we have By Fact3, Pr[|y| ≥ n/2 − 2 √ n] ≤ 2 −3·2 2 −2 = 2 −14 = 6.1035 · 10 −5 . Putting the two terms together, we get Claim 25 (Restatement of Claim 18). For all d < n/2 − 2.1√
n,
Pr[n/2 − 2.1
√
n ≤ |y| ≤ n/2 − 2
√
n] ∼ N (4) − N (4.2)
≤ φ(4)/4 − φ(4.2)(4.2 −1 − 4.2 −3 )
≤ 2.0219 * 10 −5
Pr[|y| ≥ n/2 − 2.1
√
n||y| ≤ n/2 − 2
√
n] ≤
2.0219 · 10 −5
6.1035 · 10 −5 ≤ 1/3.
√
n,
Pr[ (x 2 , y) ≥ n/2 + 2
√
n] ≥ 0.95.
AcknowledgementsWe would like to thank Anna Gal, T. S. Jayram and David Woodruff for stimulating discussions about the problem at various points of time.APPENDIX A Proofs of Technical LemmasWe begin with a proof of Claim 14, which we state here for convenience.Claim 19 (Restatement of Claim 14). For any constant c, it is possible to cover {0, 1} n with at most 2 n−O( √ n log n) Hamming balls, each with radius r = c √ n.Proof. We use the probabilistic method. Let r := c √ n. For x ∈ {0, 1} n , let B x := B(x, r) be the Hamming ball of radius r centered at x. For a t to be determined later, pick x 1 , . . . , x t independently and uniformly at random from {0, 1} n . We want to show that with nonzero probability, the universe {0, 1} n is covered by these t Hamming balls B x 1 , . . . , B x t . Now, fix any x ∈ {0, 1} n and any 1 ≤ i ≤ t. Since x i was picked uniformly at random, each x is equally likely to be in B x i . Therefore,where inequality stems from Fact 4. Let B AD x = 1≤i≤t x ∈ B x i be the event that x is not covered by any of the Hamming balls we picked at random, and let B AD = B AD x be the event that some x is not covered by the Hamming balls. WeBy the union bound,Picking t = ln 2(n + 1)2 n−θ( Proof. If |x| = |x ′ |, then w(x) = w(x ′ ) by symmetry. Further, note that GHD(x , y) = 0 if and only if GHD(−x , y) = 1. Therefore, it suffices to handle the case where |y| ≤ n/2 − c √ n and GHD( 0, y) = 0. For the rest of the proof, we assume that x i = x ′ i , except for the nth coordinate, where x n = 0 and x ′ n = 1. Thus, |x| = |x ′ | − 1. We show that w(x) < w(x ′ ); the rest of the lemma follows by induction. Let Y be the set of strings with Hamming weight |y| ≤ n/2 − c √ n. Partition Y into the following three sets:• A := {y : |y| = n/2 + c √ n ∧ y n = 0}.• B := {y : |y| < n/2 + c √ n ∧ y n = 0}.• C := {y : y n = 1}.Proof. The proof follows from the following claim, instantiated with c = 2 and α = 2.1.Claim 26.For all α > c, |x| = γ n, and all γ ≥ 1 − (1 − c/α)/4,Proof. Let m := |x| = γ n and let n 1 = n/2 − α √ n. Then, the probability that a random y with |y| = n 2 can be expressed using the hypergeometric distribution Hyp(k; n, m, n 1 ). Let the m set bits of x be the defects. The probability of k of the n 1 bits of y are defective is Hyp(k; n, m, n 1 ). Note that (x, y) = (m − k) + (n 1 − k) = m + n 1 − 2k. Therefore,We express the probability Pr |y|=n 1 [ (x, y) ≥ n/2 + c √ n] asNext, we use a concentration of measure result due to Hush and Scovel[HS05]. Here, we present a simplified version.Theorem 27 (Hush, Scovel). Let m = γ n > n 1 = n/2 − α √ n, and let β = n/m(n − m).The expected value of a random variable K distributed according to Hyp(K ; n, m, n 1 ) isSet η := (α − c) √ n/4. Note thatwhere the inequality holds because γ ≥ 1 − (1 − c/α)/4. Note also that (1 − c/α)/4 = (α − c)/4α, so 1 − (1 − c/α)/4 = (3α + c)/4α. By Theorem 27It follows that Pr[K ≤ γ n 2 − α+c 2 √ n] ≥ 1 − exp − 2(α−c)α 2 (1+o(1)) 3α+c .
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"MedFuse: Multi-modal fusion with clinical time-series data and chest X-ray images Engineering Division NYU Abu Dhabi",
"MedFuse: Multi-modal fusion with clinical time-series data and chest X-ray images Engineering Division NYU Abu Dhabi"
] | [
"Nasir Hayat [email protected] \nDepartment of Radiology\nNYU Grossman School of Medicine New York\nNYUSA\n",
"Farah E Shamout [email protected] \nDepartment of Radiology\nNYU Grossman School of Medicine New York\nNYUSA\n",
"Abu Dhabi \nDepartment of Radiology\nNYU Grossman School of Medicine New York\nNYUSA\n",
"Uae \nDepartment of Radiology\nNYU Grossman School of Medicine New York\nNYUSA\n"
] | [
"Department of Radiology\nNYU Grossman School of Medicine New York\nNYUSA",
"Department of Radiology\nNYU Grossman School of Medicine New York\nNYUSA",
"Department of Radiology\nNYU Grossman School of Medicine New York\nNYUSA",
"Department of Radiology\nNYU Grossman School of Medicine New York\nNYUSA"
] | [
"Proceedings of Machine Learning Research"
] | Multi-modal fusion approaches aim to integrate information from different data sources. Unlike natural datasets, such as in audio-visual applications, where samples consist of "paired" modalities, data in healthcare is often collected asynchronously. Hence, requiring the presence of all modalities for a given sample is not realistic for clinical tasks and significantly limits the size of the dataset during training. In this paper, we propose MedFuse, a conceptually simple yet promising LSTM-based fusion module that can accommodate uni-modal as well as multi-modal input. We evaluate the fusion method and introduce new benchmark results for in-hospital mortality prediction and phenotype classification, using clinical time-series data in the MIMIC-IV dataset and corresponding chest X-ray images in MIMIC-CXR. Compared to more complex multi-modal fusion strategies, MedFuse provides a performance improvement by a large margin on the fully paired test set. It also remains robust across the partially paired test set containing samples with missing chest X-ray images. We release our code for reproducibility and to enable the evaluation of competing models in the future. | 10.48550/arxiv.2207.07027 | [
"https://export.arxiv.org/pdf/2207.07027v2.pdf"
] | 250,526,380 | 2207.07027 | 87f2ccde9e69a2e6e11487e1ef1e8d122b406101 |
MedFuse: Multi-modal fusion with clinical time-series data and chest X-ray images Engineering Division NYU Abu Dhabi
2022
Nasir Hayat [email protected]
Department of Radiology
NYU Grossman School of Medicine New York
NYUSA
Farah E Shamout [email protected]
Department of Radiology
NYU Grossman School of Medicine New York
NYUSA
Abu Dhabi
Department of Radiology
NYU Grossman School of Medicine New York
NYUSA
Uae
Department of Radiology
NYU Grossman School of Medicine New York
NYUSA
MedFuse: Multi-modal fusion with clinical time-series data and chest X-ray images Engineering Division NYU Abu Dhabi
Proceedings of Machine Learning Research
1822022Machine Learning for Healthcare Engineering Division NYU Abu Dhabi Abu Dhabi, UAE Krzysztof J. Geras
Multi-modal fusion approaches aim to integrate information from different data sources. Unlike natural datasets, such as in audio-visual applications, where samples consist of "paired" modalities, data in healthcare is often collected asynchronously. Hence, requiring the presence of all modalities for a given sample is not realistic for clinical tasks and significantly limits the size of the dataset during training. In this paper, we propose MedFuse, a conceptually simple yet promising LSTM-based fusion module that can accommodate uni-modal as well as multi-modal input. We evaluate the fusion method and introduce new benchmark results for in-hospital mortality prediction and phenotype classification, using clinical time-series data in the MIMIC-IV dataset and corresponding chest X-ray images in MIMIC-CXR. Compared to more complex multi-modal fusion strategies, MedFuse provides a performance improvement by a large margin on the fully paired test set. It also remains robust across the partially paired test set containing samples with missing chest X-ray images. We release our code for reproducibility and to enable the evaluation of competing models in the future.
Introduction
Humans perceive the world through multi-modal data (Ngiam et al., 2011). To date, most of the successful models learning from perceptual data in healthcare are uni-modal, i.e. they rely on a single data modality (Huang et al., 2020a). Multi-modal learning has been widely explored in the context of audio-visual applications (Vaezi Joze et al., 2020) and natural image datasets (Zellers et al., 2021;Hayat et al., 2020), but less so in healthcare. The main goal of multi-modal fusion is to exploit relevant information from different modalities to improve performance in downstream tasks (Baltrušaitis et al., 2018). Multi-modal fusion strategies can be characterized as early, joint, or late fusion (Huang et al., 2020a). The joint fusion paradigm is the most promising, since its core idea is to model interactions between the representations of the input modalities.
We highlight two main challenges facing multi-modal joint fusion in healthcare. First, many of the state-of-the-art approaches make a strong assumption that all modalities are available for every sample during training, inference, or both (Pölsterl et al., 2021). Although some clinical studies follow suit of this assumption (Huang et al., 2020a), obtaining paired data is not feasible since daily clinical practice produces heterogeneous data with varying sparsity. For example, physiological data is more frequently collected than chest Xray images in the Intensive Care Unit (ICU) setting. These two modalities are the key focus of our study because they play a very important role in clinical prediction tasks (Harutyunyan et al., 2019;Lohan, 2019). Developing a unified fusion model for those two modalities also presents its own challenges as they (i) have significantly different input dimensions, (ii) require modality-specific feature extractors due to differences in information and noise content (Nagrani et al., 2021), and (iii) are not temporally aligned and hence cannot be paired easily. Considering those challenges, our primary aim is to propose a fusion architecture that can deal with partially paired data, in order to achieve favorable performance in downstream prediction tasks.
The second challenge is that there are no well-studied publicly available multi-modal clinical benchmarks. Therefore, most studies rely on a single data modality to perform clinical prediction tasks (Harutyunyan et al., 2019), or use privately curated multi-modal datasets (Huang et al., 2020a). Here, our secondary aim is to introduce new multi-modal benchmark results for two popular clinical prediction tasks using the publicly available Medical Information Mart for Intensive Care (MIMIC)-IV (Johnson et al., 2021) and MIMIC-CXR (Johnson et al., 2019) datasets, and we also release the code for reproducibility. We compare our approach to vanilla early and joint fusion as well as open-source state-of-theart joint fusion approaches (Vaezi Joze et al., 2020;Pölsterl et al., 2021). In summary, we make the following contributions:
• We propose MedFuse, a new LSTM-based (Hochreiter and Schmidhuber, 1997) multimodal fusion approach. Conventional joint fusion strategies concatenate feature representations of multiple modalities as a single feature representation, and then process that concatenated representation for downstream tasks, such as using a classifier. On the contrary, we treat the multi-modal representation as a sequence of uni-modal representations (or tokens), such that the fusion module aggregates these representations through the recurrence mechanism of LSTM. We assume a sequential structure to leverage the recurrent inductive bias of LSTM and to handle input sequences of variable length, in case of a missing modality. The fusion module is agnostic to the architecture of the modality-specific extractors and can handle missing data during training and inference.
• To evaluate the proposed approach, we link two open-access real-world datasets: MIMIC-IV (Johnson et al., 2021), which contains clinical time-series data collected in the ICU, and MIMIC-CXR (Johnson et al., 2019), which contains chest X-ray images. We pre-process the data and introduce new benchmark results for two tasks (Harutyunyan et al., 2019): in-hospital mortality prediction and phenotype classification.
The results show that the model's performance remains robust across uni-modal samples and improves for paired multi-modal samples. The model achieves state-of-the-art results without imposing any assumptions on correlation between modalities.
• Considering the lack of multi-modal learning benchmarks in healthcare, we release our data pre-processing and benchmark code to allow reproducibility of the results and enable the evaluation of competing models in the future. The code can be found at: https://github.com/nyuad-cai/MedFuse. An overview of the proposed work is shown in Figure 1. Figure 1: Overview of the proposed work. We first extract and link the datasets from MIMIC-IV and MIMIC-CXR based on the task definition (i.e., inhospital mortality prediction, or phenotype classification). The data splits of the training, validation, and test sets are summarized for each task, and the prevalence of positive and negative labels for in-hospital mortality is shown. Phenotype classification involves 25 labels as shown in Table 4.
Generalizable Insights about Machine Learning in the Context of Healthcare
State-of-the-art multi-modal fusion approaches typically investigate synchronous sources of information using natural datasets, such as audio, visual, and textual modalities. In healthcare, data is often sparse and heterogeneous and hence modalities are not always paired. Our work overcomes the challenge of missing data by proposing a flexible fusion approach that is agnostic to the modality-specific encoders. Therefore, it can be used for other types of input data, beyond chest X-ray images and clinical time-series data. It also highlights the value of processing a sequence of uni-modal representations, compared to the conventional concatenation strategy in joint fusion. Overall, the work highlights the promise of multi-modal fusion in healthcare to improve performance in downstream tasks.
Related Work
Routine clinical practice produces large amounts of data from different sources (i.e. modalities), including medical images, laboratory test results, measurements of vital signs, and clinical notes (Asri et al., 2015). Advances in deep learning have enabled building predictive models using subsets of modalities, typically clinical time-series data (Shickel et al., 2017) and medical images (Litjens et al., 2017). Here, we provide an overview of related work on multi-modal fusion in healthcare using imaging and non-imaging data.
Multi-modal Learning
Multi-modal learning has been widely explored for jointly learning representations of multiple modalities (Baltrušaitis et al., 2018). Example tasks include visual grounding , language grounding through visual cues (Zhang et al., 2021b), action recognition , video classification (Nagrani et al., 2021), image captioning , or visual-question answering (Zellers et al., 2021). Since machine learning studies typically investigate different combinations of audio, visual, and textual modalities, many of the existing methods are driven by the assumption that the modalities share intrinsic and structural information. This is not always true for heterogeneous data in healthcare. Hence, due consideration should be given to learning with multiple medical data modalities, since conventional assumptions for non-medical data are not necessarily applicable.
Multi-modal Fusion with Medical Images
There is an increasing interest in advancing the fusion of multi-modal medical images (Hermessi et al., 2021). The images usually represent different views of the same organ or lesion of interest, acquired using one or more sensors, whereby the images share the same set of labels. Proposed methods mainly focus on pixel-level fusion of complementary views acquired through multiple sensors to obtain a unified composite representation of the raw images James and Dasarathy, 2014). Various feature-and prediction-level fusion approaches were proposed for improved classification (Puyol-Antón et al., 2021;Wu et al., 2019;Zhang et al., 2021a) or segmentation performance (Hermessi et al., 2021). Since textual reports are a natural byproduct of radiology exams, they were also used as additional modalities for tasks like visual-question answering (Li et al., 2020;Sharma et al., 2021), report generation (Sonsbeek and Worring, 2020), or zero-shot image classification (Hayat et al., 2021b;.
Multi-modal Fusion with Clinical Data and Medical Images
Several studies investigated the fusion of medical images and clinical data extracted from the patient's Electronic Health Records (EHR) for various applications (Huang et al., fine-tuning during fusion Figure 2: Overview of network with MedFuse module. First, we pre-train the modality-specific encoders and classifiers independently for each input modality. Specifically, we train f ehr and g ehr using the clinical time-series data and f cxr and g cxr using the chest X-ray images. Next, we project the chest X-ray latent representation v cxr to v * cxr , in order to match the dimension of v ehr . We pass v ehr and v * cxr as an input sequence to the LSTM-based f f usion , and we classify its last hidden state h f usion to compute the overall prediction y f usion . f f usion , f ehr , f cxr , g f usion , and φ are fine-tuned together for fusion. 2020a). For example, a stream of work covers tasks pertaining to cancer, such as recurrence prediction (Ho et al., 2021), lesion detection (Shao et al., 2020), or patient survival prediction (Vale-Silva and Rohr, 2020). Other tasks include detection of pulmonary embolism (Huang et al., 2020b), predicting the progression of Alzheimer's disease (Lee et al., 2019), diagnosis of neurological disease (Xin et al., 2021), or diagnosis of cervical dysplasia (Xu et al., 2016). While these studies highlight the impact of using multiple data modalities on downstream performance, many curate datasets for specific tasks and share the assumption that the images and selected clinical features are paired.
Some studies specifically focused on the integration of clinical data and chest X-ray images. For example, the integration of the two modalities showed a favorable impact on the predictive performance in prognostication tasks among patients with COVID-19 (Shamout et al., 2021;Jiao et al., 2021). Some studies jointly refine a common latent representation after aggregating encoded features of each modality (Grant et al., 2021;Jiao et al., 2021), while others combine predictions computed by each modality through weighted averaging (i.e., late fusion) (Shamout et al., 2021;Jiao et al., 2021). While late fusion enables the computation of predictions even for incomplete samples, it requires that the two modalities are assigned the same labels, which is not always feasible. Closely related to our work is that of Hayat et al. (2021a), where they propose a dynamic training approach for partially paired clinical time-series data and chest X-ray images for the task of phenotype classification. However, their method is not scalable since it incorporates an additional classifier (and prediction) for every possible combination of input modalities.
Methodology
We define a two stage-approach (i) to learn modality-specific perceptual models to extract the latent features (Section 3.1), and (ii) integrate these features through a joint multimodal fusion module, MedFuse (Section 3.2). The overall architecture is shown in Figure 2. Without loss of generality, we focus here on two modalities only and denote the clinical time-series data as ehr and the chest X-ray images as cxr when defining the methodology.
Modality-specific Encoders
One of the main sources of heterogeneity in healthcare is the varying dimensionality of the input modalities, which makes it challenging to develop a unified encoder for all input modalities. Another difference is the target space, since we do not assume that the modalities must be assigned the same set of labels. Hence, we first define modality-specific encoders as follows.
For a given instance, let x ehr ∈ R t×d represent the clinical time-series data associated with ground-truth labels y ehr , where t is the number of time steps and d is the number of features derived from the clinical variables. We implement the encoder, f ehr , for the clinical time-series modality as two stacked layers of an LSTM network (Hochreiter and Schmidhuber, 1997) with a dropout layer. We compute a latent feature representation v ehr ∈ R m consisting of the last hidden state of the stacked LSTM, where m = 256. We then apply a classifier, g ehr , to compute the predictions, such thatŷ ehr = g ehr (v ehr ). To fine-tune the encoder, we optimize the following loss:
L ehr (y ehr ,ŷ ehr ) = BCE(y ehr ,ŷ ehr ),
where BCE is the Binary Cross-Entropy loss. Let x cxr ∈ R w×h×c represent the chest X-ray image belonging to the same instance associated with the ground-truth labels y cxr , where w is the width dimension, h is the height dimension, and c is the number of channels. In all of our experiments, h = 224, w = 224, and c = 3, as we replicate each image across three channels. We implement the encoder, f cxr , as a ResNet-34 (He et al., 2016) to compute v cxr ∈ R n , which is the feature representation after the average pooling layer of the convolutional network where n = 512. Similarly, we then apply a classifier, g cxr , to compute the predictions, such that y cxr = g cxr (v cxr ) and optimize the following loss to fine-tune the encoder:
L cxr (y cxr ,ŷ cxr ) = BCE(y cxr ,ŷ cxr ).
(
The encoders can hence be independently pre-trained using their respective labels and losses.
The MedFuse Module
To fuse the modalities, we first dismiss the classifiers, g ehr and g cxr , and keep the the pretrained modality-specific encoders, f ehr and f cxr . Since the latent space dimensions of the two modalities are different, we use a projection layer, φ, that projects v cxr to the same dimensionality as v ehr :
v * cxr = φ(v cxr )(3)
such that v * cxr ∈ R m . We then create an input sequence consisting of the the uni-modal feature representations of the sample:
v f usion = [v ehr , v * cxr ].(4)
We parameterize a multi-modal fusion network, f f usion , as a single LSTM layer with input dimension of 256 and a hidden dimension of 512, that aggregates the multi-modal sequence through recurrence. The motivation for using an LSTM is two-fold. First, it follows the intuition of decision-making, where clinicians examine information from each modality sequentially, or one at a time. This allows the LSTM module to initially learn from v ehr , and then update its internal state using information in v * cxr . Second, it can handle input sequences of variable number of modalities, so it inherently deals with missing modalities. In the case that the chest X-ray image is missing during training or inference, the network processes a single-element sequence, [v ehr ].
The last hidden state, h f usion , of f f usion is then processed using a classifier g f usion that computes the final fusion predictions, such thatŷ f usion = g f usion (h f usion ). We jointly train the encoders f ehr and f cxr , the projection layer φ, the fusion module f f usion , and the classifier g f usion , by optimizing the following loss:
L f usion (y f usion ,ŷ f usion ) = BCE(y f usion ,ŷ f usion ),(5)
where y f usion = y ehr , since we assume that the clinical time-series data modality is the base modality associated with the prediction task of interest, and is always present during training and inference. All classifiers g ehr , g cxr , and g f usion consist of a single linear layer followed by sigmoid activation.
Experiments
Datasets and Benchmark Tasks
For our experiments, we extract the clinical time-series data from MIMIC-IV (Johnson et al., 2021) along with the associated chest X-ray images in MIMIC-CXR (Johnson et al., 2019). Here we describe the two tasks and provide more details on each:
• Phenotype classification: The goal of this multi-label classification task is to predict whether a set of 25 chronic, mixed, and acute care conditions are assigned to a patient in a given ICU stay. For a given instance, x ehr contains clinical time-series data collected during the entire ICU record, and y ehr is a vector of 25 binary phenotype labels. We link each instance with the last chest X-ray image collected during the same ICU stay. MIMIC-III contains International Classification of Diseases (ICD) version 9 (ICD-9) codes, whereas MIMIC-IV contains both ICD-9 and ICD-10. In the original benchmark paper (Harutyunyan et al., 2019), the 25 phenotype labels were initially defined using the Clinical Classifications Software (CCS) for ICD-9 (WHO et al., 1988). Since ICD-9 and ICD-10 codes are aggregated to different CCS categories, we mapped all ICD-10 codes to ICD-9 using the guidelines provided by the Centers for Medicare & Medicaid Services 1 , and then map them to CCS categories.
We evaluate this task using the Area Under the Receiver Operating Characteristic (AUROC) curve and the Area Under the Precision Recall curve (AUPRC). • In-hospital mortality prediction: The goal of this binary classification task is to predict in-hospital mortality after the first 48 hours spent in the ICU. Hence, for a given instance, x ehr contains clinical time-series data collected during the first 48 hours of the ICU record, and y ehr is a binary label indicating in-hospital mortality.
Since the task requires a minimum of 48 hours, we exclude ICU stays that are shorter than 48 hours. Here, we pair each instance with the last chest X-ray image collected during the ICU stay. We evaluate this task using AUROC and AUPRC.
Pre-processing of Clinical Time-series Data
We modified the extraction and data pre-possessing pipeline of Harutyunyan et al. (2019), which was originally implemented in TensorFlow (Abadi et al., 2015), and introduce a new version for MIMIC-IV using Pytorch (Paszke et al., 2019). To make a fair comparison and illustrate the efficacy of multi-modal learning, we use the same set of 17 clinical variables. Amongst those, five are categorical (capillary refill rate, Glasgow coma scale eye opening, Glasgow coma scale motor response, Glasgow coma scale verbal response, and Glasgow coma scale total) and 12 are continuous (diastolic blood pressure, fraction of inspired oxygen, glucose, heart rate, height, mean blood pressure, oxygen saturation, respiratory rate, systolic blood pressure, temperature, weight, and pH). For all the tasks, we regularly sample the input every two hours, discretize and standardize the clinical variables to obtain the input for f ehr as in previous work (Harutyunyan et al., 2019). After data pre-processing and one-hot encoding of the categorical features, we obtain a vector representation of size 76 at each time-step of the clinical time-series data, such that for a given instance, x ehr ∈ R t×76 and t depends on the instance and task.
Data Splits
Using the patient identifier of the clinical time-series data, we randomly split the dataset into 70% for training, 10% for validation, and 20% for test set, as shown in Figure 1. We report final results on the test sets and compute 95% confidence intervals with 1000 iterations via the bootstrap method (Efron and Tibshirani, 1994). Here, we denote the clinical timeseries data as EHR and the chest X-ray images as CXR. (EHR+CXR) PARTIAL contains paired and partially paired samples (i.e. samples where chest X-ray is missing). (EHR + CXR) PAIRED contains data samples where both modalities are present. For example, the (EHR + CXR) PARTIAL training set for patient phenotyping contains 7756 samples associated with chest X-rays amongst 42628 samples. We extract from MIMIC-CXR chest X-ray images and split them based on a random patient split. We then transfer images from the training set to either the validation or test set, in case are were associated with patients in the validation or test splits of the clinical time-series data. This procedure resulted with 325188 images in the training set, 15282 images in the validation set, and 36625 images in the test set. We define y cxr as a vector of 14 binary radiology labels extracted from radiology reports through CheXpert (Irvin et al., 2019). We denote this uni-modal dataset as CXR UNI and it is fixed across all tasks. We introduce an additional notation for CXR PAIRED , which includes only chest X-ray images within (EHR+CXR) PAIRED , and EHR PARTIAL , which includes only clinical time-series data within (EHR + CXR) PARTIAL .
Training Strategy with the MedFuse Module
The training strategy consists of two steps: pre-training of the modality-specific encoders followed by jointly fine-tuning the encoders and fusion module. During the pre-training stage, we train the image encoder using the full uni-modal training dataset CXR UNI with the 14 radiology labels. We also pre-train the clinical time-series data encoder for each task independently using the training sets EHR PARTIAL , since each task is associated with its own set of inputs and labels. After pre-training the modality-specific encoders, we discard the uni-modal classifiers and fine-tune the encoders, projection layer, and MedFuse using (EHR + CXR) PARTIAL . We compare this training strategy to fine-tuning the fusion module with randomly initialized feature extractors.
Baseline Models
We compare the performance of our proposed multi-modal approach to several existing baselines:
• Early fusion: The vanilla early fusion approach commonly used in recent work (Huang et al., 2020a) (Figure 3 (left)) assumes the presence of paired data modalities during training and inference. We train two versions. In the first version, we pre-train modality-specific networks independently: f cxr and g cxr with the CXR PAIRED training set, and f ehr and g ehr with the EHR PAIRED training set. We then freeze the encoders f cxr and f ehr , concatenate their latent feature representations, and finetune a projection layer and a fully connected classification network, denoted as g cl using the (EHR + CXR) PAIRED training set. In the second version, we use the (EHR + CXR) PARTIAL training set for fine-tuning the projection layer and g cl . Inspired by Kyono et al. (2021), we learn a vector to substitute for missing chest X-ray images. • Joint fusion: In this setting, we train a network end-to-end including the modalityspecific encoders (f cxr and f ehr ) and a classification network applied to the concate- nated latent representations of the two encoders (Figure 3 (right)). We train two versions. In the first version, we train a randomly initialized network end-to-end using (EHR + CXR) PAIRED . In the second version, we train a randomly initialized network end-to-end using (EHR + CXR) PARTIAL with a learnable vector to substitute for any missing chest X-ray images. 2020), this approach also assumes paired input data. We apply an MMTM module after the first LSTM layer in the clinical time-series modality, and either the third or the fourth ResNet layer. We train a randomly initialized network with the MMTM module end-to-end using the (EHR+CXR) PAIRED training set, and closely follow the training strategy described in the original paper. 2 • Dynamic Affine Feature Map Transform (DAFT): Also requiring paired input data, we use the general purpose DAFT module (Pölsterl et al., 2021) to rescale and shift the feature representations after the first LSTM layer using the chest X-ray representation computed either through the third or fourth layer of ResNet. Similarly, we use (EHR+CXR) PAIRED , and follow the training approach in the original work's respository. 3
We also compare it with a uni-modal two-layer LSTM network trained with clinical timeseries data only, and the method proposed by Hayat et al. (2021a) and AUPRC results for our proposed approach with MedFuse and the baseline models. We include results for early and joint fusion when trained with either (EHR + CXR) PAIRED or (EHR + CXR) PARTIAL , where the latter uses a learnable vector in the case of a missing chest X-ray image. We also show results of our proposed approach when we finetune the fusion module with (EHR + CXR) PARTIAL and randomly initialized encoders (RI) or pre-trained encoders (PT), and the best version of the latter when using the optimal number of uni-modal samples during fine-tuning (OPTIMAL). Best results are shown in bold.
Model Training and Selection
We perform hyperparameter tuning over 10 runs for our proposed network with MedFuse and each of the baseline models and their different versions. In each run, we randomly sample a learning rate between 10 −5 and 10 −3 , and then choose the model and learning rate that achieve the best AUROC on the respective validation set. For the baselines with architectural choices (i.e. MMTM and DAFT), we choose the architecture that achieves the best performance on the validation set, and report its results on the test set. We use the Adam optimizer (Kingma and Ba, 2014) across all experiments with a batch size of 16. We set the maximum number of epochs to 50 and use early stopping if the validation AUROC does not improve for 15 epochs. We also apply image augmentations as described in Appendix A.1.
With the best learning rate chosen via hyperparameter tuning, we vary the percentage of samples with EHR only data in the (EHR + CXR) PARTIAL training set, fine-tune MedFuse accordingly and evaluate it on the validation set. We select the best model based on the best AUROC performance on the (EHR + CXR) PARTIAL validation set, and report its results on the test set. We denote this chosen model as MedFuse (OPTIMAL).
Results
In this section, we describe the results for a number of experiments to provide insights on our proposed approach. The learning rates that achieved the best results are summarized in Appendix A.2 for all models. The results on the validation set in the experiments where we vary the percentage of uni-modal samples during training are shown in Appendix A.3. The optimal percentages are 10% for in-hospital mortality prediction, and 20% for phenotype classification. Table 3: Performance results on the (EHR + CXR) PARTIAL test set. We compare our proposed approach with MedFuse with early and joint fusion when trained with (EHR + CXR) PARTIAL , including samples with missing chest X-ray images (substituted with a learnable vector). All methods were trained with the full (EHR + CXR) PARTIAL training set, except for MedFuse (OPTIMAL) which uses the optimal number of uni-modal samples during fine-tuning. Best results are shown in bold.
Task
Phenotyping In-hospital mortality
Performance Results in the Uni-modal & Multi-modal Settings
In Table 1, we compare our proposed approach to the uni-modal stacked LSTM. As expected, we first observe that the performance of the uni-modal LSTM improves on the EHR PAIRED test set, in terms of AUROC and AUPRC for both tasks, when using the larger EHR PARTIAL training set. Our proposed approach using MedFuse achieves the best performance on the paired test set when the chest X-ray images are used during training and inference as an auxiliary modality (0.770 AUROC and 0.481 AUPRC for phenotype classification, and 0.865 AUROC and 0.594 AUPRC for in-hospital mortality). We note similar, but less significant trends, in the larger partially paired test set, which may be due to the fact that only 18.8% and 26.2% of samples are paired in the phenotyping and in-hospital mortality test sets, respectively.
Performance Results in the Paired Setting
Since the baseline models were originally designed for paired input, we evaluate all models on the (EHR + CXR) PAIRED test set as shown in Table 2. First, we observe that early fusion and joint fusion perform comparably across both tasks when trained with (EHR + CXR) PAIRED , with early fusion achieving a slightly better performance in terms of AUROC. We also note that training early fusion using (EHR + CXR) PARTIAL leads to a drop in AUROC and AUPRC across both tasks, while joint fusion only improves for phenotype classification. Second, we observe that the Unified approach by Hayat et al. (2021a) achieves the best performance amongst all baseline approaches, with 0.765 AU-ROC and 0.461 AUPRC for phenotype classification, and 0.835 AUROC and 0.495 AUPRC for in-hospital mortality prediction. Third, we observe that our proposed approach with MedFuse (OPTIMAL) achieves the best performance across both tasks, with 0.770 AUROC and 0.481 AUPRC for phenotype classification, and 0.865 AUROC and 0.594 AUPRC for in-hospital mortality prediction. We also performed an ablation study where we randomly dropped the chest X-ray modality in the paired test set. The results are shown in Appendix A.4. We also compared the use of substituting the missing modality with zeros or with a learnable vector for early and joint fusion and the results are shown in Appendix A.5. The two techniques perform comparably. Figure 4: Performance results across different subsets of labels for the phenotype classification task in the (EHR + CXR) PAIRED test set. The multi-modal approach with MedFuse achieves the highest AUROC and AUPRC gains for the mixed conditions followed by chronic conditions, compared to the results achieved by the uni-modal stacked LSTM with EHR PAIRED ).
Performance Results in the Partially Paired Setting
In Table 3, we evaluate our proposed approach with MedFuse as well as early and joint fusion on the partially paired test set. Compared to early fusion, our proposed approach trained with the full (EHR + CXR) PARTIAL training set achieves a better performance for phenotype classification (0.758 compared to 0.748 AUROC and 0.418 compared to 0.394 AUPRC). It performs comparably with early fusion in the in-hospital mortality prediction task, although early fusion achieves a better AUPRC. Our approach outperforms joint fusion in the in-hospital mortality setting (0.861 compared to 0.841 AUROC and 0.501 compared to 0.482 AUPRC), and performs comparably for phenotype classification. Overall, MedFuse (OPTIMAL), fine-tuned with paired samples and only 10% of uni-modal samples for inhospital mortality prediction and 20% of uni-modal samples for phenotype classification, achieves the best performance (0.768 AUROC and 0.429 AUPRC for phenotype classification and 0.874 AUROC and 0.567 AUPRC for inhospital mortality prediction). We also performed an ablation study where we varied the percentage of uni-modal samples in the partially paired setting. The results are shown in Appendix A.6.
We also compared the performance of MedFuse to an ensemble consisting of (i) MedFuse for paired samples, and (ii) a uni-modal LSTM for samples with missing chest X-rays. While the results are comparable, as shown in Appendix A.7, the results imply that an ensemble of strong models may be better suited for some tasks, such as phenotyping. This however requires the training of two models.
Phenotype-wise Analysis
In Figure 4, we show the AUROC (left) and AUPRC (right) results across different categories of phenotype labels: acute, mixed, and chronic conditions. The label types and their prevalence are listed in Table 4. We note that our approach mostly improves the performance in terms of AUROC and AUROC for mixed and chronic conditions, which are generally hard to predict through uni-modal clinical time-series data (Harutyunyan et al., 2019). In particular, across mixed conditions, the AUROC increases from 0.749 to 0.800, and the AUPRC increases from 0.458 to 0.565. For chronic conditions, the AUROC increases from 0.717 to 0.745 and the AUPRC increases from 0.487 to 0.512. We observe Table 4: Performance results across the different phenotype labels on (EHR + CXR) PAIRED test set, compared to the uni-modal stacked LSTM with EHR PAIRED . We report the performance results for the individual phenotypes using AUROC and AUPRC, and show the prevalence of labels in the (EHR + CXR) PARTIAL training set, and the (EHR + CXR) PAIRED test set. The labels and results in bold indicate that MedFuse achieved a performance improvement. Table 4, we report the performance across all 25 labels for the paired test set using uni-modal and multi-modal data. We observe an improvement across a number of thorax-related phenotypes, such as pneumonia and pleurisy, which are usually clinically assessed using chest imaging (Long et al., 2017). This further highlights the importance of using the chest X-ray images as auxiliary information along with the clinical time-series data.
In-hospital Mortality Age-wise Analysis
We evaluate the performance of our approach across different age groups, as shown in Table 5, and compare it to the uni-modal stacked LSTM. We observe that the AUROC and AUPRC improve across age groups 40-60, 60-80, and >80 years, while the AUROC decreases for the 18-40 years. The latter result needs further investigation with a larger dataset, since the test sets only contain 11 positive samples for the youngest age group. Additionally, there are variations in the relative improvements. For example, the AUPRC increases by 24% for the 40-60 years group, compared to 1.3% in the 60-80 years group.
Discussion
In this paper, we present a multi-modal fusion approach, named MedFuse, and new benchmark results for integrating partially paired clinical time-series data and chest X-ray images. We evaluate it for two popular benchmark tasks, namely in-hospital mortality prediction and phenotype classification, using publicly available datasets MIMIC-IV and MIMIC-CXR. Our study has several strengths. First, our approach is simple and easy to implement. The results show that the proposed approach performs better than the uni-modal LSTM baseline, as it considers chest X-ray images, when available, as an additional source of information. In addition, the approach outperforms several baselines, and the phenotypewise and age-wise analysis provide some insight as to where it improves performance. We conclude that the proposed method is overall a better choice than the baseline methods because (i) the LSTM-based fusion module can inherently deal with missingness (i.e., partially paired data), and (ii) the combination of the architecture and the training procedure provides performance gains. Otherwise, the size of the partially paired training set does not seem to be correlated with the performance improvements, as illustrated with the validation set results in Appendix A.3. The results overall highlight the promise of multi-modal fusion in improving the performance of clinical prediction models. Multi-modal learning is also generally more closely aligned with the decision-making process of clinicians, who consider multiple sources of information when assessing a patient.
Moreover, in contrast with conventional multi-modal approaches that assume paired input, our proposed method is more flexible since it can process samples with missing chest X-ray images. There is a rising interest in learning cross-modal interactions between modalities during training time and in reconstructing missing modalities (Ngiam et al., 2011;Xin et al., 2021;Ma et al., 2021;Sylvain et al., 2021;Ma et al., 2021). In contrast with natural multi-modal datasets, assuming a high degree of correlation in such settings is not a trivial task in healthcare especially when the modalities do not necessarily share the same labels, and this is an area of future work. The difficulty stems from the sparse and asynchronous nature of medical data, i.e. it would be difficult to use a biopsy report for skin tissue to reconstruct common thorax diseases features (Hayat et al., 2021a). Additionally, some of the existing work for learning cross-modal interactions assumes the presence of all modalities during training (Sylvain et al., 2021).
Another strength is that the approach can be easily scaled to more than two modalities with no amendments to the fusion loss function, compared to existing work where the complexity of the computation increases with the number of modalities (Hayat et al., 2021a).
However, this requires evaluation and is an area of future work. We also do not assume any correlation among the input modalities, in terms of information content or assigned labels. Furthermore, we formalize and introduce new benchmark results for two popular tasks that are typically evaluated in the context of clinical time-series data only (Harutyunyan et al., 2019). By gaining access to the MIMIC-IV and MIMIC-CXR datasets (Johnson et al., 2021(Johnson et al., , 2019, researchers can utilize our open-access data pre-processing pipeline and introduce new results for direct comparison.
Limitations. The study also has its own limitations. To begin with, we focus on tasks pertaining to the integration of clinical time-series data and chest X-ray images from a single data source, and we evaluate our work on two benchmark tasks due to limited resources. The original work by Harutyunyan et al. (2019) includes two other tasks, decompensation prediction and length of stay prediction, which we would like to evaluate our method on in the future. The in-hospital mortality task should also be investigated in the setting where chest X-rays collected beyond the first 48 hours of the ICU stay are excluded. We also do not run any experiments on settings where the clinical time-series data may be missing, but the chest X-ray image is available. In future work, this requires the definition of additional benchmark tasks where the chest X-ray image is the primary modality. Since we currently evaluate our method with two input modalities only, another interesting next step would be to use more than two to further evaluate the robustness of the model, considering its scalability. In its current formulation, the model also lacks interpretability, since we mainly focus on fusion within the scope of this paper. We later plan to explore incorporating attention layers (Vaswani et al., 2017) at the input level of the feature encoders to evaluate the importance of features within each modality, and within the fusion module to evaluate the overall informativeness of each modality. On a related note, our work can benefit from performing instance-level analysis. However, this requires clinical expertise that bridges between chest X-ray image and clinical time-series analysis, which we are currently missing. To realize the full potential of multi-modal learning, there is more work to be done to understand the clinical underpinnings of multi-modal fusion. Overall, the study highlights an extremely worthwhile direction to further leverage the value of multi-modal learning in healthcare, especially as the diversity and quantity of medical data continues to increase. Appendix A.
A.1. Image Augmentations
For the chest X-ray images, we apply a series of transformations during pre-training and fine-tuning across all experiments and tasks. Specifically, we resize each image to 256 × 256 pixels, randomly apply a horizontal flip, and apply a set of random affine transformations, such as rotation, scaling, shearing, and translation. We then apply a random crop to obtain an image of size 224 × 224 pixels. During validation and testing, we perform image resizing to 256 × 256 and apply a center crop to 224 × 224 pixels.
A.2. Hyperparameter Search Results
The results of hyperparameter tuning are shown in Table A1. We summarize the learning rates that achieved the best performance for each model. Table A1: Learning rates that achieved the best results during hyperparameter search.
We conducted 10 runs for each model with randomly sampled learning rates between 10 −5 and 10 −3 . For MMTM and DAFT, we additionally selected the version that achieved the best validation set AUROC.
Task
Phenotyping In-hospital mortality
Method
Learning rate LSTM trained with EHR PAIRED 8.866 × 10 −5 1.000 × 10 −4 LSTM trained with EHR PARTIAL 5.399 × 10 −4 5.399 × 10 −4 Early trained with (EHR + CXR) PARTIAL 9.084 × 10 −5 3.095 × 10 −4 Early trained with (EHR + CXR) PAIRED 3.833 × 10 −5 9.515 × 10 −5 Joint trained with (EHR + CXR) PARTIAL 3.831 × 10 −5 7.565 × 10 −4 Joint trained with (EHR + CXR) PAIRED 5.652 × 10 −5 4.032 × 10 −5 MMTM * 5.326 × 10 −5 4.355 × 10 −5 DAFT * * 6.493 × 10 −5 6.493 × 10 −5 Unified 2.042 × 10 −4 2.606 × 10 −4 MedFuse (Randomly initialized encoders) 4.741 × 10 −5 9.382 × 10 −5 MedFuse (Pre-trained encoders) 7.347 × 10 −5 1.452 × 10 −5 * We trained two versions of MMTM for each task, where the MMTM module is placed after the third or fourth ResNet layer. Placing it after the fourth layer achieved the best performance for both tasks. * * We trained two versions of DAFT for each task, where we transform the LSTM representation either after the third or fourth ResNet layer. Placing it after the third layer achieved the best performance for phenotype classification, whereas placing it after the fourth layer achieved the best performance for in-hospital mortality.
A.3. Percentage of Uni-modal Samples within the Training Set
We also run experiments where we vary the percentage of uni-modal samples during finetuning. The best AUROC results for both tasks on the validation set are shown in Figure A1. For in-hospital mortality (shown in red), we notice that a relatively smaller portion of uni-modal samples (10%) achieves the best performance. For patient phenotyping (shown in blue), we observe a similar trend where the best AUROC is achieved with only 20% of uni-modal samples. We fix the sampling percentage that achieves the best validation AUROC across all experiments, unless noted otherwise. Hence, this highlights that the best performance gains of MedFuse are achieved even with a small percentage of uni-modal samples.
A.4. Percentage of Uni-modal Samples within the Paired Test Set
We also performed an ablation study where we randomly dropped the chest X-ray modality for a percentage of samples in the paired test set. The results are shown in Figure A2. We observe that the as the percentage of dropping increases, the AUROC decreases for both tasks. Figure A2: Performance on the test set with randomly dropped CXR modality in the paired test set. The plot shows the AUROC on the paired test set for different percentages of randomly dropped CXR modality from paired test samples.
A.5. Missing Modality with Early and Joint Fusion
We also ran initial experiments to compare the learnable vector with imputing zeros for a missing chest X-ray modality. The results are shown in Table A2. We note that the results are comparable with no obvious differences. Table A2: Missing modality with early and joint fusion. We report the AUROC and AUPRC results on the entire test set (EHR PARTIAL ), including samples with missing chest X-ray images (substituted with a zeros or a learnable vector) All methods below were pre-trained using the (EHR PARTIAL ) training set and a fixed learning rate of 0.0001. We performed another ablation study where we varied the number of uni-modal samples in the partially paired test set. The results are shown in Figure A3. Hence, including 0% of uni-modal test samples is equivalent to the fully paired test set. We observe an increase in the AUROC in the in-hospital mortality task, as the percentage of uni-modal samples increase, but a more a consistent AUROC in the phenotyping task. We do however observe that the widths of the confidence intervals decrease as the percentage of uni-modal samples increases across both tasks. Figure A3: Performance on the test set when varying the percentage of uni-modal samples. The plot shows the AUROC on the partial test set for different percentages of randomly selected uni-modal samples.
A.7. Ensemble of uni-modal and multi-modal models
We ran another experiment to compare the performance of MedFuse to that of an ensemble of two models: an EHR only model that computes predictions for partial input (i.e., not associated with a chest X-ray) using LSTM, and a paired model that computes predictions for paired input using MedFuse. The results are shown in Table A3. We observe that the ensemble slightly outperforms MedFuse for phenotyping only. This implies that an ensemble of strong models may be better suited for some tasks, such as phenotyping, which however requires the training of two models.
Figure 3 :
3Architecture of early and joint fusion baselines. In early fusion (left), the encoders are first pre-trained. Then, we freeze them and fine-tune the projection layer and fusion classification module. In joint fusion (right), the encoders and classification module are randomly initialized and trained end-to-end.
•
Multi-modal Transfer Module (MMTM): Originally proposed by Vaezi Joze et al. (
MedFuse (OPTIMAL) 0.770 (0.745, 0.795) 0.481 (0.436, 0.531) 0.865 (0.837, 0.889) 0.594 (0.526, 0.655)
MedFuse (OPTIMAL) 0.768 (0.756, 0.779) 0.429 (0.408, 0.452) 0.874 (0.860, 0.888) 0.567 (0.529, 0.607)
improvements for acute conditions, where the AUROC increases from 0.761 to 0.772 and the AUPRC increases from 0.432 to 0.433. In
Figure A1 :
A1Performance on the validation set when varying the sampling percentage for uni-modal training samples. The plot shows the AUROC on the validation set for different percentages of randomly selected uni-modal training samples.
Table 1 :
1Performance results in the uni-modal vs multi-modal setting. Here, we compare the stacked LSTM network for the clinical time-series data only, with our network using MedFuse. In the first four rows, we summarize the AUROC and AUPRCresults in the paired setting with uni-modal (EHR PAIRED ) and multi-modal data
((EHR + CXR) PAIRED ). In the last two rows, we show the results on the par-
tially paired test set, with the uni-modal subset (EHR PARTIAL ) and multi-modal data
((EHR + CXR) PAIRED ). All results shown below are for MedFuse (OPTIMAL). Best
results are shown in bold.
Modalities
Phenotyping
In-hospital mortality
Model
Training set
Test set
AUROC
AUPRC
AUROC
AUPRC
LSTM
EHR PAIRED
EHR PAIRED
0.716
0.407
0.818
0.460
(0.688, 0.743) (0.367, 0.453) (0.787, 0.845) (0.395, 0.535)
LSTM
EHR PARTIAL
EHR PAIRED
0.746
0.453
0.825
0.500
(0.720, 0.772) (0.409, 0.502) (0.793, 0.852) (0.428, 0.576)
MedFuse (EHR + CXR) PARTIAL
EHR PAIRED
0.740
0.441
0.833
0.514
(0.713, 0.767) (0.398, 0.489) (0.802, 0.861) (0.443, 0.584)
MedFuse (EHR + CXR) PARTIAL (EHR + CXR) PAIRED
0.770
0.481
0.865
0.594
(0.745, 0.795) (0.436, 0.531) (0.837, 0.889) (0.526, 0.655)
LSTM
EHR PARTIAL
EHR PARTIAL
0.765
0.425
0.861
0.522
(0.754, 0.777) (0.404, 0.447) (0.846, 0.876) (0.482, 0.564)
MedFuse (EHR + CXR) PARTIAL (EHR + CXR) PARTIAL
0.768
0.429
0.874
0.567
(0.756, 0.779) (0.408, 0.452) (0.860, 0.888) (0.529, 0.607)
Table 2 :
2Performance results on the (EHR + CXR) PAIRED test set. We show the AUROC
Table 5 :
5Performance of MedFuse across different age groups for in-hospital-mortality onthe (EHR + CXR) PAIRED ) test set, compared to the uni-modal stacked LSTM
with EHR PAIRED . We compare the AUROC and AUPRC for the different age groups.
The results in bold indicate improved performance with multi-modal data.
A.6. Percentage of Uni-modal Samples within the Partially Paired Test SetTask
Phenotyping
In-hospital mortality
Method Missing Vector AUROC AUPRC AUROC
AUPRC
Joint
Zeros
0.756
0.406
0.843
0.466
Joint
Learnable
0.752
0.402
0.853
0.486
Early
Zeros
0.743
0.392
0.842
0.481
Early
Learnable
0.742
0.388
0.851
0.489
Table A3 :
A3MedFuse compared to an ensemble evaluation. We report the AUROC and AUPRC results on the partially paired test set.Task
Phenotyping
In-hospital mortality
Method
AUROC
AUPRC
AUROC
AUPRC
Ensemble 0.770 (0.759, 0.782) 0.431 (0.410, 0.454) 0.870 (0.857, 0.884) 0.547 (0.509, 0.589)
MedFuse
0.768 (0.756, 0.779) 0.429 (0.408, 0.452) 0.874 (0.860, 0.888) 0.567 (0.529, 0.607)
† Currently at G42 Healthcare. © 2022 N. Hayat, K.J. Geras & F.E. Shamout.
Acknowledgements. This work is supported in part by the NYUAD Center for Artificial Intelligence and Robotics, funded by Tamkeen under the NYUAD Research Institute Award CG010. We would also like to thank the High Performance Computing (HPC) team at NYUAD for their support.
. Centers for Medicare & Medicaid Services. Centers for Medicare & Medicaid Services, https://www.cms.gov/Medicare/Coding/ICD10/ 2018-ICD-10-CM-and-GEMs
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"https://github.com/nyuad-cai/MedFuse."
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[
"Suboptimal s-union familes and s-union antichains for vector spaces *",
"Suboptimal s-union familes and s-union antichains for vector spaces *"
] | [
"Yunjing Shan \nSchool of Mathematics and Statistics\nBeijing Jiaotong University\n100044BeijingChina\n",
"Junling Zhou [email protected] \nSchool of Mathematics and Statistics\nBeijing Jiaotong University\n100044BeijingChina\n"
] | [
"School of Mathematics and Statistics\nBeijing Jiaotong University\n100044BeijingChina",
"School of Mathematics and Statistics\nBeijing Jiaotong University\n100044BeijingChina"
] | [] | Let V be an n-dimensional vector space over the finite field F q , and let L(The optimal s-union families in L(V ) have been determined by Frankl and Tokushige in 2013. The upper bound of cardinalities of s-union (s < n) antichains in L(V ) has been established by Frankl recently, while the structures of optimal ones have not been displayed. The present paper determines all suboptimal s-union families for vector spaces and then investigates s-union antichains. For s = n or s = 2d < n, we determine all optimal and suboptimal s-union antichains completely. For s = 2d + 1 < n, we prove that an optimal antichain is either V d or contained in V d V d+1 which satisfies an equality related with shadows. | 10.1016/j.disc.2023.113505 | [
"https://arxiv.org/pdf/2207.06727v1.pdf"
] | 250,526,401 | 2207.06727 | 5275572d19bd30b578c915c80cbcc812101f5056 |
Suboptimal s-union familes and s-union antichains for vector spaces *
Jul 2022
Yunjing Shan
School of Mathematics and Statistics
Beijing Jiaotong University
100044BeijingChina
Junling Zhou [email protected]
School of Mathematics and Statistics
Beijing Jiaotong University
100044BeijingChina
Suboptimal s-union familes and s-union antichains for vector spaces *
Jul 2022s-union antichain cross-intersecting shadow vector space
Let V be an n-dimensional vector space over the finite field F q , and let L(The optimal s-union families in L(V ) have been determined by Frankl and Tokushige in 2013. The upper bound of cardinalities of s-union (s < n) antichains in L(V ) has been established by Frankl recently, while the structures of optimal ones have not been displayed. The present paper determines all suboptimal s-union families for vector spaces and then investigates s-union antichains. For s = n or s = 2d < n, we determine all optimal and suboptimal s-union antichains completely. For s = 2d + 1 < n, we prove that an optimal antichain is either V d or contained in V d V d+1 which satisfies an equality related with shadows.
Introduction
Let X be an n-element set and let X k denote the set of all k-element subsets of X. For the power set of X, we use the notation 2 X . We say that a family of subsets F ⊆ 2 X is s-union if |F ∪F ′ | ≤ s holds for all F , F ′ ∈ F. A family F is called t-intersecting if for all F , F ′ ∈ F, we have |F ∩ F ′ | ≥ t. Since F ⊆ 2 X is s-union if and only if {X \ F : F ∈ F} is an (n − s)-intersecting family, the two concepts are essentially the same.
Let F ⊆ 2 X be s-union. An s-union family is said to be optimal if it has the largest possible cardinality. It is obvious that F = 2 X is the optimal s-union family if s = n. For s = n − 1, there are many optimal s-union families achieving the maximum cardinality 2 n−1 [4]. For s ≤ n − 2, Katona [9] showed that |F| ≤ d where y is a fixed element of X. In 2017, Frankl [5] investigated the suboptimal sunion families (s < n), meaning that they have the largest possible cardinality under the condition that they are not contained in any of the optimal s-union families. Frankl established the following theorem. Theorem 2]) Let X be an n-element set and 2 ≤ s < n. Suppose that F ⊆ 2 X is s-union and F K(n, s). Then the following hold. where D ∈ X d+1 , y ∈ X are fixed with y / ∈ D or
Theorem 1.1. ([5,F = {F ⊆ X : |F | ≤ 2} {F ∈ X 3 : |F ∩ D| ≥ 2},
where D is a fixed 3-subset of X if s = 5.
The problems in extremal set theory have natural extensions to families of subspaces over a finite field. Throughout the paper we always let V be an n-dimensional vector space over the finite field F q . Let V k denote the family of all k-dimensional subspaces of V . For m ∈ R, k ∈ Z + , define the Gaussian binomial coefficient by
m k q := 0≤i<k q m−i − 1 q k−i − 1 .
Obviously, the size of V k is n k q . If k and q are fixed, then m k q is a continuous function of m which is positive and strictly increasing when m ≥ k. If there is no ambiguity, the subscript q can be omitted. Let
L(V ) = 0≤k≤n V k .
For any two subspaces A, B ∈ L(V ), let A ≤ B denote that A is a subspace of B and A + B denote the linear span of A, B. Let F ⊆ L(V ) be a family of subspaces, we say that F is s-
union if dim(F + F ′ ) ≤ s holds for all F , F ′ ∈ F. A family F ⊆ L(V ) is called t-intersecting if for all F , F ′ ∈ F, we have dim(F ∩ F ′ ) ≥ t.
In particular, we say that F is an intersecting family if t = 1. Endow V with the usual inner product ·, · . For a subspace U of V , let
U ⊥ = {v ∈ V : u, v = 0 for all u ∈ U } be the orthogonal complement of U . For a family F ⊆ L(V ), denote F ⊥ = {F ⊥ : F ∈ F}. Since (F + F ′ ) ⊥ = F ⊥ ∩ F ′⊥ holds for all F, F ′ ∈ F, it is obvious that F is s-union if and only if F ⊥ is (n − s)-intersecting.
Obviously, F = L(V ) is the optimal s-union family if s = n. For s = n − 1, the optimal s-union families in L(V ) were determined by Blokhuis et al. [1]. For s ≤ n − 2, we record the optimal families K[n, s] as follows, which were displayed in the form of t-intersecting families in [8]. For s = 2d, let
K[n, 2d] = {F ≤ V : dim(F ) ≤ d}. For s = 2d + 1, let K[n, 2d + 1] = {F ≤ V : dim(F ) ≤ d} {F ∈ V d + 1 : E ≤ F },
where E is a fixed 1-dimensional subspace of V . (ii) When s = 2d + 1,
|F| ≤ d i=0 n i + n − 1 d .
Moreover, for s ≤ n − 2, equality holds if and only if F = K[n, 2d + 1].
Excluding the optimal families provided in Theorem 1.2, we will consider the suboptimal s-union families in L(V ) and establish a vector space version of Theorem 1.1. Let us define the families T [n, s]. For s = 2d, define
T [n, 2d] = ({F ≤ V : dim(F ) ≤ d} \ {F ⊆ V d : dim(F ∩ U ) = 0}) {U }, where U is a fixed (d + 1)-dimensional subspace of V . For s = 2d + 1, define T [n, 2d+1] = {F ≤ V : dim(F ) ≤ d} {F ∈ V d + 1 : E ≤ F, dim(F ∩U ) ≥ 1} E + U d + 1 , where E ∈ V 1 and U ∈ V d+1 are fixed subspaces of V with E U . For s = 5, define also J [n, 5] = {F ≤ V : dim(F ) ≤ 2} {F ∈ V 3 : dim(F ∩ D) ≥ 2},
where D is a fixed 3-dimensional subspace of V . A main result of this paper in the next theorem shows that T [n, 2d], T [n, 2d + 1] and J [n, 5] (for s = 5) are suboptimal s-union families.
Theorem 1.3. Let 2 ≤ s < n. Suppose that F ⊆ L(V ) is s-union and F K[n, s].
Then the following hold.
(i) If s = 2d, then |F| ≤ d i=0 n i − q d(d+1) n−d−1 d + 1.(1)
Moreover for s ≤ n − 2, equality holds if and only if F = T [n, 2d].
(ii) If s = 2d + 1, either q ≥ 3 and n ≥ 2d + 3, or q = 2 and n ≥ 2d + 4, then
|F| ≤ d i=0 n i + n−1 d − q d(d+1) n−d−2 d + q d+1 .(2)
Moreover for s ≤ n−2, equality holds if and only if F = T [n, 2d+1] or alternatively F = J [n, 5] if s = 5.
A family F ⊆ L(V ) is an antichain if F F ′ holds for any two distinct F, F ′ ∈ F. In 2021, Frankl [6] obtained the upper bound of the cardinalities of s-union antichains for vector spaces.
Theorem 1.4. ([6, Theorem 3.5]) If F ⊆ L(V ) is an s-union antichain with 2 ≤ s < n, then |F| ≤ n ⌊ s 2 ⌋ .(3)
The second main objective of the paper is to determine the structures of all optimal s-union antichains and then the suboptimal ones. Let us define the families A[n, s],
B[n, s]. For 2 ≤ s ≤ n, define A[n, s] = ( V ⌈ s 2 ⌉ \ {F ∈ V ⌈ s 2 ⌉ : U ≤ F }) {U }, where U is a fixed (⌈ s 2 ⌉ − 1)-dimensional subspace of V ; define B[n, s] = ( V ⌊ s 2 ⌋ \ {F ∈ V ⌊ s 2 ⌋ : F ≤ W }) {W },
where W is a fixed (⌊ s 2 ⌋ + 1)-dimensional subspace of V . It is obvious that an s-union antichain is just an antichain if s = n. We determine the structures of all optimal and suboptimal antichains in this paper.
if F = V ⌊ n 2 ⌋ or F = V ⌈ n 2 ⌉ . (ii) If F V ⌊ n 2 ⌋ and F V ⌈ n 2 ⌉ , then |F| ≤ n ⌊ n 2 ⌋ − q ⌊ n 2 ⌋ 1 .
Moreover, equality holds if and only if
F = A[n, n] or B[n, n].
For 0 ≤ u ≤ n, let us define the u-shadow of H ⊆ L(V ) by
△ u (H) = {G ∈ V u : G ≤ H for some H ∈ H}.
In particular, the (u − 1)-shadow of the family H ⊆ V u is denoted by △(H) for convenience.
For s-union antichains with s < n, we establish another main theorem as follows.
Theorem 1.6. Let F ⊆ L(V ) be an s-union antichain with s < n. Then the following hold.
(a) F = V ⌊ s 2 ⌋ ; (b) F = F d F d+1 for s = 2d + 1, where F d+1 ⊆ V d+1 , F d = V d \ △(F d+1 ) and | △ (F d+1 )| = |F d+1 |. (ii) Suppose s = 2d and F V d . Then (a) or (b) holds. (a) If d = 1, then |F| ≤ n 1 − q.
Moreover, equality holds if and only if
F = B[n, 2]. (b) If d ≥ 2, then |F| ≤ n d − q n − d 1 .
Moreover, equality holds if and only if
F = A[n, 2d].
The main objective of the paper is to prove Theorems 1.3, 1.5 and 1.6. For Theorem 1.6, we first consider the case d = 1, s = 2 < n. It is obvious that if d = 1, the optimal 2-union antichain is V 1 . Let F be a suboptimal 2-union antichain. We can easily find that if |F| > 1 then any i-dimensional subspace with i = 0 or i ≥ 3 does not belong to F and that |F
Preliminaries
In this section, we recall a number of basic theorems and establish several new lemmas in the vector spaces, which are essential for our proofs. Firstly, we introduce the celebrated Erdős-Ko-Rado theorem and Hilton-Milner theorem for vector spaces.
Theorem 2.1. ([7, Theorem 1], [10, Theorem 3]) Let 1 ≤ t ≤ k. Suppose H ⊆ V k is a t-intersecting family. Then we have |H| ≤ n−t k−t , if n ≥ 2k, 2k−t k , if 2k − t < n ≤ 2k.
Moreover, equality holds if and only if one of the following holds:
(i) If n > 2k, then H = {H ∈ V k : T ≤ H} for some T ∈ V t . (ii) If 2k − t < n < 2k, then H = Y k for some Y ∈ V 2k−t . (iii) If n = 2k, then H = {H ∈ V k : T ≤ H} for some T ∈ V t or H = Y k for some Y ∈ V 2k−t .|H| ≤ n−1 k−1 − q k(k−1) n−k−1 k−1 + q k .
Moreover, equality holds if and only if
H = {H ∈ V k : E ≤ H, dim(H ∩ U ) ≥ 1} E + U k , where E, U are fixed 1-dimensional, k-dimensional subspaces of V (respectively) with E U or for k = 3, H = {F ∈ V 3 : dim(F ∩ D) ≥ 2}, where D is a fixed 3-dimensional subspace of V .
Shadow is an important notion in extremal set theory. We will make use of the following two vector space version of theorems on shadows.
Theorem 2.3. ([8, Theorem 3]) Let 1 ≤ t ≤ k ≤ n and let H ⊆ V k be t-intersecting. Then for k − t ≤ u ≤ k, we have |△u(H)| |H| ≥ [ 2k−t u ] [ 2k−t k ] .(4)
Note that for k − t < u < k, the RHS of (4) is strictly greater than 1.
Theorem 2.4. ([3, Theorem 1.4]) Let H ⊆ V k and let m ≥ k be the real number which satisfies |H| = m k . Then | △ (H)| ≥ m k − 1 .
Moreover, equality holds if and only if
H = M k for some M ∈ V m , m ∈ Z + .
The following result is an analog of [5, Proposition 1] for finite sets.
Lemma 2.5. Suppose F ⊆ L(V ) is s-union and 2 ≤ s < n. Let F i = F ∩ V i for 0 ≤ i ≤ ⌊ s 2 ⌋. Then |F i | + |F s+1−i | ≤ n i .(5)
Moreover, for s ≤ n − 2, equality holds if and only if F s+1−i = ∅.
Proof. When F s+1−i = ∅, the theorem holds trivially. We suppose F s+1−i = ∅ in the following. For a fixed i ≤ ⌊ s 2 ⌋, define the family
H = {F ⊥ : F ∈ F s+1−i } ⊆ V n + i − s − 1 . We claim that △ i (H) ∩ F i = ∅. Suppose F ∈ △ i (H) ∩ F i . Then there exists H ∈ H such that F ≤ H, H ⊥ ∈ F s+1−i . So we have dim(F ∩ H ⊥ ) = 0, i.e., dim(F + H ⊥ ) = i + (s + 1 − i) = s + 1
, a contradiction to the s-union property of F. By the claim, we have
| △ i (H)| + |F i | ≤ n i .(6)
Since F s+1−i ⊆ F is s-union, H is an (n − s)-intersecting family. In Theorem 2.3, setting u = i, t = n − s, k = n + i − s − 1, yields the following inequality for s ≤ n − 1: (4). Hence, we have |H| + |F i | ≤ n i by (6). It is clear that |H| = |F s+1−i |. Therefore, the desired result follows.
| △ i (H)| ≥ |H|. When s ≤ n − 2, since F s+1−i = ∅, i.e., H = ∅, we have | △ i (H)| > |H| by
We introduce a counting formula for vector spaces, which is further interpreted by applying the q-analog of inclusion-exclusion principle in [2]. Lemma 2.6. Let Z be an m-dimensional subspace of the n-dimensional vector space V over F q . For a positive integer l with m+l ≤ n, let x denote the number of l-dimensional subspaces W of V such that dim(Z ∩ W) = 0. Then the following hold.
(i) x = q lm n−m l = 0≤t≤min{m,l} (−1) t q t(t−1) 2 m t n−t l−t . (ii) x ≥ n l − m 1 n−1 l−1 .
Proof. (i) This is the result of Propositions 2.2 and 2.3 in [2].
(ii) Let a = min{m, l}, if a = 1, then the equality in (ii) holds by (i). If a ≥ 2, we have
x = n l − m 1 n − 1 l − 1 + a t=2 (−1) t q t(t−1) 2 m t n − t l − t .
It suffices to prove that q
t(t−1) 2 m t n−t l−t ≥ q (t+1)t 2 m t+1 n−t−1 l−t−1 , where a ≥ t ≥ 2. Since n ≥ m + l ≥ 2a, we have q n−t −1 q l−t −1 · q t+1 −1 q m−t −1 > (q n−t −1)·(q t+1 −1) q l+m−2t = q n+1 −q n−t −q t+1 +1 q l+m−2t > q n+1 −q n−2 −q a+1 q l+m−2t > q n q l+m−2t > q t ,
which implies the desired result.
Two families A and B in L(V ) are said to be cross-Sperner if there exist no A ∈ A and B ∈ B with A ≤ B or B ≤ A. Wang and Zhang [11] obtained the upper bound of sizes of a pair of cross-Sperner families of finite vector spaces.
|A| + |B| ≤ max{ n b − n − a b − a + 1, n a − b a + 1}.
Moreover equality holds if and only if one of the following holds:
(i) n a ≤ n b and A = {A} f or some A ∈ V a and B = {B ∈ V b : A B}; (ii) n a ≥ n b and B = {B} f or some B ∈ V b and A = {A ∈ V a : A B}.
We say that two families A and
B in L(V ) are cross-t-intersecting if dim(A∩B) ≥ t for all A ∈ A, B ∈ B, where t ≥ 1.
In particular, A and B are said to be cross-intersecting if t = 1. Wang and Zhang [11] also obtained the upper bound of sizes of cross-t-intersecting families.
|A| + |B| ≤ n b − t−1 i=0 q (a−i)(b−i) a i n − a b − i + 1.(7)
Moreover equality holds if and only if one of the following holds:
(i) A = {A} f or some A ∈ V a and B = {B ∈ V b : dim(A ∩ B) ≥ t};
(ii) n a = n b and B = {B} f or some B ∈ V b and A = {A ∈ V a : dim(A∩B) ≥ t}.
We will sharpen the upper bound of (7) in a special case in the following lemma, which will be very useful in the proofs of Theorems 1.3 and 1.6. Lemma 2.9. Suppose A ⊆ V k and B ⊆ V k+1 are cross-intersecting families. Further suppose B is 2-intersecting. Then for n ≥ 2k + 1,
|A| + |B| ≤ n k − q k(k+1) n−k−1 k + 1.(8)
Moreover for n ≥ 2k + 2, equality holds if and only if B = {B} f or some B ∈ V k+1 and
A = {A ∈ V k : dim(A ∩ B) ≥ 1}.
Proof. If k = 1, since B ⊆ V 2 is 2-intersecting, then |B| = 1 and (8) holds by Lemma 2.6 (i). We always let k ≥ 2 in the following proof.
First we consider the case n = 2k + 1. Since n k = n k+1 , we can set a = k + 1, b = k, t = 1 in Theorem 2.8. Then (8) holds by (7).
Next we let n ≥ 2k + 2. Since B ⊆ V k+1 is 2-intersecting and n ≥ 2k + 2, then by Theorem 2.1, we have by Lemma 2.6 (i), which implies (8) by using (10). Moreover, the equality in (8) holds if and only if the equality in (10) holds, that is
1 ≤ |B| ≤ n − 2 k − 1 .(9)A = {A ∈ V k : dim(A ∩ B) ≥ 1}. Case b: Suppose |B| ≥ 2. Let B = {B 1 , B 2 , . . . , B r } ⊆ V k + 1 ,
where r ≥ 2. We claim that
|Γ(B)| ≥ q k(k+1) n − k − 1 k + q (k−1)(k+1) n − k − 2 k − 1 .(11)
Since
|Γ 1 | = q k(k+1) n − k − 1 k .(12)
Obviously, there exists
E ∈ B 1 1 , E B 2 such that B 2 ≤ W ∈ V n−1 , where V = E ⊕ W (namely V = E + W and E ∩ W = {0}). Define a subfamily of Γ 2 by Γ 3 = {T ∈ V k : dim(T ∩ B 2 ) = 0, E ≤ T }, which has a one-to-one correspondence to Γ 4 = {R ∈ W k−1 : dim(R ∩ B 2 ) = 0}.
There is a natural injective map ϕ : Γ 3 → Γ 4 by T → T ∩ W . For any R ∈ Γ 4 , since R + B 2 ≤ W, E W , we have E + R ∈ Γ 3 . So ϕ is also surjective. By Lemma 2.6 (i), we have
|Γ 2 | ≥ |Γ 3 | = |Γ 4 | = q (k−1)(k+1) n − k − 2 k − 1 .
Together with (12), we complete the proof of (11). Combining (9), (10), and (11), we have
|A| + |B| ≤ n k − q k(k+1) n−k−1 k − q (k−1)(k+1) n−k−2 k−1 + n−2 k−1 .(13)
Hence, we only need to show that
n − 2 k − 1 < 1 + q (k−1)(k+1) n − k − 2 k − 1(14)
to prove the final conclusion. If k = 2, (14) clearly holds. If k ≥ 3, by Lemma 2.6 (ii), we have the inequality:
q (k−1)(k+1) n − k − 2 k − 1 > n − 1 k − 1 − k + 1 1 n − 2 k − 2 .
Since q n−k ≤ q n −1 q k −1 ≤ q n−k+1 and n ≥ 2k + 2, we have
n−1 k−1 − k+1 1 n−2 k−2 + 1 − n−2 k−1 > n−1 k−1 − k+1 1 n−2 k−2 − n−2 k−1 = q n−1 −1 q k−1 −1 n−2 k−2 − q k+1 −1 q−1 n−2 k−2 − q n−k −1 q k−1 −1 n−2 k−2 ≥ (q n−k − q k+1 − q n−2k+2 ) n−2 k−2 ≥ (2q n−k−1 − q k+1 − q n−2k+2 ) n−2 k−2 ≥ 0,
meaning that (14) holds as well. So we obtain that |A| + |B| < n k − q k(k+1) n−k−1 k + 1 by (13) and (14).
Reviewing the whole proof, we get that the families A and B attaining the equality in (8) are just those stated in the lemma.
3
Proof of Theorem 1.3
In this section, we will prove Theorem 1.3. Let F ⊆ L(V ) be an s-union family of maximum size with F K[n, s]. We will calculate the cardinality
|F| = 0≤i≤s |F i |, where F i = F ∩ V i .(15)
We consider the singularity of s.
(1) Let s = 2d + 1. It is easily seen that F i = ∅ for i ≥ 2d + 2. Adding up (5) for
0 ≤ i ≤ d, we have d i=0 (|F i | + |F 2d+2−i |) ≤ d i=0 n i .(16)
Since F d+1 ⊆ F is (2d + 1)-union, we have that F d+1 is an intersecting family. Since for G ≤ F ∈ F, the family F ∪ {G} is also s-union. So we can assume that G ≤ F and F ∈ F implies G ∈ F. Then we distinguish two cases. Case a: Suppose there exists G ∈ F with dim(G) ≥ d + 2. Then G d+1 ⊆ F d+1 , which implies that dim( F ∈F d+1 ) = 0. By Theorem 2.2, we have
|F d+1 | ≤ n−1 d − q d(d+1) n−d−2 d + q d+1 .(17)
Moreover, there exists F 2d+2−i = ∅ for 0 ≤ i ≤ d in this case. Then (5) is a strict inequality for some i, which implies the inequality in (16) is strict as well. Hence, by (15)-(17), we have
|F| = d i=0 (|F i | + |F 2d+2−i |) + |F d+1 | < d i=0 n i + n − 1 d − q d(d+1) n − d − 2 d + q d+1 .
Case b: Suppose that F i = ∅ for all i ≥ d + 2. Now (5) holds trivially, that is (16) holds trivially as well. We have dim( F ∈F d+1 ) = 0, because otherwise F ⊆ K[n, 2d + 1]. We use Theorem 2.2 again. The upper bound in (2) can be obtained by (16) and (17). Moreover, if the equality in (2) holds, then F i = V i for all 0 ≤ i ≤ d and the equality in (17) holds as well. So applying Theorem 2.2, we have
F d+1 = {F ∈ V d + 1 : E ≤ F, dim(F ∩ U ) ≥ 1} E + U d + 1 ,
where E, U are fixed 1-dimensional, (d + 1)-dimensional subspaces of V with E U . If d + 1 = 3, the equality is also attained by taking
F 3 = {F ∈ V 3 : dim(F ∩ D) ≥ 2}, where D is a fixed 3-dimensional subspace of V .
(2) Let s = 2d. For 0 ≤ i < d, we use (5) as well. Then
0≤i<d (|F i | + |F 2d+1−i |) ≤ 0≤i<d n i .(18)F = {F ≤ V : dim(F ) ≤ d} \ {F ⊆ V d : dim(F ∩ U ) = 0} {U }.
where U is a fixed (d + 1)-dimensional subspace of V .
4
Proofs of Theorems 1.5 and 1.6
In this section, we will prove Theorems 1.5 and 1.6. The main approach adopts a series of replacement in an s-union antichain by shadows or shades. First, we will define the concept of shade and disclose a new relationship between a family of k-dimensional subspaces and its shadows or shades. For a family H ⊆ V k , we define its shade by
▽(H) = {G ∈ V k + 1 : H ≤ G for some H ∈ H}.| △ (H)| − |H| ≥ x k−1 − x k = q k −q x−k+1 q k −1 x k−1 = q k −q x−k+1 q k −1 k−2 i=0 q x−i −1 q k−1−i −1 .
Let f (x) be the RHS of the above inequality. By setting y = q x , we can rewrite f (x) as a polynomial g(y) of degree k, namely
g(y) = q k −yq −k+1 q k −1 k−2 i=0 yq −i −1 q k−1−i −1 .
Because the polynomial g(y) has k simple roots 1, q, . . . , q k−2 , q 2k−1 , f (x) has k simple roots 0, 1, . . . , k − 2, 2k − 1. It is clear that f ′ (x) has a simple root in each interval between these roots. Since f (x) < 0 if x > 2k −1, then in [k −2, 2k −1], f (x) is increasing up to some value and then decreasing. Hence for k ≤ x ≤ n ≤ 2k − 2,
f (x) ≥ min{f (k), f (2k − 2)}.
Clearly, we have
f (2k − 2) − f (k) = 2k−2 k−1 − 2k−2 k − k k−1 + 1 = q k −q k−1 q k−1 −1 2k−2 k−2 − q k−1 1 > 2k−2 k−2 − q k > 0. Therefore, f (x) ≥ f (k) = q k − 1 1 .
The equality holds if and only if
x = k, that is |H| = 1. Equivalently, H = {U }, where U is a fixed k-dimensional subspace of V . (ii) We claim that | ▽ (H)| = | △ (H ⊥ )|. For any G ∈ ▽(H), there exists H ∈ H such that H ≤ G. Then G ⊥ ≤ H ⊥ , where dim(G ⊥ ) = n − k − 1, dim(H ⊥ ) = n − k, and H ⊥ ∈ H ⊥ . Thus G ⊥ ∈ △(H ⊥ )
. This gives an injective map ϕ : ▽(H) → △(H ⊥ ) by G → G ⊥ , which is obviously surjective.
Let k ≤ ⌊ n 2 ⌋ − 1. Then n − k ≥ ⌈ n 2 ⌉ + 1. We can obtain the following inequality by the above claim and the result of (i):
| ▽ (H)| − |H| = | △ (H ⊥ )| − |H ⊥ | ≥ q n − k − 1 1 .
Moreover, the equality holds if and only if H
⊥ = {U ⊥ }, that is H = {U }, where U is a fixed k-dimensional subspace of V .
Throughout the remainder of the section, let F ⊆ L(V ) be a given s-union antichain and let d = For short l(F), m(F) are briefly denoted by l and m respectively.
Denote F i = F ∩ V i for m ≤ i ≤ l.F = ( V d \ {F ∈ V d : U ≤ F }) {U }, where U is a fixed (d − 1)-dimensional subspace of V . Proof. (1) Suppose m = l < d. Then |F| ≤ n l ≤ n d−1 . If d = 1, |F| = 1 = n 1 − q n−1 1 . If d ≥ 2, we have n d − q n−d 1 − n d−1 = q n−d+1 −q d q d −1 n d−1 − q n−d 1 > n d−1 − q n−d 1 > q n −1 q d−1 −1 − q n−d+1 > 0.
Hence,
|F| ≤ n d − 1 < n d − q n − d 1 .|G 1 | = |F| − |H| + | ▽ (H)| ≥ |F| + q n − m − 1 1 .(20)
If m(G 1 ) = m + 1 < l, then let H 1 = G 1 ∩ V m(G 1 ) and G 2 = (G 1 \ H 1 ) ∪ ▽(H 1 ). After this we obtain an s-union antichain G 2 for which by Lemma 4.1
|G 2 | = |G 1 | − |H 1 | + | ▽ (H 1 )| ≥ |G 1 | + q n−m(G 1 )−1 1 ≥ |F| + q n−m−1 1 + q n−m−2 1 .
Repeat doing like this until we raise the minimum dimension of the spaces in F to l, and we obtain an s-union antichain G l−m satisfying
|G l−m | ≥ |F| + q n − m − 1 1 + q n − m − 2 1 + · · · + q n − l 1 . Since G l−m ⊆ V l , l ≤ d, then |G l−m | ≤ n d . Since q n−l 1 ≤ q n−i 1 for m + 1 ≤ i ≤ l and m < l ≤ d, we have |F| ≤ n d − q n−m−1 1 − q n−m−2 1 − · · · − q n−l 1 ≤ n d − q n−m−1 1 ≤ n d − q n−l 1 ≤ n d − q n−d 1 .(21)F = (G 1 \ ▽(H)) H = ( V d \ {F ∈ V d : U ≤ F }) {U }.
Combining (1) Note that the inequality is strict if either s = 2d or s = 2d + 1 and l ≥ d + 2. If l(F 1 ) = l − 1 ≥ max{d + 1, m + 1}, then let
(ii) If m = d < l, then |F| ≤ n d − q d(d+1) n−d−1 d + 1, if s = 2d < n, n d , if s = 2d + 1 ≤ n. (iii) If d < m ≤ l, then |F| ≤ s d+1 , if n ≤ 2m, n+s−2d−2 s−d−1 , if n > 2m.D 1 = F 1 ∩ V l(F 1 ) , F 2 = (F 1 \ D 1 ) ∪ △(D 1 )
. After this we obtain an s-union antichain F 2 for which by Theorem 2.3
|F 2 | = |F 1 | − |D 1 | + | △ (D 1 )| ≥ |F 1 | ≥ |F|.
(i) Suppose m < d < l. Repeat the above process until we decrease the maximum dimension of the spaces in F to d.
If s = 2d, we obtain a 2d-union antichain F l−d satisfying
|F l−d | > |F l−d−1 | > · · · > |F 1 | > |F|.
Since m = m(F l−d ) < l(F l−d ) = d < l, then replacing F with F l−d in (3) of the proof of Lemma 4.2 and applying (21), we have
|F| < |F l−d | ≤ n d − q n − d 1 .
(1) Suppose m < d < l. By Lemma 4.3 (i), if n = 2d, then |F| < n
d − q d 1 ; if n = 2d + 1, |F| ≤ n d − q n−d 1 < n d − q d 1 . (2) Suppose m = d < l.
Similarly as the proof of Lemma 4.3 (ii), we obtain an
antichain F l−d−1 satisfying |F| ≤ |F l−d−1 |, m(F l−d−1 ) = d and l(F l−d−1 ) = d + 1. Let G = F l−d−1 ∩ V d+1 . Case a: If n = 2d, let M = (F l−d−1 \ G) ∪ △(G).
Then we obtain an antichain M satisfying |M| ≤ n d .
By Lemma 4.1 (i), we have
| △ (G)| − |G| ≥ q d 1 .
Hence,
|F| ≤ |F l−d−1 | = |M| + |G| − | △ (G)| ≤ n d − q d 1 .
Moreover, the equality holds if and only if
l = d + 1, M = V d , G = {W }, where W is a fixed (d + 1)-dimensional subspace of V , that is F = (M \ △({W }) {W } = B[n, n]. Case b: If n = 2d + 1, let H = F l−d−1 ∩ V d . Since F l−dF = V d+1 . Further, if F V d+1 , then |F| ≤ 2d+1 d+2 < 2d+1 d − q d 1 .
If d < m < l, similarly as the proof of Lemma 4.3 (iii) but we decrease the maximum dimension of the spaces in F to m + 1 if l > m + 1, then we obtain an antichain F l−m−1 satisfying |F l−m−1 | > · · · > |F 1 | > |F|.
If l = m + 1, then let F l−m−1 = F. Let G = F l−m−1 ∩ V m+1 and N = (F l−m−1 \ G) ∪ △(G). By Lemma 4.1 (i), we have
| △ (G)| − |G| ≥ q m 1 .
Hence,
|F| ≤ |F l−m−1 | = |N | + |G| − | △ (G)| ≤ 2d + 1 m − q m 1 < 2d + 1 d − q d 1 .
To sum up, an optimal antichain satisfies |F| ≤ n ⌊ n 2 ⌋ and equality occurs in (3) Case b or Lemma 4.2 (i); a suboptimal antichain has |F| ≤ n ⌊ n 2 ⌋ − q ⌊ n 2 ⌋ 1 and equality occurs in (2) or Lemma 4.2 (ii) if n = 2d. This completes the proof. Proof of Theorem 1.6. We divide the proof into two parts, according to the singularity of s.
(1) Suppose s = 2d. The case of d = 1 is trivial and has been explained in Section 1. Hence, we only need to consider d ≥ 2 in the following. By
q d(d+1) n−d−1 d − 1 − q n−d 1 = q d(d+1) n−d−1 d − n−d+1 1 ≥ q d(d+1) n−d−1 d − q n−d+1 ≥ q d(d+1) (q n−d−1 −1) q d −1 − q n−d+1 ≥ q 2d+2 (q n−d−1 −1) q d −1 − q n−d+1 > q d+2 (q n−d−1 − 1) − q n−d+1 > 0. Hence, n d − q d(d+1) n − d − 1 d + 1 < n d − q n − d 1 .
Case b: Noting n > s = 2d, n d−1 > q n−d+1 > q n−d 1 , we have
n d − q n−d 1 − 2d d+1 = q n−d+1 −1 q d −1 n d−1 − 2d d−1 − q n−d 1 > q n−d+1 −q d q d −1 n d−1 − q n−d 1 > 0.
Hence, 2d d + 1 < n d − q n − d 1 .
Case c: Since a k = q a−k a−1 k−1 + a−1 k for a ≥ k + 1 and note d ≥ 2, we have Concluding remarks
In the present paper, we determine all suboptimal s-union families for vector spaces. For s = n or s = 2d < n, we determine all optimal and suboptimal s-union antichains completely. For s = 2d + 1 < n, we prove that an optimal s-union antichain is either V
d or F = F d F d+1 , where F d+1 ⊆ V d+1 , F d = V d \ △(F d+1 )
and | △ (F d+1 )| = |F d+1 |. It is very interesting to display all optimal (2d + 1)-union antichains of the latter type. Obviously, F = ( V d \ S d ) S d+1 satisfies the above condition, where S is a fixed (2d + 1)-dimensional subspace of V . Consulting the situation of s-union antichains in an n-element set, we conjecture that this is the unique desired structure (of the latter type).
Let F ⊆ L(V ) be a (2d + 1)-union antichain with F not contained in any optimal (2d + 1)-union antichain. By Lemma 4.2, if l ≤ d, we have
|F| ≤ n d − q n − d 1 < n d − q d 1 .
It is obvious that the upper bounds provided in Lemma 4.3 (i) and (iii) are strictly smaller than n d − q d 1 . From the proof of Lemma 4.3 (ii), we know that a suboptimal (2d + 1)union antichain has the form F = F d F d+1 , where F d+1 ⊆ V d+1 , F d = V d \ △(F d+1 ) and | △ (F d+1 )| > |F d+1 |. In view of Theorem 1.5, we make the following conjecture.
Conjecture 5.1. Let F ⊆ L(V ) be a (2d + 1)-union antichain with F not contained in any optimal (2d + 1)-union antichain and 2d + 1 < n. Then
|F| ≤ n d − q d 1 .
Moreover, equality holds if and only if F = B[n, 2d + 1].
s = 2d + 1. The optimal s-union families are proved to be isomorphic to the Katona familes K(n, s) defined as follows. For s = 2d, let K(n, 2d) = {F ⊆ X : |F | ≤ d}.For s = 2d + 1, letK(n, 2d + 1) = {F ⊆ X : |F | ≤ d} {F ⊆ X : |F | = d + 1, y ∈ F },
Moreover for s ≤ n − 2, equality holds if and only ifF = ({F ⊆ X : |F | ≤ d} \ {F ∈ X d : F ∩ D = ∅}) {D},where D is a fixed (d + 1)-subset of X.(ii) When s = 2d + Moreover for s ≤ n − 2, equality holds if and only if F = {F ⊆ X : |F | ≤ d} {F ∈ X d+1 : y ∈ F, F ∩ D = ∅} {D},
Theorem 1 . 5 .
15Let F ⊆ L(V ) be an antichain and n > 1. Then the following hold. . Moreover, equality holds if and only
2
⌋ . Moreover, equality holds if and only if either (a) or (b) holds.
V 2 |
2≤ 1. Thus F = B[n, 2] and |F| = n 1 − 2 1 + 1 = n 1 − q.
Theorem 2. 2 .
2([1, Theorem 1.4]) Suppose k ≥ 2, and either q ≥ 3 and n ≥ 2k+1, or q = 2 and n ≥ 2k + 2. Let H ⊆ V k be an intersecting family with dim( H∈H H) = 0. Then
Theorem 2.7. ([11, Theorem 1.5]) Let a, b, t be positive integers with a < b < n. If A ⊆ V a and B ⊆ V b are cross-Sperner, then
Theorem 2.8. ([11, Theorem 1.4]) Let n ≥ 4, a, b, t be positive integers with a, b ≥ 2, t < min{a, b}, a+b < n+t, and n a ≤ n b . If A ⊆ V a and B ⊆ V b are cross-t-intersecting, then
For
any C ⊆ V k+1 , define Γ(C) = {T ∈ V k : dim(T ∩ C) = 0 for some C ∈ C}. Since dim(A ∩ B) ≥ 1 for all A ∈ A and B ∈ B, we have A ∩ Γ(B) = ∅. Hence, |A| + |Γ(B)| ≤ n k . (10) Case a: Suppose B = {B} ⊆ V k+1 . Then Γ(B) = {T ∈ V k : dim(T ∩ B) = 0}. Hence we have |Γ(B)| = q k(k+1) n−k−1 k
{B 1 , B 2 } ⊆ B, then Γ({B 1 , B 2 }) ⊆ Γ(B) by the definition of Γ(B). Let Γ 1 = {T ∈ V k : dim(T ∩ B 1 ) = 0}, Γ 2 = {T ∈ V k : dim(T ∩ B 2 ) = 0, dim(T ∩ B 1 ) > 0}. It is clear that Γ 1 ∪ Γ 2 ⊆ Γ({B 1 , B 2 }) and Γ 1 ∩ Γ 2 = ∅.Then |Γ(B)| ≥ |Γ 1 | + |Γ 2 |. By Lemma 2.6 (i), we have
Lemma 4. 1 . 1 .
11Suppose that H ⊆ V k , n ≥ 3. Then the following hold.(i) If k ≥ ⌈ n 2 ⌉ + 1, then | △ (H)| − |H| ≥ q k−1 1 . Moreover, equality holds if and only if H = {U }, where U is a fixed k-dimensional subspace of V . (ii) If k ≤ ⌊ n 2 ⌋ − 1, then | ▽ (H)| − |H| ≥ q n−k−1 Moreover, equality holds if and only if H = {U }, where U is a fixed k-dimensional subspace of V . Proof. (i) Let |H| = x k , where k ≤ x ≤ n ≤ 2k − 2. Then by Theorem 2.4, | △ (H)| ≥ x k−1 . So we have
⌊ s 2
2⌋. For any family G ⊆ L(V ), define l(G) = max{dim(G) : G ∈ G}, m(G) = min{dim(G) : G ∈ G}.
Lemma 4. 2 .
2If l ≤ d, then the following hold.
( i ).
i|F| ≤ n d . Moreover, equality holds if and only if F = V d . Moreover, equality holds if and only if
( 2 )
2Suppose m = l = d. Then |F| ≤ n d , and equality holds if and only if F = V d . (3) Suppose m < l ≤ d. Let H = F m , G 1 = (F \ H) ∪ ▽(H). It is clear that G 1 is also an s-union antichain in this case and (F \ H) ∩ ▽(H) = ∅. Then by Lemma 4.1, we have
Moreover, equality holds if and only if m + 1 = l = d, G 1 = V d and the equality in (20) holds. That is H = {U }, where U is a fixed (d − 1)-dimensional subspace of V by Lemma 4.1. Thus
-(3) yields part (i) of the lemma. Note that if F V d then m = l < d or m < l ≤ d and the arguments in(1)and(3)apply. This proves part (ii).
Lemma 4 . 3 .
43If l ≥ d + 1, then the following hold.
( i )
iIf m < d < l, then |F| ≤ n d − q n−d 1and equality holds only if s is odd.
Proof.
Let D = F l , F 1 = (F \ D) ∪ △(D). It is obvious that F 1 is also an s-union antichain and (F \ D) ∩ △(D) = ∅. Since F is an s-union antichain and l ≥ d + 1, then for any D, D ′ ∈ D, we have dim(D ∩ D ′ ) = 2l − dim(D + D ′ ) ≥ 2l − s ≥ 1. By Theorem 2.3, we have | △ (D)| ≥ |D|. Then |F 1 | = |F| − |D| + | △ (D)| ≥ |F|.
Lemmas 4.2 and 4.3, it suffices to show that the upper bounds provided in (ii) and (iii) of Lemma 4.3 are strictly smaller than n d − q n−d 1 . Case a: It is readily checked that
( 2 )
2Suppose s = 2d + 1. It is clear that the upper bounds provided in Lemma 4.3 (i) and (iii) are strictly smaller than n d ; and the equality in Lemma 4.3 (ii) holds if and only if (22) holds, that is F = H G, where G ⊆ V d+1 , H = V d \ △(G) and | △ (G)| = |G|. Then by Lemma 4.2, we complete the proof.
Moreover, for s ≤ n − 2, equality holds if and only if F = K[n, 2d].Theorem 1.2. ([8, Theorem 4]) Suppose that F ⊆ L(V ) is s-union, 2 ≤ s < n. Then
the following hold.
(i) When s = 2d,
|F| ≤
d
i=0
n
i
.
When i = d, we have a stronger result. For convenience, we set A = F d , B = F d+1 in the following. Since F K[n, 2d], there exists G ∈ F with dim(G) ≥ d + 1. HenceG
d+1 ⊆ B = ∅.
Since B ⊆ F is 2d-union, then for all B, B ′ ∈ B, we have
dim(B ∩ B ′ ) = dimB + dimB ′ − dim(B + B ′ ) ≥ 2d + 2 − 2d = 2,
i.e., B is a 2-intersecting family. Similarly, we have that A and B are cross-intersecting.
Then we apply Lemma 2.9 for k = d to obtain
|A| + |B| ≤
n
d
− q d(d+1) n − d − 1
d
+ 1.
(19)
Since for i ≥ 2d + 1, F i = ∅, we can add up (18) along with (19) to obtain (1) by
(15). Moreover, if the equality in (1) holds, then the equalities in (18) and (19) hold as
well. Then by Lemmas 2.5 and 2.9, we have
−1 is an antichain, we have that H, G are cross-Sperner. Then by Theorem 2.7, |F| ≤ |F l−d−1 | = |H| + |G| ≤ Moreover, equality holds if and only if l = d + 1 and either F = A[n, n] or F = B[n, n]. (3) Suppose d < m ≤ l. Case a: Let n = 2d. Now we have n < 2m. Then by Lemma 4.3 (iii), |F| ≤ 2d d+1 < Case b: Let n = 2d + 1. If d < m = l, then |F| ≤ 2d+1 m ≤ 2d+1 d+1 . Moreover, equality holds if and only ifn
d
− q
d
1
.
2d
d − q d
1 .
If s = 2d + 1, we obtain a (2d + 1)-union antichain F l−d satisfyingHere note that if l = d + 1, then we have |F l−d | = |F 1 | ≥ |F|. Similarly by (21), we have(ii) Suppose m = d < l. Similarly as above but we decrease the maximum dimension of the spaces in F to d + 1 if l > d + 1. Then we obtain an s-union antichain F l−d−1 satisfying, then by the 2d-union property of F l−d−1 , we have that H and G are cross-intersecting families and G is a 2-intersecting family. Whenever n ≥ 2d + 1, by Lemma 2.9, we haveIf s = 2d + 1, then by the (2d + 1)-union property of F l−d−1 , we have that G is an intersecting family. By Theorem 2.3, we have(iii) Suppose m > d. As before but we decrease the maximum dimension of the spaces in F to m if l > m. Now we obtain an s-union antichain F l−m ⊆ V m satisfyingIf l = m, then let F l−m = F. By the s-union property of F l−m , for any F,If n ≤ 2m, then by Theorem 2.1 and noting m > d, we haveIf n > 2m, then by Theorem 2.1 and noting s ≥ l ≥ m > d, we haveProof of Theorem 1.5. Obviously, we have the two assertions in Lemma 4.2 for the case l ≤ d. Next, we give new upper bounds of |F| for the case l > d by similar approach of Lemma 4.3. Now d = ⌊ n 2 ⌋.
A Hilton-Milner theorem for vector spaces. A Blokhuis, A E Brouwer, A Chowdhury, P Frankl, T Mussche, B Patkós, T Szőnyi, Electron. J. Combin. 1771A. Blokhuis, A.E. Brouwer, A. Chowdhury, P. Frankl, T. Mussche, B. Patkós, T. Szőnyi, A Hilton-Milner theorem for vector spaces, Electron. J. Combin. 17 (2010) R71.
Rota, q-Analogs of the inclusion-exclusion principle and permutations with restricted position. W Y C Chen, G C , Discrete Math. 104W. Y. C. Chen, G. C. Rota, q-Analogs of the inclusion-exclusion principle and permutations with restricted position, Discrete Math. 104.1 (1992) 7-22.
Shadows and intersections in vector spaces. A Chowdhury, B Patkós, J. Combin. Theory Ser. A. 117A. Chowdhury, B. Patkós, Shadows and intersections in vector spaces, J. Combin. Theory Ser. A 117 (2010) 1095-1106.
Intersecting theorems for systems of finite sets. P Erdős, C Ko, R Rado, Quart. J. Math. Oxf. 212P. Erdős, C. Ko, R. Rado, Intersecting theorems for systems of finite sets, Quart. J. Math. Oxf. 2 (12) (1961) 313-320.
A stability result for the Katona Theorem. P , J. Combin. Theory Ser. B. 122P. Frankl, A stability result for the Katona Theorem, J. Combin. Theory Ser. B 122 (2017) 869-876.
Analogues of Milner's Theorem for families without long chains and of vector spaces. P , European J. Combin. 93103279P. Frankl, Analogues of Milner's Theorem for families without long chains and of vector spaces, European J. Combin. 93 (2021) 103279.
The Erdős-Ko-Rado theorem for vector spaces. P Frankl, R M Wilson, J. Combin. Theory Ser. A. 43P. Frankl, R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A 43 (1986) 228-236.
The Katona theorem for vector spaces. P Frankl, N Tokushige, J. Combin. Theory Ser. A. 120P. Frankl, N. Tokushige, The Katona theorem for vector spaces, J. Combin. Theory Ser. A 120 (2013) 1578-1589.
Intersection theorems for systems of finite sets. G O H Katona, Acta Math. Acad. Sci. Hung. 15G. O. H. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hung. 15 (1964) 329-337.
Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs. H Tanaka, J. Combin. Theory Ser. A. 113H. Tanaka, Classification of subsets with minimal width and dual width in Grass- mann, bilinear forms and dual polar graphs, J. Combin. Theory Ser. A 113 (2006) 903-910.
Nontrivial independent sets of bipartite graphs and crossintersecting families. J Wang, H Zhang, J. Combin. Theory Ser. A. 120J. Wang, H. Zhang, Nontrivial independent sets of bipartite graphs and cross- intersecting families, J. Combin. Theory Ser. A 120 (2013) 129-141.
| [] |
[
"Imputation Strategies Under Clinical Presence: Impact on Algorithmic Fairness Europe PMC Funders Group",
"Imputation Strategies Under Clinical Presence: Impact on Algorithmic Fairness Europe PMC Funders Group"
] | [
"Vincent Jeanselme :[email protected] ",
"Maria De-Arteaga arteaga:[email protected] ",
"Zhe Zhang zhezhang:[email protected] ",
"Jessica Barrett ",
"Brian Tom ",
"Vincent Jeanselme ",
"\nMcCombs School of Business\nMRC Biostatistics Unit\nUniversity of Cambridge\nCambridgeUK\n",
"\nRady School of Management\nUniversity of Texas at Austin\nAustinUSA\n",
"\nMRC Biostatistics Unit\nUniversity of California\nSan DiegoUSA\n",
"\nMRC Biostatistics Unit\nUniversity of Cambridge\nCambridgeUK\n",
"\nUniversity of Cambridge\nCambridgeUK\n"
] | [
"McCombs School of Business\nMRC Biostatistics Unit\nUniversity of Cambridge\nCambridgeUK",
"Rady School of Management\nUniversity of Texas at Austin\nAustinUSA",
"MRC Biostatistics Unit\nUniversity of California\nSan DiegoUSA",
"MRC Biostatistics Unit\nUniversity of Cambridge\nCambridgeUK",
"University of Cambridge\nCambridgeUK"
] | [
"Proc Mach Learn Res"
] | Biases have marked medical history, leading to unequal care affecting marginalised groups. The patterns of missingness in observational data often reflect these group discrepancies, but the algorithmic fairness implications of group-specific missingness are not well understood. Despite its potential impact, imputation is too often an overlooked preprocessing step. When explicitly considered, attention is placed on overall performance, ignoring how this preprocessing can reinforce groupspecific inequities. Our work questions this choice by studying how imputation affects downstream algorithmic fairness. First, we provide a structured view of the relationship between clinical presence mechanisms and groupspecific missingness patterns. Then, through simulations and real-world experiments, we demonstrate that the imputation choice influences marginalised group performance and that no imputation strategy consistently reduces disparities. Importantly, our results show that current practices may endanger health equity as similarly performing imputation strategies at the population level can affect marginalised groups differently. Finally, we propose recommendations for mitigating inequities that may stem from a neglected step of the machine learning pipeline. | 10.48550/arxiv.2208.06648 | [
"https://export.arxiv.org/pdf/2208.06648v2.pdf"
] | 251,564,797 | 2208.06648 | 9958d05dc3e9f5cabe92f11a61098ebcdd0e9ea8 |
Imputation Strategies Under Clinical Presence: Impact on Algorithmic Fairness Europe PMC Funders Group
2022
Vincent Jeanselme :[email protected]
Maria De-Arteaga arteaga:[email protected]
Zhe Zhang zhezhang:[email protected]
Jessica Barrett
Brian Tom
Vincent Jeanselme
McCombs School of Business
MRC Biostatistics Unit
University of Cambridge
CambridgeUK
Rady School of Management
University of Texas at Austin
AustinUSA
MRC Biostatistics Unit
University of California
San DiegoUSA
MRC Biostatistics Unit
University of Cambridge
CambridgeUK
University of Cambridge
CambridgeUK
Imputation Strategies Under Clinical Presence: Impact on Algorithmic Fairness Europe PMC Funders Group
Proc Mach Learn Res
1932022# These authors contributed equally to this work. This work is licensed under a CC BY 4.0 International license. Author Manuscript Proc Mach Learn Res. Author manuscript; available in PMC Published in final edited form as: Europe PMC Funders Author Manuscripts Europe PMC Funders Author ManuscriptsClinical PresenceFairnessImputation
Biases have marked medical history, leading to unequal care affecting marginalised groups. The patterns of missingness in observational data often reflect these group discrepancies, but the algorithmic fairness implications of group-specific missingness are not well understood. Despite its potential impact, imputation is too often an overlooked preprocessing step. When explicitly considered, attention is placed on overall performance, ignoring how this preprocessing can reinforce groupspecific inequities. Our work questions this choice by studying how imputation affects downstream algorithmic fairness. First, we provide a structured view of the relationship between clinical presence mechanisms and groupspecific missingness patterns. Then, through simulations and real-world experiments, we demonstrate that the imputation choice influences marginalised group performance and that no imputation strategy consistently reduces disparities. Importantly, our results show that current practices may endanger health equity as similarly performing imputation strategies at the population level can affect marginalised groups differently. Finally, we propose recommendations for mitigating inequities that may stem from a neglected step of the machine learning pipeline.
Introduction
Machine learning models for healthcare often rely on observational data. At the core of observational data generation is a complex interaction between patients and the healthcare system, which we refer to as clinical presence (Jeanselme et al., 2022). Each observation, from orders of laboratory tests to treatment decisions, reflects access to medical care, patients' medical states, and also practitioners' expertise and potential biases. Historically, healthcare access, treatment and outcomes have been marked by inequalities (Chen et al., 2021;Freeman and Payne, 2000;Jeanselme et al., 2021;Kim et al., 2016;Norris and Nissenson, 2008). For instance, Price-Haywood et al. (2020) hypothesised that the disproportionate mortality rate from Covid-19 among Black patients can, in part, be explained by longer waiting times before accessing care.
Clinical presence patterns can, therefore, reflect disparities. Specifically, observation and missingness can vary across groups. Developing machine learning models on these data raises ethical concerns about automating and reinforcing injustices.
Current practices for handling missing data often rely on imputing data with overall performance in mind (Emmanuel et al., 2021), without consideration of the algorithmic fairness consequences associated with this choice. Despite the risk of aggravating inequities reflected in group-specific missingness patterns, the effect of this imputation step remains understudied. In this work, we explore the impact of imputation on data imprinted by groupspecific missingness patterns emerging from medical practice and historical biases. First, we identify scenarios of clinical presence that could result in group-specific missingness patterns, grounded on historical evidence of these phenomena in medicine. Then, we explore the downstream impact on group performance of standard imputation strategies on simulated data affected by this clinical missingness. Finally, we study group performances of different imputation strategies in real-world data.
This work provides empirical evidence that machine learning pipelines differing solely in their handling of missingness may result in distinct performance gaps between groups, even when population performances present no difference. The choice of imputation strategy may therefore impact performance in a way that reinforces inequities against historically marginalised groups. Moreover, our experiments show that no imputation strategy consistently outperforms the others and current recommendations may harm marginalised groups. Finally, we emphasise the relevance of this analysis by providing real-world evidence of clinical missingness patterns and echo the previous results in the MIMIC III dataset.
Related work
This work explores the link between missingness and algorithmic fairness in machine learning for healthcare. In this section, we review related literature across domains.
Clinical missingness
Clinical missingness is a medical expression of the well-studied missingness patterns (Little and Rubin, 2019): Missing Completely At Random (MCAR) -random subsets of patients and/or covariates are missing, Missing At Random (MAR) -missing data patterns are a function of observed variables, and Missing Not At Random (MNAR) -missing patterns depend on unobserved variables or the missing values themselves.
Traditional statistical models are not adapted to handle missing covariates. Consequently, practitioners may rely on single imputation strategies such as mean, median, nearest neighbours (Batista et al., 2002;Bertsimas et al., 2021) or the preferred multiple imputation methods (Newgard and Lewis, 2015;Rubin, 2004;White et al., 2011). Typically, these imputation approaches assume MCAR and/or MAR patterns. They may be ill-adapted to handle informative missingness, particularly as MNAR and MAR are non-identifiable from observational data alone and require domain expertise for adequate modelling. The recommended strategy to tackle this non-identifiability issue is to control the imputation model on additional covariates to render the MAR assumption more plausible (Haukoos and Newgard, 2007). Our work shows the potential shortcomings of this covariate-adjusted imputation strategy under group-specific missingness patterns.
Algorithmic fairness in medicine
The risk of reinforcing historical biases is of critical concern in medicine, where inequalities can have life-threatening implications. Measuring and mitigating this risk is the aim of algorithmic fairness (Chouldechova and Roth, 2020). In this paper, we follow the 'equal performance' group definition of algorithmic fairness (Rajkomar et al., 2018), which evaluates if the model performs comparably across groups (Chouldechova et al., 2018;Flores et al., 2016;Noriega-Campero et al., 2019).
Definition 1 (Equal Performance)-A pipeline p is fairer than another q with regard to group g if its performance gap is the smallest, i.e. |Δ g (p)| < |Δ g (q)| with Δ g (p) ≔ d(p({X i }G i =g)) -d(p({X i }G i ≠g)) for some performance metric d, a pipeline p and (X i , G i ), the covariates and associated group for patient i. This metric has been leveraged to quantify models' impact on algorithmic fairness in medicine (Chen et al., 2018(Chen et al., , 2019Pfohl et al., 2019;Seyyed-Kalantari et al., 2020;Zhang et al., 2020). For instance, Seyyed-Kalantari et al. (2020) demonstrates X-ray classifiers' performance gap between marginalised groups. However, the link between imputation and algorithmic fairness has received limited attention despite the risk of clinical missingness disparities. Our work aims to fill this gap.
Algorithmic fairness and missingness
As a community, we need to understand how to best handle clinical missingness when imprinted by biases. Martínez-Plumed et al. (2019); Fricke et al. (2020) show that mean imputation presents better fairness properties compared to complete case analysis. These works focus on one imputation strategy and ignore the potential variability of the impact of different strategies. Closer to our work, Zhang and Long (2021) show that the choice of imputation may lead to different fairness gaps when enforcing synthetic missingness patterns. However, these works do not discuss how the different missingness patterns may arise in medicine, and how a specific group may be impacted differently by different imputation strategies. In our work, we study different missingness patterns that may arise as a result of the data-generating process in healthcare. Finally, Ahmad et al. (2019); Ghassemi et al. (2020); Rajkomar et al. (2018) describe multiple challenges linked to medical data, among which they state that historical biases may lead to missingness patterns that could impact fairness, but they do not empirically study this. While informative missingness has recently received revived attention (Jeanselme et al., 2022;Getzen et al., 2022), no work has studied its potential association with fairness. Our work aims to address these gaps in the literature by demonstrating the existence of this problem, characterising different types of group-specific missingness patterns in medicine, and exploring the impact of different imputation strategies under different clinical presence scenarios. In addition to showing the impact of imputation choice on fairness gaps, we highlight that the same imputation strategy may benefit a group under one missingness pattern but hurt this same group in another. Importantly, we also show that a given group may benefit under one imputation and suffer under another imputation in the same setting, even if the two strategies perform identically at the population level. These are novel findings that invite practitioners to perform careful sensitivity analysis of imputation choice on fairness gaps.
Clinical missingness scenarios
This section shows how group-specific missingness can result from clinical presence. Figure 1 introduces the following scenarios:
Limited access to quality care (S1) When certain groups do not have access to the same health services, this results in more missing covariates for these groups.
Socioeconomic factors resulting from structural injustices (Barik and Thorat, 2015;Nelson, 2002;Szczepura, 2005;Yearby, 2018) such as insurance, work schedule flexibility, distance to hospitals (Barik and Thorat, 2015) or mobility, result in inconsistent medical history (Gianfrancesco et al., 2018), additional waiting time before looking for care (Weissman et al., 1991), avoidance of preventing care (Smith et al., 2018), and limited access to advanced diagnostic tools (Lin et al., 2019). This diminished access to care is potentially reflected as missing data. For instance, patients may have no annual checkup data if their insurance does not cover or encourage this service.
(Mis)-informed collection (S2)
Often, medical research has focused on a subset of the population. The resulting guidelines may be ill-adapted to other groups and relevant covariates may be missing due to standard recommendations.
Historically researchers focused on (perceived) highest-risk groups: breast cancer predominantly studied in women (Arnould et al., 2006;Giordano, 2018), cardiovascular disease in men (Vogel et al., 2021), skin cancers in whiter skins (Gloster Jr and Neal, 2006), and autism in men (Gould and Ashton-Smith, 2011). Resultant medical practices and guidelines target these groups. However, substantial evidence shows the prevalence of these diseases among other groups. Stemming from biological differences, different groups may present different symptoms and expressions for the same condition. The difference in disease expression and the absence of adapted tests result in missing covariates necessary to identify the disease. For instance, screening recommendations may only be prescribed conditioned on observation of "standard" symptoms. If the symptoms considered are not the expected disease expression for a marginalised subgroup, this will result in more missing screening procedures for this group.
Confirmation bias (S3)
Practitioners collect data based on expertise and informative proxies that are not recorded, e.g. patient feeling unwell.
For instance, practitioners may record the value of a test only if they suspect it will be abnormal. The literature presents evidence of this phenomenon where the presence of a specific medical test is more informative of the outcome than the test result itself (Agniel et al., 2018;Sisk et al., 2020). Wells et al. (2013) also suggest that missing laboratory tests correspond to healthy results, e.g. doctors do not collect or record data if they are irrelevant. Similarly, sicker patients present more complete data (Rusanov et al., 2014;Sharafoddini et al., 2019;Weiskopf et al., 2013).
Formalisation
Consider two covariates (X 1 , X 2 ) influenced by the underlying condition Y and the group membership G. Note that the disease prevalence may also depend on G. One covariate X 1 is observed for all patients, while X 2 is potentially missing. Following the notations from Mohan and Pearl (2021), let O 2 be the indicator of observation of X 2 such that the observed value is defined as:
X 2 * = ∅ if O 2 = 0 X 2 otherwise
In (S1), G informs O 2 because of group socioeconomic differences. In (S2) and (S3), G impacts the observation process through group-specific disease expression. While the influence of medical covariates on the missingness patterns characterises both (S2) and (S3), (S2) describes how guidelines may depend on observed covariates, whereas (S3) reflects how the observation process may depend on X 2 itself or unobservable covariates correlated with X 2 . For instance, (S2) may consist of a guideline recommending to measure X 2 if X 1 is within a given range. However, if a patient is a member of a group for which X 1 is not informative-or for which the informative range is different-X 2 might not be observed as X 1 is not in the guideline test-triggering range. This may lead to more missing data for X 2 in the group with different characteristics for X 1 . (S3) differs as practitioners would record the value of X 2 only if this one is abnormal.
These dependencies result in three distinct patterns between missingness, group and covariates, summarised with directed acyclic graphs (DAGs) in Figure 2.
Experiments
In this section, we explore how the choice of imputation affects group-specific performance, and potentially reinforces disparities in data marked by clinical missingness. We first present simulation studies in which we enforce specific missingness patterns. This analysis allows us to control clinical missingness patterns and measure the potential impact of imputation on algorithmic fairness. We accompany these results with real-world evidence of group-specific missingness patterns and show the impact of different imputation strategies on marginalised group performance. For reproducibility, all experiments' code is available on Github 1 .
Datasets
Assume a population of N patients with associated covariates X, marginalised group membership G, and outcome of interest Y.
Simulation-We introduce a bidimen-sional (X ∈ ℝ 2 ) synthetic population (N = 10,100) divided into two groups (G ∈ {0,1}), and assume the marginalised group is a minority in the population with ratio 1:100. These groups differ in disease expression, i.e. positive cases across groups differ in how they express the disease. Then clinical missingness patterns are enforced on the second dimension X 2 following the scenarios introduced in Section 3. Figure 3 provides a graphical summary of how clinical missingness is enforced on the synthetic data. The associated predictive task is to classify between positives and negatives.
(See Appendix A.1 for full data generation protocol reflecting the enforcement of the previously-introduced scenarios).
MIMIC III-
The real-world analysis relies on the laboratory tests from Medical Information Mart for Intensive Care (MIMIC III) dataset (Johnson et al., 2016). Following data harmonisation (Wang et al., 2020), we select adults who survived 24 hours or more after admission to the intensive care unit, resulting in a set of 36,296 patients sharing 67 laboratory tests. The goal is to predict short-term survival (7 days after the observation period -Y) using the most recent value of each laboratory test observed in the first 24 hours of observation (X). We select short-term survival as it is a standard task in the machine learning literature (Jeanselme et al., 2022;Nagpal et al., 2021;Tsiklidis et al., 2022;Xu et al., 2019) and the associated labels are less likely to suffer from group-specific misdiagnosis, and, therefore, disentangles our analysis from potential biases in labelling (Chen et al., 2020). In practice, deploying this model could be used for care prioritisation of patients with predicted elevated risk.
Handling missing data
The simulation and MIMIC III datasets present missing data that are traditionally imputed for analysis. We consider the following common imputation strategies:
Single median imputation (Median)-Missing data are replaced by the population median of each covariate. Due to its straightforward implementation, this methodology remains predominant in the literature despite known shortcomings (Rubin, 1976;Sinharay et al., 2001;Crawford et al., 1995).
Multiple Imputation using Chained Equation (MICE)-Missing data are iteratively
drawn from a regression model built over all other available covariates after median initialisation. This approach is repeated I times with an associated predictive model for each imputed draw. At test time, the same imputation models generate I imputed points for which models' predictions are averaged. MICE is recommended in the literature (Janssen et al., 2010;Newgard and Haukoos, 2007;Wood et al., 2004;Zhou et al., 2001;White et al., 2011) as it quantifies the uncertainty associated with missingness. In the experiments, we used 10 iterations repeated 10 times resulting in I =10 datasets with associated predictive models.
Group MICE-The previous MICE methodology assumes a MAR mechanism. To make this assumption more plausible, Haukoos and Newgard (2007) recommend the addition of potentially informative covariates. In our experiment, we, therefore, rely on both group membership and covariates for imputing the missing data (X ∼ X, G with X representing the imputed covariates).
Group MICE Missing-Encoding
Experimental setting
After imputation, each pipeline relies on a logistic regression model -a pillar in medicine (Nick and Campbell, 2007;Goldstein et al., 2017) -to discriminate between positive and negative cases (Y ∼ X).
Adopting the equal performance across groups definition (Rajkomar et al., 2018) This metric quantifies algorithmic fairness but does not quantify how deployment can hurt subgroups at a fixed threshold on the predicted risk. In the MIMIC III study, we measure the False Negative Rate (FNR) assuming the availability of priority care for 30% of the population (sensitivity to this threshold is presented in Appendix A.2). In the 30% highest-risk population, we measure the prioritisation -the group-specific proportion of patients who would receive care under this policy -and misclassification rates in the groups of interest. In this setting, FNR corresponds to the non-prioritisation of high-risk patients. The gap in FNR between groups answers the question: how marginalised groups would be incorrectly deprioritized? Additional experimental design descriptions and results are provided in Appendix A.
Results
This section presents the insights obtained through both simulations and real-world experiments.
Simulations
We conduct 100 simulations in which the three clinical presence scenarios are independently enforced. We apply the imputation strategies described in Section 4.2 and train a logistic regression with l2 penalty (λ = 1). Results are computed on a 20% test set and averaged over the 100 simulations. Figure 4 presents the AUC gap (Δ defined in Section 2.2) between the majority and the minority, and group-specific AUCs.
Insight 1: Equally-performing imputation strategies at the population level can result in different marginalised group performances-Consider (S1), all imputation methodologies result in similar population AUCs, as shown by the grey dots. However, note how the AUC evaluated on the marginalised group presents a gap of 0.1 between MICE and Group MICE. This phenomenon is explained by how imputation strategies result in different imputed covariate distributions. The logistic regressions built on these imputed data would weigh covariates differently and then have different predicted values.
Insight 2: No strategy consistently outperforms the others across clinical presence scenarios-Population-level performances remain stable between Group
MICE and MICE over all scenarios, but these strategies have contrasting marginalised group AUCs. Importantly, Group MICE should be preferred in (S1) as it minimises the performance gap. For the same reason, MICE should be used in (S2), whereas both methodologies present inconclusive fairness differences in (S3). While this result is specific to this simulation, this exemplifies how no methodology consistently reduces the performance gap across groups.
Insight 3: Current recommendation of leveraging additional covariates to satisfy MAR assumption, or using missingness indicators can harm marginalised group's performance-Note how Group MICE presents worse performance than MICE in (S2). The recommendation of including additional covariates to make the MAR assumption more plausible is not always suitable as it may add noise and lead to poorer performance. In another example, see how the model considering missingness provides an edge in (S3) compared to Group MICE but hurts performance in (S1). This observation reinforces the necessity of measuring the performance sensitivity to imputation. Additionally, it underlines how understanding the missingness process is essential to control for relevant covariates.
MIMIC III
In this real-world experiment, we consider groups defined by the following attributes: ethnicity (Black vs non-Black), sex (female vs male), and insurance (publicly vs privately insured). Table 1 shows the number of orders and the number of distinct laboratory tests (out of the 67 possible tests) performed during the first-day post-admission for each subgroup. This last number reflects the missingness of the vector used for prediction.
For this experiment, patients are split into three sets: 80% for training, 10% for hyperparameter tuning and 10% for testing. We perform a l2 penalty search for the logistic regression among λ ∈ [0.1,1,10,100]. Table 2 presents predictive performances at the population level averaged on the bootstrapped test set over 100 iterations. Assuming capacity for additional care for the 30% highest risk, we explore care prioritisation. Figure 5 displays our main results: the gaps in prioritisation and the false negative rates stratified by groups of interest under the different imputation strategies.
Insight 4: Real-world data presents group-specific clinical presence patterns -While the causes of clinical missingness cannot be distinguished from observational data alone, one can observe evidence of non-random missingness patterns in the MIMIC III dataset, as shown in Table 1. Specifically, note the larger number of orders for patients who die during their stay compared with the ones who survive. This pattern is consistent with a possible confirmation bias scenario (S3), if doctors are monitoring sicker patients more closely. Another example of non-random missingness is that there are fewer test orders for female, Black, and publicly insured patients, but little difference in the diversity of tests prescribed. While this may be explained by the underlying conditions or other medically relevant factors, the combination of similar diversity of tests but less frequent observations results in a less up-to-date patient's health status for modelling. Thus, even though the cause of testing differences is unclear, these observations show the connection between testing patterns, group membership, and outcomes. This real-world evidence of non-random missingness patterns among subgroups of patients raises concerns about increasing inequities if the fairness implications of imputation methods are not considered.
Insight 5: Marginalised groups can benefit or be harmed by equally performing imputation strategies at the population level-Note how MICE and Group MICE perform similarly at the population level in Table 2, but present different performances for marginalised groups (see Figure 5). Consider the ethnicity split: these methodologies have opposite consequences on Black patients. MICE would result in more care for Black patients and a smaller gap in FNR. By contrast, Group MICE would halve prioritisation and double the FNR gap in favour of non-Black patients. Crucially, this difference solely results from the imputation strategy adopted in these two pipelines.
Insight 6: Different marginalised groups may be impacted oppositely by the same imputation strategy-Female and publicly insured patients have higher prioritisation rates under all imputation methods. However, these groups show opposite gaps in their FNR compared to their counterparts (men and privately insured patients):
women have more false negative cases missed while those publicly insured have fewer false negatives.
In another case of opposite impacts of imputation, Group MICE presents the smallest FNR performance gap for sex, but the largest gaps for both ethnicity and insurance. Group MICE also results in better FNR performance for publicly insured but worse for Black patients. This observation underlines the importance of identifying marginalised groups in development and deployment populations. The optimal trade-off between group and population performances, and between marginalised groups, needs to be considered as different pipelines could have opposite impacts.
Discussion
This paper is motivated by how interactions between patients and the healthcare system can result in group-specific missingness patterns. We show that resultant inequities in clinical missingness can impact downstream algorithmic fairness under different imputation strategies. This analysis demonstrates that no imputation strategy consistently provides better performances for marginalised groups. In particular, a model providing an edge in one setting can underperform in another, or even harm a different group. Moreover, the experiments conducted using the MIMIC-III dataset demonstrate the relevance of the identified problem as more than a merely theoretical concern, showing that it is present in a widely used electronic health record dataset.
Note that our work does not claim that the specific patterns we observe will necessarily be present in other datasets. As we have emphasised, different combinations of missingness processes may lead to different fairness gaps and interactions between imputation and group performance. It may even lead to equal fairness performance of all imputation strategies, but one cannot know this a priori.
Learning from medical data without sufficient attention to the potential entanglement of clinical missingness and historical biases could reinforce and automatise inequities, and further harm historically marginalised groups. This work calls for caution in the use of imputation to reach health equity. We invite practitioners to:
• Record protected attributes and identify marginalised groups.
• Explore the practitioner-patient interaction process to identify clinical missingness disparities.
• Report the assumptions made at each stage of the pipeline.
• Perform sensitivity analysis on imputation to understand its impact on algorithmic fairness.
Future work will theoretically define in which settings the presented results stand and how model choice could mitigate discrepancies in the missingness patterns. Moreover, clinical missingness is only one dimension of how clinical presence shapes the data-generating process. The temporality and irregularity of medical time series may convey group-specific disparities that machine learning methods may amplify. Directed Acyclic Graphs (DAGs) associated with the identified clinical missingness scenarios. Full circled covariates are observed, dotted ones unobserved. Y is the condition, G, the group membership, X 1 and X 2 the two covariates. O 2 is the observation process associated to X 2 . Red dependencies underline the differences between scenarios. Graphical summary of clinical missingness enforcement in the simulation experiments. Note that our simulations' choices result in missingness in the marginalised group only in (S1) and (S2), but in the majority only in (S3). AUC performance gaps Δ and group-specfic AUCs across scenarios on 100 synthetic experiments. If Δ < 0, the marginalised group has worse AUC than the majority. Prioritisation performance gaps Δ across marginalised groups in MIMIC III experiment. If Δ > 0, the marginalised group has a larger value of the given metric than the rest of the population.
missingness has been shown to improve performance when the patterns of missingness are informative(Groenwold, 2020; Lipton et al., 2016;Saar-Tsechansky and Provost, 2007; Sperrin et al., 2020). As clinical missingness can contain informative patterns(Jeanselme et al., 2022; Lipton et al., 2016), we concatenate missingness indicators to the imputed data from Group MICE (Appendix A explores the concatenation of missing indicators with the other strategies).
of algorithmic fairness, we measure each pipeline's discriminative performances for the different groups. We use the Area Under the Curve for the Receiver Operating Characteristic curve (AUC -ROC, i.e. d in Section 2.2) as proposed in Röösli et al. (2022); Larrazabal et al. (2020); Zhang et al. (2022).
Figure 2 .
2Figure 2.
Figure 3 .
3Figure 3.
Figure 4 .
4Figure 4.
Figure 5 .
5Figure 5.
Proc Mach Learn Res. Author manuscript; available in PMC 2023 January 03. Europe PMC Funders Author Manuscripts Europe PMC Funders Author Manuscripts Europe PMC Funders Author Manuscripts Europe PMC Funders Author ManuscriptsMean (std) number of orders and observed tests performed during the first postadmission stratified by marginalised group and outcomes.+ By the 8 th day after admission. * Significant t-test p-value (< 0.001).Jeanselme et al.
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Jeanselme et al.
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Table 1
Orders
Distinct tests
Alive
+ 5.68 (4.64)
*
40.80 (6.73)
*
Dead
+ 7.57 (5.44)
37.22 (7.50)
Black 5.24 (4.08)
*
40.94 (6.94)
*
Other 5.86 (4.77)
40.52 (6.84)
Female 5.54 (4.45)
*
40.75 (6.89)
*
Male 6.03 (4.91)
40.41 (6.80)
Public 5.67 (4.57)
*
40.46 (6.76)
*
Private 6.11 (5.01)
40.75 (7.01)
Jeanselme et al.
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Table 2
2Predictive performance under different imputation strategies. Mean (std) computed on the test set bootstrapped 100 times.AUC ROC
Group MICE Missing 0.786 (0.009)
Group MICE 0.738 (0.012)
MICE 0.742 (0.012)
Median 0.748 (0.011)
Proc Mach Learn Res. Author manuscript; available in PMC 2023 January 03.
Proc Mach Learn Res. Author manuscript; available in PMC 2023 January 03.Europe PMC Funders Author ManuscriptsEurope PMC Funders Author Manuscripts
AcknowledgmentsThe authors would like to thank Changjian Shui (McGill Univeristy) for constructive feedback on the manuscript. This work has been partially funded by UKRI Medical Research Council (MC_UU_00002/5 and MC_UU_00002/2) and the NIH through grant R01NS124642.Supplementary MaterialRefer to Web version on PubMed Central for supplementary material.
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| [] |
[
"Perform Like an Engine: A Closed-Loop Neural-Symbolic Learning Framework for Knowledge Graph Inference",
"Perform Like an Engine: A Closed-Loop Neural-Symbolic Learning Framework for Knowledge Graph Inference"
] | [
"Guanglin Niu \nInstitute of Artificial Intelligence\nBeihang University\nBeijingChina\n",
"Bo Li \nInstitute of Artificial Intelligence\nBeihang University\nBeijingChina\n\nHangzhou Innovation Institute\nBeihang University\nHangzhouChina\n",
"Yongfei Zhang [email protected] \nBeijing Key Laboratory of Digital Media\nBeihang University\nBeijingChina\n\nState Key Laboratory of Virtual Reality Technology and Systems\nBeihang University\nBeijingChina\n",
"Shiliang Pu [email protected] \nHikvision Research Institute\nHangzhouChina\n"
] | [
"Institute of Artificial Intelligence\nBeihang University\nBeijingChina",
"Institute of Artificial Intelligence\nBeihang University\nBeijingChina",
"Hangzhou Innovation Institute\nBeihang University\nHangzhouChina",
"Beijing Key Laboratory of Digital Media\nBeihang University\nBeijingChina",
"State Key Laboratory of Virtual Reality Technology and Systems\nBeihang University\nBeijingChina",
"Hikvision Research Institute\nHangzhouChina"
] | [
"Proceedings of the 29th International Conference on Computational Linguistic"
] | Knowledge graph (KG) inference aims to address the natural incompleteness of KGs, including rule learning-based and KG embedding (KGE) models. However, the rule learningbased models suffer from low efficiency and generalization while KGE models lack interpretability. To address these challenges, we propose a novel and effective closed-loop neuralsymbolic learning framework EngineKG via incorporating our developed KGE and rule learning modules. KGE module exploits symbolic rules and paths to enhance the semantic association between entities and relations for improving KG embeddings and interpretability. A novel rule pruning mechanism is proposed in the rule learning module by leveraging paths as initial candidate rules and employing KG embeddings together with concepts for extracting more high-quality rules. Experimental results on four real-world datasets show that our model outperforms the relevant baselines on link prediction tasks, demonstrating the superiority of our KG inference model in a neural-symbolic learning fashion. The source code and datasets of this paper are available at https://github.com/ngl567/EngineKG. | null | [
"https://www.aclanthology.org/2022.coling-1.119.pdf"
] | 251,718,718 | 2112.01040 | 40e5e518e6a8e9468e257e610638b4adaec7af6e |
Perform Like an Engine: A Closed-Loop Neural-Symbolic Learning Framework for Knowledge Graph Inference
1400 October 12-17, 2022
Guanglin Niu
Institute of Artificial Intelligence
Beihang University
BeijingChina
Bo Li
Institute of Artificial Intelligence
Beihang University
BeijingChina
Hangzhou Innovation Institute
Beihang University
HangzhouChina
Yongfei Zhang [email protected]
Beijing Key Laboratory of Digital Media
Beihang University
BeijingChina
State Key Laboratory of Virtual Reality Technology and Systems
Beihang University
BeijingChina
Shiliang Pu [email protected]
Hikvision Research Institute
HangzhouChina
Perform Like an Engine: A Closed-Loop Neural-Symbolic Learning Framework for Knowledge Graph Inference
Proceedings of the 29th International Conference on Computational Linguistic
the 29th International Conference on Computational Linguistic13911400 October 12-17, 20221391
Knowledge graph (KG) inference aims to address the natural incompleteness of KGs, including rule learning-based and KG embedding (KGE) models. However, the rule learningbased models suffer from low efficiency and generalization while KGE models lack interpretability. To address these challenges, we propose a novel and effective closed-loop neuralsymbolic learning framework EngineKG via incorporating our developed KGE and rule learning modules. KGE module exploits symbolic rules and paths to enhance the semantic association between entities and relations for improving KG embeddings and interpretability. A novel rule pruning mechanism is proposed in the rule learning module by leveraging paths as initial candidate rules and employing KG embeddings together with concepts for extracting more high-quality rules. Experimental results on four real-world datasets show that our model outperforms the relevant baselines on link prediction tasks, demonstrating the superiority of our KG inference model in a neural-symbolic learning fashion. The source code and datasets of this paper are available at https://github.com/ngl567/EngineKG.
Introduction
Typical knowledge graphs (KGs) store triple facts and some of them also contain concepts of entities (Bollacker et al., 2008). The KGs have proven to be incredibly effective for a variety of applications such as dialogue system (Zhou et al., 2018) and question answering (Huang et al., 2019). However, the existing KGs are always incomplete which restricts the performance of knowledge-based applications. Thus, KG inference plays a vital role in completing KGs for better applications of KGs.
The existing KG inference approaches are usually classified into two main categories: (1) Rule learning-based models such as AMIE+ (Galárraga Figure 1: The brief architecture of our closed-loop framework for KG inference EngineKG that performs like a four-stroke engine. et al., 2015) and AnyBurl (Meilicke et al., 2019) mine rules from KGs and employ these rules to predict new triples by deduction. However, rule learning-based models suffer from low efficiency of the rule mining process and the poor generalization caused by the limited coverage of inference patterns.
(2) KGE technique learns the embeddings of entities and relations to predict the missing triples via scoring each triple candidate, including TransE (Bordes et al., 2013), HAKE and DualE (Cao et al., 2021). The previous KGE models perform in a data-driven fashion, contributing to good efficiency and generalization but lacking interpretability.
Some recent researches attempt to combine the advantages of rule learning-based and KGE-based models to complement each other in a neuralsymbolic learning fashion. An idea is to introduce logic rules into KGE models, such as RUGE (Guo et al., 2018) and its advanced model IterE (Zhang et al., 2019b). These approaches all convert the rules into formulas by t-norm based fuzzy logic to obtain newly labeled triples. However, these models cannot maintain the interpretability which is a vital feature of symbolic rules. On the other hand, some rule learning-based models succeed in leveraging KG embeddings to extract rules via numerical calculation rather than discrete graph search, including RNNlogic (Qu et al., 2021), RLvLR (Omran et al., 2019), DRUM (Sadeghian et al., 2019) and RuLES (Ho et al., 2018). Although the efficiency of mining rules is improved, the performance especially generalization of purely employing rules to implement KG inference is still limited.
To address the above challenges, we propose a closed-loop neural-symbolic learning framework EngineKG via combining an embedding-based rule learning and a rule-enhanced KGE, in which paths and concepts are utilized. Our model is named EngineKG because it performs like an engine as shown in Figure 1: (1) Intake Stroke. The closed-path rules (or named chain rules) are injected into the KGE module (analogous to intake) to guide the procedure of learning KG embeddings, where the initial seed rules are mined by any rule learning tool, and the rule set would grow via our designed rule learning module from the first iteration. (2) Compression Stroke. The KGE module leverages the rules and paths to learn the low-dimensional embeddings (analogous to compression) of entities and relations, improving interpretability and accuracy. (3) Expansion Stroke. The novel rule learning module outputs newly learned rules (analogous to exhaust) by the effective rule pruning strategy based on paths, relation embeddings and concepts. (4) Exhaust Stroke. Update the rule set (analogous to exhaust) by merging the previous rule set and the newly learned rules for boosting KGE and KG inference in the next iteration.
Our research makes three contributions:
• We propose a novel and effective closed-loop neural-symbolic learning framework that performs embedding-based rule learning and ruleenhanced KGE iteratively, balancing good accuracy, interpretability and efficiency.
• Paths and ontological concepts are well exploited for supplementing the valuable semantics to both KGE and rule learning, facilitating the better performance of KG inference.
• The link prediction results and the effectiveness of rule learning on four datasets illustrate that our model outperforms various state-ofthe-art KG inference approaches.
2 Related Work
Rule Learning-Based Models
According to the symbolic characteristics of KG, some rule learning techniques specific to KGs are applied to KG inference with relatively good accuracy and interpretability, including AMIE+ (Galárraga et al., 2015), Anyburl (Meilicke et al., 2019), DRUM (Sadeghian et al., 2019), RLvLR (Omran et al., 2019) and RNNLogic (Qu et al., 2021). AMIE+ (Galárraga et al., 2015) introduces optimized query writing techniques into traditional inductive logic programming algorithms to generate horn rules efficiently. Anyburl learns closed-path rules from KGs in a reinforcement learning framework. DRUM, RLvLR and RNNLogic employ KG embeddings for enhancing the efficiency and scalability of rule learning. Whereas, all the previous rule learning algorithms lack generalization since the number of rules mined at one time is limited.
KG Embedding Models
The typical KG embedding (KGE) models learn the embeddings of entities and relations to measure the plausibility of each triple. TransE (Bordes et al., 2013) regards the relations as translation operations from head to tail entities. ComplEx (Trouillon et al., 2016) embeds the KG into a complex space while DualE (Cao et al., 2021) embeds relations into the quaternion space to model the symmetric and antisymmetric relations. HAKE embeds entities into the polar coordinate system and is able to model the semantic hierarchies of KGs. RUGE (Guo et al., 2018) and IterE (Zhang et al., 2019b) both convert rules into formulas by t-norm fuzzy logic to infer newly labeled triples. Particularly, IterE iteratively conducts rule learning and KG embedding, but the significant distinctions between our model EngineKG and IterE include: (1) Usage of rules: our model leverages rules to compose paths for learning KG embeddings while IterE uses rules to produce labeled triples. Meanwhile, we maintain the interpretability of symbolic rules, while IterE does not.
(2) Additional information: our model introduces paths and concepts into both rule learning and KG embedding while IterE simply depends on triples.
Path-Enhanced Models
In terms of the graph structure of KGs, paths denote the associations between entities apart from relations and are applied to multi-hop reasoning (Lin et al., 2018;Xiong et al., 2017;Neelakantan et al., 2015). PTransE (Lin et al., 2015) extends TransE by measuring the similarity between relation and path embeddings. MultiHopKG (Lin et al., 2018) explores the answer entities via searching corre-sponding paths with reinforcement learning. However, these models represent paths in a data-driven fashion, lacking interpretability and accuracy.
Methodology
In this section, we first describe the problem formulation and notation of our work in section 3.1. Then, following the workflow of EngineKG as shown in Figure 2, we introduce the rule-enhanced KGE module in section 3.2 and the embedding-based rule learning module in section 3.3.
Problem Formulation and Notation
Definition of Closed-Path Rule. The closed-path (CP) rule or named chain rule is a fragment of the horn rule, which we are interested in for the KGE module and the inference. A CP rule is of the form
Head(x, y) ⇐ B 1 (x, z 1 ) ∧ B 2 (z 1 , z 2 )∧ · · · ∧ B n (z n−1 , y) (1)
where B 1 (x, z 1 ), B 2 (z 1 , z 2 ), · · · , B n (z n−1 , y) denote the atoms in the rule body Body(x, y), and Head(x, y) is the rule head. B i and Head indicate relations. Standard confidence (SC) and head coverage (HC) are two predefined statistical measurements to assess rules (Galárraga et al., 2015;Omran et al., 2019), which are defined as follows:
where #(e, e ′ ) indicates the number of entity pairs (e, e ′ ) that satisfy the condition on the right side of the colon. In general, the rules with SC and HC both higher than 0.7 are regarded as high-quality rules (Zhang et al., 2019b).
Definition of Path.
A path between an entity pair (h, t) is in the form of [h → r 1 → e 1 → · · · → r n → t] where r i and e i are the intermediate relation and entity, and the length of a path is the number of the intermediate relations.
Rule-Enhanced KGE Module
We aim to learn the entity and relation embeddings from triple facts, rules and paths via neuralsymbolic learning. Firstly, we extract the paths via PCRA algorithm (Lin et al., 2015). Apart from other path-finding approaches such as PRA (Lao et al., 2011), PCRA algorithm could measure the reliability of each path for KGE module. Particularly, we develop a joint logic and data-driven path representation mechanism to learn path embeddings. Logic-Driven Path Representation (Intake Stroke). The CP rules could compose paths into shorter and more accurate ones for enhancing the representation of paths. For instance, a length-2 path [T he P ursuit of Happiness Figure 2 could be composed into a shorter path (actually a triple)
CastActor −−−−−−→ W ill Smith P ersonLanguage − −−−−−−−−−− → English] as shown in[T he P ursuit of Happiness T V Language − −−−−−−− → English] via the CP rule T V Language(x, y) ⇐ CastActor(x, z) ∧ P ersonLanguage(z, y).
Furthermore, the relation T V Language could signify the original multi-hop path.
Data-Driven Path Representation.
For the scenario that the path cannot be further composed by rules such as the path Figure 2, we represent this path by summing all the relation embeddings along the path. With the entity pair (h, t) together with the linking path set P, the energy function for measuring the plausibility of the path-specific triple (h, t, P) is designed as (5) where h and t are the head and tail entity embeddings. p i denotes the i-th path in the path set P and p i is the embedding of p i achieved by the joint logic and data-driven path representation. R(p i |h, t) indicates the reliability of path p i between the given entity pair (h, t) obtained by the PCRA algorithm. Optimization Objective (Compression Stroke). Along with the translation-based KGE models, the energy function for formalizing the plausibility of a triple fact (h, r, t) is given as
[T he P ursuit of Happiness CountryOf Origin −−−−−−−−−−−→ U.S.A. LanguageSpoken − −−−−−−−−−− → English] inEp(h, t, P) = pi∈P R(pi|h, t) pi∈P R(pi|h, t) ∥h + p i − t∥E t (h, r, t) = ∥h + r − t∥ (6)
in which r is the embedding of the relation r.
The existing KGE techniques neglect the semantic association between relations. Remarkably, the length-1 rules model the causal correlations between two relations. As shown in Figure 2, the relation pair in the rule F ilmLanguage(x, y) ⇐ T V Language(x, y) should have higher similarity Figure 2: The overall architecture of our developed KG inference model EngineKG in a closed-loop neural-symbolic learning framework. Specific to the rule-enhanced KG embedding module, the green highlighted parts contain the triples and the composed paths via rules, indicating the inputs of the KGE module. than other relations. Thus, we measure the association between relation pairs as
E r (r 1 , r 2 ) = ∥r 1 − r 2 ∥(7)
where r 1 and r 2 are the embeddings of relations r 1 and r 2 . E r (r 1 , r 2 ) should be closer to a small value if r 1 and r 2 appear in a length-1 rule at the same time.
With the energy functions specific to the factual triple, the path representation and the relation correlation, the joint loss function for training is designed as follows:
L = (h,r,t)∈T (Lt + α1Lp + α2Lr) (8) Lt = (h ′ ,r,t ′ )∈T ′ [γ1 + Et(h, r, t) − Et(h ′ , r, t ′ )]+ (9) Lp = (h ′ ,t ′ )∈T ′ [γ2 + Ep(h, t, P ) − Ep(h ′ , t ′ , P )]+ (10) Lr = rp∈S rn∈S ′ [γ3 + Er(r, rp) − Er(r, rn)]+(11)
where L is the whole training loss consisting of three components: the triple-specific loss L t , the path-specific loss L p , and the relation correlationspecific loss L r . α 1 and α 2 are the weights of paths and relation correlation, respectively. γ 1 , γ 2 and γ 3 are three margins in each loss function.
[x] + is the function returning the maximum value between 0 and x. T is the set of triples observed in the KG and T ′ is the set of negative samples obtained by random negative sampling. S is the set of positive relations that are correlated with relation r by length-1 rules and S ′ is the set of negative relations beyond S and relation r.
We employ mini-batch Stochastic Gradient Descent (SGD) algorithm to optimize the joint loss function for learning entity and relation embeddings. The entity and relation embeddings are initialized randomly and constrained to be unit vectors by the additional regularization term with L2 norm.
Embedding-Based Rule Learning Module
We develop an embedding-based rule learning (Expansion Stroke) to mine high-quality CP rules via conducting the rule searching and the rule pruning efficiently. Remarkably, a path can naturally represent the body of a CP rule. Motivated by this observation, we firstly reuse the paths extracted in section 3.2 and regard these paths as candidate CP rules, which improves the efficiency of rule searching. For instance, given an entity pair (h, t) connected by a relation r and a path [h → r 1 → e 1 → r 2 → e 2 →, · · · → e n−1 → r n → t], it can be deduced as a CP rule r(x, y) ⇐ r 1 (x, z 1 )∧r 2 (z 1 , z 2 )∧· · ·∧r n (z n−1 , y), where x, y and z i (i = 1, · · · , n − 1) are the variables in the rule, and r i (i = 1, · · · , n) is a relation.
To evaluate the plausibility of candidate CP rules efficiently, we develop a novel rule pruning strategy consisting of two components: Embeddingbased Semantic Relevance and Concept-based Co-occurrence. It should be noted that the Concept-based Co-occurrence is available when the KG contains concepts. For the KGs without concepts, employing Embedding-based Semantic Relevance solely is still valid to learn rules. Embedding-based Semantic Relevance. Intuitively, a candidate rule is plausible if the rule body corresponding to a path p is semantically relevant to the rule head corresponding to the relation r. We focus on the paths and the CP rules with lengths no longer than 2 for the trade-off of efficiency and performance. Based on the KG embeddings learned in our KGE module, we could measure the semantic relevance between the body and the head of a candidate rule by the path embedding and relation embedding as well as the score function as
E sr (r, p) = exp(−∥r − p∥)(12)
where p denotes the embedding of the path p. The embedding-based semantic relevance indicates a global plausibility of a rule from the perspective of relations. Furthermore, a concept-based co-occurrence is proposed to evaluate the local relevance of the arguments in a rule. Concept-based Co-occurrences. The neighbor arguments in a high-quality CP rule are expected to share as many same concepts as possible. Given a CP rule N ationality(x, y) ⇐ BornIn(x, z) ∧ LocatedIn(z, y), the tail argument of relation BornIn and the head argument of relation LocatedIn should share the concept Location. Considering there are far fewer concepts than entities, we encode each concept as a one-hot representation to maintain the precise concept features. The concept embedding of the head or tail argument of an atom can be formalized as
AC h (r) = 1 |C h (r)| c∈C h (r) OH(c) (13) AC t (r) = 1 |C t (r)| c∈Ct(r) OH(c)(14)
where AC h (r) and C h (r) are the concept embedding and concept set in the head argument of an atom containing relation r while AC t (r) and C t (r) are that of in the tail argument. OH(c) denotes the one-hot representation of the concept c. Specific to a CP rule in the form of r(x, y) ⇐ r 1 (x, z 1 )∧r 2 (z 1 , z 2 )∧· · ·∧r n (z n−1 , y), three types of co-occurrence score functions are designed according to the different positions of the overlapped arguments:
E h co (r, r 1 ) = sim(AC h (r), AC h (r 1 )) (15) E t co (r, r n ) = sim(AC t (r), AC t (r n ))(16)E i co (r i , r i+1 ) = sim(AC t (r i ), AC h (r i+1 ))(17)
where E h co (r, r 1 ) and E t co (r, r n ) respectively denote the co-occurrence similarities specific to the head arguments and the tail arguments between the rule head and the rule body. E in co (r i , r i+1 ) represents the co-occurrence similarity between the adjacent arguments in the rule body. sim(x, y) represents the cosine distance function for measuring the similarity between x and y.
Then, the whole co-occurrence score function can be achieved by composing all the scores in Eqs.
15-17 as
E co (r, p) = E h co (r, r 1 ) + E t co (r, r n ) + n−1 i=1 E i co (r i , r i+1 )(18)
Consequently, the overall score function for evaluating candidate rules is defined as:
E cg = E sr (r, p) + βE co (r, p)(19)
where β is the weight of the co-occurrence score. We set a threshold and select the candidate rules with the scores calculated by Eq. 19 above the threshold as filtered candidate rules. Afterward, we output the high-quality rules from the filtered candidate rules that satisfy the thresholds of the precise quality criteria namely standard confidence and head coverage defined in Eqs. 2-4. Then, the updated rule set is obtained via fusing the newly learned rules and the previous rule set (Exhaust Stroke) for the KGE module in the next iteration.
Algorithm Flow and Complexity
It is noteworthy that from the first iteration of En-gineKG, our rule learning module could potentially achieve sustainable growth of rules. The entire iteration process will keep running until no fresh rules can be generated. Then, the learned KG embeddings learned in the last iteration are exploited for the KG inference. The Algorithm 1 summarizes the whole closed-loop KG inference procedure of our EngineKG model. To evaluate the complexity of our EngineKG model, we denote n e , n r , n p , n c and n t as the amount of entities, relations, paths, concepts and Generate the initial candidate rules from the paths in P;
10
Load the learned entity and relation embeddings e and r;
11
Calculate the coarse-grained evaluation score E cg of each candidate rule according to 12 if E cg of a rule is smaller than st then 13 Eliminate this rule;
14
Pick out the candidate rules that satisfy the thresholds of the standard confidence and the head coverage;
15
Output the newly learned rules;
16
Update the rule set by merging the newly learned rules and the rule set in the last iteration;
triples in a KG. The average length of paths is l p . The embedding dimension of both entities and relations is represented as d. The embedding dimension of concepts is n c due to the one-hot encoding applied for concept representations. Our model complexity of parameter sizes is O(n e d+n r d+n 2 c ). For each iteration in training, the time complexity of our model is O(n t n p l p d).
Experiments
Experimental Setup
Datasets. Four datasets containing ontological concepts are employed for our experiments, including FB15K (Bordes et al., 2013), FB15K237 (Toutanova and Chen, 2015), NELL-995 and DBpedia-242. Particularly, NELL-995 here is a re-split of the original dataset (Xiong et al., 2017) into training/validation/test sets. DBpedia-242 is generated from the commonly-used KG DBpedia (Lehmann et al., 2015) to ensure each entity in the dataset has a concept. The statistics of the experimental datasets are listed in Table 1.
Baselines. We compare our model EngineKG with two categories of baselines:
(1) The traditional KGE models depending on triple facts: TransE (Bordes et al., 2013), Com-plEx (Trouillon et al., 2016), RotatE (Sun et al., 2019), QuatE (Zhang et al., 2019a), HAKE and DualE (Cao et al., 2021).
(2) The models using paths or rules: the pathbased model MultiHopKG (Lin et al., 2018), the rule learning-based models RNNLogic (Qu et al., 2021) and RPJE (Niu et al., 2020), and the model combining rules with KG embeddings IterE (Zhang et al., 2019b).
The evaluation results of these baselines are obtained by employing their open-source codes with the suggested hyper-parameters.
Training Details. We implement our model in C++ and on an Intel i9-9900 CPU with a memory of 64G. For a fair comparison, the embedding dimension of all the models is fixed as 100, the batch size is set to 1024 and the number of negative samples is set to 10. Specific to our model, during each iteration, the maximum training epoch is set to 1000, and the standard confidence and the head coverage are selected as 0.7 and 0.1 for better performance. The entity and relation embeddings are initialized randomly. We employ grid search for selecting the best hyper-parameters on the validation dataset. Evaluation Metrics. Take the head entity prediction for an instance, we fill the missing head entity with each entity e in the KG, and score a candidate triple (e, r, t) according to the following energy function together with the path information:
E e (e, r, t, P) = E t (e, r, t) + α 1 E p (e, t, P) (20)
in which we reuse the energy functions in Eq. 5 and Eq. 6, and P is the path set consisting of all the paths between entities e and t. We rank the scores of the candidate triples in ascending order. Tail entity prediction is similar way. We employ three frequently-used metrics: (1) Mean rank (MR) and (2) Mean reciprocal rank (MRR) of the triples containing the correct entities.
(3) Hits@n is the proportion of the correct triples ranked in the top n. The lower MR, the higher MRR and the higher Hits@n declare the better performance. All the results are "filtered" by wiping out the candidate triples that are already in the KG (Wang et al., 2014).
Results of Link Prediction
The evaluation results of link prediction are reported in Table 2. Firstly, our model En-gineKG significantly and consistently outperforms all the state-of-the-art baselines on all the datasets and all the metrics. Compared to best-performing models RotatE and RPJE on MR, EngineKG achieves performance gains of 95.0%/42.4%/3.7%/83.5% compared to RotatE and 100.0%/71.1%/38.8%/20.0% against RPJE on datasets FB15K/FB15K237/DBpedia-242/NELL-995, respectively. Particularly, on FB15K and FB15K237, the difference between the best performing baseline RPJE and our developed model is statistically significant under the paired at the 99% significance level. Secondly, our model achieves better performance than the traditional models that utilize triples alone, indicating that EngineKG is capable of taking advantage of extra knowledge including rules and paths as well as concepts, which all benefit to improving the performance of the whole model. Thirdly, EngineKG further beats IterE, illustrating the superiority of exploiting both rules and paths for KG inference in a joint logic and data-driven fashion.
Evaluation on Various Relation Properties
The relations can be classified into four categories: One-to-One (1-1), One-to-Many (1-N), Many-to-
FB15K
Head Entities Prediction Tail Entities Prediction 1-1 1-N N-1 N-N 1-1 1-N N-1 N-N TransE (Bordes et al., 2013) 0 Figure 4: The performance curves of MRR, Hits@10 and Hits@1 over iterations on four datasets.
Performance Evaluation Over Iterations
We evaluate the performance of our rule learning module on the learning time and the number of rules compared to the excellent rule learning tool AMIE+. For generating high-quality rules, our model takes 6.29s/2.26s/1.55s/10.50s in an iteration on average while AMIE+ takes 79.19s/26.83s/5.35s/105.53s on datasets FB15K/FB15K237/NELL-995/DBpedia, illustrating the higher efficiency of EngineKG. In Figure 3, the amount of rules mined by AMIE+ is shown as that at the initial iteration. Thus, we can discover that the quantity of rules generated in the first iteration and the third iteration is twice and three times the number of rules obtained by AMIE+. More specifically, Figure 3 exhibits the number of rules and Figure 4 indicates the performance curves on the four datasets over iterations. Notably, the number of rules and the performance continue to grow as the iteration goes on and they both converge after three iterations on all the datasets. These results illustrate that: (1) Rule learning and KGE modules in our model indeed complement each other and benefit in not only producing more highquality rules but also obtaining better inference results.
(2) More rules are beneficial to improving the performance of KG inference. (3) The iteration process will gradually converge along with the rule learning.
Ablation Study
To evaluate each contribution in our whole model EngineKG, we observe the performance on FB15K237 as to the five different ablated settings:
(1) Omitting paths (-Path).
(2) Omitting rules (-Rule).
(3) Omitting concepts (-Concept) by removing concept-based co-occurrences in rule learning.
(4) Replacing rule mining tool AMIE+ with Any-Burl (Meilicke et al., 2019) for obtaining the seed rules (+AnyBurl). (5) Employing only half of the seed rules without iteration (-HalfRule). Figure 5 shows that the performance of our whole model is better than that of all the ablated models except for "+AnyBurl", demonstrating that all the components in our designed model are valid and our model is free of any rule mining tool for obtaining the seed rules. Besides, removing paths and rules both have significant impacts on the performance, which suggests the paths and rules in our model play a more vital role in KG inference.
Case Study
As shown in Figure 6, although the head entity Jonathan and the candidate tail entity Y ork are not linked by any direct relation in the KG, there is an explicit path between them. This path can be represented as the relation P ersonBornInCity
The triple with tail entity missing: (Jonathan, PersonBornInCity, ?) The correct tail entity: York The candidate triple with the path from the KG: PersonBornInCity PersonBornInCity York Jonathan Simon
The CP rule matching the path: PersonBornInCity(x,y) <= HasSibling(x,z)^PersonBornInCity(z,y) Figure 6: An example of the interpretable tail entity prediction via path and rule on NELL-995.
deduced by a matched CP rule. The standard confidence of the rule shown in Figure 6 is 0.9, which explains that although one's sibling being born in a certain city does not necessarily mean that that person was born in the same city, we can employ this trustworthy rule to explain the reliability of the predicted result obtained by our model. The path and the rule together boost the score of the correct candidate entity Y ork calculated by Eq. 20, and especially provide the interpretability of the result.
Conclusion and Future Work
In this paper, we develop a novel closed-loop neural-symbolic learning framework EngineKG for KG inference by jointly rule learning and KGE while exploiting paths and concepts. In the KGE module, both rules and paths are introduced to enhance the semantic associations and interpretability for learning the entity and relation embeddings. In the rule learning module, paths and KG embeddings together with entity concepts are leveraged in the designed rule pruning strategy to generate high-quality rules efficiently and effectively. Extensive experimental results on four datasets illustrate the superiority and effectiveness of our approach compared to some state-of-the-art baselines. In the future, we will investigate combining other semantics such as contextual descriptions of entities, and attempt to apply our model to dynamic KGs.
= #(e, e ′ ) : Body(e, e ′ ) ∧ Head(e, e ′ ) (2) SC = Support #(e, e ′ ) : Body(e, e ′ ) (3) HC = Support #(e, e ′ ) : Head(e, e ′ )
Algorithm 1 :
1Training framework of our model EngineKG Input: G: Training set C h , C t : The head and tail concept set associated with relations P: The set of paths extracted from G via PCRA γ 1 , γ 2 , γ 3 : The margins in loss functions α 1 , α 2 : The weights for trade-off st: The score threshold for coarse-grained evaluation of rules M ax e : The maximum epochs 1 Initialize entity embeddings e and relation embeddings r randomly and encode the concept embeddings from C h and C t by one-hot representation; 2 Mine rules by a rule mining tool such as AMIE+; 3 while new rules can be learned do 4 for epoch=1,2,. . . , M ax e do 5 Sample a minibatch of triples T from G;6Compose the paths between the entity pairs in T by the logic and data-driven path representation described in section 3.2;7Generate the set of negative samples T ′ by the random negative sampling as in TransE(Bordes et al., 2013);8 Update e and r by optimizing the loss functions in Eqs. 5-11; 9
Figure 5 :
5Ablation study on FB15K237. The dash lines indicate the performance of our whole model on MRR (red), Hits@10 (blue) and Hits@1 (green).
Table 2 :
2Link prediction results on four datasets. Bold numbers are the best results, and the second best is underlined.
Table 3 :
3Link prediction results on FB15K and FB15K237 on various relation properties (Hits@10). MultiHopKG could only predict tail entities rather than head entities.NELL-995
0.443
0.496
0.516
0.555
0.555
0.579
0.632
0.668
0.707
0.707
0.374
0.437
0.453
0.479
0.479
0.576
0.609
0.625
0.647
0.647
0.521
0.523
0.523
0.523
0.523
0.487
0.496
0.500
0.501
0.501
0.360
0.423
0.454
0.454
0.454
0.496
0.502
0.506
0.506
0.506
0.288
0.291
0.293
0.293
0.293
Performance
AcknowledgementsThis work was partially supported by Zhejiang Science and Technology Plan Project (No. 2022C01082), the National Natural Science Foundation of China (No. 62072022, 61772054) and the Fundamental Research Funds for the Central Universities.One (N-1), and Many-to-Many (N-N). We select some well-performing models observed inTable 2as the baselines in this section.
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| [
"https://github.com/ngl567/EngineKG."
] |
[
"Inferring the dense matter equation of state from neutron star observations via artificial neural networks",
"Inferring the dense matter equation of state from neutron star observations via artificial neural networks"
] | [
"Ameya Thete \nDepartment of Physics\nBirla Institute of Technology and Science\nPilani -KK Birla Goa Campus\n403726ZuarinagarGoaIndia\n",
"Kinjal Banerjee [email protected] \nDepartment of Physics\nBirla Institute of Technology and Science\nPilani -KK Birla Goa Campus\n403726ZuarinagarGoaIndia\n",
"Tuhin Malik \nCFisUC\nDepartment of Physics\nUniversity of Coimbra\n3004-516CoimbraPortugal\n"
] | [
"Department of Physics\nBirla Institute of Technology and Science\nPilani -KK Birla Goa Campus\n403726ZuarinagarGoaIndia",
"Department of Physics\nBirla Institute of Technology and Science\nPilani -KK Birla Goa Campus\n403726ZuarinagarGoaIndia",
"CFisUC\nDepartment of Physics\nUniversity of Coimbra\n3004-516CoimbraPortugal"
] | [] | The difficulty in describing the equation of state (EoS) for nuclear matter at densities above the saturation density (ρ 0 ) has led to the emergence of a multitude of models based on different assumptions and techniques. These EoSs, when used to describe a neutron star (NS), lead to differing values of observables. An outstanding goal in astrophysics is to constrain the dense matter EoS by exploiting astrophysical and gravitational wave measurements. Nuclear matter parameters appear as Taylor coefficients in the expansion of the EoS around the saturation density of symmetric and asymmetric nuclear matter, and provide a physically-motivated representation of the EoS. In this paper, we introduce a deep learningbased methodology to predict key neutron star observables such as the NS mass, NS radius, and tidal deformability from a set of nuclear matter parameters. Using generated mock data, we confirm that the neural network model is able to accurately capture the underlying physics of finite nuclei and replicate inter-correlations between the symmetry energy slope, its curvature and the tidal deformability arising from a set of physical constraints. We study the validity of our trained model using Bayesian inference and show that the performance of our model is on par with physics based models with the added benefit of much lower computational cost. | null | [
"https://export.arxiv.org/pdf/2208.13163v2.pdf"
] | 251,905,969 | 2208.13163 | 871d6420b8ea65a02d916294ec6e1cba9f8f62d5 |
Inferring the dense matter equation of state from neutron star observations via artificial neural networks
4 Jun 2023
Ameya Thete
Department of Physics
Birla Institute of Technology and Science
Pilani -KK Birla Goa Campus
403726ZuarinagarGoaIndia
Kinjal Banerjee [email protected]
Department of Physics
Birla Institute of Technology and Science
Pilani -KK Birla Goa Campus
403726ZuarinagarGoaIndia
Tuhin Malik
CFisUC
Department of Physics
University of Coimbra
3004-516CoimbraPortugal
Inferring the dense matter equation of state from neutron star observations via artificial neural networks
4 Jun 2023Prepared for submission to JCAPNeutron starsMachine learningBayesian reasoning
The difficulty in describing the equation of state (EoS) for nuclear matter at densities above the saturation density (ρ 0 ) has led to the emergence of a multitude of models based on different assumptions and techniques. These EoSs, when used to describe a neutron star (NS), lead to differing values of observables. An outstanding goal in astrophysics is to constrain the dense matter EoS by exploiting astrophysical and gravitational wave measurements. Nuclear matter parameters appear as Taylor coefficients in the expansion of the EoS around the saturation density of symmetric and asymmetric nuclear matter, and provide a physically-motivated representation of the EoS. In this paper, we introduce a deep learningbased methodology to predict key neutron star observables such as the NS mass, NS radius, and tidal deformability from a set of nuclear matter parameters. Using generated mock data, we confirm that the neural network model is able to accurately capture the underlying physics of finite nuclei and replicate inter-correlations between the symmetry energy slope, its curvature and the tidal deformability arising from a set of physical constraints. We study the validity of our trained model using Bayesian inference and show that the performance of our model is on par with physics based models with the added benefit of much lower computational cost.
Introduction
Neutron Stars (NSs) are some of the most fascinating astrophysical objects in multi-messenger astronomy. They contain matter at extreme conditions far beyond the ones accessible in a terrestrial laboratory. The core of a NS is believed to contain matter at a few times nuclear saturation density, ρ 0 [1][2][3][4]. 1 While the structure of a NS can be determined using the Tolman-Oppenheimer-Volkoff (TOV) equations [5,6], this requires the knowledge of the dense matter Equation of State (EoS). Understanding the internal structure of neutron stars in terms of fundamental interactions between its constituents is an open problem in nuclear physics. To understand the physics of dense matter, at low densities of ρ ∼ 1 − 2ρ 0 , we can use ab initio approaches derived from chiral effective field theory (χEFT) [7][8][9][10][11]. At large densities of ρ ≥ 40ρ 0 , perturbative quantum chromodynamics (QCD) calculations converge and provide reliable estimates [11][12][13][14]. However, in the intermediate density region around ρ ∼ 2 − 10ρ 0 which is relevant for most structural descriptions of neutron stars, reliable calculations from first principles are currently unavailable [11,15]. Quantum field theory calculations based on lattice QCD are challenging at these densities due to the sign problem that arises in Monte Carlo simulations [16]. As a result, structural descriptions of neutron stars rely on relativistic and non-relativistic phenomenological models for the EoS. The nuclear matter parameters (NMPs), which form the basis of the construction of these equations of state for neutron star matter, are not directly accessible. While lower-order NMPs can be empirically extracted through finite nuclei nuclear physics experiments [17][18][19][20], in order to constrain higher-order NMPs, we need to rely on astrophysical observations [21][22][23].
Recent developments in multi-messenger astronomy have provided important information about high-density nuclear matter physics relevant for NSs. Constraining the EoS is a joint task between nuclear physics and astrophysics. Measured astrophysical quantities such as NS observables can uncover properties of dense nuclear matter. Narrowing the constraints on NS observables, therefore, has the ability to constrain the behavior of matter under extreme conditions. It is expected that precise and simultaneous measurements of NS properties like mass, radius, moment of inertia and tidal deformability may help constrain the EoS to a narrow range [24][25][26][27]. High mass pulsars like PSR J1614-2230 (M = 1.908 ± 0.016M ⊙ ) [28][29][30], PSR J0348-0432 (M = 2.01 ± 0.04 M ⊙ ) [31], PSR J0740+6620 (M = 2.07 ± 0.07 M ⊙ [32], and very recently PSR J1810+1714 (M = 2.13± 0.04 M ⊙ ) [33] have already placed tight constraints on the EoS. GW signals emitted from a NS merger event depend on the behaviour of the neutron star matter at high densities [34,35]. Therefore, the values of tidal deformability obtained from GW events such as GW170817 associated with a binary NS merger, as well as the simultaneous measurement of NS masses and radii from high-precision X-ray space missions, such as NICER (Neutron star Interior Composition ExploreR), may help further constrain the EoS. Some current observational evidences are the simultaneous measurements of NS mass 1.34 +0. 15 −0.16 M ⊙ and radius 12.71 +1.14 −1.19 km for the pulsar PSR J0030+0451 by NICER [36]. Other independent analyses show that the radius is 13.02 +1. 24 −1.06 km and the mass 1.44 +0. 15 −0.14 M ⊙ [37]. However the recent measurement of the equatorial circumferential radius of the pulsar PSR J0740+6620 with mass M = 2.072 +0.067 −0.066 M ⊙ and R = 12.39 +1.30 −0.98 km (68 % CI) [38] cannot further constrain the EoS which already predicts a NS with maximum mass more than 2M ⊙ [39].
The EoS -to a good approximation -can be expressed in terms of NMPs at saturation density. The NMPs usually considered for constructing the EoS are the incompressibility coefficient, the skewness parameter of the symmetric nuclear matter, the symmetry energy coefficient, its slope, and the curvature parameters characterizing the density dependence of the symmetry energy. Recently, there has been a comprehensive analysis of correlations of tidal deformability and other NS properties with NMPs [21,40,41]; however, these correlations are found to be model-dependent [41,42]. Ref. [43] shows that the correlations are sensitive to finite nuclei properties, which are accessible to laboratories. Therefore, finite nuclei properties are important quantities to consider in a model while determining the correlations between NS properties and NMPs. Of late, the EoSs obtained by several meta-models have gained popularity owing to their cost-effectiveness in big simulations [13,24,44]. These models are constrained by ab initio theoretical calculations of nucleon-nucleon chiral potentials for low-density neutrons and nuclear matter [45,46], and perturbative QCD for asymptotically high-density regimes [13]. In the intermediate density region, the EoS is evolved in a thermodynamically consistent manner with either piece-wise polytropic segments [47][48][49], a speed-of-sound interpolation, or a spectral interpolation [50,51]. These meta-models are limited to incorporating finite nuclei properties and differ on results in establishing a bridge between NS properties and NMPs. Non-parametric models of the NS EoS have also been proposed based on Gaussian processes (GPs) [52,53] which use Bayesian methods to infer the EoS from multi-messenger data. These models are usually computationally expensive to implement or are highly sensitive to the choice of the training data sets and therefore, might be limited by the current knowledge of the EoS [54]. Consequently, there is a need to search for alternative approaches to construct a model-independent EoS.
In recent years, deep learning (DL) has been extensively applied to a wide range of technological and scientific tasks. DL algorithms, which are a class of machine learning (ML) algorithms, are highly scalable and distributed computational techniques with the ability to learn intricate relationships from raw data using units called neurons arranged in a stacked fashion. The advent of high-performance computing and the development of parallel devices like graphics processing units have rendered DL as the primary choice of algorithms for tasks such as computer vision [55], or natural language processing [56]. DL models have been used as alternatives to conventional statistical framework and have been successfully applied to many problems in physics. Most applications of ML and DL in physics have been in analyzing data obtained from data-intensive experiments like LIGO for the detection and denoising of GW signals [57][58][59], and the Large Hadron Collider for particle track reconstruction or anomaly detection [60][61][62]. Significant progress has also been made in using these algorithms in the context of nuclear physics [63] and neutron star physics [54,64,65]. While these applications are certainly promising, it remains to be seen up to what extent a DL model can supplement existing physical models. Recent research also aims to address whether a trained DL model can produce correct predictions from experimental data alone and whether such predictions are comparable to mesoscopic phenomenological physics models [66]. Such ML and DL-based models do not have the feature richness possessed by physics-based models but they offer other benefits like cost-effectiveness while dealing with a large amount of experimental or observational data. Recent works [15,64,67] have studied the applications of machine learning methods to the neutron star EoS. They employ a feedforward neural network (FFNN) to map neutron star data to EoS parameters. Instead of considering the FFNN as merely an interpolation tool between EoS NMPs and neutron star observables, we adopt an approach similar to [54] by treating the FFNN as a representation of the EoS itself.
In this paper, our focus is on implementing an artificial neural network to emulate a physics model and assess its accuracy through validation and inference tasks. Our primary goal is to thoroughly examine the neural network's ability to mimic the physics model, and subsequently utilize the trained network for computationally expensive Bayesian inference tasks. To the best of our knowledge, the use of neural networks in Bayesian inference framework has not been employed in nuclear astrophysics literature. We show that we can indeed obtain results comparable to physics models at a fraction of the computational cost. This points out a novel utilization of machine learning and neural network models in research. Let us however clarify that we are not claiming that neural networks can replace physics based models. Our claim is the such models can help us capture some of the features of the EoS at much lower compuational costs.
The paper is structured as follows: in Section 2, we discuss the parameterization of the EoS which leads to the emergence of NMPs, followed by a brief review of the theory of artificial neural networks and the DL approach used to map NMPs to NS observables. This is followed by a description of the Bayesian statistical framework. In 3 we test how well does the best fit neural network model capture correlations between EoS parameters. We also perform a Bayesian analysis to compare our models with a physics based model and show that the computational cost is much less for our model. We finally end with our conclusions in Section 4.
Framework
In this section, we outline the different facets of the adopted framework for our analysis. In Sec. 2.1, we briefly describe the neutron star EoS and the nuclear matter parameters involved in its construction. Then, in Sec. 2.2, we describe the generation of the data set used to train the neural network model. Sec. 2.3 describes at length the DL approach used to construct a deep neural network model which accepts nuclear matter saturation parameters as inputs and produces neutron star properties obtained from a set realistic nuclear physics EoSs as targets. Finally, in Sec. 2.4 we present a Bayesian inference framework that is applied on the trained neural network model to verify the performance of our model in comparison with Skryme models.
Nuclear matter parameters
The structure of neutron stars is obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equation with a given EoS for the nuclear matter. The EoS is expressed as the variation of pressure e with density ρ, over a wide range of densities. To a good approximation, any EoS calculated from phenomenological nuclear models can be decomposed into two parts, (i) the EoS for symmetric nuclear matter e(ρ, 0); and (ii) a term involving the symmetry energy coefficient S(ρ) and the asymmetry δ,
e(ρ, δ) ≃ e(ρ, 0) + S(ρ)δ 2 ,(2.1)
where ρ = ρ n + ρ p is the baryon density, ρ n and ρ p are the neutron and proton densities respectively, and the asymmetry δ = (ρ n − ρ p )/ρ. We can then characterize the density dependence of the energy density of symmetric matter around the saturation density ρ 0 in terms of a few bulk parameters by constructing a Taylor expansion around ρ 0 . That is,
e(ρ, 0) = e 0 + K 0 2 ρ − ρ 0 3ρ 0 2 + Q 0 6 ρ − ρ 0 3ρ 0 3 + O(4) (2.2)
The coefficients e 0 , K 0 , Q 0 denote the energy per particle, the incompressibility coefficient, and the third derivative of symmetric matter at saturation density, respectively. Similarly, the behaviour of the symmetry energy around saturation can also be characterized in terms of a few bulk parameters,
S(ρ) = J sym,0 + L sym,0 ρ − ρ 0 3ρ 0 + K sym,0 2 ρ − ρ 0 3ρ 0 2 + O(3) (2.3)
where J sym,0 = S(ρ 0 ) is the symmetry energy at saturation density. The incompressibility K 0 , the skewness coefficient Q 0 , the symmetry energy slope L sym,0 , and its curvature K sym,0 evaluated at saturation density, are defined in [68]. These quantities are the key nuclear matter parameters (NMPs) that describe any equation of state (EoS). Hence, an EoS can be represented by a point in the seven-dimensional parameter space of NMPs {e 0 , ρ 0 , K 0 , Q 0 , J sym,0 , L sym,0 , and K sym,0 } [43]. Symbolically, the j th EoS in this space is written as
EoS j = {e 0 , ρ 0 , K 0 , Q 0 , J sym,0 , L sym,0 , and K sym,0 } j ≈ N (µ, Σ) (2.4)
where N (µ, Σ) is a multivariate Gaussian distribution with µ being the mean value of the nuclear matter parameters p and a covariance matrix Σ. The diagonal elements of Σ represent the variance or the squared error for the parameters p i . The off-diagonal elements of Σ are the covariances between different parameters p i and p j , and denote the correlation coefficient between them. Hence, given a mean µ and covariance matrix Σ, a large number of EoSs can be obtained. While the Taylor expansion of the symmetric and asymmetric energy is only truly accurate around the saturation density, we can treat these expansions as a parameterization of the EoS -similar to other adopted parameterizations -with the condition that this representation asymptotically approaches the Taylor expansion in the limit ρ → ρ 0 [21,69]. This lets us ignore any issue arising with the convergence of the approximation. However, the higher-order NMPs obtained via this parameterization might markedly diverge from the actual nuclear matter expansion coefficients. They can be thought of as effective parameters that incorporate the effects of missing higher-order terms. Moreover, Refs. [26,69] indicate that an EoS obtained from the Skyrme framework can be well-reproduced by considering Taylor coefficients until the third or fourth order.
Data set generation
In our analysis, the input data used to train an artificial neural network are the seven key nuclear matter parameters that govern the equation of state {e 0 , ρ 0 , K 0 , Q 0 , J sym,0 , L sym,0 , K sym,0 }. Six neutron star properties are the target variables: the maximum NS mass, M max ; the maximum NS radius, R max ; the radius for 1.4 M ⊙ NS, R 1.4 ; and the tidal deformability Λ M for NS having mass M ∈ [1.0, 1.4, 1.8]M ⊙ . We generate our data set by sampling points from the multivariate Gaussian distribution N (µ, Σ), where µ is the mean vector with
components µ i = E[p i ] and Σ is the covariance matrix with entries Σ ij = E[(p i −µ i )(p j −µ j )],
for a NMP p. 2 This method closely follows the procedure for generating the Case-II data set in Ref. [43]. We assume an a priori inter-correlation coefficient between L sym,0 and K sym,0 of 0.8, which is reasonable choice for nuclear physics models that satisfy finite nuclear properties. Models which satisfy finite nuclear properties exhibit different correlations between NMPs and NS properties as compared to meta-models or nuclear physics model for infinite nuclear matter that do not respect finite nuclear properties [41,42]. Therefore, we consider Case-II data of Ref. [43] to mimic the microphysics information of finite nuclear properties. Figure 1 presents the 7 × 7 matrix for the correlation coefficients between the NMPs in the sampled data. The central values and uncertainties on each NMP in the constructed distribution are listed in Table 1. We then construct the Skyrme EoS from a drawn NMP sample and check whether the EoS (a) predicts a NS maximum mass above 2 M ⊙ ; (b) predicts a tidal deformability for a NS with M = 1.4M ⊙ below 800; and (c) satisfies the causality condition, i.e., the speed of sound c s = dp de ≤ c at the center of maximum mass NS, where c is the speed of light in vacuum. Any samples that generate EoSs which do not satisfy these conditions are discarded. For each EoS, the TOV and deformability equations [70] are solved to obtain the six aforementioned NS observables. Naturally, realistic NS observations accrue experimental and instrumental errors, which result in corresponding uncertainties while reconstructing the NMPs. For simplicity, we choose to disregard NS observational errors and uncertainties while training the model. Following this procedure, we obtain 2106 filtered NMPs and corresponding NS observables which form the data set.
NMP MVGD Tranning Set p i µ p i Σ p i p i µ p i Σ p i p i e 0 −
Artificial neural networks
Artificial Neural Networks (ANN) are a class of machine learning algorithms that have found widespread use over the past decade. The popularity of ANNs arises from their ability to model complex nonlinear relationships within data and their potential to generalize to a wide class of functions. An ANN with sufficiently many layers or neurons is theoretically capable of representing any continuous function [71,72]. Therefore, as a general rule, a more complex network with a larger number of parameters is able to learn more abstract features from the data. Feedforward Neural Networks (FFNN) are the simplest type of neural network architecture. For an input x, the objective of a feedforward neural network is to approximate some true mapping f * (x) by a mapping f (x; θ), parameterized by a set of weights θ. A typical feedforward neural network consists of a number of processing units called neurons, arranged into one or many layers composed in a sequential fashion. A neuron performs a linear operation by aggregating weighted inputs received from neurons in the previous layer. A FFNN generally consists of an input layer, followed by one or more hidden layers, and a final output layer consisting of one or more neurons. Computation in an FFNN flows in a linear fashion, starting at the input layer and moving successively through the hidden layers until it reaches the output layer. Figure 2 provides an illustration of a simple feedforward neural network with one input layer, two hidden layers and an output layer.
The parameters θ typically represent the weights assigned to a connection between neurons in adjacent layers. The training data provides noisy, approximate examples of f * (x) and each x is paired with a corresponding label y. The goal of a learning algorithm is to learn a particular value of θ that results in the best function approximation. During the training procedure, the learning algorithm does not say what each layer does but instead decides how to use these layers to produce an optimal approximation of f * . As the training data does not show the desired output for intermediate layers, they are called hidden layers. The number of hidden layers decides the depth of the network, and the dimensionality or the number of neurons in each hidden layer determines the width of the network. To introduce nonlinearity in the computation between two successive layers within the model, a nonlinear function, called the activation function, acts element-wise on the output of one hidden layer, and the output of the function is passed to the next layer in the computation. For most modern neural networks, the default recommendation is to use the rectified linear unit, or ReLU [73,74] activation, defined as g(z) = max{0, z}. Other commonly used choices for the activation function include the tanh function or the logistic function. For a textbook review of neural networks and training algorithms, see Ref. [75].
The ability of neural networks to model highly nonlinear relationships between input and output variables makes them ideal for estimating neutron star properties from the equation of state parameters. This is important because the relationships between these two quantities are expected to be nontrivial and involve multiple intermediate steps composed of nonlinear operations. Moreover, a neural network offers two major advantages over a conventional approach with traditional physics models:
1. An ANN can efficiently map the sample NMPs to NS properties without calculating the EoS from nuclear physics models. Similar works [76,77] demonstrate that ANNs offer up to a two-fold speedup over conventional astrophysical models 2. ANNs can also accurately capture finite nuclear information, which can computationally expensive to verify with a traditional physics model in a Bayesian setting.
At this step, we wish to note that other machine learning models can also be used for an identical purpose, albeit with varying degrees of success. We performed a preliminary comparison of FFNNs with other ML models, specifically linear regression, support vector regression and eXtreme Gradient Boosting (XGBoost) [78]. We observed that FFNNs outperformed all the other models in the study, and therefore, we chose to use FFNNs for this work.
In supervised learning instances, the data set is generally partitioned into non-overlapping training, testing, and validation sets. The training data set is used by the neural network to learn, the validation set is used to see whether the network is learning properly during the training procedure and to select the optimal values for the hyperparameters, and the testing data set is used to assess the performance of the trained model. It should be noted that the model does not see any instances from the testing data set until after the training is completed. Accordingly, we randomly partition the original generated data set following a 70%-20%-10% split into a training set, a validation set, and a testing set consisting of 1515, 422, and 169 instances, respectively. Before training, the features in the split datasets are standardized by removing the mean and scaling them to unit variance. Standardization of data is a common requirement for many ML algorithms as they tend to perform poorly if the individual features are not standard normally distributed, i.e. Gaussian with unit variance and zero means. Standardization is performed using Scikit-Learn's [79] StandardScalar method. The output of the network is scaled back to the original distribution before computing the loss.
x 1 x 2 x 3 x n . . . a (1) 1 a (1) 2 a (1) m . . . a (2) 1 a (2) 2 a (2) m . . . a (3) 1 a (3) k . . .
The choice of the ANN architecture, such as the number of layers and the number of neurons in each layer, represents a trade-off between the quality of fit and the overfitting problem. The model should contain sufficiently many trainable parameters to learn the ground truth function accurately. At the same time, the ANN must avoid overfitting the training data by losing its ability to generalize well. As there is no panacea for choosing an optimal model configuration, the final architecture for the ANN was chosen based on empirical tests performed on the data and selecting the set of configurations that performed best on the validation set. Our network consists of two hidden layers with 15 neurons each and uses the rectified linear unit (ReLU) activation function, which is a standard choice for most deep learning applications and is known to mitigate the vanishing gradients problem [80]. The ReLU activation function behaves as an identity function for positive inputs and saturates at 0 for negative inputs. No activation function is applied to the input or the output layer. The design of our feedforward architecture is summarized in Table 2.
The neural network is implemented and trained using the Python Keras library [81] with a TensorFlow [82] backend. Neural network parameters are initialized with the Glorot uniform distribution [83]. We use the Adam optimizer [84] to update the weights of the ANN. For training, we use an initial learning rate of 1 × 10 −3 and the training data is batched into batches of size 16. Training is performed for a total of 40 epochs, or until the validation loss stops decreasing. The model is trained on a 2-core Intel Xeon CPU @ 2.20GHz. The network seeks to optimize a root mean squared error (RMSE) loss between the predictions and actual Layer Number of neurons Activation function 0 (Input) 7 None 1, 2 15 ReLU 3 (Output) 6 None Table 2. Neural network architecture used in this work. The input layer consists of seven neurons corresponding to nuclear matter parameters. In the output layer, the six neurons correspond to the six neutron star properties.
neutron star properties.
Bayesian statistical framework
The Bayesian statistical framework allows us to carry out a detailed analysis of the parameters of a model for a given set of fit data [85][86][87]. The hypothesis or the prior knowledge of the model parameters and various constraints on them is encoded through the prior distributions. The technique yields a joint posterior distribution of the model parameters by updating the probability for the hypothesis using the available observational data according to Bayes' theorem. The posterior distribution of the model parameters θ can be written as
P (θ|D) = L(D|θ)P (θ) Z (2.5)
where θ and D denote the set of model parameters and the fit data, respectively. P (θ|D) is the joint posterior probability of the parameters, L(D|θ) is the likelihood function, P (θ) is the prior for the model parameters and Z is the evidence. The posterior distribution of a given parameter an be obtained by marginalizing P (θ|D) over remaining parameters. The marginalized posterior distribution for a parameter θ i is obtained as
P (θ i |D) = P (θ|D) j̸ =i dθ j (2.6)
A major contribution of this study is to demonstrate how a neural network can be effectively integrated within a Bayesian inference framework for astrophysical use cases. In particular, we demonstrate the effect of intercorrelations between symmetry energy parameters on NS properties. We sample NMPs within a Bayesian framework in two different ways: (i) sampling all the NMPs randomly without any intercorrelations among them, (ii) sample all the NMPs randomly, except for the pair L sym,0 -K sym,0 which is set to a correlation coefficient of 0.9. Then, we compare the posterior distribution of the NMPs for both cases under certain constraints on NS properties.
For the sampling process, we use the Gaussian likelihood function defined as,
L(D|θ) = j 1 √ 2πσ j exp − 1 2 N d i=1 d i − m j (θ) σ j 2 (2.7)
Here the index j runs over all the data, d j and m j are the data and the corresponding model values, respectively. The model m is parameterized by a set of parameters, θ. σ j are the adopted uncertainties. The evidence is used to compare the compatibility of different models with the available data. In our present work, the evidence Z is not relevant and thus can be NMP P1 P2 ignored. To obtain the marginalized posterior distributions of the NMPs within the Bayesian framework, we require a set of fit data, a model, and a set of priors for the nuclear matter parameters. The likelihood function for a given set of fit data is evaluated for a sample of NMPs populated according to their prior distributions. The joint probability distribution of the NMPs is obtained using the product of the likelihood function and the prior distribution. The fit data for the likelihood function is provided by the neural network for a sample drawn from the prior distribution. To compute the likelihood, we use the maximum neutron star mass M max , the maximum radius R max , the radius R 1.4 , and the tidal deformability Λ 1.4 , from the set of outputs generated by the neural network for a given input of NMPs. Instead of using a distinct value for each data point, we fix d i in Eq. (2.7) to a mean value µ j . Therefore for our study, Eq. (2.7) is modified to for a set of NMPs p and the neural network ANN(·) which accepts NMPs as inputs. We define the set of priors over the NMPs as a multivariable Gaussian distribution. The mean and standard deviation on each NMP in the distribution are listed in Table 3.
p i µ p i Σ p i p i µ p i Σ p i p i e 0 −
Log Likelihood
The log-likelihood for mass, radius at maximum mass, and radius at M MCMC approaches can lead to issues with converging to a stable posterior. To overcome this problem, we use the dynamic nested sampling algorithm [88][89][90]. In dynamic nested sampling, the posterior is broken into many nested "slices" with an initial n-live number of points that vary dynamically as the sampling progresses; samples are generated from each of them and then recombined to construct the posterior distribution. We use the Dynesty dynamic nested sampling algorithm interfaced in BILBY [87] with 5000 n-live points to sample from the posterior distributions of the nuclear matter parameters. Dynesty enables flexible Bayesian inference over complex, multi-modal distributions without needing to converge to the posterior before generating valid samples [90].
Results
In this section, we present the main findings of our analysis. After selecting the best model, which we call the NS-ANN, by performing a grid search over hyperparameters such as the model depth, model width, learning rate, etc., we determine its performance on the test set. We wish to reemphasize that the test set is never used during training or the validation phase. Evaluating the model's performance on this set quantifies the generalization capacity of the model, i.e., its predictive power on unseen data. The RMSE values obtained for each NS observable on the testing data set are summarized in Table 5. We also include the root mean squared relative error as it provides a scale-independent measure of the generalization capacity and eases comparison between multiple dependent variables. In Figure 3, we plot the loss as a function of training time, which is called the learning curve. The learning curve plots the root mean squared error on the y-axis against the number of elapsed epochs -or, the number of full passes over the training data set -on the x-axis. We plot two different learning curves for a single training instance: one for the training loss (in blue) and the other for the validation loss (in red). Figure 3 also plots a 1σ (68% confidence interval) region centered around the training curves computed for 10 independent runs to indicate the degree of variability observed by training the model on different subsets of the original data set. The training and validation losses are the loss functions computed for the training set and validation sets, respectively. The former is a metric of how well the ANN learns the training data, while the latter indicates how well the model is able to predict by generalizing the learned information. Once we have successfully and efficiently learned the non-linear mapping between the empirical nuclear matter parameters and the NS observables, we must first prove the correctness of the NS-ANN model before using it in the context of a Bayesian inference framework to constrain the NMPs. In Figure 4 we plot the 1σ confidence ellipses for Λ M versus L sym,0 and K sym,0 for NS mass M = 1.0, 1.4, and 1.8M ⊙ predicted by the NS-ANN for three different pseudo-sets of NMPs. These pseudo-NMP distributions were constructed as a multivariate Gaussian distribution by setting the inter-correlation between L sym,0 and K sym,0 to r = 0.3, 0.6 and 0.9, and all other parameters of the distribution to values listed in Table 1. Once again, all NMPs that generate EoSs which do not satisfy the conditions from Sec.
N i=1 (ŷ i − y i ) 2 , where N is
2.3 are discarded. We use the NS-ANN model for each NMP in the dataset to generate a corresponding set of NS observables. Through this analysis, we wish to establish that a neural network is capable of replicating the underlying microphysical information conveyed by a conventional EoS model. The values of the correlation coefficients for the results illustrated in the figure are summarized in Table 6. For the first set, we observe that the correlations of Λ 1.0,1.4,1.8 with K sym,0 are χ ∼ 0.5 − 0.8, and those with L sym,0 are χ ∼ 0.2 − 0.8. For the second set, we see a non-trivial narrowing of the confidence ellipses, indicating stronger correlations of Λ 1.0 and Λ 1.4 with L sym,0 and K sym,0 , χ ∼ 0.7−0.8, while these correlations become moderate for Λ 1.8 . We also observe that the Λ M −L sym,0 correlations decrease with increasing mass of NS, while an opposite trend is observed for Λ M − K sym,0 correlations. Moreover, for the same NS mass, Λ M − L sym,0 correlations are much more sensitive to L sym,0 − K sym,0 correlations over Λ M − K sym,0 correlations. These results help emphasize that the correlations of the tidal deformability with L sym,0 and K sym,0 are sensitive to the physical correlations among the empirical NMPs arising from a set of physical constraints [26,27]. Additionally, the observations made in this analysis are in excellent agreement with results from previous studies performed on observing the effect of correlations of the tidal deformability with the slope of the symmetry energy, and its curvature [40,43,92]. At this point we also wish to note that there exists a strong model-independent intercorrelation between L sym,0 and J sym,0 , but this correlation does not disrupt the relationships between some NS properties, such as tidal deformability, radius, and NMPs [43], and as a result is not covered in this study. Table 6. The values of the correlation coefficients for Λ 1.0,1.4,1.8 with L sym,0 and K sym,0 for three different L sym,0 − K sym,0 correlation coefficients, r
As a next step, we apply our trained NS-ANN model to a dataset acquired from a different class of physics models in order to assess the model-dependent uncertainty related to neutron star properties.
The nuclear matter parameters are known to be model-dependent and hence also the properties of stars or EoSs [93][94][95]. In Table 7, we compare the NS properties obtained using a dataset generated by a density-dependent meson coupling framework called the DDB model. This model is constructed using a Bayesian inference approach in conjunction with minimal constraints on the nuclear saturation properties, the maximum mass of a neutron star exceeding 2M ⊙ , and a low-density equation of state (EoS) calculated using chiral effective field theory. In the first horizontal panel, we show that NS properties predicted by the NS-ANN ML model match well with the theoretical predictions of the Skyrme model. In the second horizontal panel, then we apply our trained NS-ANN model to a data set generated by DDB. It can be seen that the residual errors on the NS properties are much larger in this case. We posit that this wider uncertainty might arise due to input parameters (NMPs) in the DDB dataset lying outside the parameter space covered by the training set. To avoid any extrapolation error, we then filter the DDB according to the parameter space of the training sets. We call this set DDB-FL and apply the NS-ANN on this set. It can be seen that the dispersion is still large. The NS-ANN model is trained on data obtained from the non-relativistic Skryme model, so it is unsurprising that it fails to completely capture the features of the relativistic DDB model.
Validity of NS-ANN model in Bayesian Inference
In this section, we apply our NS-ANN model in a Bayesian inference framework. The primary objective is to validate the trained model's effectiveness in computationally expensive calculations of this type. The results will be analyzed in two parts: i) The posterior of neutron star properties will be compared between trained NS-ANNs and Skyrme models in Bayesian Inference, ii) the CPU time will be compared between the two models.
Comparison between NS-ANN and Skyrme
As discussed above, Bayesian inference requires three ingredients: a) the model, b) the fit data or likelihood, and c) prior. We will perform identical Bayesian inference calculations with both Skyrme and NS-ANN models and compare their posteriors. The input priors are the nuclear matter parameters, namely e 0 , ρ 0 , K 0 , Q 0 , J sym,0 , L sym,0 , and K sym,0 . In Table 3, we have defined two multivariate Gaussian priors, P1 and P2. P1 represents the prior validity in which the neural network is originally trained, while in P2, we have doubled the uncertainty to evaluate the performance of the NS-ANN model. The likelihood for our calculation is defined in section 2.4.1 for the fit data described in 4. In the likelihood, we have only considered observed NS properties because we are motivated to compare the performance of the NS-ANN trained model with the Skyrme nuclear physics model within observational constraints.
We used the Pymultinest sampler to sample our parameter. In Figure 5, we compared the Bayesian posterior results for the neutron star properties obtained from the prior set P1 with the NS-ANN and Skyrme model. This included the NS maximum mass (Mmax), radius Table 7. The median and the minimum (min) and maximum (max) values of the associated 90% confidence intervals for the NMPs and the NS properties obtained from Skyrme, DDB, and DDB-FL datasets. We also indicate these quantities for maximum relative residual errors σ rr on the NS properties.
for a 2.07 M⊙ NS, radius (R 1.4 ), tidal deformability (Λ 1.4 ) for a 1.4 M ⊙ NS, and the square of the speed of sound (C 2 s ) at the center of the maximum mass NS. We also compared the same for the prior set P2 in Figure 6. In both figures, the joint distribution of NS properties is represented by corner plots, which are obtained from Bayesian Inference posteriors. Each variable in a corner plot is plotted on one of the axes, and its distribution is shown by its diagonal. Off-diagonal plots show the joint distributions of each pair of variables. Using it to understand how variables are related is a useful tool.
In Table 8 Table 9. CPU inference time estimates for the ANN model and a Skyrme model to infer NS observations from a set of NMPs for prior sets P1 and P2. The timing tests were performed on a 12-core Intex i7-8700K CPU @ 3.70 GHz. The inference is performed with a batch size of one.
of the speed of sound is ∼ 3% at the center of the maximum mass NS. However, in the case of P2, all the residuals are noticeably increased, with the tidal deformability of 1.4 M ⊙ NS and the square of the speed of sound at the center of the maximum mass NS being around ∼ 5%. It should be emphasized that the neural network model can be trusted for interpolation but used with caution for extrapolation. The main advantage of this type of ANN model is that it can produce very fast draft results for computationally expensive calculations.
CPU Time
In Table 9, we compared the CPU time for NS-ANN and Skyrme model for both prior sets. The timing tests were performed on a 12-core Intel i7-8700K CPU @ 3.70 GHz. It is noteworthy that the NS-ANN model is approximately 500 times faster than the Skyrme model. It is exciting that the neural network model can be employed in computationally expensive calculations to obtain fast draft results. This will help in reducing carbon emissions and help reduce the environmental cost of using high computational power for the preliminary testing of any model.
Conclusions
We have demonstrated the application of ANNs to analyze NS observables like the radius, mass, and tidal deformability from a set of seven parameters, or nuclear saturation properties that characterize the equation of state of cold dense matter. We have shown that a neural network that models such a mapping is able to learn the microphysics information of finite nuclei, which are the intercorrelations arising between the NMPs. Moreover, we delved into the utilization of Artificial Neural Networks (ANN) as a complement to theoretical models, enabling them to undertake computationally demanding tasks, such as Bayesian Inference studies to constrain nuclear models. This study demonstrates that in situations where speed and computational efficiency are desired, a trained neural network can function as a surrogate for traditional physics-based EoS models, and possibly also provide insights that might be difficult to obtain otherwise. However, it is essential to clarify that this methodology only complements other approaches, and does not seek to replace them in any form.
Let us briefly summarize what we have done in this work. First, we generate a set of pseudo-NMP data using the Skyrme model. A suitable set of NMPs are chosen so that the resulting neutron star EoSs are consistent with the currently observed maximum mass of ∼ 2M ⊙ and satisfy causality constraints. Next, we train an artificial neural network on this data set to learn a non-linear mapping between the set of nuclear matter parameters and NS observables. We demonstrate that the ANN is capable of inferring, with reasonable accuracy, NS observables from empirical parameters, as compared to conventional physics models which require the computation of an EoS followed by tedious equation-solving. This NS-ANN model is then validated to ensure that it satisfies the microphysics information of Figure 6. Comparison of neutron star properties obtained from Bayesian posterior for prior set P2 with ANN and Skyrme model, including NS maximum mass (M max ), radius for a 2.07 M ⊙ NS, radius (R 1 .4) and tidal deformability (Λ 1.4 ) for a 1.4 M ⊙ NS, and square of the speed of sound (C 2 s ) at the center of the maximum mass NS. The vertical lines represent the 68% CIs, and the light and dark intensities represent the 1σ, 2σ, and 3σ CIs, respectively. finite nuclear matter. More specifically, we study whether a trained ANN model is able to capture correlations between the tidal deformability for a NS with mass 1.4M ⊙ , Λ 1.4 and the symmetry energy slope L sym,0 as well as its curvature K sym,0 . We find that the NS-ANN model learns a mapping that is sensitive to L sym,0 − K sym,0 correlations in agreement with previous studies. Using the Bayesian inference framework, we determine the extent to which the neural network can replicate the physics model, and then utilize the trained neural network for inference tasks, which typically involve computationally expensive calculations. By doing so, we are able to achieve results that are comparable to the physics model while significantly reducing the computational time required.
Presently, our framework is a proof-of-concept that demonstrates the applicability of ANNs to NS physics. For a more realistic application of our framework, empirical uncertain-ties ought to be considered. This can be achieved, for example, by using a class of ANNs called Bayesian Neural Networks to perform the inference task. Bayesian networks have the ability to cast the problem into a probabilistic domain by inferring probability distributions over a prior of NMP inputs. A model which predicts uncertainties will also be able to potentially further reveal the effects of NS observational constraints on the NMPs. Presently, the entire domain of validity of the ANN cannot be automatically ensured. Due to a lack of training data for some parts in the modeled output space, the ANN might be unable to accurately predict regions that have not been learned.
This work is limited only to hadronic NS compositions, i.e, the set of β-equilibriated EoSs employed in this work are composed of neutrons, protons, electrons, and muons. Present observational constraints on NS properties cannot rule out the possibility of exotic degrees of freedom or deconfined quark phases inside the NS core. In future work, a detailed and systematic analysis with an ANN trained on a set of EoSs with different compositions of particles is required to investigate the uncertainties on higher-order NMPs with available observational constraints. The ANN map can also be further extended to determine the required number of NS observations and their precision to conclude the existence of quark phases inside the NS core.
Data availability
The trained network and corresponding data sets are made publicly available through a GitHub repository. 3
Figure 1 .
1Correlations among the various NMPs for the sampled data set. Correlations among the off-diagonal pairs corresponding to L sym,0 − K sym,0 are the only nontrivial values. Negligible correlations between other NMPs are visible as the figure plots sample statistics.
Figure 2 .
2An example of a fully-connected feedforward neural network with two hidden layers. Each hidden layer consists of m neurons, denoted by a (l) j , where l denotes an index over the hidden layers and the index j runs over the dimension of the hidden layer. This particular network accepts an n-dimensional input through the input layer (in green), passes the input through the hidden layers (in lavender), and produces a k-dimensional output at the output layer (in red). Solid lines between individual nodes denote weighted connections that are learned during the training procedure. Activation functions are not shown in this figure.
the a total number of samples in the test set. We also show the root mean squared relative error, (RMSE/ȳ) × 100, whereȳ = (1/N ) N i=1 y i is the mean of the true value. All quantities are dimensionless except R max and R 1.4 , which are in units of km, and M max , which is in units of M ⊙ .
Figure 3 .
3Learning curves for the training (blue, solid) and validation (red, dashed) data sets plotted against the number of elapsed epochs. The shaded region around the solid curve indicates a 1σ or 68% CI region around the central value computed for 10 independent training runs.
Figure 4 .
4The 1σ confidence ellipses in the planes of Λ M -L sym,0 (top) and Λ M -K sym,0 (bottom) with M = 1.0, 1.4 and 1.8M ⊙ generated by the NS-ANN model for the correlation coefficient, r, between L sym,0 and K sym,0 set to 0.3, 0.6 and 0.9.
Figure 5 .
5Comparison of neutron star properties obtained from Bayesian posterior for prior set P1 with ANN and Skyrme model, including NS maximum mass (M max ), radius for a 2.07 M ⊙ NS, radius (R 1 .4) and tidal deformability (Λ 1.4 ) for a 1.4 M ⊙ NS, and square of the speed of sound (C 2 s ) at the center of the maximum mass NS. The vertical lines represent the 68% CIs, and the light and dark intensities represent the 1σ, 2σ, and 3σ CIs, respectively.
Table 1. The mean value µ pi and error Σ pipi for the nuclear matter parameters p i employed for the multivariate Gaussian distribution. All quantities are in units of MeV except for ρ 0 which is in units of fm −3 .16.0
0.25
−15.99
0.25
ρ 0
0.16
0.005
0.158
0.005
K 0
230
20
242
15
Q 0
−300
100
−307
70
J sym,0
32
3
32.65
2.8
L sym,0
60
20
73
12
K sym,0 −100
100
−20
50
Table 3 .
3The mean value µ pi and error Σ pipi for the nuclear matter parameters p i in the prior multivariate Gaussian distribution. All quantities are in units of MeV except for ρ 0 which is in units of fm −3 .
MCMC) algorithm. This algorithm updates the existing parameters with a new set of parameters with a probability proportional to the ratio of the two points. However,1.4 are defined as
follows:
log χ(M max ) = log
1
exp M cal −M obs
−0.01
+ 1
(2.8)
log χ(R 2.07 ) = −0.5
R 2.07cal − R 2.07obs
∆R 2.07
2
+ log 2π∆R 2
2.07
(2.9)
log χ(R 1.4 ) = −0.5
R 1.4cal − R 1.4obs
∆R 1.4
2
+ log 2π∆R 2
1.4
(2.10)
The calculations are performed for a set of NS observational properties, such as maximum
mass (M max ), the radius at NS mass (R 2.07 ), and at 1.4M ⊙ (R 1.4 ) listed in Table 4.
Bayesian parameter estimation is commonly carried out using the Markov Chain Monte
Carlo (Constraints
Quantity
Value/Band
Reference
M max
> 2.0 M ⊙
[91]
R 2.07
12.4 ± 1.0 km
[91]
R 1.4
13.02 ± 1.24 km
[37]
Table 4 .
4The constraints imposed in the Bayesian inference: Observed maximum mass of NS, Radius of 2.07 M ⊙ NS, Radius of 1.4 M ⊙ NS.
For a NS mass of 1.4M ⊙ , we have a prediction error below 2% for the radius R 1.4 and 5% for the tidal deformability Λ 1.4 . This implies that using the ANN, we can infer Λ 1.4 and R 1.4 with an average error of 19.239 and 0.194 km, respectively. The NS maximum mass, M max , and the maximum radius R max , are also predicted with an error below 2%. Moreover, the saturation of the loss curves for the training and validation sets indicates that the training has converged to a minimum and that the model can generalize sufficiently well.ŷ
RMSE (RMSE/ȳ) ×100
M max
0.024
1.1
R max
0.088
0.8
R 1.4
0.194
1.5
Λ 1.0
123.800
3.7
Λ 1.4
19.239
4.3
Λ 1.8
5.344
8.2
Table 5. Root Mean Squared Error (RMSE) on the test set, defined as (1/N )
, we compared the relative % residuals of the NS-ANN model with the Skyrme model for both P1 and P2. It is worth noting that the results for P1 are very comparable to the original physics model. This is expected because P1 is the prior range in which the NS-ANN was originally trained. The relative % residuals for the NS maximum mass, radius for 2.07 and 1.4 M⊙ NS, and tidal deformability for 1.4 M⊙ NS are ∼ 1%, while the square NS M max R 2.07 R 1.4 Λ 1.4 C 2Table 8. The maximum relative residual for the maximum mass of a neutron star (NS), denoted as Mmax, is attained. Additionally, we determine the radius R2.07 for a 2.07 solar mass (M⊙) NS, the radius R1.4 and the dimensionless tidal deformability Λ 1.4 for a 1.4 M ⊙ NS, as well as the squared speed of sound at the core of the NS with maximum mass. These quantities are obtained using the prior sets P1 and P2.s
Units
M ⊙
km
km
...
c 2
Maximum relative % residual
P1
0.3
0.6
0.3
1.2 2.5
P2
0.8
1.6
1.5
5.3 5.1
Model
P1
P2
ANN
2.23 min
3.08 min
Skyrme 16h 27 min 19h 55 min
ρ0 = 0.16 fm −3
E[X] is the expectation value of a random variable X and is defined as E[X] = ∞ −∞ xp(x), where p(x) is the probability density function.
https://github.com/ameya1101/NS-ANN
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Spanning the full range of neutron star properties within a microscopic description. T Malik, M Ferreira, M B Albino, C Providência, 10.1103/PhysRevD.107.103018Phys. Rev. D. 1071030182301.08169T. Malik, M. Ferreira, M.B. Albino and C. Providência, Spanning the full range of neutron star properties within a microscopic description, Phys. Rev. D 107 (2023) 103018 [2301.08169].
Bayesian inference of neutron-star observables based on effective nuclear interactions. J Zhou, J Xu, P Papakonstantinou, 10.1103/PhysRevC.107.055803Phys. Rev. C. 107558032301.07904J. Zhou, J. Xu and P. Papakonstantinou, Bayesian inference of neutron-star observables based on effective nuclear interactions, Phys. Rev. C 107 (2023) 055803 [2301.07904].
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"Sanfeng Wu \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Hongyi Yu \nDepartment of Physics\nCenter of Theoretical and Computational Physics\nThe University of Hong Kong\nHong KongChina\n",
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"Grant Aivazian \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
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"David G Mandrus \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n",
"Di Xiao \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburgPAUSA\n",
"Wang Yao \nDepartment of Physics\nCenter of Theoretical and Computational Physics\nThe University of Hong Kong\nHong KongChina\n",
"Xiaodong Xu \nDepartment of Material Science and Engineering\nUniversity of Washington\n98195SeattleWashingtonUSA\n\nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Jason S Ross \nDepartment of Material Science and Engineering\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Sanfeng Wu \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Hongyi Yu \nDepartment of Physics\nCenter of Theoretical and Computational Physics\nThe University of Hong Kong\nHong KongChina\n",
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"Aaron M Jones \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Grant Aivazian \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n",
"Jiaqiang Yan \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n",
"David G Mandrus \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n",
"Di Xiao \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburgPAUSA\n",
"Wang Yao \nDepartment of Physics\nCenter of Theoretical and Computational Physics\nThe University of Hong Kong\nHong KongChina\n",
"Xiaodong Xu \nDepartment of Material Science and Engineering\nUniversity of Washington\n98195SeattleWashingtonUSA\n\nDepartment of Physics\nUniversity of Washington\n98195SeattleWashingtonUSA\n"
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] | [] | Monolayer group VI transition metal dichalcogenides have recently emerged as semiconducting alternatives to graphene in which the true two-dimensionality (2D) is expected to illuminate new semiconducting physics. Here we investigate excitons and trions (their singly charged counterparts) which have thus far been challenging to generate and control in the ultimate 2D limit. Utilizing high quality monolayer molybdenum diselenide (MoSe 2 ), we report the unambiguous observation and electrostatic tunability of charging effects in positively charged (X + ), neutral (X o ), and negatively charged (X -) excitons in field effect transistors via photoluminescence. The trion charging energy is large (30 meV),enhanced by strong confinement and heavy effective masses, while the linewidth is narrow (5 meV) at temperatures below 55 K. This is greater spectral contrast than in any known quasi-2D system. We also find the charging energies for X + and Xto be nearly identical implying the same effective mass for electrons and holes. | 10.1038/ncomms2498 | [
"https://arxiv.org/pdf/1211.0072v2.pdf"
] | 9,872,370 | 1211.0072 | e265d065c74d788d155dd8fc65e788bd0604a074 |
Electrical Control of Neutral and Charged Excitons in a Monolayer Semiconductor
Jason S Ross
Department of Material Science and Engineering
University of Washington
98195SeattleWashingtonUSA
Sanfeng Wu
Department of Physics
University of Washington
98195SeattleWashingtonUSA
Hongyi Yu
Department of Physics
Center of Theoretical and Computational Physics
The University of Hong Kong
Hong KongChina
Nirmal J Ghimire
Department of Physics and Astronomy
University of Tennessee
37996KnoxvilleTennesseeUSA
Materials Science and Technology Division
Oak Ridge National Laboratory
37831Oak RidgeTennesseeUSA
Aaron M Jones
Department of Physics
University of Washington
98195SeattleWashingtonUSA
Grant Aivazian
Department of Physics
University of Washington
98195SeattleWashingtonUSA
Jiaqiang Yan
Materials Science and Technology Division
Oak Ridge National Laboratory
37831Oak RidgeTennesseeUSA
Department of Materials Science and Engineering
University of Tennessee
37996KnoxvilleTennesseeUSA
David G Mandrus
Department of Physics and Astronomy
University of Tennessee
37996KnoxvilleTennesseeUSA
Materials Science and Technology Division
Oak Ridge National Laboratory
37831Oak RidgeTennesseeUSA
Department of Materials Science and Engineering
University of Tennessee
37996KnoxvilleTennesseeUSA
Di Xiao
Department of Physics
Carnegie Mellon University
15213PittsburgPAUSA
Wang Yao
Department of Physics
Center of Theoretical and Computational Physics
The University of Hong Kong
Hong KongChina
Xiaodong Xu
Department of Material Science and Engineering
University of Washington
98195SeattleWashingtonUSA
Department of Physics
University of Washington
98195SeattleWashingtonUSA
Electrical Control of Neutral and Charged Excitons in a Monolayer Semiconductor
1 * These authors contributed equally to the work. †
Monolayer group VI transition metal dichalcogenides have recently emerged as semiconducting alternatives to graphene in which the true two-dimensionality (2D) is expected to illuminate new semiconducting physics. Here we investigate excitons and trions (their singly charged counterparts) which have thus far been challenging to generate and control in the ultimate 2D limit. Utilizing high quality monolayer molybdenum diselenide (MoSe 2 ), we report the unambiguous observation and electrostatic tunability of charging effects in positively charged (X + ), neutral (X o ), and negatively charged (X -) excitons in field effect transistors via photoluminescence. The trion charging energy is large (30 meV),enhanced by strong confinement and heavy effective masses, while the linewidth is narrow (5 meV) at temperatures below 55 K. This is greater spectral contrast than in any known quasi-2D system. We also find the charging energies for X + and Xto be nearly identical implying the same effective mass for electrons and holes.
Introduction
Above bandgap photo-excitation creates electrons and holes in the conduction and valence bands respectively. If the screening is weak enough the attractive Coulomb interaction between one electron and one hole creates a bound quasi-particle known as a neutral exciton (X o ) which has an energy structure similar to a neutral hydrogen atom. Excitons can further become charged by binding an additional electron (X -) or hole (X + ) to form charged 3-body excitons analogous to Hor H 2 + respectively [1][2][3] . These exciton species are elementary quasi-particles describing the electronic response to optical excitation in solids and are integral to many optoelectronic applications from solar cells and light emitting diodes (LEDs) 4 to optical interconnects 5 and quantum logical devices 6,7 .
The main arena for the exploration of excitonic physics has been 3D semiconductors and their heterostructures that form quasi-2D quantum wells where the carrier wavefunction typically occupies a few tens to thousands of atomic layers. Observation and control of excitons in truly 2D systems has been a long pursued goal, largely motivated by the enhancement in exciton and trion binding energies in the strict 2D limit 8 . In addition, unlike semiconductor heterostructures, the close proximity of external stimuli to the exciton wavefunction can offer unprecedented tunability and diversifies the applications possible in devices made with 2D semiconductors.
Here we report the experimental observation and control of the fascinating excitonic physics in a 2D semiconductor by utilizing high quality monolayer molybdenum diselenide (MoSe 2 ). MoSe 2 belongs to the group-VI transition metal dichalcogenides which form in layers weakly bound to each other by Van der Waals forces. The monolayers have received much attention recently as they make up a new class of 2D semiconductors with a direct bandgap in the visible frequency range and are predicted to exhibit coupled spin-valley physics 9 . Recent progresses focus on MoS 2 , including the demonstration of having a direct bandgap 10,11 , high mobility electronics 12 , optical generation of valley polarization [13][14][15] , and electrical control of Berry phase properties 16 .
Using a back-gated field effect transistor (FET) device, we demonstrate the reversible electrostatic tunability of the exciton charging effects from positive (X + ) to neutral (X o ) and to negative (X -). We observe a large trion binding energy of 30 meV with a narrow emission linewidth of 5 meV. These narrow, well separated features have temperature dependence of typical 2D excitons and exist at high temperature suggesting remarkable stability. Interestingly, the binding energies of X + and Xare similar implying that low energy electrons and holes in MoSe 2 have the same effective mass. Our work demonstrates that monolayer MoSe 2 is a true 2D
semiconductor opening the door for the investigation of phenomena such as exciton condensation [17][18][19] and the Fermi-edge singularity 20,21 , as well as for a new generation of optoelectronic devices such as LEDs and excitonic circuits 5 .
Results
Crystal structure and spectral features of monolayer MoSe 2 . In monolayer MoSe 2 , Mo and Se atoms form a 2D hexagonal lattice with trigonal prismatic coordination (Fig. 1a). First principles calculations show that it has a direct bandgap at the corners (K points) of the first Brillouin zone (Fig. 1b). The curvature of the bands suggests comparable effective mass for low energy electrons and holes at K points (Fig. 1c) 9 . These band edge electrons and holes near K points are predominantly from the d-orbitals of Mo atoms. Their wavefunctions are calculated to be strongly confined in the Mo layer within a length scale of ~0.2 nm in the out-of-plane direction. Monolayer MoSe 2 is thus an ideal nanomaterial for exploring excitonic physics in the ultimate 2D limit.
We use mechanical exfoliation to obtain monolayer MoSe 2 on 300 nm SiO 2 on n+ doped Si and atomic force microscopy (AFM) to identify the layer thickness 22 . Figure 1d,e show the optical micrograph and the corresponding AFM image of a representative sample, in which the monolayer thickness of ~0.7 nm is identified 12 (Fig. 1f). Standard electron beam lithography (EBL) is used to fabricate monolayer field effect transistors (Fig. 1g). With the contacts simply grounded, the n+ Si functions as a back gate providing uniform electrostatic doping in the MoSe 2 ( Fig. 1h).
The excitonic features of MoSe 2 are investigated by differential reflectance and microphotoluminescence (µ-PL) measurements (Methods). Figure 2 shows the results from an unpatterned MoSe 2 sample, S1. At 20 K, we observe two main features associated with the A and B excitons in the differential reflection spectrum 10,11,14,[23][24][25] (Fig. 2a). The presence of A and B excitons has been attributed to spin-orbit coupling induced valence band splitting in bulk 24 .
The observed energy difference of ~200 meV agrees well with the calculated splitting (180 meV) in monolayers (Fig. 1c).
With the same sample and temperature under 2.33 eV laser excitation, the PL spectrum does not show a measurable feature which can be attributed to the B exciton, likely since it is not the lowest energy transition. Instead, we observe two pronounced peaks at 1.659 and 1.627 eV in the vicinity of the A exciton (Fig. 2b). Note that the PL spectrum lacks the broad low energy peak observed in MoS 2 which has been attributed to defect-related, trapped exciton states 10,11,14,25 .
The striking spectral features demonstrate the high quality of our MoSe 2 samples (Methods) and provide strong evidence for monolayer MoSe 2 being a direct bandgap semiconductor (Supplementary Figure S1) where the two distinct transitions are excitons. The higher energy emission at 1.659 eV is the neutral exciton, X o , and the lower energy peak is a trion 3 . In unpatterned samples, we assume the trion to be Xsince all measured devices show n-doped characteristics (Supplementary Figure S2). All measured unpatterned samples show a binding energy, which is the energy difference between trion and X o , of ~30 meV (Fig. 2b inset). This is more than twice typical numbers reported in GaAs quantum wells 3,26,27 and similar to a recent mention in MoS 2 14 .
Gate dependence of MoSe 2 photoluminescence. In order to confirm the above assignment and control the exciton charging effects, we performed gate dependent PL measurements using monolayer MoSe 2 FETs. Here, the excitation laser is at 1.73 eV for better resonance with the luminescent states. Figure 3a shows a color map of the PL spectrum of device D1 at 30 K as a function of back-gate voltage, V g , in which we clearly observe four spectral features whose intensities strongly depend on V g . Near zero V g the spectrum shows a broad low energy feature around 1.57 eV and a narrow high energy peak at 1.647 eV. With large V g of either sign, these peaks disappear and a single emission peak dominates the spectrum. Both peaks (at negative or positive V g ) have similar energies and intensities with the latter increasing with the magnitude of V g .
This observed gate dependence confirms the assignment of states as labeled in Figure 3a.
Since the broad low energy peak does not show up in unpatterned samples before FET fabrication, we attribute it to exciton states trapped to impurities (X I ) 10,11,14,25 which are likely introduced during EBL processing and are not the focus of this paper. The sharp peak at 1.647 eV is the X o , slightly red-shifted compared to unpatterned samples. From the gate dependence, we identify the peaks near 1.627 eV as the Xand X + trions when V g is largely positive and negative respectively. Remarkably, these two distinct quasi-particles (X + and X -) exhibit a nearly identical binding energy. The difference is within 1.5 meV over the whole applied V g range.
Since the binding energy of a trion is dependent on its effective mass, this observation implies that the electron and hole have approximately the same effective mass.
The gate dependent measurements unambiguously demonstrate the electrical control of exciton species in a truly 2D semiconductor, as illustrated in Figure 3b. The conversion from X o to trion can be represented as e(h) + X o → X -(X + ), where e and h represent an electron or a hole respectively. By setting V g to be negative, the sample is p-doped, favoring excitons to form lower energy bound complexes with free holes. As V g decreases, more holes are injected into the sample and all X o turn into X + to form a positively charged hole-trion gas. With positive V g a similar situation occurs with free electrons to form an electron-trion gas. In the following, we
show that a standard mass action model can be used to describe the conversion dynamics. In the simulation, we first fit the X o curve to obtain our two free parameters: the maximum background electron concentration ݊ ୫ୟ୶ = 3.6 × 10 ଵ cm ିଶ when the trion intensity saturates and the photo-excited electron concentration ݊ = 1.5 × 10 ଵ cm ିଶ . We then fit the trion gate dependence with these parameters held fixed (Supplementary Note 1). The deviation in the trion experimental data from the calculated curve near zero V g is artificial due to the mutual background from X o and X I . We note that this inferred electron concentration is much smaller than the product of the gate capacitance and V g . This discrepancy can be attributed to the large contact resistance (Supplementary Figure S4) of the sample, which prevents the carrier concentration from reaching equilibrium on the experimental time scale at 30 K. We expect that future improved contact technologies will eliminate this effect.
Temperature dependence of MoSe 2 photoluminescence. The observed exciton states also
show fine features consistent with 2D excitons, such as temperature dependent line shape, peak energy, and relative weight of X o and trion, which further supports the excitonic nature of this monolayer system (Fig. 4). Figure 4a shows However, the Xpeak shows a slightly asymmetric profile with a long low-energy tail consistent with electron recoil effects 27 : The recombination of a Xwith momentum k will emit a photon and leave a free electron with the same momentum k due to momentum conservation. From energy conservation, the emitted photon has an energy ℏ߱
= ℏ߱ ୭ − ℏ మ ଶ * మ ெ ெ ష ,
where ℏ߱ ୭ is the energy of trions with ݇ = 0, and ܯ ଡ଼ and ܯ ଡ଼ ష are the X o and Xeffective masses, respectively.
The lineshape of trion PL will thus be the convolution of a symmetric peak function (hyperbolic secant) and an exponential low-energy tail function (Supplementary Figure S5 and Supplementary Note 2). When the temperature is above 70K, we find homogenous broadening dominates over electron recoil and the PL spectrum is fit well by two hyperbolic secant functions (Fig. 4c).
From the fits we extract the Xand X o peak position (Fig. 4d) and the ratio of the integrated intensity of the Xto the X o (Fig. 4e) where we do not present trion data above 150 K because it becomes negligible. We find that the peak positions are fit well (solid line in Fig. 4d) using a standard semiconductor bandgap dependence 29
of ܧ ሺܶሻ = ܧ ሺ0ሻ − ܵۦℏ߱ۧ ቂcoth ቀ ۦℏఠۧ ଶ் − 1ቁቃ
where ܧ ሺ0ሻ is the ground state transition energy at 0 K, ܵ is a dimensionless coupling constant, and ۦℏ߱ۧ is an average phonon energy. From the fits we extract for X o (X -) the ܧ = 1.657 (1.625) eV, ܵ = 1.96 (2.24) and ۦℏ߱ۧ = 15 meV for both. Applying our mass action model (Supplementary Figure S3 and Supplementary Note 1) with a trion binding energy of 30 meV results in a good fit to the X -:X o intensity ratio (solid line in Fig. 4e).
Discussion
In summary, we have shown that monolayer MoSe 2 is a true 2D excitonic system which exhibits strong electrostatic tuning of exciton charging via a standard back-gated FET. The observed narrow, well separated spectral features are within the Ti-Sapphire laser spectral range and thus provide remarkable opportunities to selectively probe and control specific excitons using current continuous wave and ultrafast Ti-Sapphire laser technologies. Our results further demonstrate that high quality monolayer dichalcogenides can serve as a platform for investigating excitonic physics and photonic applications in the truly 2D limit with the potential Electron recoil effects result in an exponential lineshape on one side the photoluminescence peaks of trions. Details are described in Supplemental Note 2.
Supplementary Note 1: Mass action model.
To determine the relative intensity of the photoluminescence (PL) signals from the various exciton species we adopt a steady state (dynamical equilibrium) model of all the particles in our system. Here, for simplicity, we consider electron-trion and assume that the free hole in the system is negligible. We denote ݊ ଡ଼ , ݊ ଡ଼ ష , and ݊ ୣ for the concentration of X o , X -, and free electrons. ݊ ≡ ݊ ଡ଼ + ݊ ଡ଼ ష denotes the number of photoexcited electrons, and ݊ ≡ ݊ ୣ + ݊ ଡ଼ ష denotes the background electrons (doping level) before light excitation. ݊ and ݊ are the initial conditions controlled by gate voltage and laser intensity, while ݊ ଡ଼ , ݊ ଡ଼ ష , and ݊ ୣ are steady state variables.
To establish the relationship between these quantities we first write the reaction rate equation for trion formation, X o + e -→ X -. From the law of mass action with trions we have 28,32,33 :
݊ ଡ଼ ݊ ୣ ݊ ଡ଼ ష = ܣ ݇ ܶ exp (− ܧ ݇ ܶ )
Here ܶ is the temperature, k B is Boltzmann constant, ܧ is the trion binding energy and ܣ = ସெ πℏ మ ெ ష = 6.18 × 10 ଵଵ ଵ ୡ୫ మ ୫ୣ in MoSe 2 in which ܯ ଡ଼ = ݉ ୣ + ݉ ୦ and ܯ ଡ଼ ష = 2݉ ୣ + ݉ ୦ are the exciton and trion effective masses respectively. Next, from charge conservation we obtain:
݊ ୣ + ݊ ଡ଼ + 2݊ ଡ଼ ష = ݊ + ݊ .
Solving the above equations gives
݊ ଡ଼ ష = ݊ + ݊ + ݊ − ඥሺ݊ + ݊ + ݊ ሻ ଶ − 4݊ ݊ 2 Here, ݊ = ܣ ݇ ܶ exp (− ா ా ் )
. By plotting these quantities as a function of T and ݊ , we are able to model the full range of our experimental data (see Supplemental Figure S3). We note that the entire picture and the values we have here for electron-trion can be applied directly to the hole-trion thanks to the same effective mass of electron and hole in our massive Dirac Fermion system. Therefore we have also explained the data in the hole-domain, which is labeled by the negative value of ݊ in Supplemental Figure S3.
Supplementary Note 2: Electron recoil effects.
Because of energy and momentum conservation, the radiative recombination of an electron-trion with center of mass wave vector k results in a conduction electron with wave vector k and a photon with energy ℏ߱ = ℏ߱ +
ℏ మ మ ଶெ − ℏ మ మ ଶ * = ℏ߱ − ℏ మ ଶ * మ ெ ெ
, where ܯ ் = 2݉ * + ݉ * is the trion mass, ܯ = ݉ * + ݉ * is the exciton mass, and ℏ߱ is the energy of trion with k = 0. The photon emission rate is given by the optical matrix element )ܓ(ܯ and trion distribution ,)ܓ(݂ which is
ܴሺ߱ሻ = න |ሻܓሺܯ|ܓ݀ ଶ ݂ሺܓሻߜ(ℏ߱ − ℏ߱ + ܧሺ݇ሻ) where ܧሺ݇ሻ ≡ ℏ మ ଶ * మ ெ ெ .
In the low density limit, the distribution function )ܓ(݂ can be approximated by the Boltzmann distribution ݂ሺܓሻ ∝ exp ቀ− ℏ మ మ ଶெ ಳ ் ቁ. The optical matrix element |ܯሺܓሻ| ଶ has already been numerically evaluated by Stébé et al. 34 . Although for the 2D case they only give results for ߪ ≡ * * = 0, 0.1, 0.2, and 0.3, we find those curves can all be well approximated by an exponentially decaying function |ܯሺܓሻ| ଶ = |ܯሺሻ| ଶ ݁ ିଶ.ଷா()/ா బ . With ܧ ≡ 2ሺ1 + ߪሻ|ܧ ଷ |, and ܧ ଷ is the 3D exciton binding energy. We expect this equation applies for any ߪ ∈ ሾ0,1ሿ.
The photon emission rate as a function of ߱ is then
ܴሺ߱ሻ = න |ሻܓሺܯ|ܓ݀ ଶ ݂ሺܓሻߜ൫ℏ߱ − ℏ߱ + ܧሺ݇ሻ൯ = ܴሺ߱ ሻ exp − ൬ 27.3 ܧ + ݉ * ܯ 1 ݇ ܶ ൰ ሺℏ߱ − ℏ߱ሻ൨ Θሺ߱ − ߱ሻ
Here Θ is the Heaviside step function. In the MoSe 2 system, ߪ = 1 and the 3D exciton binding energy is ܧ ଷ = ܴ ௬ * = 50 meV. We have plotted the line shape ܴሺ߱ሻfor several temperatures in Supplemental Figure S5. Thus the trion line shape can be fit through convolution of the above contribution with a symmetric broadening effect.
Figure 3c
3cshows the extracted X o (black) and trion (red) peak intensity as a function of V g where we have adjusted the negative V g data due to background signal. The plot shows that the maximum X o intensity is about equal to the saturated trion PL when X o vanishes. This observation indicates conservation of the total number of X o and trion in the applied voltage range and similar radiative decay rates for both quasi-particles. Thus the PL intensity represents the amount of the corresponding exciton species. Since the dynamic equilibrium of free electrons, holes, and excitons are governed by the rate equation and law of mass action 27 , we calculate the gate dependent X o and trion abundance (Supplementary Figure S3), shown by the solid lines in Figure 3c, which agrees with the data.
the evolution of Xand X o (normalized PL) as a function of temperature in an unpatterned sample, S2, under 1.96 eV laser excitation. At low temperatures, we again observe a binding energy of 30 meV. As the temperature rises we see the Xsignal drop significantly at about 55 K which we attribute to electrons escaping their bound trion state due to thermal fluctuations (Supplementary Note 1).Figure 4bis the zoom-in plot at 15K where we observe slightly different line shapes for Xand X o . The X o peak is symmetric showing homogenous thermal broadening effects and is well fit by a hyperbolic secant function which yields a full width half maximum of 5 meV27,28 .
Figure
to outperform quasi-2D systems. The results represent unique prospects for this burgeoning class of 2D materials in addition to the recent attention received for their valley physics. Our work expands the horizons for using 2D semiconductors for diverse fundamental studies and technical applications. Note: During the review process, we became aware of the independent observation of negatively charged exciton in monolayer MoS 2 30 and investigation of PL intensity of thin film MoSe 2 as a function of temperature 31 . Methods Sample growth. MoSe 2 single crystals were grown by vapor transport. First, MoSe 2 powder was prepared by heating a stoichiometric mixture of Mo (Alfa, 99.999%) powder and Se (Alfa, 99.999%) pieces sealed in a quartz tube under 1/3 atmosphere of pure argon. The ampoule was slowly heated up to 850 o C and kept at this temperature for 24 hours. Then, 3.5g of MoSe 2 powder was mixed with 0.5g of iodine and sealed in a quartz tube. The sealed ampoule was loaded into a tube furnace for 10 days with the hot zone kept at 1050 o C and the growth zone at 1000 • C. Plate-like crystals with typical dimensions 6 × 10 × 0.1 mm 3 were obtained at the cold end. Room temperature X-ray diffraction confirmed the samples are single phase with 2H structure. Elemental analysis was performed using a Hitachi TM-3000 tabletop electron microscope equipped with a Bruker Quantax 70 energy dispersive x-ray (EDX) system. The analysis confirmed the expected stoichiometric Mo:Se ratio in the crystals. Device fabrication. After mechanical exfoliation of MoSe 2 monolayers on SiO2 substrates, a JEOL JBX-6300FS electron beam lithography system was used to pattern contacts in a 300nmbilayer PMMA resist. An e-beam evaporator was used to deposit 6/60 nm of Ti/Au followed by liftoff in hot acetone, cleaning in several IPA baths, and drying under N2.Optical measurements. All spectra were collected via a Princeton Instruments Acton 2500i grating spectrometer and an Andor Newton CCD with samples in a helium flow cryostat. For differential reflectance measurements an Ocean Optics tungsten halogen white light source is reflected into a 40X ultra-long working distance objective using a neutral density (ND), achromatic beam splitter and then focused on the sample (~2 um spot size). The reflected signal from the sample is collected by the objective and sent back through the beam splitter into the spectrometer. For PL measurements, a similar setup is used except the ND beam splitter is replaced by a dichroic beam splitter appropriate for the laser wavelength and a laser line notch filter is inserted before the spectrometer.30. Mak, K. F. et al. Tightly bound trions in monolayer MoS 2 . Nature Materials (2012).doi:10.1038/nmat3505 31. Tongay, S. et al. Thermally Driven Crossover from Indirect toward Direct Bandgap in 2D Semiconductors: MoSe 2 versus MoS 2 . Nano letters 12, 1| MoSe 2 characteristics and devices. a, Coordination structure and top view of monolayer MoSe 2 . b, Density function theory calculated band structure. c, band structure at K point shows 180 meV valence band splitting due to spin-orbit coupling. d, Optical micrograph of exfoliated MoSe 2 flake on 300nm SiO 2 . Scale bar: 5 µm. e, atomic force microscope (AFM) image of area highlighted in (d). Scale bar: 1 µm. f, AFM line scan along dashed line in (e). g, optical micrograph of MoSe 2 device. Scale bar: 5 µm. h, schematic of back-gated MoSe 2 device.
Figure 2| Figure 3|
2|3|Differential reflectance and photoluminescence spectra of monolayer MoSe 2 at 20 K. a, Differential reflectance shows A and B excitons. b, Photoluminescence (PL) excited by 2.33 eV laser shows neutral exciton (X o ) and the lower energy charged exciton (X -). PL from the B exciton has not been observed. Inset: PL of the exciton peaks. The Xshows a charging energy of about 30 meV. Electrostatic control of exciton charge. a, MoSe 2 photoluminescence (color scale in counts) is plotted as a function of back-gate voltage. Near zero doping, we observe mostly neutral and impurity-trapped excitons. With large electron (hole) doping, negatively (positively) charged excitons dominate the spectrum. b, Illustration of the gate dependent trion and exciton quasi-particles and transitions. c, trion and exciton peak intensity vs. gate voltage at dashed arrows in (a). Solid lines are fits based on the mass action model.
Figure 4| Figure S1 :Figure S3 :
4|S1S3Temperature dependence of photoluminescence spectrum. a, Normalized photoluminescence of monolayer MoSe 2 vs. temperature. b, Line shape fitting at 15 K. Black is data. Red and blue curves are fits with and without considering the electron-recoil effect. c, Data and fit at 70 K using two symmetric peaks. d, Neutral exciton (black) and trion (red) peak position vs. temperature with fits (solid lines). e, Integrated area ratio of trion:exciton vs.temperature with mass action model fitting (red). Photoluminescence (PL) of single and bilayer MoSe 2 . Both spectrums were taken at 20 K with 532 nm excitation laser. Intensity is normalized to laser power and integration time. Under the same experimental conditions, bilayer samples (confirmed by atomic force microscopy) exhibit suppressed photoluminescence intensity and an overall redshift compared to single layer samples. Like MoS 2 , these PL features suggest the crossover to direct bandgap at the single layer 10,11 and offer a method of determining the layer thickness. Mass action model plots. Calculated densities of a, exciton, b, trion and c, electron 2D MoSe 2 as a function of background electron density and temperature with a fixed amount of absorbed photons of 3.2 ൈ 10 10 cm -2 during exciton life time. The white and blue lines correspond to the experimental data in the main text for gate and temperature dependence respectively. Details on the model are found in Supplementary Note 1. Supplementary Figure S4: Low temperature transport of single layer MoSe 2 . a. Sourcedrain bias vs. current for various gate voltages. b. Gate voltage vs. current for various sourcedrain biases. The gate dependent photoluminescence (PL) presented in the main text was measured at zero source-drain bias. These figures illustrate that at any gate voltage, there isminiscule conductance for this experimental condition. Specifically, we find the resistance near zero bias to be on the order of 100 GΩ for all gate voltages. Only with appreciable biases above 100 mV do devices show significant conduction. With this in mind, it will be interesting to investigate the gate dependent PL with good metal contacts.
Figure S5 :
S5Electron recoil effect on photoluminescence lineshape in trions.
Acknowledgments:The authors thank David Cobden and Ming Gong for helpful discussions. This work is mainlySince a maximized X o corresponds to a minimized carrier density (see SupplementalFigure S3and Supplementary Note 1), this gate dependence suggests that samples are originally n-doped. This is also consistent with observations 12,14 in MoS 2 .
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| [] |
[
"Strongest model-independent bound on the lifetime of Dark Matter",
"Strongest model-independent bound on the lifetime of Dark Matter"
] | [
"Benjamin Audren [email protected] \nInstitut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n",
"Julien Lesgourgues [email protected] \nInstitut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n\nCERN\nTheory DivisionCH-1211Geneva 23Switzerland\n\nLAPTh\nUniv. de Savoie\nCNRS\nB.P.110, Annecy-le-VieuxF-74941France\n",
"Gianpiero Mangano [email protected] \nIstituto Nazionale di Fisica Nucleare -Sezione di Napoli\nComplesso Universitario di Monte S. Angelo\nI-80126NapoliItaly\n",
"Pasquale Dario Serpico [email protected] \nLAPTh\nUniv. de Savoie\nCNRS\nB.P.110, Annecy-le-VieuxF-74941France\n",
"Thomas Tram [email protected] \nInstitut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n"
] | [
"Institut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland",
"Institut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland",
"CERN\nTheory DivisionCH-1211Geneva 23Switzerland",
"LAPTh\nUniv. de Savoie\nCNRS\nB.P.110, Annecy-le-VieuxF-74941France",
"Istituto Nazionale di Fisica Nucleare -Sezione di Napoli\nComplesso Universitario di Monte S. Angelo\nI-80126NapoliItaly",
"LAPTh\nUniv. de Savoie\nCNRS\nB.P.110, Annecy-le-VieuxF-74941France",
"Institut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland"
] | [] | Dark Matter is essential for structure formation in the late Universe so it must be stable on cosmological time scales. But how stable exactly? Only assuming decays into relativistic particles, we report an otherwise model independent bound on the lifetime of Dark Matter using current cosmological data. Since these decays affect only the low-multipoles of the CMB, the Dark Matter lifetime is expected to correlate with the tensor-to-scalar ratio r as well as curvature Ω k . We consider two models, including r and r + Ω k respectively, versus data from Planck, WMAP, WiggleZ and Baryon Acoustic Oscillations, with or without the BICEP2 data (if interpreted in terms of primordial gravitational waves). This results in a lower bound on the lifetime of CDM given by 160 Gyr (without BICEP2) or 200 Gyr (with BICEP2) at 95% confidence level. | 10.1088/1475-7516/2014/12/028 | [
"https://arxiv.org/pdf/1407.2418v1.pdf"
] | 33,308,106 | 1407.2418 | 2a22b0326d94c4de584dd41d0d359bad884c7b28 |
Strongest model-independent bound on the lifetime of Dark Matter
9 Jul 2014
Benjamin Audren [email protected]
Institut de Théorie des Phénomènes Physiques
École Polytechnique Fédérale de Lausanne
CH-1015LausanneSwitzerland
Julien Lesgourgues [email protected]
Institut de Théorie des Phénomènes Physiques
École Polytechnique Fédérale de Lausanne
CH-1015LausanneSwitzerland
CERN
Theory DivisionCH-1211Geneva 23Switzerland
LAPTh
Univ. de Savoie
CNRS
B.P.110, Annecy-le-VieuxF-74941France
Gianpiero Mangano [email protected]
Istituto Nazionale di Fisica Nucleare -Sezione di Napoli
Complesso Universitario di Monte S. Angelo
I-80126NapoliItaly
Pasquale Dario Serpico [email protected]
LAPTh
Univ. de Savoie
CNRS
B.P.110, Annecy-le-VieuxF-74941France
Thomas Tram [email protected]
Institut de Théorie des Phénomènes Physiques
École Polytechnique Fédérale de Lausanne
CH-1015LausanneSwitzerland
Strongest model-independent bound on the lifetime of Dark Matter
9 Jul 2014Prepared for submission to JCAP
Dark Matter is essential for structure formation in the late Universe so it must be stable on cosmological time scales. But how stable exactly? Only assuming decays into relativistic particles, we report an otherwise model independent bound on the lifetime of Dark Matter using current cosmological data. Since these decays affect only the low-multipoles of the CMB, the Dark Matter lifetime is expected to correlate with the tensor-to-scalar ratio r as well as curvature Ω k . We consider two models, including r and r + Ω k respectively, versus data from Planck, WMAP, WiggleZ and Baryon Acoustic Oscillations, with or without the BICEP2 data (if interpreted in terms of primordial gravitational waves). This results in a lower bound on the lifetime of CDM given by 160 Gyr (without BICEP2) or 200 Gyr (with BICEP2) at 95% confidence level.
Introduction
Stability and particle physics
Although the existence of dark matter (DM) is well established by a large number of observations in cosmology and astrophysics, we have presently very few clues on its particle physics nature. This is mostly due to the purely gravitational origin of the evidence collected so far, which does not provide any handle for particle identification. While a number of strategies are ongoing to constrain or detect different classes of models, it is worth remarking that already some of the most basic DM properties can help shedding light on its nature. One such example is provided by the high stability that this species must possess. If one thinks of the Standard Model (SM) of particle physics, most of its particles are unstable: exact stability is in fact the exception and must be enforced by some exact symmetry, such as the unbroken QED gauge symmetry for the electron or Lorentz symmetry for the lightest neutrino. Much more frequent are examples of meta-stability due to some approximate symmetries, such as for the heavier neutrino states, for the ones often found in nuclear physics (including the neutron decay) due to mass quasi-degeneracies, and, possibly, for the proton itself if the accidental baryon number symmetry is broken at some very high energy scale as in Grand Unified gauge theories. In fact, this kind of considerations provides a useful guideline in DM model building, see e.g. [1].
Loosely speaking, one knows that the DM lifetime should be at least comparable to the lifetime of the universe, otherwise it could not fulfil its role in structure formation and astrophysical observations. However, inferring from that phenomenological condition an infinite lifetime is a strong prejudice dictated by simplicity, but with very little empirical or theoretical justification. For example, for typical WIMP candidates one often assumes a discrete Z 2 symmetry under which the SM particles and DM have opposite charge, but it is easily conceivable that this symmetry is broken at a more fundamental level, with the only requirement that the lifetime of the DM particle is sufficiently long. Stringent bounds on the lifetime τ of WIMP DM candidates with electroweak scale masses come, for example, from the diffuse gamma ray flux , at the level of τ 10 26 s, see for instance [2]. Hence, allowed timescales for the decay should be longer than a billion times the lifetime of the universe, which would exclude any plausible effect on gravitational structures.
Gravitational effects
The drawback of these considerations is their model-dependence. In particular, the bounds depend on the nature and energy distribution of the by-products of the decay. Interestingly, however, looser but way more general and robust constraints can be obtained again from purely gravitational considerations. The key property that allows one to constrain the DM lifetime gravitationally is that in the decay process, a non-relativistic (usually cold) DM component is replaced by a combination of radiation and of massive particles, which in turn have a finite velocity dispersion. This alters notably the growth of structures. More specifically, if significant DM decay takes place, the background evolution of the universe can show departure from the standard case and affect several cosmological observables (e.g. the size of the sound horizon at recombination). At the perturbation level, one also expects an enhancement of the Late Integrated Sachs-Wolfe (LISW) effect, beyond the one due to the cosmological constant, as shown in [3] and described in section 3 below.
Previous works
In the past decade, several studies have derived constraints on the DM lifetime using cosmological data, starting from the study of decaying hot neutrino DM in [4]. The case of decaying cold DM was first analysed by Ichiki et al. [3], who found a 95% C.L. bound of 52 Gyr using WMAP-1yr temperature C data, and assuming decay into fully relativistic species. Ref. [5] developed the formalism to describe the cosmological effects of an unstable relic and its relativistic decay products, both at the background and perturbation levels. Since then, the bounds have been refined in two ways. First, more data sets on CMB temperature/polarization and on large scale structures have been included in the analysis. For example, by including WMAP-5yr, Type Ia supernova data, Lyman−α forest, large scale structure and weak lensing observations, Ref. [6] obtained a bound of 100 Gyr (and also updated or corrected previous bounds from [7][8][9]). Second, some more general bounds have been obtained by allowing the daughter particles to be massive and thus non-relativistic or only mildly relativistic, see for instance [10][11][12][13][14]. Most recently, a detailed formulation of the problem, both in presence of massless or massive decay products, has been given in [15]. In this reference, it has been additionally shown that the impact of σ 8 constraints are also important, and that a possible tension between the value of σ 8 inferred from Planck SZ cluster data and the one extrapolated from CMB temperature data could be resolved by assuming τ ∼ 200 Gyr and relativistic daughter particles. However, this estimate did not account for parameter degeneracies, and relied on the assumption that Planck SZ cluster results are not affected by systematic errors.
Scope and outline of this paper
In this paper, we aim at updating cosmological bounds on the DM lifetime with a proper statistical analysis, accounting for degeneracies and correlations with other cosmological parameters, as well as estimating the cosmological model dependence of the bound thus obtained.
In particular, we will check for degeneracies between decaying DM and spatial curvature, since both can have somewhat similar effects on the CMB. We also consider the impact of including or not BICEP2 results [16] on B-mode polarisation interpreted in terms of r.
In the following, we limit ourselves to the case of relativistic decay products, leaving the case of non-relativistic species for future investigation. Note that this case is nonetheless representative of several DM candidates, for which the decay products are either massless, or at least well inside the relativistic regime. This is usually the case, provided that the produced particles have a much smaller mass than the decaying DM matter particle, and that the decay happens reasonably late. The decay products could consist either in non-standard particles, or in standard model neutrinos produced with typical momenta much larger than their mass. A notable case of such a DM candidate is represented by the majoron J, with mass in the keV range [7,8,[17][18][19]. In the simplest see-saw-like models, the leading decay channel is in two relativistic neutrinos. The majoron lifetime is then inversely proportional to the square of standard active neutrino masses m ν ,
τ J = 16π m J v 2 m 2 ν . (1.1)
Here m J is the Majoron mass, and v the lepton number breaking scale [20]. Bounds on τ J can be used to constrain the value of v as function of the standard neutrino mass scale. Note that while the results of our study apply also to heavier DM candidates producing energetic neutrinos, these scenarios are better constrained using e.g. limits on the neutrino flux in the Milky Way, leading to stronger bounds (exceeding 10 6 Gyr, see for instance [21]) than what is found by using cosmological data only. On the other hand, the constraints discussed here are basically the only limits applying to dark matter decaying into unspecified, non-standard forms of dark radiation. This paper is structured as follows. In Section 2 we recall the formalism describing a cosmological scenario with a decaying DM candidate, both for the background and perturbation evolution. Note that we present perturbation equations both in the synchronous gauge (the only case treated in the previous literature) and in Newtonian gauge, which allowed us to double-check the numerical results we obtained. We then describe their implementation in the public numerical code class 1 [22,23]. Section 3 contains a short description of data sets used in the analysis and our results, and in Section 4 we conclude and give our outlooks.
Equations and implementation
Background equations
The background density of the decaying cold DM (dcdm) and of the produced decay radiation (dr) is governed by the two equations
ρ dcdm = −3 a a ρ dcdm − a Γ dcdm ρ dcdm , (2.1) ρ dr = −4 a a ρ dr + a Γ dcdm ρ dcdm , (2.2)
where Γ dcdm is the decay rate defined with respect to proper time, and primes denote derivatives with respect to conformal time. In the language of class, ρ dr and ρ dcdm fall into the category of {B}-variables since they must be evolved alongside the scale factor 2 . Choosing the fractional energy density in decaying DM plus decay radiation today, Ω dcdm + Ω dr , class then finds the corresponding initial condition by using a shooting method. However, since the initial scale factor is set dynamically by the code, we must formulate our initial condition such that it is independent of a in the infinite past. Hence, the target of the shooting method is to find the correct value of the DM energy in a typical comoving volume, E ini ≡ a 3 ini ρ dcdm (a ini ). At the same time, we fix the initial condition for the density of decay radiation using the asymptotic solution of Eqs. (2.1, 2.2) for a going to zero.
Perturbation equations in synchronous gauge
At the level of scalar perturbations, the transfer of energy between the dcdm and dr species is encoded into the continuity and Euler equations of the type
T µ0 dcdm;µ = −C , T µ0 dr ;µ = C , (2.3) ∂ i T µi dcdm;µ = −D , ∂ i T µi dr ;µ = D . (2.4)
The coupling terms C, D accounting for the decay of non-relativistic particles take a trivial form in the synchronous gauge comoving with the decaying species dcdm, i.e. in the gauge such that the metric perturbations δg 00 , δg i0 and the velocity divergence θ dcdm vanish. In this gauge, denoted by the index (s), C (s) is given by the product of the conformal decay rate, the dcdm particle rest mass and the local value of the number density of these particles. Expanding this quantity in background and perturbations, one gets
C (s) = a Γ dcdm ρ dcdm (1 + δ dcdm ) . (2.5)
In the same gauge, the decays do not create any additional flux divergence, and D (s) = 0. Note that assuming similar expressions for C and D in other gauges would lead to wrong results. The Euler equation derived from (2.4) for dcdm in the synchronous gauge (s) then reads
θ (s) dcdm = − a a θ (s) dcdm = 0 . (2.6)
Given adiabatic initial conditions there is no reason for ordinary DM (cdm) and dcdm not to be aligned at early times. Hence, one can fully specify the synchronous gauge by choosing an initial equal-time hypersurface such that θ dcdm = θ cdm = 0. It follows that they will remain zero at any time and we conclude that the synchronous gauge comoving with cold DM is simultaneously comoving with dcdm. Therefore, one can refer to a single synchronous gauge (s), in which the Euler equations for both cdm and dcdm can be omitted.
Perturbation equations in Newtonian gauge
Since the class code is written in both synchronous and Newtonian gauge, we wish to derive the full set of equations in both gauges, while the previous literature only presented synchronous equations. Implementing both gauges allows for a useful consistency check, since one must recover the same observables in the two gauges. After writing the continuity and Euler equations in the synchronous gauge, we gauge-transform them using Eqs. (27a-27b) of Synchronous Newtonian [24], which take a slightly more complicated form in presence of a decay rate:
m contḣ /2 −3φ m ψ 0 ψ m shear (ḣ + 6η)/2 0δ (s) dcdm = δ (n) dcdm + 3 a a + a Γ dcdm α , (2.7) δ (s) dr = δ (n) dr + 4 a a − a Γ dcdm ρ dcdm ρ dr α , (2.8) θ (s) dr = θ (n) dr − k 2 α = 0 , (2.9)
with k the wavenumber, α ≡ (h + 6η )/2k 2 and where we address the reader to [24] for the (by now standard) notation of the different potentials. The final set of equations in both gauges can be written as
δ dcdm = −θ dcdm − m cont − a Γ dcdm m ψ , (2.10) θ dcdm = − a a θ dcdm + k 2 m ψ , (2.11) δ dr = − 4 3 (θ dcdm + m cont ) + a Γ dcdm ρ dcdm ρ dr (δ dcdm − δ dr + m ψ ) , (2.12) θ dr = k 2 4 δ dr − k 2 σ dr + k 2 m ψ − a Γ dcdm 3ρ dcdm 4ρ dr 4 3 θ dr − θ dcdm ,(2.13)
where the metric source terms m cont and m ψ are given in Table 1.
Boltzmann hierarchy for decay radiation
The full perturbations of the decay radiation distribution function can be written in different ways. We adopt here the same set of equations as in [5], in which the perturbations of the (integrated) phase-space distribution function are defined as
F dr ≡ dqq 3 f 0 dr Ψ dr dqq 3 f 0 dr r dr ,(2.14)
with r dr defined as
r dr ≡ ρ dr a 4 ρ cr,0 ,(2.15)
where the the critical energy density today, ρ cr,0 , has been introduced to make r dr dimensionless. The derivative of r dr is given by
r dr = a Γ dcdm ρ dcdm /ρ dr ,(2.16)
so that r dr is constant in absence of a source. The point of introducing such a factor in the definition of F dr is to cancel the time-dependence F dr due to the background distribution function f 0 dr in the denominator of equation (2.14). This simplifies the Boltzmann hierarchy for the Legendre multipoles F dr, , which obey the following equations
F dr,0 = −kF dr,1 − 4 3 r dr m cont + r dr (δ dcdm + m ψ ) , (2.17) F dr,1 = k 3 F dr,0 − 2k 3 F dr,2 + 4k 3 r dr m ψ + r dr k θ dcdm ,(2.F dr, = k 2 + 1 ( F dr, −1 − ( + 1)F dr, +1 ) , > 2 . (2.20)
The expression for m shear can be found in Table 1. For the sake of simplicity, we have reported these equations in a spatially flat universe, but for our analysis we implemented the equations in a general curved FLRW model, following [25]. The Boltzmann hierarchy is truncated at some max following the prescription of [24] generalised to spatial curvature [25].
Comparison with data
Observable effects
When discussing the effect of a given parameter on the CMB describing some new physics, one should specify which other parameters are kept fixed. The best choice is the one allowing to cancel all trivial effects, in order to isolate the distinct residual effect associated to the new physics.
Here the focus is on the effect of the DM decay rate Γ dcdm . If we were varying Γ dcdm while keeping the DM density fixed today (either the physical density ω dcdm = Ω dcdm h 2 or fractional density Ω dcdm ), the code would automatically adjust initial conditions in the early universe. The direct effect of Γ dcdm on the perturbations would then be mixed with that of changing the early cosmological evolution, and in particular the redshift of equality.
Hence, a better choice is to fix all initial conditions, so that varying Γ dcdm only affects the late cosmological evolution. In order to do this easily, we implemented an alternative parametrisation in class. Instead of providing ω dcdm+dr or Ω dcdm+dr as input and letting the code compute the initial dcdm density, the user can choose to pass the initial density of decaying DM (in dimensionless units, as Ω ini dcdm ≡ (ρ ini dcdm a 3 /ρ cr,0 ) or ω ini dcdm ≡ Ω ini dcdm h 2 ), and the code will find the correct density today. Note however, that this procedure also involves a shooting method in order to satisfy the closure equation i Ω i = 1 − Ω k . With this approach, we preserve the full cosmological evolution at least until photon decoupling, since for realistic values of Γ dcdm allowed by observations, the DM decay is only significant at late time, long after photon decoupling. In particular, the effects of Γ dcdm on the CMB are the following: i) a change in the angular diameter distance to decoupling, shifting the whole CMB spectra in multipole space;
ii) a late Integrated Sachs-Wolfe (ISW) effect, since a modification of the homogeneous and perturbed density of DM at late times affects the evolution of metric fluctuations through the Poisson equation;
iii) a different amount of CMB lensing, affecting the contrast between maxima and minima in the lensed CMB spectra. [26]. For all models except the "Decaying CDM" one, the decay rate Γ dcdm is set to zero, implying that the "dcdm" species is equivalent to standard cold DM with a present density ω cdm = ω ini dcdm = 0.12038. The "Decaying CDM" model has Γ dcdm = 20 km s −1 Mpc −1 , the "Tensors" model has r = 0.2, and the "Open" ("Closed") models have Ω k = 0.02 (−0.2). The main differences occur at low multiples and comes from either different late ISW contributions or non-zero tensor fluctuations.
To check (ii), we plot in Figure 1 the unlensed temperature spectrum of models with Γ dcdm set either to 0 or 20 km s −1 Mpc −1 3 . To keep the early cosmological evolution fixed, we stick to constant values of the density parameters (ω ini dcdm , ω b ), of primordial spectrum parameters (A s , n s ) and of the reionization optical depth τ reio . Of course, for Γ dcdm = 0, the dcdm species is equivalent to standard cold DM with a current density ω cdm = ω ini dcdm . We need to fix one more background parameter in order to fully specify the late cosmological evolution. Possible choices allowed by class include h, or the angular scale of the sound horizon at decoupling, θ s = r s (t dec )/d s (t dec ). We choose to stick to a constant value of θ s , in order to eliminate the effect (i) described above, and observe only (ii). We see indeed in Figure 1 that with such a choice, the spectra of the stable and decaying DM models overlap everywhere except at small multipoles. To check that this is indeed due to a different late ISW effect, we show in Figure 2 the decomposition of the total spectrum in individual contribution, for the stable model and a dcdm model in which the decay rate was pushed to 100 km s −1 Mpc −1 .
Since the dominant effect of decaying DM is a modification of the small-part of the CMB temperature spectrum, in the rest of the analysis, it will be relevant to investigate de- generacies between Γ dcdm and other parameters affecting mainly the large-angle CMB spectra, like the spatial curvature parameter Ω k or the tensor-to-scalar ratio r (defined throughout this paper at the pivot scale k * = 0.05/Mpc). We show examples of such models in Figure 1, from which it is not obvious that very small variations of Γ dcdm , Ω k and r can be distinguished, given the cosmic variance uncertainty on low 's. It is useful to plot the matter power spectrum P (k) of the same models, to see whether CMB lensing or direct measurements of P (k) can help to reduce the degeneracy. This is done in Figure 3. We see that all the parameters discussed here have a different effect on P (k). Playing with tensor modes leaves the matter power spectrum invariant, since it is related to scalar perturbations only. Varying Γ dcdm changes P (k) slightly for several reasons:
• the different background evolution of ρ dcdm leads to an overall vertical shift of the spectrum;
• the different values of h needed to get the same θ s changes the ratio of the Hubble scale at equality and today, hence shifting the spectrum horizontally;
• on top of these shifting effects, the different evolution of δ dcdm is such that dcdm has a reduced linear growth factor, affecting the actual shape of the matter power spectrum. When introducing the curvature parameter, one gets a combination of the first two effects only. Moreover, variations of Γ dcdm and Ω k leading to an effect in the CMB of the same amplitude give effects on the P (k) with very different amplitudes. This comparison shows that, at least in principle, CMB lensing effects and direct constraints on P (k) may help to break degeneracies, and to measure Γ dcdm independently of Ω k and r. This can only be confirmed by a global fit to current observations.
The data
The parameter extraction is done using a Metropolis Hastings algorithm, with a Cholesky decomposition to better handle the large number of nuisance parameters [27]. We investigate two combinations of experiments which we denote by A and B. Both share the Planck likelihoods, consisting of the low-, high-, lensing reconstruction and low-WMAP polarisation, as well as the WiggleZ data [28], and the BOSS measurement of the Baryon Acoustic Oscillation scale at z = 0.57 [29]. The set B adds the BICEP2 public likelihood code [16]. We used the publicly available Monte Python 4 code [30] for the analysis. We performed the analysis selecting flat priors for the following set of parameters
{ω b , H 0 , A s , n s , τ reio , ω dcdm+dr , Γ dcdm , r, Ω k } ,
in addition to the other nuisance parameters for the Planck likelihood, omitted here for brevity. The first five cosmological parameters stand respectively for the baryon density, the Hubble parameter, the amplitude at k * = 0.05/Mpc and tilt of the initial curvature power spectrum, and the optical depth to reionisation. The next parameter ω dcdm+dr denote the physical density of decaying dark matter plus its decay product today (in practise, ω dcdm+dr is extremely close to ω dcdm up to typically 4%). Finally, the last two parameters are the dcdm decay rate and the tensor-to-scalar ratio, also measured at the pivot scale k * = 0.05/Mpc. In some of our runs, we vary the curvature parameter Ω k = 1 − Ω tot . The tensor tilt n t is set to satisfy the self-consistency condition from inflation, i.e n t = −r/8(2 − r/8 − n s ), whereas the tensor running α t is neglected. For the neutrino sector, for simplicity, we performed the same assumption as in [26] (two relativistic neutrinos and one with a mass of 0.06 eV).
Results
The results are summarized in Table 2 For the ΛCDM + {Γ dcdm , r} model, we find that the best-fit model has a negligible decay rate. Using the A dataset, the upper bound is Γ dcdm < 5.9 km s −1 Mpc −1 (95% CL). The decay rate is not significantly correlated with any other cosmological parameter, except ω dcdm+dr and r, as can be seen in Figure 4. Indeed, the data prefer a certain amount of DM at early times, corresponding to the correct redshift of equality. Hence models with a large decay rate have a smaller DM density today, explaining the negative correlation between Γ dcdm and ω dcdm+dr . There is also a correlation between Γ dcdm and r: both parameters can enhance the small-l CMB temperature spectrum, so larger values of r lead to a stronger bound on Γ dcdm . Still, since r is peaked in zero (as usual using Planck data), we know that the bound on Γ dcdm that we would obtain under the assumption r = 0 would be very similar to what we get here. For the same model and the B dataset, the bounds on the tensor-to-scalar ratio moves close to r 0.17 at k * = 0.05/Mpc (slightly lower than in the ΛCDM + r model, because of the correlation with Γ dcdm ), pushing the bound on the DDM decay rate down to 4.8 km s −1 Mpc −1 .
For the ΛCDM + {Γ dcdm , r, Ω k } model, and using either the A or B data set, we see in Figure 5 that Γ dcdm is not correlated with Ω k , and that the bounds on Γ dcdm are nearly the same as in the flat model. Indeed, the combination of CMB and LSS data allow us to distinguish between the effects of these two parameters. Since r and Ω k are the two parameters most likely to be degenerate with Γ dcdm within the simplest extensions of ΛCDM, we conclude that current CMB and LSS data provide very robust limits on the dcdm decay rate, depending on the data set, but not on the assumed cosmological model.
Conclusions
We have shown that the lifetime of CDM must be above 160 Gyr (or 200 Gyr assuming that BICEP2 has detected gravitational waves), even for the most conservative case where CDM decays entirely into Dark Radiation. This is a model independent bound, since it relies only on the gravitational interactions of CDM and its decay product which can not be avoided in any particle physics model. If the decay product is allowed to have a mass, we would expect For the ΛCDM + {Γ dcdm , r, Ω k } model, comparison of the results for {ω dcdm+dr , Γ dcdm , r, Ω k } using the dataset set A (blue contours) and B (yellow/orange contours), for the 1d and 2d posterior distributions. The contours represent 68% and 95% confidence levels.
this bound to worsen depending on the mass of the daughter particles. We will consider this scenario in a future publication.
The bound on Γ dcdm has relevant implications on particle physics model buildings. Depending on the specific scenario containing a decaying massive particle which may act as a DM contribution, the lifetime constraint can typically be translated into a lower bound on the particular new mass scale which enters the decay process via a non standard interaction. As a key example, consider the already mentioned Majoron scenario. In this case the pseudo-scalar Goldstone boson related to the breaking of lepton number conservation, acquires a finite mass due to non-perturbative quantum gravity effects that explicitly break global symmetries, and decays into (mainly) neutrino pairs. From its decay rate of Eq. GeV . (4.1) This is just an example of how a strong constraint on DM stability can provide relevant information on its yet unknown nature and constrain models of new non-standard interactions. Finally, we would like to remark that these bounds are expected to become even stronger in the near future. Indeed, a key role in their improvement will be played by future weak lensing surveys, which will also help in reducing degeneracies with massive neutrinos, see e.g. [31,32].
Figure 1 .
1CMB temperature power spectrum for a variety of models, all with the same parameters {100 θ s , ω ini dcdm , ω b , ln(10 10 A s ), n s , τ reio } = {1.04119, 0.12038, 0.022032, 3.0980, 0.9619, 0.0925} taken from the Planck+WP best fit
Figure 2 .
2The single contributions to the CMB temperature spectrum (Sachs-Wolfe, early and late Integrated Sachs-Wolfe, Doppler and polarisation-induced) for a stable model (solid) and a dcdm model (dashed) with Γ dcdm = 100 km/s/Mpc. The value of other parameters is set as inFigure 1. We see that only the late ISW effect is sensitive to the decay rate (for other contributions, solid and dashed lines are indistinguishable).
Figure 3 .
3Matter power spectrum P (k) (computed in the Newtonian gauge) for the same models considered inFigure 1. The black curve (Stable CDM) is hidden behind the red one (Tensors).
Figure 4 .
4Comparison of the results for {ω dcdm+dr , Γ dcdm , r} for the ΛCDM + {Γ dcdm , r} model for the 1-d and 2-d posterior distributions, using the dataset set A (blue contours) and B (yellow/orange contours). The contours represent 68% and 95% confidence levels.
Figure 5 .
5Figure 5. For the ΛCDM + {Γ dcdm , r, Ω k } model, comparison of the results for {ω dcdm+dr , Γ dcdm , r, Ω k } using the dataset set A (blue contours) and B (yellow/orange contours), for the 1d and 2d posterior distributions. The contours represent 68% and 95% confidence levels.
(1.1), a lifetime larger than 200 Gyr translates into the following lower bound on the lepton number breaking scale v v > 4.4 · 10 8 m ν eV m J keV 1/2
Table 1 .
1Metric source terms for scalar perturbations in synchronous and Newtonian gauge.
Table 2. Marginalised Bayesian credible intervals for the cosmological parameters of the models considered in our analysis. We quote either mean values and 68% confidence levels or 95% upper/lower bounds. The last lines show the results for the derived parameter τ dcdm = 1/Γ dcdm representing the dcdm lifetime (assuming a flat prior on the rate Γ dcdm , and not on the lifetime).and Figures 4 and 5.
Model
ΛCDM + {Γ dcdm , r}
ΛCDM + {Γ dcdm , r, Ω k }
Data
A
B
A
B
100 ω b
2.231 +0.025
−0.024
2.226 +0.024
−0.024
2.247 +0.028
−0.030
2.247 +0.028
−0.029
H 0 [km/s/Mpc]
68.89 +0.62
−0.61
68.92 +0.61
−0.62
68.21 +0.79
−0.79
68.07 +0.83
−0.80
10 9 A s
2.145 +0.044
−0.050
2.143 +0.044
−0.047
2.157 +0.046
−0.054
2.156 +0.045
−0.052
n s
0.9643 +0.0055
−0.0056
0.9666 +0.0055
−0.0056
0.9705 +0.0071
−0.0077
0.9742 +0.0072
−0.0076
τ reio
0.082 +0.012
−0.011
0.082 +0.011
−0.011
0.08676 +0.012
−0.013
0.08792 +0.011
−0.013
ω dcdm+dr
0.1142 +0.0016
−0.0014
0.1142 +0.0017
−0.0014
0.1117 +0.0026
−0.0023
0.1113 +0.0025
−0.0023
Γ dcdm [km s −1 Mpc −1 ]
< 5.9
< 5.0
< 6.0
< 4.9
r
< 0.13
0.164 +0.032
−0.040
0.05273 +0.012
−0.053
0.1713 +0.033
−0.039
10 2 Ω k
-
-
−0.3517 +0.28
−0.26
−0.4405 +0.30
−0.27
τ dcdm [Gyr]
> 160
> 200
> 160
> 200
www.class-code.net
www.cern.ch/lesgourg/class-tour/lecture1.pdf
It is useful to bear in mind the conversion factor 1 km s −1 Mpc −1 = 1.02 × 10 −3 Gyr −1 .-7 -
https://github.com/baudren/montepython_public
AcknowledgementsGM acknowledges support by the Istituto Nazionale di Fisica Nucleare I.S. TASP and by MIUR, PRIN Fisica Teorica Astroparticellare. BA, JL and TT received support from the Swiss National Foundation. At LAPTh, this activity was developed coherently with the research axes supported by the ANR Labex grant ENIGMASS.
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Fast and accurate CMB computations in non-flat FLRW universes. J Lesgourgues, T Tram, arXiv:1312.2697astro-ph.COJ. Lesgourgues and T. Tram, "Fast and accurate CMB computations in non-flat FLRW universes," arXiv:1312.2697 [astro-ph.CO].
P Ade, Planck Collaboration CollaborationarXiv:1303.5076Planck 2013 results. XVI. Cosmological parameters. astro-ph.COPlanck Collaboration Collaboration, P. Ade et al., "Planck 2013 results. XVI. Cosmological parameters," arXiv:1303.5076 [astro-ph.CO].
Efficient sampling of fast and slow cosmological parameters. A Lewis, 10.1103/PhysRevD.87.103529arXiv:1304.4473Phys.Rev. 8710103529astro-ph.COA. Lewis, "Efficient sampling of fast and slow cosmological parameters," Phys.Rev. D87 (2013) no. 10, 103529, arXiv:1304.4473 [astro-ph.CO].
The WiggleZ Dark Energy Survey: Final data release and cosmological results. D Parkinson, S Riemer-Sorensen, C Blake, G B Poole, T M Davis, 10.1103/PhysRevD.86.103518arXiv:1210.2130Phys.Rev. 86103518astro-ph.COD. Parkinson, S. Riemer-Sorensen, C. Blake, G. B. Poole, T. M. Davis, et al., "The WiggleZ Dark Energy Survey: Final data release and cosmological results," Phys.Rev. D86 (2012) 103518, arXiv:1210.2130 [astro-ph.CO].
The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Release 10 and 11 galaxy samples. L Anderson, BOSS Collaboration CollaborationarXiv:1312.4877astro-ph.COBOSS Collaboration Collaboration, L. Anderson et al., "The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Release 10 and 11 galaxy samples," arXiv:1312.4877 [astro-ph.CO].
Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code. B Audren, J Lesgourgues, K Benabed, S Prunet, 10.1088/1475-7516/2013/02/001arXiv:1210.7183JCAP. 13021astro-ph.COB. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, "Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code," JCAP 1302 (2013) 001, arXiv:1210.7183 [astro-ph.CO].
Weak Gravitational Lensing as a Method to Constrain Unstable Dark Matter. M.-Y Wang, A R Zentner, 10.1103/PhysRevD.82.123507arXiv:1011.2774Phys.Rev. 82123507astro-ph.COM.-Y. Wang and A. R. Zentner, "Weak Gravitational Lensing as a Method to Constrain Unstable Dark Matter," Phys.Rev. D82 (2010) 123507, arXiv:1011.2774 [astro-ph.CO].
Effects of Unstable Dark Matter on Large-Scale Structure and Constraints from Future Surveys. M.-Y Wang, A R Zentner, 10.1103/PhysRevD.85.043514arXiv:1201.2426Phys.Rev. 8543514astro-ph.COM.-Y. Wang and A. R. Zentner, "Effects of Unstable Dark Matter on Large-Scale Structure and Constraints from Future Surveys," Phys.Rev. D85 (2012) 043514, arXiv:1201.2426 [astro-ph.CO].
| [
"https://github.com/baudren/montepython_public"
] |
[
"Circular birefringence in crystal optics",
"Circular birefringence in crystal optics"
] | [
"R J Potton \nSchool of Computing, Science and Engineering\nMaterials and Physics Research Centre\nJoule Physics Laboratory\nUniversity of Salford\nGreater ManchesterM5 4WTUK\n"
] | [
"School of Computing, Science and Engineering\nMaterials and Physics Research Centre\nJoule Physics Laboratory\nUniversity of Salford\nGreater ManchesterM5 4WTUK"
] | [] | In crystal optics the special status of the rest frame of the crystal means that spacetime symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors of crystals as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time reversal. a) Electronic mail: [email protected] | null | [
"https://export.arxiv.org/pdf/1407.6797v1.pdf"
] | 119,185,454 | 1407.6797 | 3a7b85875c7ff6a7a7c2b6b2f2f055ed150b6ea6 |
Circular birefringence in crystal optics
R J Potton
School of Computing, Science and Engineering
Materials and Physics Research Centre
Joule Physics Laboratory
University of Salford
Greater ManchesterM5 4WTUK
Circular birefringence in crystal optics
1
In crystal optics the special status of the rest frame of the crystal means that spacetime symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors of crystals as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time reversal. a) Electronic mail: [email protected]
Introduction
To account for optical activity in terms of the dielectric response in crystal optics is more difficult than might reasonably be expected [1]. Consequently, recourse is typically had to a phenomenological account. In the simplest cases the normal modes are assumed to be circularly polarized so that forward and backward waves of the same handedness are degenerate. If this is so, then the circular birefringence can be expanded in even powers of the direction cosines of the wave normal [2]. The leading terms in the expansion suggest that optical activity is an allowed effect in the crystal classes having second rank property tensors with non-vanishing symmetrical, axial parts.
To reconcile the phenomenological approach with a local theory of dielectric response one needs to examine carefully the symmetry restriction on permittivity tensor elements. In doing this one must recognise the special status of the rest frame of the crystal and its connection with time-reversal symmetry [3]. The upshot of this examination is that one should focus attention on subgroups of the proper Poincaré group, containing space-time inversion PT but not P and T separately, rather than on subgroups of the full Poincaré group.
In section 2, Maxwell's equations for a polarisable medium are set out, following Berreman, in a form that lends itself to detailed analysis of the phenomenon in question. Electromagnetic constitutive equations for the dynamic response in gyrotropic media are introduced in section 3 by way of extension of Birss's formulation [4] of similar relations for static effects. A justification for the inclusion of additional elements in the dynamic permittivity, permeability and conductivity tensors is provided in an appendix. In section 4 wave equations that exhibit the PT symmetry responsible for the degeneracy of forward and backward waves of the same handedness are developed. Section 5 contrasts the present approach, based on the space-time symmetry of the permittivity tensor, with phenomenological treatments that expand circular birefringence, expressed as a pseudo-scalar, in direction cosines of the wave normal. The corresponding listings of gyrotropic crystal classes do not exactly match providing the opportunity of experimentally testing what is set out here.
Maxwell's equations in a polarizable medium
The aim is to explain the interaction of plane polarized components of a circularly polarized wave in terms of the constitutive relations of a less than fully symmetric medium. To facilitate this it is expedient to employ the 4×4 matrix formulation developed by Teitler [5] and Berreman [6].
B E (2.5) 0 μ 0 t 0 1 r σE E ε B μ (2.6)
To illustrate the matrix method, the medium is initially taken to be dielectric and isotropic:
0 σ (2.7) I ε (2.8) I μ (2.9)
and a refractive index, n, is defined by 0 0 c n . When particularized to plane waves travelling along the z-axis Maxwell's curl equations take the form:
y t x z B c n E c n (2.10) x t y z E c n c n B (2.11) x t y z B c n E c n (2.12) y t x z E c n c n B (2.13)
or as a matrix equation with basis The equation is diagonal in the basis
x y y x B , E c n , B , E c n : 0 0 0 0 0 0 0 0 0 y x x y x y y x B E c n B E c n B E c n B E c n , ,, 2 1 :0 0 0 0 0 0 0 0 0 0 0 0 0 y x
x y
x y y x respectively as may be verified by consideration of the Poynting vectors of the respective field superpositions.
In an isotropic medium, circularly polarized modes could equally serve as a basis (the helicity basis).
Constitutive relations for harmonic fields
The discrete symmetries of space and time are variously denoted by P and T or I and I t [7] defined as follows:
z , y , x z y x , , : P (3.1) t t : T (3.2)
are elements of the full inhomogeneous Lorentz (full Poincaré) group [8].
When applied to static fields the dielectric constitutive relation (2.2) relates fields that have definite transformation properties under discrete space-time symmetries [4]. Both E and D are, in fact, odd under P (polar) and even under T (-i) [9] † . Consequently, ε is a polar-i tensor. The polar-i designation is the one used by Birss. It is germane to ask whether any discrete space-time symmetry persists generally (that is to say, independently of any particular crystalline point symmetry) for dynamic fields (i.e. waves) in crystalline condensed matter. In pursuance of this it is prudent to examine how the application of full Poincaré group symmetry operations must differ when light propagates in a crystalline medium. Appendix A shows that, in a crystalline medium, an association can be made between oriented spatial entities and ordering in time. This association is invariant under combined space and time inversion which is a symmetry operation of the proper Poincaré group:
PT : z , y , x , t z , y , x , t (3.1)
but not under separate space inversion, P, and time inversion, T, which are not. Consequently the ruling symmetry in crystal optics is found to be the proper Poincaré group depleted of all Lorentzian boosts and of all but a discrete set of proper and improper spatial rotations. Matrix and tensor representation of the Poincaré group give rise to complex amplitudes which are affected by time re-ordering because of the anti-unitary character of space-time reversal.
It is well accepted that time-reversal:
* c c : t t (3.2)
where c is any complex amplitude, is a prevalent symmetry (for example of Schrodinger's wave equation) [10] whereas time inversion:
t t : T (3.3)
is not [7]. By analogy with the former it is possible to introduce space-time reversal: † It will be seen shortly that linear boosts are excluded from the set of symmetry operations so fields can be treated as vectors rather than components of second rank tensors.
: R α α α α α α t t t
where α is a tensor or a pseudo(axial)-tensor and c is a complex amplitude.
In section 4 the constitutive equation:
εE D (3.5)
is the subject of attention. In accordance with the discussion in Appendix A the terms in this equation are taken to be classifiable under space-time reversal, R. D and E are odd [9], hence, ε is even. However, in addition to the usual polar-i part, represented by a real, symmetric second rank tensor, ε has an axial-c part. Its tensor representative is antisymmetric and, in non-magnetic crystal classes, pure imaginary. This part of ε is non-zero when fields are not static and the sequence in which x-and y-components change matters i.e. in the presence of gyrotropy.
The essence of the argument in section 4 is that the degeneracy of circularly polarized waves has its origin in the space-time reversal covariance of the wave equation.
Circular birefringence
If the permittivity tensor ε has off-diagonal elements The axial-c part of the permittivity tensor yields a pure imaginary and antisymmetric ε . The antisymmetry of ε arises because of the dependence of dielectric response on the sequence in which changes in different components of the electric field occur. This is a feature of PT symmetry which is appropriate for gyrotropic media. In non-magnetic crystal classes time reversal is a symmetry that stands on its own (in contrast to magnetic classes in which it occurs only in combination with certain point symmetries). For non-magnetic classes property tensors are either even in time (pure real) or odd in time (pure imaginary). where I is the 2 2 unit matrix:
0 0 0 0 0 0 0 y x x y x y y x t z t i i i : R (4.6)
The space-time reversal, R, covariance of the wave equations for forward and backward right-handed waves follows from the identification of z as first rank axial-i and of t c n as zeroth rank polar-c so that:
z z z : R ' (4.7) t t t c n c n c n : R ' . (4.8)
Moreover the coefficient, a, of t c n , is R invariant:
a 2 1 a 2 1 a : R yx xy i ' i (4.9)
It is to this covariance that the degeneracy of waves of the same handedness is attributable.
The off-diagonal terms in (4.5) suggest that diagonalization might yield superpositions of forward right circular and backward left circular waves and vice-versa. It is straightforward to investigate such superpositions using Jones calculus [11]. They are strange linear polarized standing waves with nodes spaced by /k but in which the time dependence of the fields is phase modulated with position and time. Fortunately we are rescued from such monsters by a selection rule. The matrix elements in question occur in positions that could only possibly admix forward and backward waves of opposite handedness. However, circular birefringence means that translational symmetry precludes any such admixture.
The crystal classes expected to exhibit circular birefringence on the basis of this analysis are listed in the next section and comparison is made with phenomenological analyses employing second and higher rank tensors.
Predictions of gyrotropic classes
Tables 2 and 4 of reference [4] show that the crystal classes 1, 2, 222, 4, 4mm, 3, 3m, 6 and 6mm have non-vanishing second rank antisymmetric axial-c tensors. On the basis of the analysis of section 4 they are therefore expected to exhibit circular birefringence. The conventional phenomenological theory of birefringence [2] introduces a pseudoscalar, G, related to the changes in ordinary and extraordinary refractive indices brought about by gyrotropy. It is then supposed that G can be expanded using a symmetric second rank axial tensor, g ij , in direction cosines of the wave normal:
j i ij g G (6.1)
For waves in an isotropic medium it amounts to:
2 L R G . (6.2)
Of the classes mentioned above, three, 4mm, 3m and 6mm, have vanishing symmetric second rank axial tensors and are not expected from the phenomenology as presented. This is not a necessarily a serious problem since a more precise phenomenological expression is:
l k j i ijkl j i ij h g G (6.3)
and fourth rank axial tensors are non-vanishing in classes 4mm, 3m and 6mm. (The expansion of G as a polynomial of even degree in the direction cosines enshrines the assumption of reciprocal propagation of similarly circularly polarized waves.)
On the other hand the classes m, mm2,4, 422,42m, 32, 622, 23 and 432 predicted as optically active by phenomenology are not anticipated as such from section 4. The implication of this is that dielectric response of higher tensor rank is a possible cause of the effect † † . This leads naturally to the other school of thought [13] which is that gyrotropy is an intrinsically nonlocal phenomenon and that the dielectric properties require at least third rank tensors for their expression:
l ijl ij ij s c in k , 1 1 . (6.4)
One feature of this approach is that it causes attention to be focused on non-symmorphic space-group elements in the form of screw axes [1].
Three non-enantiomorphous classes have attracted attention in connection with optical activity; m, mm2 and42m [12]. Given that4 is in fact enantiomorphous the claim that optical activity may be allowed in a crystal, the symmorphic space group of which has a mirror plane, rests on a single case, namely42m. The absence of enantiomorphs may in fact be quite a good guide to the absence optical activity in crystals with symmorphic space groups since it does not depend on arbitrarily truncated series expansions. The nonenantiomorphous classes m, mm2 and42m are not predicted as optical active by the analysis of section 4.
10 † † Another possibility is that the axial-c part of the magnetic permeability or the axial-i part of the conductivity tensor is responsible.
Further work
The association of time reversal with certain point symmetry operations that exists in magnetic groups [4] must, by Neumann's principle, influence the allowed elements of the dielectric tensor ε . Particularization of symmetry restrictions in these cases will require further study.
The generalization of equation (4.2) to 0 σ suggests a way of investigating CPT symmetry [3]. σ connects odd in C current density with even in C electric field E and is therefore odd in C and presumably, therefore, also odd in PT [9]. This connection seems to result from "internal" source terms that are present in conductive media but not in dielectrics.
Conclusion
It has been shown that, for certain crystal classes, a wave equation exhibiting covariance under space-time reversal (but not under separate space inversion or time reversal) can account for the birefringence and degeneracies of circularly polarized waves. Proof that is non-associative:
In figure 1 C B C B A B A B A C if is associative
is non-associative by reduction ad absurdum
One could define C B A to be the most recent event the future light cone of which contains A and B and C, but a new law of composition would then be needed for the l.u.b. of each subset of two or more events. It is a moot point as to whether the poset with l.u.b. defined for each pair of events just introduced may be regarded as a semi-lattice [14,15]. If so, it is a semilattice that is non-associative. Nevertheless, a meagre algebraic allowance of commutativity and idempotency of is sufficient for what is to follow.
It is now possible to show that a PT invariant association can be made between oriented spatial simplexes and oriented time displacements, though only in the absence of linear boosts as space-time symmetries. O is an event in Minkowski space-time and A, B and C are neighbouring mutually space-like separated events chosen to be simultaneous with O in some inertial frame. The spatial displacements from O to A, B and C are a, b and c and are entities with attitude, magnitude and orientation. OABC defines a simplex which has the same attributes. There is a family of such objects related by the translations and spatial rotations of the restricted form of the Poincaré group appropriate to the crystal class in question (the B A C AB BC 13 proper Poincaré group depleted of linear boosts and of all but a finite set of static rotations). A sign (+ or -) can be attached to each of the two orientations of the simplex OABC by forming the triple scalar product a.b×c provided only that an ordering of a, b and c to within cyclic permutation can be established. In the restricted version of the Poincaré group appropriate to crystal optics such an ordering is provided by the arrangement of a, b and c according to magnitude. Note again that by virtue of Lorentz contraction such an ordering is precluded when linear boosts are allowed space-time symmetries.
The sign of the orientation of the 3-space simplex is connected with ordering in time in the following way. The magnitudes of a, b and c are the spatial separations of pairs of simultaneous events that have l.u.b.s as already discussed. Furthermore a, b and c not being tied to any particular spatial location, their l.u.b.s can be brought by spatial translation to the same 3-space position and their future light cones can be nested somewhat in the manner of Russian dolls. Thus, the orientation of the spatial simplex is associated with an order relation in time. This association is invariant under space-time inversion PT. The fact that the association is not maintained under P and T separately means that these symmetries are less significant in crystal optics than elsewhere [16,17].
The incorporation of a vector product in the triple scalar product and its connection with time ordering means that the polar/axial and i/c characterization of tensors acquires a new level of subtlety. The governing PT symmetry is less restrictive of allowed phenomena than the separately applied P and T symmetries that work well in the analysis of static effects [5]. Specifically, fields and their response functions, and terms in equations in general, will be either PT-even or PT -odd, the former having axial-c as well as polar-i parts, the latter both polar-c and axial-i parts. This follows the general precept that all terms be allowed in dynamical equations apart from those specifically excluded on symmetry grounds.
basis of linearly polarized forward and backward modes.
Given the above considerations (4.4) is put into nearly block diagonal form by the
Figure 1
1shows the intersection of the future light cones in 2+1 dimensional space-time of three events, A, B and C, with the plane 0 t t for some observer. More recent events are represented by smaller light cone sections in the figure with remoteness in time proportional to the radius of the circle.
second) two rows of the matrix. By way of explanation ε , in nonmagnetic classes, is a pure imaginary antisymmetric axial tensor so that:basis being: forward right,
x
y
y
x
B
E
c
n
i
B
E
c
n
, backward right,
y
x
x
y
B
E
c
n
i
B
E
c
n
, forward left,
x
y
y
x
B
E
c
n
i
B
E
c
n
and backward left,
y
x
x
y
B
E
c
n
i
B
E
c
n
, waves. The first two basis functions (like the second two) are
degenerate by virtue of the invariance under space-time reversal of the factor that precedes
t
c
n
in the first (yx
xy
xy
t r
Appendix A A connection between space and time orientations in crystal opticsThis appendix shows that in crystal optics PT symmetry supersedes separate P and T symmetries. There are significant implications for the derivation of crystal symmetry forbidden effects. The argument adduced hinges on the existence of crystalline space-time structures that are invariant under PT but not under P or T separately.The setting of the discussion is Minkowski space-time and the various homogeneous Lorentz and inhomogeneous Lorentz (Poincaré) groups[8]under which its structures are invariant or covariant. It is first necessary to recognise that crystal optics does not admit Lorentzian linear boosts because crystal symmetry restrictions on property tensors are meaningless except in the rest frame of the crystal by virtue of the phenomenon of Lorentz contraction. A cubic crystal is no longer cubic in a boosted frame. Crystal optics necessarily resides in the rest frame of the crystal. Moreover, this frame is related to the laboratory frame by a unique linear boost. Nevertheless, frame transformations, including finite static rotations and crucially space-time translations remain under which all equations must be covariant. A significant part of the space-time structure that remains in the crystal rest frame relates to causality. The crystal space-time may be regarded as a partially ordered set (poset) of events by virtue of the fact that, if event B lies within the future light cone of A, we may write A B . The partial order relation may in turn be used to define a least upper bound (l.u.b.s, ) for pairs of events. The l.u.b. of C and D may be defined as the most recent event, the future light cone of which contains both C and D. This definition gives precision to the everyday concept of proximate cause and might be used elsewhere to extend discussion of causation, which is often approached by way of Boolean algebra, to the relativistic domain. However, the properties of this algebra are different in crucial ways from Boolean algebra. Differences emerge when different dimensions of space-time are considered. In 2+1 and 3+1 dimensional space-time, though not in 1+1dimension, l.u.b. may be shown to be nonassociative contrary to the situation in Boolean algebra where union of sets is associative. (It is emphasised that the partial ordering and l.u.b. have been defined respectively for events and pairs of events and not for arbitrary subsets of all events).
G Brooker, Modern Classical Optics. OxfordOUP327Brooker G 2003 Modern Classical Optics (OUP: Oxford) p327
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| [] |
[
"Dual CFT on Dyonic Kerr-Sen black hole and its gauged and ultraspinning counterparts",
"Dual CFT on Dyonic Kerr-Sen black hole and its gauged and ultraspinning counterparts"
] | [
"Muhammad F A R Sakti \nDepartment of Physics\nFaculty of Science\nHigh Energy Physics Theory Group\nChulalongkorn University\n10330BangkokThailand\n",
"Piyabut Burikham \nDepartment of Physics\nFaculty of Science\nHigh Energy Physics Theory Group\nChulalongkorn University\n10330BangkokThailand\n"
] | [
"Department of Physics\nFaculty of Science\nHigh Energy Physics Theory Group\nChulalongkorn University\n10330BangkokThailand",
"Department of Physics\nFaculty of Science\nHigh Energy Physics Theory Group\nChulalongkorn University\n10330BangkokThailand"
] | [] | We demonstrate a strong evidence of entropy matching that rotating dyonic black holes in Einstein-Maxwell-Dilaton-Axion (EMDA) theory is holographically dual to a 2D conformal field theory. We first investigate the duality on dyonic Kerr-Sen black hole with non-vanishing dilaton and axion charges. The near-horizon geometry of extremal dyonic Kerr-Sen spacetime possesses the SL(2, R) × U (1) isometry where the asymptotic symmetry group method can be used to find the corresponding central charge. We find two different branches of masses which correspond to CFT with two different central charges cL = 12am+ and cL = 12am−. The exact agreement between the Bekenstain-Hawking entropy and entropy from CFT is then found also in two different branches of extremal entropy. Furthermore, we demonstrate that this duality is robust insofar for non-zero AdS length. The duality holds for both dyonic Kerr-Sen-AdS black hole and its ultraspinning counterpart. In both cases, we obtain the expected entropy from CFT which matches exactly with the Bekenstein-Hawking entropy. Since dyonic and axion charges are proportional to 1/m, we note that there are possibly more than two branches of the central charge for non-zero AdS length in terms of mass. When we turn off dyonic charge, the axion charge vanishes giving the results of Kerr-Sen-AdS black hole. Moreover, when we assume the equal electromagnetic charges, it recovers the results when the dilaton charge vanishes. Lastly, we compare the results of dyonic Kerr-Sen-AdS black hole and its ultraspinning counterpart to those of the dyonic Kerr-Newman-AdS black hole and the ultraspinning counterpart. Depending on the dyonic charge parameters, it is found that extremal ultraspinning dyonic Kerr-Sen-AdS black hole is not always superentropic. | 10.1103/physrevd.106.106006 | [
"https://export.arxiv.org/pdf/2206.10868v2.pdf"
] | 253,476,036 | 2206.10868 | 552fbe0ffe4eb1806c674b2493e29643f2b362f4 |
Dual CFT on Dyonic Kerr-Sen black hole and its gauged and ultraspinning counterparts
12 Nov 2022
Muhammad F A R Sakti
Department of Physics
Faculty of Science
High Energy Physics Theory Group
Chulalongkorn University
10330BangkokThailand
Piyabut Burikham
Department of Physics
Faculty of Science
High Energy Physics Theory Group
Chulalongkorn University
10330BangkokThailand
Dual CFT on Dyonic Kerr-Sen black hole and its gauged and ultraspinning counterparts
12 Nov 2022
We demonstrate a strong evidence of entropy matching that rotating dyonic black holes in Einstein-Maxwell-Dilaton-Axion (EMDA) theory is holographically dual to a 2D conformal field theory. We first investigate the duality on dyonic Kerr-Sen black hole with non-vanishing dilaton and axion charges. The near-horizon geometry of extremal dyonic Kerr-Sen spacetime possesses the SL(2, R) × U (1) isometry where the asymptotic symmetry group method can be used to find the corresponding central charge. We find two different branches of masses which correspond to CFT with two different central charges cL = 12am+ and cL = 12am−. The exact agreement between the Bekenstain-Hawking entropy and entropy from CFT is then found also in two different branches of extremal entropy. Furthermore, we demonstrate that this duality is robust insofar for non-zero AdS length. The duality holds for both dyonic Kerr-Sen-AdS black hole and its ultraspinning counterpart. In both cases, we obtain the expected entropy from CFT which matches exactly with the Bekenstein-Hawking entropy. Since dyonic and axion charges are proportional to 1/m, we note that there are possibly more than two branches of the central charge for non-zero AdS length in terms of mass. When we turn off dyonic charge, the axion charge vanishes giving the results of Kerr-Sen-AdS black hole. Moreover, when we assume the equal electromagnetic charges, it recovers the results when the dilaton charge vanishes. Lastly, we compare the results of dyonic Kerr-Sen-AdS black hole and its ultraspinning counterpart to those of the dyonic Kerr-Newman-AdS black hole and the ultraspinning counterpart. Depending on the dyonic charge parameters, it is found that extremal ultraspinning dyonic Kerr-Sen-AdS black hole is not always superentropic.
Introduction
It has been pointed out by Bekenstein and Hawking that black holes must have an entropy to prevent the violation of second law of thermodynamics. It also corresponds to the existence of radiation from the black holes with finite temperature. These thermodynamic behaviors of black holes indicate that there exists the underlying micro structure within the black holes. One big question then arises from this theory questioning the explanation of the origin of the black hole microstates. It has been a main curiosity for decades to explain the Bekenstein-Hawking entropy from this microscopic point of view. Nonetheless, there is yet no complete answer to this question. However, for a large class of extremal supersymmetric black holes, this question has been answered in which the Bekenstein-Hawking entropy can be computed also by counting the degeneracy of Bogomol'nyi-Prasad-Sommerfield soliton bound states [1,2].
Another answer to the previous question is the Kerr/CFT duality. This duality states that there is a correspondence between the associated physical quantities of the extremal four-dimensional Kerr black hole and almost similar physical quantities in a chiral conformal field theory (CFT) [3]. This statement is parallel to the general one given by Brown and Hanneaux for AdS 3 [4]. However, the AdS 3 spacetime is replaced by near-horizon extremal black hole metric. In the near-horizon region of extremal black holes, it is shown that there is a conformal invariance implying the presence of an AdS structure on the spacetime with SL(2, R) × U (1) isometry. The generalization of this correspondence to the other extremal rotating black holes, in four and higher dimensions, and beyond Einstein theory have been extensively studied during the last decade [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The 2D chiral CFT has been carried out by considering the asymptotic symmetry group with some proposed boundary conditions, from which we can generate a class of diffeomorphisms of the near-horizon Kerr geometry. After defining charges associated with the diffeomorphisms and evaluating the Dirac brackets of the charges, we can produce a Virasoro algebra with non-vanishing (quantum) central charge which is the extension of U (1) isometry of the near-horizon extremal black hole geometry. The entropy is then calculated using the Cardy's growth of states for a 2D CFT which is the function of central charge and temperature. The temperature can then be calculated by assuming Frolov-Thorne vacuum. From this duality calculation, the exact matching between entropy from CFT and Bekenstein-Hawking entropy of extremal Kerr black hole is established.
Interestingly, the Kerr/CFT correspondence may apply not only to the extremal black holes, but also to non-extremal black ones. The hidden conformal symmetry of the Kerr black hole has been revealed for the first time in Ref. [22]. The conformal symmetry appears on the solution space of the scalar probe in the black hole background. Further studies show that not only scalar probe, yet higher spin particles can also be the probe to reveal the hidden conformal symmetry [23]. Similar to the Kerr/CFT in extremal black holes, this also can be extended to various black hole solutions in four and five dimensions, and also in different gravitational theories [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47]. The revelation of the hidden conformal symmetry shows that in the near-horizon region and low-frequency limit, the phase space possesses SL(2, R)×SL(2, R) isometry similar to AdS 3 spacetime. In these papers, the central charges are computed using numerical observation in order to match the Bekenstein-Hawking entropy with the Cardy entropy. Another observation for non-extremal Kerr black hole is that we can identify two sets of Virasoro diffeomorphisms acting on the horizon leading to non-vanising left-and right-moving central charges [48]. This remarkable calculation fills up the gap left from the first revelation of hidden conformal symmetry in the non-extremal black hole. Another intriguing result is provided in Ref. [49] that one can also extend the conventional CFT to the warped CFT in order to calculate the entropy for non-extremal Kerr black holes. Therein the non-trivial diffeomorphisms lead to a Virasoro-Kac-Moody algebra with non-trivial central extensions.
Inspired by the Kerr/CFT correspondence, it is natural to extend the similar calculation to the family of black holes in Einstein-Maxwell-Dilaton-Axion (EMDA) theory. We want to study the thermodynamic properties of dyonic Kerr-Sen black hole, its gauged, and ultraspinning counterparts. Non-dyonic Kerr-Sen black hole has been investigated in Ref. [6] for extremal case and in Ref. [41] for non-extremal case showing that the central charge of this black hole is identical to Kerr black hole which is the solution to Einstein vacuum field equation, although Kerr-Sen also possesses non-vanishing electric and dilaton charges. The solution of dyonic case is provided in Ref. [50] which is the extension of Ref. [51] for non-dyonic case, yet with non-zero cosmological constant (AdS length). This black hole family emerges in the low-energy heterotic string theory that includes dilaton and axion. Since there exist scalar fields representing dilaton and axion, this solution will differ from dyonic Kerr-Newman (-AdS) black hole solution.
One fascinating limit of gauged dyonic black hole is the ultraspinning limit, i.e. the spin of the black hole is boosted to the value of AdS length. In the ultraspinning limit, the Kerr-Newman-AdS black hole solution possesses non-compact horizon and becomes superentropic. The non-compact horizon occurs due to the presence of conical singularity while superentropic means that for a given volume V , the entropy will be maximized [52]. So far, the study for Kerr-Newman-AdS black hole has been done in Ref. [16] in the context of Kerr/CFT correspondence. In the present paper, it will be demonstrated that the Kerr/CFT correspondence is also valid for dyonic black hole family in EMDA theory and indeed robust. First, we consider dyonic Kerr-Sen black hole and show that there are similar features between dyonic and non-dyonic cases. This work is then extended to the gauged counterpart, then in the situation where the cosmological constant is non-zero, and finally when the ultraspinning limit is reached. It is intriguing also to compare the results with the dyonic Kerr-Newman-AdS black hole family.
The paper is organized as follows. After the introduction, we briefly review the dyonic Kerr-Sen black hole and its thermodynamic properties in Section 2. In Section 3, we investigate the near-horizon extremal dyonic Kerr-Sen black hole and derive its entropy using the Kerr/CFT correspondence. In Section 4 and 5, we extend the work from previous section to the gauged dyonic Kerr-Sen-AdS black hole and its ultraspinning counterpart, respectively. In Section 6, we provide the comparison between the dual CFT of dyonic Kerr-Sen-AdS and dyonic Kerr-Newman-AdS black holes. Section 7 concludes our work.
Dyonic Kerr-Sen black holes
The dyonic NUT generalization of Kerr-Sen black hole solution was first given in Ref. [53] in ungauged case. However, in this paper, we set the NUT parameter to zero, and consider only the dyonic Kerr-Sen black hole. The dyonic generalization of Kerr-Sen solution without NUT parameter is re-written in Ref. [50] in a simpler form using certain coordinate transformations. In Ref. [50], the gauged solution of EMDA theory is also provided in a simpler way coming from the original derivation in Refs. [54,56]. In this section, we will briefly review the spacetime metric of the dyonic Kerr-Sen black holes which is given in Ref. [50]. The Lagrangian density of the EMDA theory that is considered to find dyonic Kerr-Sen black hole is given by [55,56]
L = √ −g R − 1 2 (∂φ) 2 − 1 2 e 2φ (∂χ) 2 − e −φ F 2 + χ 2 ǫ µνρλ F µν F ρ λ , (2.1)
where ǫ µνρλ is the Levi-Civita antisymmetric tensor density in 4D. The dual of the gauge potential B is defined by dB = −e −φ ⋆ F − χF . The theory presented in Lagrangian density (2.1) has an SL(2, R) global symmetry that can transform the dyonic black hole to purely electric or magnetic black hole. The dyonic Kerr-Sen black hole solution to that Lagrangian density is given in Eq. (4) in Ref. [50]. However, as mentioned in Ref. [50], the the dyonic Kerr-Sen black hole spacetime metric can also be written in a more symmetric way by shifting the radial coordinate as r → r + d. In this coordinate, the spacetime metric together with the electromagnetic potential, the dual electromagnetic potential, dilaton field, and axion field, are given by
ds 2 = − ∆ ̺ 2X 2 + ̺ 2 ∆ dr 2 + ̺ 2 dθ 2 + sin 2 θ ̺ 2Ŷ 2 , (2.2) A = q(r + d − p 2 /m) ̺ 2X − p cos θ ̺ 2Ŷ , B = p(r + d − p 2 /m) ̺ 2X + q cos θ ̺ 2Ŷ ,(2.
3)
e φ = (r + d) 2 + (k + a cos θ) 2 ̺ 2 , χ = 2 kr − da cos θ (r + d) 2 + (k + a cos θ) 2 ,(2.4)
respectively, whereX = dt − a sin 2 θdφ,Ŷ = adt − (r 2 − d 2 − k 2 + a 2 )dφ, (2.5)
̺ 2 =r 2 − d 2 − k 2 + a 2 cos 2 θ, ∆ =r 2 − 2mr − d 2 − k 2 + a 2 + p 2 + q 2 .
We have to note that m, a, q, p, d, k are mass, spin, electric charge, magnetic (dyonic) charge, dilaton charge, and axion charge of the black hole, respectively. The dilaton and axion charges clearly depend on the electric and magnetic charges by the following relations
d = p 2 − q 2 2m , k = pq m . (2.6)
One also can write
d 2 + k 2 = p 2 + q 2 2m 2 . (2.7)
This black hole possesses inner and outer horizons as given by
r ± = m ± m 2 + d 2 + k 2 − a 2 − p 2 − q 2 . (2.8)
The dyonic Kerr-Sen black hole satisfies the following thermodynamic relation
dM = T H dS + Ω H dJ + Φ H dQ + Ψ H dP. (2.9)
The quantities on above equation are given by
M = m, J = ma, Q = q, P = p, T H = r + − m 2π(r 2 + − d 2 − k 2 + a 2 ) , S BH = π(r 2 + − d 2 − k 2 + a 2 ), (2.10) Ω H = a r 2 + − d 2 − k 2 + a 2 , Φ H = q(r + + d − p 2 /m) r 2 + − d 2 − k 2 + a 2 , Ψ H = p(r + + d − p 2 /m) r 2 + − d 2 − k 2 + a 2 , (2.11)
where those are mass, angular momentum, electric charge, magnetic charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity, electric potential, and magnetic potential, respectively. Now we consider the extremal limit of dyonic Kerr-Sen black hole. In this limit, m 2 +d 2 +k 2 = a 2 +p 2 +q 2 , and both horizons in Eq. (2.8) coincide as r ± = m. Note that one can also write d 2 + k 2 in terms of q 2 + p 2 to reduce the number of parameters,
m 2 = a 2 + q 2 + p 2 − q 2 + p 2 2m 2 .
(2.12)
Based on this relation, there are two branches of extremal dyonic Kerr-Sen black hole where the mass is given by
m 2 = 1 2 (a 2 + p 2 + q 2 ) 1 ± 1 − p 2 + q 2 a 2 + p 2 + q 2 2 .
(2.13)
We identify each branch m +(−) with the plus (minus) sign of the square root in (2.13) respectively. For this extremal black hole, the Hawking temperature in Eq. (2.10) vanishes while other thermodynamic quantities that contain mass term remain non-zero and possess two branches. These branches will reduce into one only when the spin approaches zero. In this case, the black hole solution will reduce to extremal dyonic GMGHS (Gibbons-Maeda-Garfinkle-Horowitz-Strominger) black hole with zero entropy [61]. This condition is distinctive from extremal (dyonic) Reissner-Nordström solution which has non-zero entropy. Zero entropy is related to zero central charge in CFT. Vanishing c L is trivial and exactly denotes that the Virasoro algebra reduces to the classical Witt algebra. In the next section, we will study the thermodynamic properties of this extremal dyonic Kerr-Sen black hole using the Kerr/CFT correspondence.
CFT Dual of Dyonic Kerr-Sen Black Hole
The main upshot of this section is to provide the derivation of Cardy entropy [3],
S CF T = π 2 3 (c L T L + c R T R ) , (3.1)
for extremal black holes in EMDA theory given in this paper. We will derive the corresponding central charges (c L , c R ) using the asymptotic symmetry group (ASG) and the temperatures (T L , T R ) in order to compute the Cardy entropy of the black holes given in this paper. We apply the Cardy entropy formula for the black hole solutions in this paper for extremal case. First, in order to apply the asymptotic symmetry group calculation, we need to find the near-horizon geometry satisfying the SL(2, R) × U (1) isometry group. We will present explicitly the near-horizon geometry of the extremal rotating dyonic Kerr-Sen black hole. Then we will derive the non-trivial diffeomorphisms which are associated with non-vanishing conserved surface charges. It will be shown that the four-dimensional rotating dyonic Kerr-Sen black hole belongs to phase space representing one copy of the Virasoro algebra with a particular central charge. After finding the temperatures using the generalized Frolov-Thorne vacuum, one can compute the Cardy formula to match with the Bekenstein-Hawking entropy for black holes.
Near-horizon extremal black hole metric
The near-horizon form of the spacetime metric can be obtained using the specific coordinate transformation representing near-horizon region approximation. First, we will find the near-horizon geometry of the extremal dyonic Kerr-Sen black hole. To find the near-horizon geometry of the extremal dyonic Kerr-Sen black hole (2.2), we consider the following coordinate transformations [5,57]
r = r + + ǫr 0 r,t = r 0 ǫ t,φ = φ + Ω H r 0 ǫ t, (3.2)
where r 0 is scaling constant where we may define as r 2 0 = r 2 + − d 2 − k 2 + a 2 . In near-horizon limit, we have ǫ → 0. We can obtain the near-horizon extremal metric of Eq. (2.2) which is given by
ds 2 = Γ(θ) −r 2 dt 2 + dr 2 r 2 + dθ 2 + γ(θ) (dφ + erdt) 2 , (3.3)
where the metric functions are given by
Γ(θ) = ̺ 2 + , γ(θ) = r 4 0 sin 2 θ ̺ 2 + , ̺ 2 + = m 2 − d 2 − k 2 + a 2 cos 2 θ, e = 2am r 2 0 .
One can see that the AdS 2 factor emerges in this near-horizon metric denoting that the metric now has AdS 2 × S 2 structure. This is the origin how we infer that the AdS/CFT correspondence for black holes, namely Kerr/CFT may apply.
Since there exists several non-vanishing scalar and vector fields, it is also applicable to use the nearhorizon coordinate transformation to those fields. In the near-horizon limit, we find that the gauge field and its dual are given by
A + = f a (θ) (dφ + erdt) + q (m + d) 2 + k 2 − a 2 − 2p 2 e(m 2 − d 2 − k 2 + a 2 ) dφ, (3.4) B + = f b (θ) (dφ + erdt) + p (m + d) 2 + k 2 − a 2 − 2p 2 e(m 2 − d 2 − k 2 + a 2 ) dφ, (3.5)
where the functions f (θ) for each field are given by
f a (θ) = q 2p 2 − (m + d) 2 − k 2 + a 2 cos 2 θ + 2apm cos θ 2am̺ 2 + r 2 0 , (3.6) f b (θ) = p 2p 2 − (m + d) 2 − k 2 + a 2 cos 2 θ − 2aqm cos θ 2am̺ 2 + r 2 0 , (3.7)
where we can gauge away the second term on the electromagnetic potential (3.4) and its dual (3.5). Moreover, after finding the gauge field in the near-horizon limit, we can also find the dilaton and axion field in this limit which are given as
φ + = exp (m + d) 2 + (k + a cos θ) 2 ̺ 2 + , χ + = 2 mk − da cos θ (m + d) 2 + (k + a cos θ) 2 . (3.8)
It is worth noting that these near-horizon forms of the fields are needed in the calculation of the central charge. However, basically their contribution will not appear directly to the central term.
In order to find the isometry group of the near-horizon extremal metric (3.3), one can apply the Killing equation and solve it. In the end, by solving Killing equation, it is found that the spacetime metric (3.3) possesses SL(2, R) × U (1) isometry group which are represented by the following Killing fields
ζ 0 = ∂ φ , (3.9)
which denote the rotational U (1) isometry and
X 1 = ∂ t , X 2 = t∂ t − r∂ r , X 3 = 1 2r 2 + t 2 2 ∂ t − tr∂ r − e r ∂ φ . (3.10)
which generates SL(2, R). The Killing vectors (3.9) and (3.10) fully denote an enhanced SL(2, R) × U (1) isometry group. This isometry resembles the isometry of AdS 2 spacetime, so the asymptotic symmetry group as used by Brown and Henneaux [4] can be employed to compute the central charge. Note that only U (1) that can be extended to a Virasoro algebra while the SL(2, R) is taken to be frozen at extremality [57]. Enticingly, for non-extremal black hole, one can construct similar SL(2, R) × U (1) isometry where SL(2, R) isometry can be extended to Kac-Moody algebra [49]. In this case, one can extend the calculation to the entropy in warped CFT. However, we will not consider this case within this paper.
Charges
The approach of Brown and Henneaux [4] can be employed to find the central charge of the holographic dual CFT description of an extremal rotating black hole in EMDA theory. To compute the charges that associate with asymptotic symmetry group (ASG) of near-horizon extremal dyonic Kerr-Sen black hole, we should consider all possible contributions from all different fields in the action. Nonetheless, it has been pointed out by Compére [57] and also shown in Ref. [6] for Kerr-Sen black hole that the contributions from electromagnetic and scalar fields, except gravity are zero. Their contributions emerge only through the black hole's parameters in the central term from gravity part. So, asymptotic symmetry of the general black hole family in EMDA theory includes diffeomorphisms ξ that satisfy
δ ξ g µν = L ξ g µν = ξ σ (∂ σ g µν ) + g µσ (∂ ν ξ σ ) + g σν (∂ µ ξ σ ), (3.11)
where the metric deviation is denoted by δ ξ g µν = h µν . The associated conserved charge is
Q ξ = 1 8π ∂Σ k g ζ [h; g]. (3.12)
The given integral is over the boundary of a spatial slice. The contribution of the metric tensor on the central charge is given explicitly by
k g ζ [h; g] = − 1 4 ǫ ρσµν ζ ν D µ h − ζ ν D λ h µλ + h 2 D ν ζ µ − h νλ D λ ζ µ + ζ λ D ν h µλ + h λν 2 (D µ ζ λ + D λ ζ µ ) dx ρ ∧ dx σ . (3.13)
We should note that the last two terms in Eq. (3.13) vanish for an exact Killing vector and an exact symmetry, respectively. The charge Q ζ generates symmetry through the Dirac brackets. The ASG possesses algebra which is given by the Dirac bracket algebra of the following charges [59] {Q ζ ,
Qζ} DB = 1 8π k g ζ Lζg; g = Q [ζ,ζ] + 1 8π k g ζ Lζḡ;ḡ .
(3.14)
In order to employ the ASG, we need to specify the boundary conditions on the metric deviations h µν . The boundary conditions are imposed in order to produce finite and integrable charges. There is not necessarily a unique set of consistent boundary conditions. Therefore, we adopt the boundary conditions such in most Kerr/CFT correspondence articles. In the basis (t, r, θ, φ), we impose the following boundary conditions
h µν ∼ O(r 2 ) O 1 r 2 O 1 r O(1) O 1 r 3 O 1 r 2 O 1 r O 1 r O 1 r O(1) . (3.15)
The most general diffeomorphism symmetry that preserves such boundary conditions (3.15) is generated by the following Killing vector field
ζ = c t + O r −3 ∂ t + {−rǫ ′ (φ) + O(1)} ∂ r + O r −1 ∂ θ + ǫ(φ) + O r −2 ∂ φ , (3.16)
where c t is an arbitrary constant and the prime ( ′ ) denotes the derivative respect to φ. This ASG contains one copy of the conformal group of the circle which is generated by
ζ ǫ = ǫ(φ)∂ φ − rǫ ′ (φ)∂ r ,(3.17)
that will be the part of the near-horizon extremal metric. We know that the azimuthal coordinate is periodic under the rotation φ ∼ φ + 2π. Hence we may define ǫ z = −e −izφ and ζ ǫ = ζ ǫ (ǫ z ). By the Lie bracket, the symmetry generator (3.17) satisfies the Witt algebra
i[ζ y , ζ z ] LB = (y − z)ζ y+z . (3.18)
Then by defining
Q ζ ≡ L z − xδ z,0 ,(3.19)
on (3.14) where x is a free parameter which will not change the central charge, we obtain the conserved charges algebra in quantum form, such that
[L y , L z ] = (y − z)L y+z + c L 12 (y 2 − 1)δ y+z,0 . (3.20)
From the algebra above, we can read-off the value of the left-moving central charge for the near-horizon extremal dyonic Kerr-Sen black hole. It is obtained that
c L = 12am = 12J,(3.21)
This result is identical to the central charge of Kerr-Sen [6] and Kerr [3] black holes. However, we need to note that the relation between mass, spin, and electromagnetic charges is different with to those of Kerr-Sen and Kerr black holes. For extremal dyonic Kerr-Sen black hole, since the mass is given in Eq. (2.13). Each branch corresponds to CFT with the central charge c L = 12am ± , respectively. Recall that these branches will reduce into one only when the spin approaches zero denoting that c L also vanishes. This means that the entropy from CFT will also vanish. Vanishing c L is trivial and exactly denotes that the Virasoro algebra reduces to the classical Witt algebra.
Temperatures
Before going to match the entropies, it is required to calculate the corresponding temperatures. In order to do so, we need to employ the analog of the Hartle-Hawking vacuum, i.e. Frolov-Thorne vacuum that has been used in the Kerr/CFT correspondence [3] because the angular momentum and other thermodynamic quantities are included within this vacuum. Now, we may apply the first law of black hole thermodynamics for rotating dyonic Kerr-Sen black holes (2.9) where the extremal condition satisfies
dM = Ω ex H dJ + Φ ex H dQ + Ψ ex H dP, (3.22) since T ex H = 0. So, we can write T H dS = − [(Ω H − Ω ex H )dJ + (Φ − Φ ex )dQ + (Ψ − Ψ ex )dP ] . (3.23)
For such constrained variations (3.23), we may construct
dS = dJ T L + dQ T q + dP T p . (3.24)
For Kerr black hole, it is considered a quantum scalar field with eigenmodes of the asymptotic energy E and angular momentum J which are given by the following form
Φ = E,J,sφ E,J,s e −iEt+iJφ f s (r, θ),(3.25)
In order to transform this to near-horizon quantities and take the extremal limit, we note that in the nearhorizon coordinates (3.2) we have
e −iEt+iJφ = e −i(E−Ω ex H J)tr0/ǫ+iJφ = e −inRt+inLφ ,(3.26)
where
n R = (E − Ω ex H J)r 0 /ǫ, n L = J. (3.27)
But this is only suitable when there is no contribution of other thermodynamical potentials. Using the fact that any system possesses density of state ρ = e S , where S is the entropy and the fact that there are thermodynamic potentials coming from electromagnetic fields, we may extend Eq. (3.27) to
n R = (E − Ω ex H J − Φ ex Q − Ψ ex P )r 0 /ǫ, n L = J. (3.28)
The density matrix in the asymptotic energy, angular momentum, electric charge, magnetic charge, and pressure eigenbasis now has the Boltzmann weighting factor
e − E−Ω H J−Φ ex Q−Ψ ex P T H = e − n R T R − n L T L − Q Tq − P Tp . (3.29)
When the trace over the modes inside the horizon is taken, the Boltzmann weighting factor will be a diagonal matrix. We can compare the Eqs. (3.28) and (3.29) to obtain the definition of the CFT temperatures, such that
T R = T H r 0 ǫ ex , T L = − ∂T H /∂r + ∂Ω H /∂r + ex , T q = − ∂T H /∂r + ∂Φ/∂r + ex , T p = − ∂T H /∂r + ∂Ψ/∂r + ex . (3.30)
However, we only need T L and T R in order to compute the Cardy entropy. It is obviously seen that T R = 0 for extremal black holes. We also have
T L = m 2 − d 2 − k 2 + a 2 4πam . (3.31)
We have obtained the left-moving temperatures for dyonic Kerr-Sen. Moreover, as mentioned before that this extremal black hole corresponds with two different branches of central charge, 12am ± , the left-moving temperature also behaves the same T L =
m 2 ± −d 2 −k 2 +a 2 4πam±
. From this temperature, we can also obtain the left-moving temperature of Kerr-Sen black hole by setting k = 0. When we turn off both electromagnetic charges, the temperature for Kerr black hole is recovered [3].
Entropy Matching
We have discussed the existence of an asymptotic Virasoro algebra at the boundary in infinity of the nearhorizon extremal geometry. By following semi-classical quantization rules, the operators that define quantum gravity with the given boundary conditions form a Virasoro algebra. We have also provided that scalar quantum fields in the analogue of the Frolov-Thorne vacuum restricted to extremal excitations having the non-vanishing left-moving temperature. Since we identify the left-sector with excitations along ∂ φ and the SL(2, R) isometry as the right sector is frozen, the states are described by a thermal density matrix with temperatures T L , T q , and T p . As mentioned in [57], T q , T p are better interpreted as the CFT chemical potentials as µ q L = −T L /T q and µ p L = −T L /T p . Given all quantities that we need in Cardy formula, we can now compute it in (3.1). It is remarkable that, surprisingly, using the left-moving central charge (3.21) and temperature (3.31) reproduces the Bekenstein-Hawking entropy (2.10) for extremal dyonic Kerr-Sen black hole
S CF T = π(m 2 − d 2 − k 2 + a 2 ) = S BH .
(3.32)
Note that for given values of a, p, q, there exist two branches of extremal black hole's entropy with mass m ± and central charges 12am ± . Simultaneously, we have also reproduced the Bekenstein-Hawking entropy for Kerr-Sen black hole by turning off p. Again, when we assume that q = p = 0, it reduces to the entropy of extremal Kerr black hole. For a → 0 but keeping p, q = 0, this Cardy entropy will vanish showing the entropy of extremal dyonic GMGHS black hole. This matching completes the conjecture of Kerr/CFT correspondence for extremal dyonic Kerr-Sen black holes in EMDA theory. This is clearly not a coincidence and shows that the dyonic Kerr-Sen black hole is dual to 2D CFT represented by non-vanishing left-moving sector.
Kerr/CFT for Gauged Dyonic Kerr-Sen Black Hole
Spacetime metric and thermodynamics
For the gauged case, the corresponding Lagrangian density of the EMDA theory (2.1) is given by [50,55,56]
L gauged = L + √ −g 4 + e −φ + e φ (1 + χ 2 ) l 2 . (4.1)
The gauged version of the dyonic Kerr-Sen black hole or the Kerr-Sen-AdS black is also presented in a simple form in Ref. [50] which is also the solution of above Lagrangian density. In the shifted radial coordinate, the spacetime metric including the electromagnetic potential, its dual, dilaton field, and axion field, are given by
ds 2 = − ∆ ̺ 2X 2 + ̺ 2 ∆ dr 2 + ̺ 2 ∆ θ dθ 2 + ∆ θ sin 2 θ ̺ 2Ŷ 2 , (4.2) A = q(r + d − p 2 /m) ̺ 2X − p cos θ ̺ 2Ŷ , B = p(r + d − p 2 /m) ̺ 2X + q cos θ ̺ 2Ŷ ,(4.
3)
e φ = (r + d) 2 + (k + a cos θ) 2 ̺ 2 , χ = 2 kr − da cos θ (r + d) 2 + (k + a cos θ) 2 ,(4.4)
respectively, whereX
= dt − a sin 2 θ dφ Ξ ,Ŷ = adt − (r 2 − d 2 − k 2 + a 2 ) dφ Ξ , ∆ = (r 2 − d 2 − k 2 + a 2 ) 1 +r 2 − d 2 − k 2 l 2 − 2mr + p 2 + q 2 , ∆ θ = 1 − a 2 l 2 cos 2 θ, Ξ = 1 − a 2 l 2 , ̺ 2 =r 2 − d 2 − k 2 + a 2 cos 2 θ. (4.5)
The dyonic Kerr-Sen-AdS black hole satisfies the following thermodynamic relation
dM = T H dS + Ω H dJ + Φ H dQ + Ψ H dP + V dP,(4.6)
When cosmological constant or the gauge coupling constant l arises as the vacuum expectation value, we can include this parameter in the first law of thermodynamics for black holes [58]. As common rotating AdS black holes, the cosmological constant can be considered as the source of the pressure on the black holes. Hence, it gives rise to another thermodynamic quantity. The quantities on above equations are given by
M = m Ξ , J = ma Ξ , Q = q Ξ , P = p Ξ , T H = r + (2r 2 + − 2d 2 − 2k 2 + a 2 + l 2 ) − ml 2 2π(r 2 + − d 2 − k 2 + a 2 )l 2 , (4.7) S BH = π Ξ (r 2 + − d 2 − k 2 + a 2 ), Ω H = aΞ r 2 + − d 2 − k 2 + a 2 , Φ H = q(r + + d − p 2 /m) r 2 + − d 2 − k 2 + a 2 , (4.8) Ψ H = p(r + + d − p 2 /m) r 2 + − d 2 − k 2 + a 2 , V = 4 3 r + S, P = 3 8πl 2 .
(4.9)
where those are mass, angular momentum, electric charge, magnetic charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity, electric potential, magnetic potential, volume and pressure.
Since there exists the cosmological constant, this black hole possesses more than two horizons as we can see from ∆ which is a quartic function. This means that there also exists cosmological horizons. From this dyonic Kerr-Sen-AdS black hole solution, we can find several solutions by taking certain limits. In order to find Kerr-Sen-AdS black hole, one can turn off p = 0 that will cause k = 0. This implies Ψ H = 0. When one consider equal charges q = p, this will result in d = 0 or no dilaton charge. On the other hand, we can find that Φ H = Ψ H . Another fascinating property of this solution is we can find the superentropic solution of dyonic Kerr-Sen-AdS black hole by taking a → l. Nevertheless, whether it is superentropic or not is defined by the value of l 2 over (d 2 + k 2 ).
Next, we consider the extremal limit for the gauged counterparts of the dyonic Kerr-Sen black hole with ∆ ′ = 0. We find m = r + l 2 (2r 2 + − 2d 2 − 2k 2 + a 2 + l 2 ). (4.10)
As a result, the horizon is given by
r 2 + = 2d 2 + 2k 2 − a 2 − l 2 + x 4 1 − 16x 4 2 + 2x 4 3 6 , (4.11)
where x 4 1 = a 4 + 16d 4 + 16k 4 + l 4 , x 4 2 = a 2 d 2 + a 2 k 2 + d 2 l 2 + k 2 l 2 − 2d 2 k 2 , and x 4 3 = 7a 2 l 2 + 6l 2 q 2 + 6l 2 p 2 . All the horizons now coincide into one. Similar to the ungauged case, the Hawking temperature vanishes while other thermodynamic quantities remain non-zero. Hence, there are possibly more than two branches of extremal black holes corresponding to roots of (4.10) since d, k ∼ 1/m. The physical conditions for the existence of extremal black holes are x 4 1 − 16x 4 2 + 2x 4 3 ≥ 0, r 2 + > 0 and m > 0.
Dual CFT of Dyonic Kerr-Sen-AdS Black hole
The near-horizon form of dyonic Kerr-Sen-AdS black hole can be obtained using similar transformation (3.2). In order to study the near-horizon extremal region, we need to approximate ∆ in terms of event horizon r + .
In the near-horizon of extremal black holes, the function ∆ takes form [18]
∆ = (r − r + ) 2 υ + O (r − r + ) 3 ,(4.12)
where the function υ is given by
υ = ∆ ′′ (r + ) 2 = 1 + 6r 2 + − 2d 2 − 2k 2 + a 2 l 2 . (4.13)
We can obtain the near-horizon extremal metric of Eq. (4.2). By additional scaling dt → dt/υ, the nearhorizon extremal metric is then given by
ds 2 = Γ(θ) −r 2 dt 2 + dr 2 r 2 + α(θ)dθ 2 + γ(θ) (dφ + erdt) 2 ,(4.14)
where the metric functions are given by
Γ(θ) = ̺ 2 + υ , α(θ) = υ ∆ θ , γ(θ) = r 4 0 ∆ θ sin 2 θ ̺ 2 + Ξ 2 , ̺ 2 + = r 2 + − d 2 − k 2 + a 2 cos 2 θ, e = 2ar + Ξ r 2 0 υ . (4.15)
In the near-horizon limit, we find that the gauge field and its dual are given by
A + = f a (θ) (dφ +êrdt) + q (r + + d) 2 + k 2 − a 2 − 2p 2 r + /m ê(r 2 + − d 2 − k 2 + a 2 )
dφ, (4.16)
B + = f b (θ) (dφ +êrdt) + p (r + + d) 2 + k 2 − a 2 − 2p 2 r + /m ê(r 2 + − d 2 − k 2 + a 2 ) dφ,ê = 2ar + Ξ r 2 0 ,(4.17)
where the functions f (θ) for each field are given by
f a (θ) = q 2r+p 2 m − (r + + d) 2 − k 2 + a 2 cos 2 θ + 2apr + cos θ 2ar + Ξ̺ 2 + r 2 0 , (4.18) f b (θ) = p 2r+p 2 m − (r + + d) 2 − k 2 + a 2 cos 2 θ − 2aqr + cos θ 2ar + Ξ̺ 2 + r 2 0 , (4.19)
where we can gauge away the second term on the electromagnetic potential (4.16) and its dual (4.17). Moreover, after finding the gauge field in the near-horizon limit, we can also find the dilaton and axion field in this limit which are given as e φ+ = (r + + d) 2 + (k + a cos θ) 2 ̺ 2 + , χ + = 2 kr + − da cos θ (r + + d) 2 + (k + a cos θ) 2 .
(4.20)
In order to compute the central charge of dyonic Kerr-Sen-AdS black hole, we can also employ the similar ASG calculation to the near-horizon extremal metric (4.14) since we obtain the identical metric form and isometry. From the same lengthy calculation, we can obtain the left-moving central charge for the near-horizon extremal dyonic Kerr-Sen-AdS black holes from EMDA theory. It is obtained that
c L = 12ar + υ , (4.21)
where the event horizon υ and r + are given in (4.13) and (4.11), respectively. We can also recover the central charge of Kerr-Sen-AdS and Kerr-Sen-AdS black holes with vanishing axion charge by setting p = 0 and p = q, respectively. It is also clear that by turning off p, q, we can find the central charge of Kerr-AdS black hole [7]. Similarly, before going to match the entropy, it is required to calculate the corresponding temperatures. For gauged case, the presence of the cosmological constant can be considered as the additional thermodynamic quantity in the thermodynamic equation. Hence, we can write the first law of black hole thermodynamics for this black hole where the extremal condition satisfies
dM = Ω ex H dJ + Φ ex H dQ + Ψ ex H dP + V ex dP, (4.22)
since T ex H = 0. We are required to write that
T H dS = − [(Ω H − Ω ex H )dJ + (Φ − Φ ex )dQ + (Ψ − Ψ ex )dP + (V − V ex )dP] . (4.23)
For such constrained variations (4.23), we may construct
dS = dJ T L + dQ T q + dP T p + dP T P . (4.24)
The last term in above equation is related to the cosmological constant. Similar to the dyonic Kerr-Sen black hole, there exist temperatures conjugate to electric and magnetic charges aside from the right-and left-moving temperatures. The existence of cosmological constant is related to the presence of another temperature conjugate to the cosmological pressure which is defined as
T P = − ∂TH /∂r+ ∂V /∂r+ ex
, although we only need the the left-moving temperature in this case. The explicit expression of the left-moving temperature is
T L = υ(r 2 + − d 2 − k 2 + a 2 ) 4πar + Ξ . (4.25)
From this temperature, we can recover the left-moving temperatures for Kerr-Sen-AdS and Kerr-Sen-AdS with vanishing axion charge similarly with the central charge. When we turn off both electromagnetic charges, the temperatures for Kerr-AdS black hole is then recovered. We have calculated the quantities that we need in Cardy formula, we can now compute it in (3.1). It is remarkably found that for extremal dyonic Kerr-Sen-AdS black hole possesses the following CFT entropy
S CF T = π Ξ (r 2 + − d 2 − k 2 + a 2 ). (4.26)
This result matches with the Bekenstein-Hawking entropy for dyonic Kerr-Sen-AdS black hole. The entropy of Kerr-Sen-AdS black hole and Kerr-Sen-AdS black hole with vanishing axion charge can also be found from this. When we assume that q = p = 0, it reduces to the entropy of extremal Kerr-AdS black hole [7]. It is worth noting that for Kerr-Newman-AdS black holes, there exists second dual CFT [5] in which can be studied when considering a → 0 and then proposing the electromagnetic field as the part of the geometry. In this second dual CFT, the temperatures T q and T p are useful to reveal the dual CFT of Reissner-Nordström black hole family. Nonetheless, for black hole solutions in EMDA theory, in this case is the Kerr-Sen black hole [41], the second dual CFT fails to be shown. Hence, we will not further consider the second dual CFT for the dyonic Kerr-Sen-AdS black hole family in this paper.
Ultraspinning dyonic Kerr-Sen-AdS black hole
Spacetime metric and thermodynamics
Another interesting spacetime we want to study is the ultraspinning dyonic Kerr-Sen-AdS black hole. In this circumstance, we consider ultraspinning limit a → l. This metric has been given in Ref. [50] while herein we consider the spacetime metric in shifted radial coordinate. In order to find the ultraspinning version of the dyonic Kerr-Sen-AdS black hole, we need to re-define the coordinateφ asφ →φΞ to exclude the conical singularity after taking ultraspinning limit. Since the azimuthal coordinate becomes non-compact, we need to compactify it asφ →φ+µ where µ is not 2π and dimensionless. As explained in Ref. [52], µ is proposed as another thermodynamic quantity or the chemical potential. The resulting metric, electromagnetic potential, dual of the electromagnetic potential, dilaton and axion fields are given by
ds 2 = − ∆ ̺ 2X 2 + ̺ 2 ∆ dr 2 + ̺ 2 sin 2 θ dθ 2 + sin 4 θ ̺ 2Ŷ 2 , (5.1) A = q(r + d − p 2 /m) ̺ 2X − p cos θ ̺ 2Ŷ , B = p(r + d − p 2 /m) ̺ 2X + q cos θ ̺ 2Ŷ ,(5.
2)
e φ = (r + d) 2 + (k + l cos 2 θ) ̺ 2 , χ = 2 kr − dl cos θ (r + d) 2 + (k + l cos 2 θ) , (5.3) respectively, whereX = dt − l sin 2 θdφ,Ŷ = ldt − (r 2 − d 2 − k 2 + l 2 )dφ. ∆ = (r 2 − d 2 − k 2 + l 2 ) 2 l 2 − 2mr + p 2 + q 2 , ̺ 2 =r 2 − d 2 − k 2 + l 2 cos 2 θ.
For the ultraspinning black hole, the thermodynamic quantities satisfy the following relation
dM = T H dS + Ω H dJ + Φ H dQ + Ψ H dP + V dP + Kdµ, (5.4)
where K is the conjugate of µ or the conjugate of chemical potential. The thermodynamic quantities of this black hole are given by
M = mµ 2π , J = mlµ 2π , Q = qµ 2π , P = pµ 2π , T H = 2r + (r 2 + − d 2 − k 2 + l 2 ) − ml 2 2π(r 2 + − d 2 − k 2 + l 2 )l 2 , (5.5) S BH = µ 2 (r 2 + − d 2 − k 2 + l 2 ), Ω H = l r 2 + − d 2 − k 2 + l 2 , Φ H = q(r + + d − p 2 /m) r 2 + − d 2 − k 2 + l 2 , (5.6) Ψ H = p(r + + d − p 2 /m) r 2 + − d 2 − k 2 + l 2 , V = 2µ 3 r + (r 2 + − d 2 − k 2 + l 2 ), K = m l 2 − (r 2 + − d 2 − k 2 ) 4π(r 2 + − d 2 − k 2 + l 2 )
.
(5.7)
Again, the extremal limit can be found by using conditions ∆ = 0 and ∆ ′ = 0,
m = 2r + l 2 (r 2 + − d 2 − k 2 + l 2 ). (5.8)
The event horizon of this extremal ultraspinning version is located at
r 2 + = d 2 + k 2 − l 2 + 4l 4 + 4d 4 + 4k 4 − 8l 2 d 2 − 8l 2 k 2 + 8d 2 k 2 + 3l 2 q 2 + 3l 2 p 2 3 . (5.9)
In this extremal ultraspinning limit, the Hawking temperature also vanishes. There are possibly more than two branches of extremal ultraspinning black holes corresponding to roots of (5.8). The physical conditions for the existence of extremal black holes are 4l 4 + 4d 4 + 4k 4 − 8l 2 d 2 − 8l 2 k 2 + 8d 2 k 2 + 3l 2 q 2 + 3l 2 p 2 ≥ 0, r 2 + > 0 and m > 0.
For the ultraspinning version of the dyonic Kerr-Sen-AdS black hole, the near-horizon coordinate transformations are identical. The near-horizon extremal geometry of ultraspinning dyonic Kerr-Sen-AdS black hole is given by Eq. (4.14) with the following functions
Γ(θ) = ̺ 2 + υ , α(θ) = υ sin 2 θ , γ(θ) = r 4 0 sin 4 θ ̺ 2 + , ̺ 2 + = r 2 + − d 2 − k 2 + l 2 cos 2 θ, υ = 2 + 6r 2 + − 2d 2 − 2k 2 l 2 , e = 2lr + r 2 0 υ . (5.
10)
The functions f (θ) for the gauge field and its dual, dilaton and axion fields in near-horizon region are given by
f a (θ) = q 2r+p 2 m − (r + + d) 2 − k 2 + l 2 cos 2 θ + 2lpr + cos θ 2lr + ̺ 2 + r 2 0 , (5.11) f b (θ) = p 2r+p 2 m − (r + + d) 2 − k 2 + l 2 cos 2 θ − 2lqr + cos θ 2lr + ̺ 2 + r 2 0 , (5.12) e φ+ = (r + + d) 2 + (k + l cos θ) 2 ̺ 2 + , χ + = 2 kr + − dl cos θ (r + + d) 2 + (k + l cos θ) 2 . (5.13)
For this ultraspinning version, the near-horizon geometry of the black hole with metric potential (5.10) also possesses SL(2, R) × U (1) isometry group with similar Killing vector fields (3.10) and (3.9).
Dual CFT of Ultraspinning Dyonic Kerr-Sen-AdS Black hole
By employing ASG calculation, for ultraspinning dyonic Kerr-Sen-AdS, the central charge is given by
c L = 6r + µl πυ ,(5.14)
which interestingly depends on chemical potential µ, where υ and r + are given in Eqs. (5.10) and (5.9). Even in this ultraspinning case, there are possibly more than two branches of extremal black holes related to the mass which also correspond to CFTs with different central charges. For ultraspinning black holes, since the thermodynamic relation is distinct from the normally spinning black hole because there is a chemical potential, it is required to employ the relation (5.4). Using the fact T ex H = 0 in (5.4) for extremal case, we can construct
T H dS = − (Ω H − Ω ex H )dJ + (Φ − Φ ex )dQ + (Ψ − Ψ ex )dP + (V − V ex )dP + (K − K ex )dµ , (5.15) dS = dJ T L + dQ T q + dP T p + dP T P + dµ T µ . (5.16)
The further analysis is identical with those when we do not consider the ultraspinning case. So, we can also define another temperature conjugate to the chemical potential, T µ = − ∂TH /∂r+ ∂K/∂r+ ex . Nevertheless, the most significant temperature to compute Cardy formula is T L as given by
T L = υ(r 2 + − d 2 − k 2 + l 2 ) 4πlr + . (5.17)
It is remarkable that one can also find the similar matching for ultraspinning version of dyonic Kerr-Sen-AdS black hole,
S CF T = µ 2 (r 2 + − d 2 − k 2 + l 2 ) = S BH . (5.18)
This matching so far completes the conjecture of Kerr/CFT correspondence for extremal dyonic black holes, especially in EMDA theory. This is clearly not a coincidence and shows that the dyonic Kerr-Sen black hole and its family are dual to 2D CFT represented by non-vanishing left-moving sector.
Dyonic Kerr-Sen-AdS and Kerr-Newman-AdS in Comparison
Here we will compare the properties from CFT of the black holes in Einstein-Maxwell theory, i.e. dyonic Kerr-Newman-AdS solution and black hole in EMDA theory, i.e. dyonic Kerr-Sen-AdS black hole. The study of superentropic black hole from rotating black holes with non-vanishing cosmological constant has been proposed in Ref. [52] using Kerr-Newman-AdS black hole solution. Furthermore, it has been proposed the same thermodynamical properties for similar black hole solution from the Kerr/CFT duality point of view. Using the Kerr/CFT method, authors in Ref. [16] calculated the entropy of extremal ultraspinning Kerr-Newman-AdS black holes. Both papers actually do not consider the existence of dyonic solution or non-zero magnetic charge. However, their results will not be so different with dyonic solution if we just change q 2 → q 2 + p 2 where p is the magnetic charge. In this section, we will compare the thermodynamical properties of the dyonic Kerr-Newman-AdS and dyonic Kerr-Sen-AdS black holes including its ultraspinning versions. Later on, we also compare with the result when we consider asymptotically dS spacetime.
Kerr-Newman-AdS black hole and its ultraspinning version
The dyonic Kerr-Newman-AdS black hole spacetime is given by
ds 2 = − ∆ ̺ 2X 2 + ̺ 2 ∆ dr 2 + ̺ 2 ∆ θ dθ 2 + ∆ θ sin 2 θ ̺ 2Ŷ 2 , (6.1) A = −qr ̺ 2X − p cos θ ̺ 2Ŷ , B = −pr ̺ 2X + q cos θ ̺ 2Ŷ ,(6.2)
where A and B are the electromagnetic potential and its dual, respectively. Here, we havê
X = dt − a sin 2 θ dφ Ξ ,Ŷ = adt − (r 2 + a 2 ) dφ Ξ ∆ = (r 2 + a 2 ) 1 +r 2 l 2 − 2mr + p 2 + q 2 , ∆ θ = 1 − a 2 l 2 cos 2 θ, Ξ = 1 − a 2 l 2 , ̺ 2 =r 2 + a 2 cos 2 θ. (6.
3)
The dyonic Kerr-Newman-AdS black hole satisfies the identical thermodynamic relation (5.4) as the dyonic Kerr-Sen-AdS black hole. The thermodynamic quantities for above metric are given by
M = m Ξ , J = ma Ξ , Q = q Ξ , P = p Ξ , T H = r + (2r 2 + + a 2 + l 2 ) − ml 2 2π(r 2 + + a 2 )l 2 , (6.4) S BH = π Ξ (r 2 + + a 2 ), Ω H = aΞ r 2 + + a 2 , Φ H = qr + r 2 + + a 2 , Ψ H = pr + r 2 + + a 2 , (6.5) V = 2π 3Ξ (r 2 + + a 2 )(2r 2 + l 2 + a 2 l 2 − r 2 + a 2 ) + a 2 l 2 (q 2 + p 2 ) r + l 2 Ξ . (6.6)
where those are mass, angular momentum, electric charge, magnetic charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity, electric potential, magnetic potential, volume and pressure. The event horizon of dyonic Kerr-Newman black hole is located at r 2 + = −a 2 − l 2 + a 4 + l 4 + 14a 2 l 2 + 12l 2 q 2 + 12l 2 p 2 6 .
(6.7)
The mass function can be written as
m = r + 1 + a 2 l 2 + 2r 2 + l 2 . (6.8)
Using the Kerr/CFT correspondence, they have found that the central charge and CFT temperature are given by
c L = 12ar + 1 + a 2 l 2 + 6r 2 + l 2 , T L = (r 2 + + a 2 ) 1 + a 2 l 2 + 6r 2 + l 2 4πar + Ξ . (6.9)
The central charge and temperature above reproduce exactly the Bekenstein-Hawking entropy of extremal dyonic Kerr-Newman-AdS black hole with the help of Cardy formula. The next case is the ultraspinning version of dyonic Kerr-Newman-AdS black hole where the ultraspinning limit is still given by a → l that yields the superentropic black hole. The ultraspinning version possesses the following thermodynamic volume and conjugate of the chemical potential
V = 2µ 3 r + (r 2 + + l 2 ), K = (l 2 − r 2 + ) (r 2 + + l 2 ) 2 + l 2 (q 2 + p 2 ) 8πr + (r 2 + + l 2 )
. (6.10)
We can write the mass and event horizon as
m = 2r + r 2 + l 2 + 1 , r 2 + = −l 2 + l 4l 2 + 3q 2 + 3p 2 3 . (6.11)
Superentropic black holes violates the reverse isoperimetric inequality (RII) [52], which asserts
R ≡ (D − 1)V ω D−2 1 D−1 ω D−2 A 1 D−1 ≥ 1, (6.12)
where A is the horizon area, D is the dimension, and
ω D = µπ D−1 2 Γ D+1 2 . (6.13)
For superentropic dyonic Kerr-Newman-AdS black holes, we have R = r 2 + r 2 + + l 2 1 6 . (6.14)
Since R < 1, we know that this black hole is superentropic. Using the Kerr/CFT method, for superentropic dyonic Kerr-Newman-AdS black holes, it is found that
c L = 3µr + l 3 π(3r 2 + + l 2 ) , T L = (r 2 + + l 2 )(3r 2 + + l 2 ) 2πr + l 3 ,(6.
Comparison
In the previous subsection, we have considered the dyonic Kerr-Newman-AdS black hole solution and its thermodynamic quantities. The results for ultraspinning version from Kerr/CFT correspondence are also obtained. We provide the comparison of central charge, temperature, and entropy from CFT of both black holes in EMDA and Einstein-Maxwell theories in Table 1. It is clear that the main difference is the presence of dilaton and axion charges in these quantities. It is important to recall that r + is also different for both black holes. The existence of dilaton and axion charge is important in the observation of the black holes, for example to probe the existence of beyond Einstein-Maxwell theory, in this case is EMDA theory. It is crucial that for extremal dyonic Kerr-Sen-AdS black hole, m > 0, r 2 + > 0, x 4 1 − 16x 4 2 + 2x 4 3 > 0. These conditions lead to (6r 2 + + a 2 + l 2 )/2 > (2r 2 + + a 2 + l 2 )/2 > d 2 + k 2 implying that c L is always positive. Aside from the central charge, parameters d, k also define the value of the left-moving temperature and entropy as we can see in Table 1. With the same conditions as c L , it can be shown that these quantities are always positive as well, similar to dyonic Kerr-Newman-AdS solution. Nonetheless, since d, k ∼ 1/m in dyonic Kerr-Sen-AdS black hole, it becomes the main difference from the dyonic Kerr-Newman-AdS solution because this leads to possibly more than two branches of mass with its own dual CFT.
c L 12ar + 1+ 6r 2 + −2d 2 −2k 2 +a 2 l 2 12ar + 1+ 6r 2 + +a 2 l 2 T L 1+ 6r 2 + −2d 2 −2k 2 +a 2 l 2 (r 2 + −d 2 −k 2 +a 2 ) 4πar + Ξ 1+ 6r 2 + +a 2 l 2 (r 2 + +a 2 ) 4πar + Ξ S CF T π Ξ (r 2 + − d 2 − k 2 + a 2 ) π Ξ (r 2 + + a 2 )
As explained in [52], ultraspinning black hole might produce superentropic black hole. It means that black holes has maximum upper entropy in superentropic circumstance. We have shown that ultraspinning dyonic Kerr-Newman-AdS black hole always violates RII, likewise the Kerr-Newman-AdS black hole. However, it is different from the ultraspinning dyonic Kerr-Sen-AdS black hole where the violation depends on the value of electromagnetic charges (or dilaton and axion charges), mass and AdS length. The RII is given by
R = r 2 + r 2 + − d 2 − k 2 + l 2 1 6 . (6.16)
Explicitly, if 0 ≤ q 2 + p 2 < 2ml or 0 ≤ d 2 + k 2 < l 2 , RII will be violated, then the black hole is superentropic. Otherwise, if q 2 + p 2 ≥ 2ml or d 2 + k 2 ≥ l 2 , RII will not be violated, and the black hole is subentropic. Notably, the extremal and non-extremal ultraspinning Kerr-Sen-AdS black hole [50] are not always superentropic depending on the value of d, k. This is different from extremal ultraspinning dyonic Kerr-Newman-AdS black hole and its non-dyonic counterpart which are always superentropic.
However, the similarity between the ultraspinning dyonic black holes happens to be on the positivity of the CFT quantities. Similar to the Kerr-Sen-AdS black hole with general spin, we can observe from Table 2 that the central charge for ultraspinning dyonic Kerr-Sen-AdS is always positive. Since from the positivity of the mass we have r 2 + + l 2 > d 2 + k 2 , then the denominator of c L will always be positive because 3r 2 + + l 2 > r 2 + + l 2 > d 2 + k 2 . From Table 2, since 3r 2 + + l 2 > r 2 + + l 2 > d 2 + k 2 , we can conclude that the central charge, temperature and entropy of extremal ultraspinning dyonic Kerr-Sen-AdS solution are always positive, similar to the extremal ultraspinning dyonic Kerr-Newman-AdS solution. π ( 3r 2 + −d 2 −k 2 +l 2 ) 3µr + l 3 π(3r 2 + +l 2 )
T L ( 3r 2 + −d 2 −k 2 +l 2 ) (r 2 + −d 2 −k 2 +l 2 ) 2πr + l 3 (r 2 + +l 2 )(3r 2 + +l 2 ) 2πr + l 3
S CF T µ 2 (r 2 + − d 2 − k 2 + l 2 ) µ 2 (r 2 + + l 2 )
There is an interesting fact that central charge exactly can be negative. However, it is not for this asymptotically AdS black hole family. Yet, when we change the solution to have positive cosmological constant (or asymptotically dS) by l 2 → −l 2 , c L can be negative. For this 'dS' type, note that we have the following relations m = − r + l 2 (2r 2 + − 2d 2 − 2k 2 + a 2 − l 2 ), r 2 + = 2d 2 + 2k 2 − a 2 + l 2 + x 4 1 − 16x 4 2 + 2x 4 3 6 , (6.17) where x 4 1 = a 4 + 16d 4 + 16k 4 + l 4 , x 4 2 = a 2 d 2 + a 2 k 2 − d 2 l 2 − k 2 l 2 − 2d 2 k 2 , x 4 3 = −7a 2 l 2 − 6l 2 q 2 − 6l 2 p 2 , m > 0, and r 2 + > 0. Negative c L can be obtained when (6r 2 + + a 2 − l 2 )/2 > d 2 + k 2 . This is always valid as long as black hole exists, i.e., the quantity x 4 1 − 16x 4 2 + 2x 4 3 > 0 (when the square root quantity is zero, exceptionally c L → ∞). It is known that negative central charge exists in non-unitary CFT which may appear in ghost system from string theory [60]. Moreover, there is possibly more than two branches of mass for this black hole resulting in more than two branches of CFT with different central charges just like in the AdS type. Notably, when we consider RII for l 2 → −l 2 in (6.16) it is obvious that R > 1 implying that superentropic black hole cannot be obtained.
Conclusion
We have demonstrated that the Kerr/CFT correspondence is explicitly well-defined for a dyonic black hole family in EMDA theory, i. e. dyonic Kerr-Sen black hole, its gauged and ultraspinning counterparts. Our work enlarges the family of metrics respecting Kerr/CFT correspondence for dyonic black holes in EMDA theory. For dyonic Kerr-sen black hole, we have obtained that the central charge is identical to Kerr-Sen and Kerr black holes, yet with different constraint on the mass parameter due to the presence of dilaton and axion charges. Interestingly, due to the presence of these charges, we have found that the mass possesses two different branches leading also to the central charge of the CFTs with two different branches. The main upshot to prove that the entropy from CFT matches with Bekenstein-Hawking entropy has succeeded for this dyonic black hole family in EMDA theory.
We then extend the duality calculation to the gauged and ultraspinning counterparts. Since there exist more than two horizons, we have had to approximate the single event horizon for its extremal case. Since for non-gauged dyonic Kerr-Sen black hole the central charge could have two branches, we have argued that possibly there should be more than two branches of the mass for gauged case that leads to more than two branches of the central charge and entropy. We have found that finite central charges for both gauged and ultraspinning black hole solutions reproduce Bekenstein-Hawking entropy. It has been proven also that the central charge, temperature and entropy for these black holes are always positive. This extremal ultraspinning dyonic black hole and its non-extremal counterpart cannot always be superentropic in the ultraspinning limit depending on the value of d, k. We have also considered the asymptotically dS solution with l 2 → −l 2 substitution. Interestingly in this case, the central charge can be negative unlike the AdS type. It is well-known that negative central charge might appear in non-unitary CFT, for example in string theory. It will be interesting for future investigation.
In conclusion, our calculations have supported the Kerr/CFT correspondence conjecture. This result suggests that the extremal rotating dyonic black holes in EMDA theory is holographically dual to 2D CFT represented by non-vanishing left-moving central charge. For the future investigation, it would be intriguing to explore the hidden conformal symmetry from non-extremal dyonic black hole family in EMDA theory using the scalar field and higher spin fields.
15 )
15which reproduces the Bekenstein-Hawking entropy for extremal ultraspinnning dyonic Kerr-Newman-AdS black hole by employing Cardy formula.
Table 1 :
1Comparison of CFT Quantities of Extremal Spinning Dyonic Black Holes.Quantity
Dyonic Kerr-Sen-AdS
Dyonic Kerr-Newman-AdS
Table 2 :
2Comparison of CFT Quantities of Extremal Ultraspinning Dyonic Black Holes QuantityDyonic Kerr-Sen-AdS Dyonic Kerr-Newman-AdSc L
3µr + l 3
AcknowledgmentsWe would like to thank the anonymous referee for valuable comments which help improve our work considerably. M. F. A. R. S. is supported by the Second Century Fund (C2F), Chulalongkorn University, Thailand.
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| [] |
[
"Variation of Mass with Velocity: \"Kugeltheorie\" or \"Relativtheorie\"",
"Variation of Mass with Velocity: \"Kugeltheorie\" or \"Relativtheorie\""
] | [
"Galina Weinstein "
] | [] | [] | This paper deals with four topics: The first subject is Abraham's spherical electron, Lorentz's contracted electron and Bücherer's electron. The second topic is Einstein's 1905 relativity theory of the motion of an electron. Einstein obtained expressions for the longitudinal and transverse masses of the electron using the principle of relativity and that of the constancy of the velocity of light. The third topic is Einstein's reply to Ehrenfest's query. Einstein's above solution appeared to Ehrenfest very similar to Lorentz's one: a deformed electron. Einstein commented on Ehrenfest's paper and characterized his work as a theory of principle and reasoned that beyond kinematics, the 1905 heuristic relativity principle could offer new connections between non-kinematical concepts. The final topic is Kaufmann's experiments. Kaufmann concluded that his measuring procedures were not compatible with the hypothesis posited by Lorentz and Einstein. However, unlike Ehrenfest, he gave the first clear account of the basic theoretical difference between Lorentz's and Einstein's views. Finally, Bücherer conducted experiments that confirmed Lorentz's and Einstein's models; Max Born analyzed the problem of a rigid body and showed the existence of a limited class of rigid motions, and concluded, "The main result was a confirmation of Lorentz's formula". | null | [
"https://export.arxiv.org/pdf/1205.5951v1.pdf"
] | 119,284,893 | 1205.5951 | 0bf2eab8a9cda03d70171125f4d0ec8b4f9ea3f6 |
Variation of Mass with Velocity: "Kugeltheorie" or "Relativtheorie"
Galina Weinstein
Variation of Mass with Velocity: "Kugeltheorie" or "Relativtheorie"
This paper deals with four topics: The first subject is Abraham's spherical electron, Lorentz's contracted electron and Bücherer's electron. The second topic is Einstein's 1905 relativity theory of the motion of an electron. Einstein obtained expressions for the longitudinal and transverse masses of the electron using the principle of relativity and that of the constancy of the velocity of light. The third topic is Einstein's reply to Ehrenfest's query. Einstein's above solution appeared to Ehrenfest very similar to Lorentz's one: a deformed electron. Einstein commented on Ehrenfest's paper and characterized his work as a theory of principle and reasoned that beyond kinematics, the 1905 heuristic relativity principle could offer new connections between non-kinematical concepts. The final topic is Kaufmann's experiments. Kaufmann concluded that his measuring procedures were not compatible with the hypothesis posited by Lorentz and Einstein. However, unlike Ehrenfest, he gave the first clear account of the basic theoretical difference between Lorentz's and Einstein's views. Finally, Bücherer conducted experiments that confirmed Lorentz's and Einstein's models; Max Born analyzed the problem of a rigid body and showed the existence of a limited class of rigid motions, and concluded, "The main result was a confirmation of Lorentz's formula".
The Law of Variation of Mass with Velocity
Abraham's spherical electron
According to Max Abraham inertia was created by the electromagnetic field in the ether. Therefore, in order to obtain the law of variation of mass with velocity, one had first to calculate the electromagnetic momentum from the electron's self field.
Using the usual definition of the second law of Newton, in the quasi-stationary approximation, Abraham replaced the electromagnetic momentum in the second law of Newton, and converted it into the definition: acceleration times mass equals force. He then obtained the law for the variation of mass with velocity, while assuming rigid spherical electrons, keeping their spherical form at any velocity.
Abraham assumed that the total apparent mass of the electron is not the same when the actual force applied to the electron is parallel with its velocity and tends to accelerate its motion, as when it is perpendicular to the velocity and tends to alter its direction: 1 Accordingly one distinguishes between the longitudinal mass M l and the transverse mass M t , and:
Lorentz's deformed electron
In 1904, in his paper, "Electromagnetic Phenomena in a System Moving with any Velocity Smaller than that of Light", Lorentz calculated the electron's mass from its electromagnetic momentum too. However, Lorentz assumed that the moving electron was contracted in the direction of motion. He obtained the following equations for longitudinal mass and the transverse mass. 2 where r is the electron's radius and, Max Abraham had opposed Lorentz's expression, because Lorentz's electron would need to rely upon non-electrical internal forces to sustain it. Still this did not satisfy Abraham's desire for a wholly electromagnetic basis for dynamics. Abraham admitted that the new expression was much simpler than his from a mathematical point of view, but simplicity was not yet a reason for preferring a mathematical expression unless the simple expression was confirmed through research. Abraham adhered to a rigid electromagnetic foundation for a theory of the electron, and also chose a rigid spherical electron.
There was a third law for the variation of mass with velocity proposed by Alfred Bücherer in 1904. 3 This law, like Abraham's, was also compatible with a completely electromagnetic electron theory:
Einstein's Theory of the Motion of an Electron
Einstein's 1905 expressions for the mass of the electron
Einstein's last derivation in section §10 of the 1905 relativity paper concerned the dynamics of a (slowly accelerated) electron. 4 Einstein obtained expressions for the longitudinal and transverse masses, using the principle of relativity and that of the constancy of the velocity of light.
Consider a particle in motion with a charge e (Einstein calls it an "electron"), in an external electromagnetic field. For its law of motion, we assume: F = mass x acceleration. If the electron is at rest at a particular instant, its motion during the next instant of time will be according to:
where m is the electron's mass, e is its charge, x, y, z are its coordinates (r) and t its time relative to the system K.
Einstein assumes that at the moment we are observing the electron, it is at the origin of the coordinates and is moving with velocity v along the x axis of system K. Therefore, at the given moment t = 0, the electron is at rest relative to another system of coordinates k, which is moving in parallel motion with velocity v along the x axis of K.
Einstein is guided by the principle of relativity: the electron is instantaneously at rest in k, and the same definition of Newton's second law (F= mass x acceleration) is valid according to the principle of relativity for both K and k. The equations of motion relative to k are thus:
(primed terms refer to k). Using the Lorentz transformations for coordinates and time and the transformations for the electric and magnetic fields, and applying them to K, one obtains: 5 Einstein then takes into account "the conventional approach" and inquires as to "the 'longitudinal' and the 'transverse' mass of the moving electron". Using the above equations written in the following form:
He then notes that eE' x , eE' y , eE' z are the components of pondermotive force acting on the electron, as viewed in the moving system, moving at this moment with the same velocity as the electron.
Einstein then says, "the force acting on the electron is called", and maintains the Newtonian equation: mass times acceleration = force, while acceleration is measured in the system K, and writes the longitudinal mass: and transverse mass:
In 1905 Einstein took the ordinary point of view and wrote the 'longitudinal' and the 'transverse' mass of the moving electron.
In a 1913 reprint Einstein appended a note to the above word "called": "The definition of force given here [mass times acceleration = force] is not advantageous as was first noted by M. Planck. It is instead appropriate to define force in such a way that the laws of momentum and conservation of energy take the simplest form". 6
Relativistic Dynamics
Although apparently the 1905 paper dealt with kinematics and electrodynamics, the scope of the paper was beyond these fields and led to the inauguration of relativistic dynamics. Einstein remarked that his results are also "valid for ponderable material points, because a ponderable material point can be made into an electron (in our sense of the word) by adding to it an arbitrarily small electric charge". 7 For Einstein ponderable mater is uncharged matter; Einstein was declaring that the ether was superfluous, but he still used the conventional "ether-based" notions. This was the language he could communicate with his colleagues (as much as he could find them while writing the relativity paper as a patent clerk).
But there is much more to Einstein's last conclusion; Einstein tried to explain to his readers something very important: his results concerning the electron (the mass of the electron and the pondermotive force acting on the electron) are also valid for material points. Einstein was thus not only presenting another model for the electron; Einstein in section §10 laid down the starting point for a dynamics of a slowly moving material pointalthough he himself did not develop a relativistic dynamics.
In 1906 Max Planck inaugurated relativistic dynamics, although Planck still remained within the confines of electrodynamics. Planck defined the second law of Newton, in terms of the rate of change of a new relativistic momentum, in his paper, "The principle of Relativity and the Equations of Mechanics". 8 He extended Einstein's procedure of 1905 in section §10 in the following way:
Planck imagined a mass point, a particle, to be at the origin of the coordinate system x, y, z, t (equivalent to Einstein's K), and having velocity components, v x , v y , v z and he required as for the equations of motion of the particle.
Planck considered a new reference system (equivalent to Einstein's k), moving with the same velocity as the mass point, relative to K; that is, moving along the x axis of the original reference system with a constant velocity v, the components of which are, v x , v y , v z . Planck said that the particle moves with finite velocity of size q and parallel to x. The particle moves according to the regular Newtonian equation of motion for a free mass point (mass times acceleration = force).
However, now Planck defined the force as the electromagnetic force, and thus the mass point moves under the action of an electromagnetic field. He transformed the Newtonian equations of motion to another reference system whose x-axis coincides with the direction of the velocity q, that is, relative to which the mass point moves with the velocity q relative to k. This is actually the system x, y, z, t at rest (K), but we can call this system, system k'. Planck inserted the velocity of the particle q instead of v into Einstein's transformation for the electric and magnetic fields, and wrote for the system x, y, z, t the following equation:
ma x /√(1 -q 2 /c 2 ) = eE' x -(ev x /c 2 )(v x E' x +v y E' y + v z E' z ) + (e/c)(v y B' z + v z B' y )
, and so on Planck said that this equation confirms a general result for Einstein's relations (transformation for the electric and magnetic fields), and for the Newtonian equation of motion for a free mass point, for any value of v. "We will now bring the equations of motion to their simplest form", said Planck. 9 Planck multiplied the Newtonian equations of motion with respect to k' by v x , v y , v z :
(v x E' x +v y E' y + v z E' z ) = m(v x a x + v y a y + v z a z )/(1 -q 2 /c 2 ) 3/2 ,
and inserted this equation into the above equations. In addition, he defined,
eE' x + (e/c)(v y B' z + v z B' y ) = F x , and so on.
All this led Planck to the following equations:
d/dt • {mv x /√(1 -q 2 /c 2 )} = F x , and so on,
where these hold for a unit charged mass point moving in an electromagnetic field in k'. If q is small compared with the velocity of light c, these equations reduce to the Newtonian equations of motion. The term inside the brackets is Planck's relativistic momentum. 10 Therefore, Planck defined the second law of Newton as the rate of change of momentum in order for the principle of relativity and the Lorentz transformations to hold good for both systems k and k'.
In 1907 Einstein adopted Planck's relativistic momentum: 11
= v/√(1 -v 2 /c 2 ),
and said that Planck's above equations of motion "do not have a physical meaning, but are rather to be understood as defining equations of the force". 12 9 Planck, 1906a, in Planck, 1958, p. 118. 10 Planck's equations later led to a single expression for the mass:
m = m 0 /√(1 -v 2 /c 2 ).
Although mass could be split along the direction of motion and normally to it, scientists thought it would be conceptually preferable to look upon mass as one quantity. Einstein did not obtain a single expression for the mass, neither in 1905, or later. In later years, at least, Einstein did not talk about a variation of mass with velocity, but only of the new definition of momentum and of energy. Einstein might have rejected or disliked the concept of a single expression for the variation of mass with velocity and therefore did not obtain such an expression.
The Principles of Relativity as Heuristic Principles
Einstein's reply to Ehrenfest
Einstein had a friend, Paul Ehrenfest a Jewish physicist from Vienna. In 1907 Ehrenfest wrote a paper. 13 There were the known problems in the 19 th century electrodynamics of moving bodies. Einstein's 1905 solution appeared to Ehrenfest very similar to Lorentz's solution to these problems: a deformed electron. Ehrenfest thought that Einstein's theory of the motion of an electron could have been obtained from the good old theory of Lorentz, if we only used the method of deduction. If this was so, Ehrenfest understood that Einstein's theory was nothing but a reformulation of the electrodynamics of Lorentz. Therefore, Einstein's innovation was the following according to Ehrenfest, "In the formulation in which Mr. Einstein published it, Lorentzian relativity electrodynamics is treated rather generally as a closed system." 14 Einstein commented on Ehrenfest's paper. His 1907 reply, "Comments on the Note of Mr. Paul Ehrenfest" is important for the demarcation between his theory of relativity and Lorentz's ether-based theory. Lorentz's theory and the descendants of Lorentz's theory are not theories of relativity. Einstein characterized his work what would be later called principle of relativity as a theory of principle and reasoned that beyond kinematics, the 1905 heuristic relativity principle could offer new connections between non-kinematical concepts, "The principle of relativity, or more exactly, the principle of relativity together with the principle of the constancy of the velocity of light, is not to be conceived as a 'closed system', in fact, not as a system at all, but merely as a heuristic principle which, when considered by itself, contains only statements about rigid bodies, clocks, and light signals. The theory of relativity provides something additional only in that it requires relations between otherwise seemingly unrelated regularities". 15
In his 1916 popular book, Relativity, the Special and the General Theory, in the chapter "The Heuristic Value of The Theory of Relativity", Einstein wrote: "[…] the theory becomes a valuable heuristic aid in the search for general laws of nature". 16 Eherenfest's query dealt with the structure of the electron: "Accordingly, it [Lorentz's theory in Einstein's formulation] must also be able to provide purely deductively an answer to the question posed by transferring Abraham's problem from the rigid electron to the deformable one […]". 17 Einstein answered Ehrenfest's query by saying that the theory of the motion of an electron is obtained as follows: "one postulates the Maxwell equations for vacuum for space-time coordinate systems. By applying the space-time transformation [Lorentz transformation] derived by means of the system of relativity, one finds the transformation equations for electric and magnetic forces. Using the latter, and applying the space-time transformation, one arrives at the law for the acceleration of an electron moving at arbitrary speed from the law for the acceleration of a slowly moving electron (which is assumed or obtained from experience)". 18 Einstein explained to Ehrenfest, "We are not dealing here at all with a 'system' in which the individual laws are implicitly contained and from which they can be found by deduction alone, but only with a principle that (similarly to the second law of the thermodynamics permits the relation of certain laws to others". 19 In 1949 Einstein explained this still further: "Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results. The example I saw before me was thermodynamics. The general principle was there given in the theorem […the second law of thermodynamics]. How, then could such a universal principle be found?" 20
Theories of Principle and Constructive Theories
After 1907 Einstein made a distinction between theories of principle, such as thermodynamics and constructive theories, such as statistical mechanics. He characterized the special theory of relativity as a theory of principle, and considered it to be basically complete when the two underlying principles of the theory (the principle of relativity and that of the constancy of velocity of light) were established.
All later work would involve development of constructive theories compatible with these basic principles.
In his paper, "What is the Theory of Relativity?", written at the request of the London Times and published on November 28, 1919, for the first time Einstein formulated his views in a systematic manner: 21 "We can distinguish various kinds of theories in physics. Most of them are constructive. They attempt to build up a picture of the more complex phenomena out of the materials of a relatively simple formal scheme from which they start out. Thus the kinetic theory of gases seeks to reduce mechanical, thermal, and diffusional processes to the movements of moleculesi.e., to build them up out of the hypothesis of molecular motion. When we say that we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question.
Along with this most important class of theories there exists a second, which I will call 'principle theories'. […] The advantages of the constructive theory are completeness, adaptability, and clearness; those of the principle theory are logical perfection and security of the foundations.
The theory of relativity belongs to the latter class. In order to grasp its nature, one needs first of all to become acquainted with the principles on which it is based".
Kaufmann's Experiments: "Kugeltheorie" and "Relativtheorie"
Kaufmann's Experiments
In 1906 Planck wrote a letter to Einstein and mentioned his paper "The principle of Relativity and the Equations of Mechanics". 22 Planck told Einstein: "Herr Bücherer has already written to me a letter about his sharp opposition to my latest research, (without giving a reason, to be sure) he declares the principle of relativity incompatible with the principle of least action. It is therefore all the more gratifying to me to see from your card that, for the present, you do not share the views of Herr. B. So long as the proponents of the principle of relativity constitute such a modest group as they do at present, it is doubly important that they agree among themselves […]." 23 Planck defended the relativity principle when Walter Kaufmann performed his experiments, the result of which seemed to contradict Einstein's model of the electron. 1900 onward heralded an especially interesting period of experimental researches, conducted by Kaufmann that repeatedly confirmed Abraham's theory, at least until 1905. These and other experiments led to the discovery of cathode corpuscles in the beta radiation of radium, corpuscles that emitted and moved in velocities close to that of light. The beta particles' velocity was substantially faster than that of ordinary cathode rays. Scientists understood that these velocities were fast enough in order to ascertain whether, and to what extent, the inertia of these particles indeed changes with velocity. In 1901 and 1902 Kaufmann's experiments led to the mathematical expression Abraham had predicted, with the aid of some radium chloride, which the Curies had given to Kaufmann. This confirmed Abraham's prediction, according to which the particles owed all their energy to the fact that they were electrified.
Kaufmann wrote in 1902 that the mass of electrons was dependent on velocity; and this dependence could be described exactly by the formula of Abraham. Therefore, the mass of the electrons had a pure electromagnetic character: "The mass of electrons in Becquerel rays depends on the velocity; the dependence is exactly represented by Abraham's formula. The mass of electrons is accordingly of purely electromagnetic nature". 24
In 1905 Kaufmann concluded that his results: "[...] speak against the correctness of Lorentz's, and also consequently of Einstein's fundamental hypothesis. If one considers this hypothesis as thereby refuted, then the attempt to base the whole of physics, including electrodynamics and optics, upon the principle of relative movement is also a failure". 25 In 1906 Kaufmann concluded his paper, "On the Constitution of the Electron" by stating that Abraham had proven that the Lorentzian electron required the concept of work, therefore, in order to avoid a conflict with the energy law it was necessary to assume the existence of an "internal potential energy" of the electron. In contrast, the existence of a pure electromagnetic basis for the mechanics of the electrons that would apply to mechanics as a whole would be proven as being impossible. 26 This was so even if one contemplated the existence of a universal external pressure (as Henri Poincaré did in his theory "Dynamics of the Electron" 1905) instead of the work due to an unknown internal energy. 27 27 In 1905 Poincaré examined the three laws -Abraham's, Lorentz's, and "the hypothesis of Langevin" (equivalent to Bücherer's), and claimed in his paper, "On the Dynamics of the Electron", that Abraham's rigid spherical electron was at odds with the principle of relativity. The Lorentzian contracted or deformable electron was the only one to be compatible with the principle of relativity. that his measuring procedures were not compatible with the elementary hypothesis posited by Lorentz- Einstein. 28 Kaufmann concluded, from 1905 onwards, that the mathematical expression proposed by Alfred Bücherer could also be in accord with his measurements and that one could not definitively decide between that expression and that of Abraham as it was derived from his experiments. In the same paper, Kaufmann noted that the two theories of Lorentz and Einstein yielded the same equations of motion for the electron, and he gave the first clear account of the basic theoretical difference between Lorentz's and Einstein's views. 29 In the annual general meeting of the German Society of Scientists and Physicists (Deutsche Gesellschaft der Naturforscher und Ärrzte) in Stuttgart, on the 19 th of September 1906, scientists discussed three world pictures, the electromagnetic theories of Abraham, Bücherer, or the other picture based on Lorentz and Einstein's "Principle of Relativity". A discussion revolving around the foundations of physics was held after Planck's lecture. The participants in the discussion were, among others, Kaufmann, Planck, Bücherer, Abraham, Arnold Sommerfeld and others. Scientists did not yet distinguish between Lorentz's theory and Einstein's theory. There were two main theories relating to the electron: Abraham's and Lorentz-Einstein's. An inclination towards Einstein and Lorentz's theories, on the part of scientists such as Planck and Max Laue, was evident.
Einstein did not participate the Stuttgart annual meeting. Einstein was still sitting in the Patent office as an expert II class, and did not participate in the Kaufmann discussion, because he was absent from the Stuttgart meeting. Planck sent Einstein a report of Kaufmann's results and of the ensuing discussion, but added that this was not the updated results. Planck wrote Einstein on November 9, 1907: "In response to your request, I am sending you by the same mail my 'Postscript to the Discussion of Kaufmann's Deflection Measurements' together with the 'Discussion' itself. But I would like to add immediately that Herr Kaufmann subsequently carried out a calculation of the influence that ions produced by the -rays exert on the electrical field between the condenser plates, from which it follows that the electrical field is extraordinary close to being homogeneous. This calculation of Kaufmann's will One would gain a possible explanation for the contraction of the electron, in supposing that the deformed electron was subject to some kind of constant external pressure (the "Poincaré pressure"), the work done by it being proportional to the variations in the electron's volume. Therefore, one was led to propose an additional external potential, yielding non-electromagnetic external forces, the Poincaré pressure that would stabilize Lorentz's electron while in motion. Poincaré analyzed the mechanism of the contraction of Lorentz's electron and the configuration of the contracted electron. If we consider the configuration of the spherical electron at rest, as the result of electrostatic internal repulsion plus an external pressure caused by the external ether, then we can look at the equilibrium configuration of the moving electron. This configuration is the one that minimizes the total potential energy of the superposed actions on the electron: electrostatic repulsion and Poincaré's pressure. Poincaré, Henri, "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo 21, 1906, pp. 129-175; pp. 151-166. 28 Kaufmann, 1906, p. 553. 29 CPAE, Vol. 2, "Einstein on the Theory of Relativity", note 80, p. 267; Kaufmann, 1906. appear very soon in the Berliner der Physikalischen Gesellschaft". 30 Kaufmann repeated his experiments and validated his results.
Einstein immediately commented on the situation in a review article in 1907 that he submitted a few weeks later to Johannes Stark's Jahrbuch der Radioaktivität und Elektonik. On November 1 st , 1907 Einstein wrote Stark "I have now finished the first part of the work for your Jahrbuch; "I am working diligently on the second [part] in my, unfortunately rather scarce, free time". The first part dealt with the special theory of relativity. Once Einstein obtained Planck's letter on November 9, 1907, he sat and added a new section dealing with Kaufmann's results. Einstein estimated that the whole paper would be 40 printed pages long, and he told Stark that he hoped he would send him the manuscript "by the end of this month". 31 The paper was published on December 4, 1907. 32 Einstein like Planck was skeptical and he wrote in the paper: "Only after a more diverse body of observations becomes available will it be possible to decide with confidence whether the systematic deviations are due to a not yet recognized source of errors, or to the circumstance that the foundations of the theory of relativity do not correspond to the facts". 33 Einstein added: "It also should be mentioned that Abraham's and Bücherer's theories of the motion of the electron yield curves that are significantly closer to the observed curve than the curve obtained from the theory of relativity. However, the probability that their theories are correct is rather small, in my opinion, because their basic assumptions concerning the dimensions of the moving electron are not suggested by theoretical systems that encompass larger complexes of phenomena". 34 In the 1906 discussion following Planck's lecture, it focused on Planck's idea, which demonstrated that Kaufmann's results had indicated the need for a rapprochement to the principle of relativity, as well as towards Abraham or Bücherer's models, which were not based on the principle of relativity. Planck re-examined Kaufmann's experiments and data analysis and did not find anything seriously amiss in Kaufmann's interpretation of his data. Nevertheless, Planck believed that Kaufmann's data was not a definitive verification of Abraham's theory or a refutation of Lorentz's.
The first participant in the discussion to comment on Planck's idea was Kaufmann himself. Kaufmann thought that it would be best if he were the one to comment on Planck's suggestion, because he had performed the experiments, and therefore he felt he was in a position to evaluate Planck's attempts, "concerning the conclusions it follows from the facts of the follow-up that neither Lorentz's theory nor Abraham's theory agree with them. This is a clear conclusion. Lorentz's theory is even less in accordance than Abraham's theory. The deviations of Lorentz's theory [...] are so great that nowhere is it possible to explain them by follow-up mistakes. Following that, as far as no principled mistake has occurred in the follow-up, Lorentz's theory is invalidated." 35 To this Planck replied, "We would be able to approach Lorentz's theory closer than Abraham's theory. Depending on the fact that the deviations in one theory are fewer than in the other, we could not determine which one was preferable". 36 At the 1906 meeting Planck spoke of Relativtheorie, "relative theory". Planck demarcated between two models: "Abraham's, according to which, the electron has the shape of a rigid sphere, and the Lorentz-Einstein's, according to which the 'Principle of Relativity' obtains precise validity. For abbreviation, I will refer to the first theory as a 'Kugeltheorie', and call the second 'Relativtheorie'". 37 In the discussion afterwards this soon became Relativitätstheorie "relativity theory". In his arguments with other physicists and his comments on their work, Einstein was progressively, and reluctantly, drawn into this new terminology, though in headings and in the text of his own publications he continued to speak of the "relativity principle".
Bücherer's experiments
In the 1906 discussion following Planck's lecture, Bücherer was probably already vacillating and he said after Planck,38 "I have followed the results and have reached a few conclusions concerning Kaufmann's measurements, which I want to specify here. [...] relying on the theory of relativity we arrive at the conclusion that other forces will operate when the rays of Becquerel are not directed any more in parallel but towards the plates of the condenser. From here follows an easy access to test the principle of relativity theory depending on Maxwell's equations, only by operating Becquerel's rays in inclination towards the electric or magnetic field. In a perpendicular movement surprisingly the same forces are received as with Lorentz. I have already thought that perhaps creating an angle caused the deviation in Kaufmann's measurements".
Bücherer then was almost prepared to give up his model, 39 "I wish to repeat the lecturer's comment [Planck's] Having understood that all the theories until now, including my theory, do not answer all the requirements, I asked myself whether it was possible to reach an agreement, with the knowledge of preserving Maxwell's equations and on the basis of the principle of the equality of action and reaction".
Indeed after 1906, Bücherer renounced his electron model that had lead to his expression for the variation of mass with velocity, and he began to gravitate towards the Lorentz-Einstein model. In 1908 he conducted experiments on the beta rays of radium that were more precise than those of Kaufmann, and demonstrated that the Lorentz's model (and that of Einstein also) was a better representation of the experimental variation of the mass of the cathode rays with their velocity. Bücherer concluded his paper by stating that the experimental results of his experiments showed that scientists should incline towards the Lorentz-Einstein theory and that "this result is the confirmation of the principle of relativity". 40 Bücherer wrote Einstein on September 7, 1908, "First of all I would like to take the liberty of informing you that I have proved the validity of the relativity principle beyond any doubt by means of careful experiments. The experimental design might be known to you from my note in the Physikal. Zeitschrift." Bücherer explained in the letter to Einstein the way he undertook the test. He then wrote, "I have thereby definitively disproved my own principle". 41 Bücherer told Einstein, "I think I can rightly claim that I attained substantially greater precision in all measurements than did Kaufmann, so that I was convinced right from the outset that I must obtain definitive results. I am enclosing a photograph of one of my radiograms, from which you will immediately recognize the superiority of my method, i.e., if you have seen Kaufmann's radiograms. Kaufmann has a great many sources of error in his experimental setup, and I have already told him about one of them". 42 Planck's opinions in 1906 might have influenced Bücherer when he conducted his experiments that confirmed Lorentz's theory.
Born and the rigid body problem
In 1908, at the time of his return to Göttingen, Max Born was already thinking of the rigid body problem and entered into polemics with Abraham on his model for the mass of the electron, 43 "I frequently came into contact with Abraham, I was well informed about his controversy with Lorentz. Abraham was anti-relativist and objected to Lorentz's derivation. I also doubted it, but doubted Abraham's derivation as well. Both proceeded by calculating the self-energy of a charged rigid body in uniform motion (Lorentz with contraction, Abraham without it) as a function of velocity and using this energy as Hamilton's function for obtaining the equations of motion. This procedure assumes that the energy calculated for constant velocity also holds for accelerated motion. My doubts were concerned with this point, and I decided to derive the equations of motion for an accelerated electron in strict accord with the principle of relativity.
This led at once to a great difficulty. For if a body is accelerated, different points of it have different velocities, hence different contractions: the idea of rigidity breaks down. My first problem was therefore: how far can the concept of a rigid body be preserved in relativity? Rigidity means lack of deformation. I worked out a mathematical expression for the deformation of a moving body on relativistic principles, the so-called strain-components, which are differential expressions containing the derivatives of the coordinates as functions of their initial values and of time. Then I postulated that these strain-components should be zero; I obtained in this way differential equations for the possible strain-free motions and I found a solution of them which represents a uniformly accelerated rigid motion in a straight line, uniform in the same sense that the acceleration in the instantaneous rest-system is constant, and rigid relative to the same system […] The main result was a confirmation of Lorentz's formula for the electromagnetic mass as a function of velocity, and its dependence on acceleration. I worked on this investigation all through the winter and spring of 1909 […] I ventured to apply to the Mathematical Society for permission to give a report on it and was admitted." And then, "Abraham joined in the debate [after the lecture] to tell me that my knowledge of physics seemed to be just as scantly as that of mathematics. He was annoyed because my theory led to Lorentz's formula for the electromagnetic mass and not to his." 44 I wish to thank Prof. John Stachel from the Center for Einstein Studies in Boston University for sitting with me for many hours discussing special relativity and its history.
Planck, 1906a, in Planck, 1958.11 Einstein, Albert (1907a), "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen", Jahrbuch der Radioaktivität 4, 1907, pp. 411-462; p. 435.12 Einstein, 1907a, pp. 433-434
Therefore, Kaufmann again concluded 24 Kaufmann, Walter, "Über Die elektromagnetische Masse des Elektrons", Physikalische Zeitschrift 4, 1902, p. 56. 25 Cushing, James, "Electromagnetic Mass, Relativity, and the Kaufmann Experiments", American Journal of Physics 49, 1981, pp. 1142, 1148. 26 Kaufmann, Walter, "Über die Konstitution des Elektrons", Annalen der Physik 19, 1906, pp. 487-553.
that my theory [of the electron] is not sufficiently developed in order to continue discussing it here. I have engaged in it and in its conclusions very intensively and I have found that it does not give a more significant contribution than the previous and the later theory of Lorentz.[...] the electron theory distorted in fixed volume [Bücherer's electron model] and the distorted system accordingly contributes almost the same contribution as the new Lorentz theory. On expanding Planck's calculations on to my electron I would not be able to say what will follow from these calculations regarding my electron. [...] Certain arguments can be raised against Einstein's theory of relativity. [...]
Abraham, Max, "Prinzipien der Dynamik des Elektrons", Annalen der Physik 10 1903, pp. 105-179; p. 152.
Lorentz, Hendrik Antoon, "Electromagnetic Phenomena in a System Moving with any Velocity Smaller than that of Light", Verslagen Konignklijke Akademie Van Wetenschapen (Amsterdam). Proceedings of the section of science, 6, 1904, pp. 809-836; reprinted in Lorentz, Hendrik Antoon, Collected Papers 1935-1939, The Hague: Nijhoff, 9 Vols; Vol. 5, pp. 172-197; pp. 184-185. 3 Bücherer, Alfred, Mathematische Einführung in die Elektronentheorie, 1904, Leipzig: B.G. Teubner, pp. 57-58. Langevin had independently proposed the same hypothesis as Bücherer about the shape of a moving electron: Langevin, Paul, "La Physique des electrons", Review générale des sciences pures et appliquées 16, 1905, pp. 257-276.
Einstein, Albert, "Zur Elektrodynamik bewegter Körper, Annalen der Physik 17, 1, 1905, pp. 891-921; pp. 917-919. 5 Einstein, 1905, pp. 919.
Ehrenfest, Paul, "Die Translation deformierbarer Elektronen und der Flächensatz", Annalen der Physik 23, 1907, pp. 204-205.14 Einstein, Albert (1907b), "Bemerkungen zu der Notiz von Hrn. Paul Ehrenfest: 'Die Translation deformierbarer Elektronen und der Flächensatz' ", Annalen der Physik 23, 1907, pp. 206-208; p. 206.15 Einstein, 1907b, pp. 206-207.
Einstein, Albert, Ideas and Opinions, 1954, New Jersey: Crown publishers, p. 228; The London Times, November 28, 1919. It should be borne in mind that Einstein wrote this article after developing the General Theory of Relativity, and when he spoke about the theory of relativity and the principle theory he probably meant both special and general relativity, because he did not write explicitly the word "special".22 Planck, 1906a, in Planck, 1958 118 (see section 2.2 above).23 Planck to Einstein, July 6, 1907, The Collected Papers of Albert Einstein, Vol. 5: The Swiss Years: Correspondence, 1902-1914 (CPAE, Vol. 5), Klein, Martin J., Kox, A.J., and Schulmann, Robert (eds.), Princeton: Princeton University Press, 1993, Doc. 47.
Born, Max, Mein Leben: Die Erinnerungen des Nobelpreisträgers, 1975, München: Nymphenburger Verlagshandlung GmbH; My Life Recollections of a Nobel Laureate, 1978, New York, Charles Scribner's Sons, p. 134. 44 Born, 1975/1978, p. 135.
Uber die Spezielle und die Allgemeine Relativitätstheorie, Gemeinverständlich. Albert Einstein, ein heuristisches Prinzip"). Einstein. 206Vieweg SohnEinstein, Albert, Uber die Spezielle und die Allgemeine Relativitätstheorie, Gemeinverständlich, 1920, Braunschweig: Vieweg Sohn, p. 29 17 ("ein heuristisches Prinzip"). Einstein, 1907b, p. 206.
. Einstein, 207Einstein, 1907b, p. 207.
. Einstein, CPAE. 2412It was the first time that Einstein compared the relativity principle to the laws of thermodynamicsEinstein, 1907b, p. 207. It was the first time that Einstein compared the relativity principle to the laws of thermodynamics. CPAE, Vol. 2, p. 412, note 8.
Autobiographisches"/"Autobiographical notes. Albert Einstein, Albert Einstein: Philosopher-Scientist. Schilpp, Paul ArthurOpen CourtEinstein, Albert , "Autobiographisches"/"Autobiographical notes" In Schilpp, Paul Arthur (ed.), Albert Einstein: Philosopher-Scientist, 1949, La Salle, IL: Open Court, pp. 1-95; pp. 48-49.
. Planck To Einstein, CPAE. 564Planck to Einstein, November 9, 1907, CPAE, Vol. 5, Doc. 64.
. Einstein, Stark, Doc. 63. 32CPAE. 5Including a part on gravitationEinstein to Stark, November 1, 1907, CPAE, Vol. 5, Doc. 63. 32 Including a part on gravitation.
. Einstein, 439Einstein, 1907a, p. 439.
. Einstein, 439Einstein, 1907a, p. 439.
Die Kaufmannschen Messungen der Ablenkbarkeit von -Strahlen in ihrer Bedeutung für die Dynamik der Elektronen. Max Planck, Physikalische Zeitschrift. 7Planck, Max (1906b) "Die Kaufmannschen Messungen der Ablenkbarkeit von -Strahlen in ihrer Bedeutung für die Dynamik der Elektronen," Physikalische Zeitschrift 7, 1906, pp. 753-761; pp. 759- 760.
Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie. Alfred Bücherer, Physikalische Zeitschrift. 9760Bücherer, Alfred, "Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz- Einsteinschen Theorie", Physikalische Zeitschrift 9, 1908, p. 760.
. Bücherer To Einstein, CPAE. 5117Bücherer to Einstein, September 7, 1908, CPAE, Vol. 5, Doc. 117.
. Bücherer To Einstein, CPAE. 5119Bücherer to Einstein September 9, 1908, CPAE, Vol. 5, Doc. 119.
| [] |
[
"Entangled states in a Josephson charge qubit coupled to a superconducting resonator",
"Entangled states in a Josephson charge qubit coupled to a superconducting resonator"
] | [
"O Buisson \nCentre de Recherches sur les Très Basses Températures\nUniversité Joseph Fourier\nBP 16638042Grenoble-cedex 9C.N.R.SFrance\n",
"F W J Hekking \nLaboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nBP 16638042Grenoble-cedex 9C.N.R.SFrance\n"
] | [
"Centre de Recherches sur les Très Basses Températures\nUniversité Joseph Fourier\nBP 16638042Grenoble-cedex 9C.N.R.SFrance",
"Laboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nBP 16638042Grenoble-cedex 9C.N.R.SFrance"
] | [] | We study the dynamics of a quantum superconducting circuit which is the analogue of an atom in a high-Q cavity. The circuit consists of a Josephson charge qubit coupled to a superconducting resonator. The charge qubit can be treated as a two level quantum system whose energy separation is split by the Josephson energy E j . The superconducting resonator in our proposal is the analogue of a photon box and is described by a quantum harmonic oscillator with characteristic frequency ω r . The coupling between the charge qubit and the resonator is realized by a coupling capacitance C c . We have studied the eigenstates as well as the dynamics of the quantum circuit. Interesting phenomena occur when the Josephson energy equals the oscillator frequency, E j =hω r . Then the quantum circuit is described by entangled states. We have deduced the time evolution of these states in the limit of weak coupling between the charge qubit and the resonator. We found Rabi oscillations of the excited charge qubit eigenstate. This effect is explained by the spontaneous emission and re-absorption of a single photon in the superconducting resonator. | 10.1007/978-1-4615-1245-5_14 | [
"https://export.arxiv.org/pdf/cond-mat/0008275v1.pdf"
] | 15,537,301 | cond-mat/0008275 | 966816519444f36cb90fa9910f0fbf780e8a4267 |
Entangled states in a Josephson charge qubit coupled to a superconducting resonator
18 Aug 2000
O Buisson
Centre de Recherches sur les Très Basses Températures
Université Joseph Fourier
BP 16638042Grenoble-cedex 9C.N.R.SFrance
F W J Hekking
Laboratoire de Physique et Modélisation des Milieux Condensés
Université Joseph Fourier
BP 16638042Grenoble-cedex 9C.N.R.SFrance
Entangled states in a Josephson charge qubit coupled to a superconducting resonator
18 Aug 2000
We study the dynamics of a quantum superconducting circuit which is the analogue of an atom in a high-Q cavity. The circuit consists of a Josephson charge qubit coupled to a superconducting resonator. The charge qubit can be treated as a two level quantum system whose energy separation is split by the Josephson energy E j . The superconducting resonator in our proposal is the analogue of a photon box and is described by a quantum harmonic oscillator with characteristic frequency ω r . The coupling between the charge qubit and the resonator is realized by a coupling capacitance C c . We have studied the eigenstates as well as the dynamics of the quantum circuit. Interesting phenomena occur when the Josephson energy equals the oscillator frequency, E j =hω r . Then the quantum circuit is described by entangled states. We have deduced the time evolution of these states in the limit of weak coupling between the charge qubit and the resonator. We found Rabi oscillations of the excited charge qubit eigenstate. This effect is explained by the spontaneous emission and re-absorption of a single photon in the superconducting resonator.
I. INTRODUCTION
Recently, a substantial interest in the theory of quantum information and computing 1 has revived the physical research on quantum systems. The elementary unit of quantum information is a two-state system, usually referred to as a quantum bit (qubit). Basic operations are realized by preparation and manipulation of, as well as a measurement on, entangled states in systems which consist of several coupled qubits. However, the fabrication of physical systems which would enable the actual implementation of quantum algorithms is far from being realized in the near future and a substantial amount of fundamental research is still needed.
During the past five years, great progress has been made in the manipulation of entangled states in systems consisting of up to four qubits based on ion traps 2 and atoms in a high-Q cavity 3 . These two experiments demonstrate clearly and unambiguously the possibility to coherently control the entangled states of a limited number of qubits, as well as to perform a quantum measurement on them. In spite of this success, it seems quite difficult to realize circuits consisting of the large number of ion traps or atoms in a cavity necessary for quantum computation.
It has been suggested that small solid state devices fabricated using nanolithography technologies are promising for quantum circuit integration. However, the coherent manipulation of entangled states as well as the realization of quantum measurements remain fundamental issues to be investigated. One of the main challenges is to gain control over all possible sources of decoherence. At present the best candidates for the implementation of quantum gates based on solid state devices are circuits using small Josephson junctions. In the superconducting state, such circuits contain less intrinsic sources of decoherence. Indeed it has been experimentally demonstrated that a single Cooper pair box is a macroscopic two level system which can be coherently controlled [4][5][6] . At about the same time, theoretical works have proposed the use of the Cooper pair box as a qubit (the so-called Josephson charge qubit) in the context of quantum computers. In particular, systems consisting of several charge qubits with controlled couplings have been discussed, the quantum measurement problem has been addressed and the decoherence time has been estimated 7,8 . More recently, qubits based on superconducting loops containing small Josephson junctions have been proposed (Josephson phase qubits) 9,10 and are currently studied 11,12 . But up to now, the existence of entangled states, which are at the heart of quantum information processing, has been demonstrated neither for charge qubits nor for phase qubits.
In this article we propose to study one of the simplest Josephson circuits in which entangled states can be realized. It consists of a charge qubit coupled to a superconducting resonator and can be described theoretically by a two level system coupled to a quantum harmonic oscillator. After a description of this quantum circuit in the next section, the Hamiltonian describing it will be derived in Sec. III. In Sec. IV, the time evolution of the eigenstates is obtained and we demonstrate the existance of entanglement. In the last Section, we discuss the dynamics of the quantum circuit for typical experimental values of the system parameters.
II. QUANTUM CIRCUIT
The Josephson circuit we study hereafter is depicted in Fig. 1. It consists of three different elementary circuits: a Cooper pair box, an LC-resonator and a "coupling" capacitor.
For small enough junction capacitance C j , gate capacitance C g , and coupling capacitance C c , the charging energy of the box is large compared to thermal fluctuations and the excess charge of the box is quantized. On the other hand, we assume the charging energy to be smaller than the superconducting gap ∆, such that no quasiparticles are present in the box. Thus the excess charge is entirely due to the presence of Cooper pairs and charge quantization occurs in units of 2e. The gate voltage V g is used as an external control parameter. When the gate charge N g = −C g V g /e is equal to unity, the Cooper pair box can be viewed as a macroscopic two-level quantum system whose energy separation is split by the Josephson energy E j . The two eigenstates |− and |+ correspond to a coherent superposition of the two different charge states of the box 4-6 . When N g ≈ 1, the Cooper pair box will be referred to as a Josephson charge qubit.
The LC-resonator system can be described by a quantum harmonic oscillator whose characteristic frequency is given by ω r . This system is the analogue of a high-Q cavity.
The capacitance C c plays a crucial role in our proposed circuit since it couples the charge qubit and the resonator to each other. These two circuits are no longer independent and the system must be considered in its totality. Thus the proposed quantum circuit of Fig. 1 realizes the simple situation in which a two level system is coupled to a harmonic oscillator. In spite of its simplicity, such a system describes a great variety of interesting situations [13][14][15] .
III. HAMILTONIAN
The circuit depicted in Fig. 1 can be characterized mechanically by two generalized coordinates, φ j and φ r . These coordinates are associated with the voltage drop δV j over the junction and δV r over the resonator, respectively, according to the Josephson relation φ i = 2eδV i t/h (i = j, r). We seek the Lagrangian L(φ j , φ r ,φ j ,φ r ) = T − V describing the dynamics of these variables. The potential energy V is a function of the coordinates only,
V (φ j , φ r ) = −E j cos φ j + E r 2 φ 2 r ,(1)
where E r = (1/L r )(h/2e) 2 is the energy associated with the inductance L r of the resonator. The kinetic energy T is quadratic in the velocitiesφ j andφ r . It is just the free electrostatic energy stored in the capacitators present in the circuit. This free energy can be written as
T = 1 2 C Σj (hφ j /2e) 2 + C Σr (hφ r /2e) 2 + 2C Σc (h/2e) 2φ jφr .(2)
Here we introduced the capacitances C Σc = C c + C g , C Σj = C j + C Σc , C Σr = C r + C Σc . Note that the Lagrangian contains an interaction between the resonator and the junction:
L int = C Σc (h/2e) 2φ
jφr . The effective coupling between these two parts of the circuit is determined by the sum of the gate capacitance and the coupling capacitance. We also note that the equations of motion d(∂L/∂φ i )/dt + ∂L/∂φ i = 0 express current conservation in the circuit.
Through the Josephson junction, Cooper pairs can tunnel from or onto the island. The number of excess Cooper pairs on the island, n, depends on the gate voltage. Charge neutrality leads us to relation
2ne = C Σj (hφ j /2e) + C Σc (hφ r /2e) + N g e.(3)
It is always possible to add a term to the Lagrangian which is a total time derivative. Let us add the termhφ j N g /2. As a result, the momenta conjugate to φ j and φ r are
p j = ∂L/∂φ j = (h/2e)[C Σj (hφ j /2e) + C Σc (hφ r /2e) + N g e] =hn, p r = ∂L/∂φ r = C Σr (h/2e) 2φ r + C Σc (h/2e) 2φ j .
Note in particular that the momentum p j is proportional to n. The Hamiltonian is obtained with the help of the Legendre transform H = p jφj +p rφr −L. We find H = H j + H r + H c , where
H j = E C,j (2n − N g ) 2 − E j cos φ j , (4) H r = E C,r (2p r /h) 2 + E r φ 2 r /2,(5)H c = −E C,c (2n − N g )(2p r /h).(6)
The charging energies E C,i (i = j, r, c) appearing here are given by
E C,i = e 2 /2C i,eff , with C j,eff = C j + (1/C Σc + 1/C r ) −1 , C r,eff = C r + (1/C Σc + 1/C j ) −1 , and C c,eff = [C Σc + (1/C j + 1/C r ) −1 ][(C j + C r )/2C Σc ].
The Hamiltonian equations of motion,ṗ i = −∂H/∂φ i ,φ i = ∂H/∂p i lead us again to current conservation. In order to obtain the quantum mechanical HamiltonianĤ, we replace p i , φ i by the corresponding operators, with [p k , φ m ] = (h/i)δ k,m . In particular, in φ−representation we have p k = (h/i)∂/∂φ k . Note also that [n, φ j ] = −i and n = −i∂/∂φ j . Below we discuss the various contributions toĤ in some detail.
Josephson junction. The commutation relation [n, φ j ] = −i implies [n, e iφ j ] = e iφ j . Using the basis states |n , where n corresponds to the number of excess Cooper pairs on the island, we thus have e iφ j |n = |n + 1 . Similarly, e −iφ j |n = |n − 1 . Therefore we can writeĤ j aŝ
H j = E C,j n (2n − N g ) 2 |n n| − E j 2 n (|n + 1 n| + |n − 1 n|) .(7)
If the gate-voltage is such that N g ≃ 1, the states with n = 0 and n = 1 are almost degenerate. At low temperatures, the HamiltonianĤ j involves only these two states, and thus can be presented as a matrix
H j ≃ E C,j N 2 g −E j /2 −E j /2 E C,j (2 − N g ) 2 .(8)
This matrix can be diagonalized. The eigenvalues are
E ∓ = E C,j [1 + (δN g ) 2 ] ∓ 1 2 (δE g ) 2 + E 2 j ,(9)
where δN g = N g − 1 and δE g = −4E C,j δN g . The corresponding eigenstates are
|− = α|0 + β|1 (10) |+ = β|0 − α|1(11)
where α 2 = 1 − β 2 = [1 + δE g / (δE g ) 2 + E 2 j ]/2. Resonator. Since the LC-circuit constitutes just a harmonic oscillator with a characteristic frequency ω r = 1/L r C r,eff , the HamiltonianĤ r can be written in the standard wayĤ r =hω r (a † a + 1/2),
where φ r = 2
E C,r hω r (a † + a),(13)p r = ih 4 hω r E C,r a † − a .(14)
Coupling term. The coupling term can also be written using the operators a, a † :
H c = −i E C,c 2 hω r E C,r (2n − N g ) a † − a .(15)
Note that the characteristic coupling energy is E c = hω r /E C,r E C,c /2.
A general analysis of the HamiltonianĤ is beyond the scope of the present paper and will be presented elsewhere 16 . In the next section we will discuss an explicit matrix form ofĤ, which can be obtained under certain simplifying conditions which are nevertheless experimentally relevant.
IV. EIGENSTATES AND ENTANGLEMENT
Throughout this section we will work in the zero-temperature limit. We are interested in the situation N g ≃ 1, such that we have to consider the charge qubit states |− and |+ only. Furthermore, as far as the resonator is concerned, we will considerhω r = E j and work only with the ground state |0 and the first excited state |1 . In the limit of weak coupling, E c ≪hω r , the coupled system can be characterized by the four basis states |−, 0 , |−, 1 , |+, 0 , |+, 1 . The Hamiltonian matrix for this low-energy subspace readŝ
H = E 0 iE β 0 −2iαβE c −iE β E 1 2iαβE c 0 0 −2iαβE c E 2 iE α 2iαβE c 0 −iE α E 3 ,(16)
which is a hermitian matrix describing the two-level system coupled to the lowest states of the resonator. Here,
E 0 = E − +hω r /2, E 1 = E − +3hω r /2, E 2 = E + +hω r /2, E 3 = E + +3hω r /2, E α = E c (2α 2 − N g ), and E β = E c (2β 2 − N g )
. Suppose that the system has been prepared in the state |ψ(t = 0) = |+, 0 at time t = 0. This situation can be achieved by a suitable manipulation of the gate voltage V g at times prior to t = 0 6,16 . At times t > 0, the time evolution of |ψ(t) describing the system is governed by the Hamiltonian (16). We keep V g fixed such that N g = 1 at t > 0. Thus we have α 2 = β 2 = 1/2, and hence E α = E β = 0. Moreover, ashω r = E j , we have E 1 = E 2 = E C,j + E j ≡Ē: without coupling, the state |+, 0 would be degenerate with the state |−, 1 . Thus the Hamiltonian takes the simple form
H = E 0 0 0 −iE c 0Ē iE c 0 0 −iE cĒ 0 iE c 0 0 E 3 .(17)
Note in particular that the state |+, 0 couples to the state|−, 1 ; as a result, the degeneracy between them is lifted and the states become entangled. The precise form of the entanglement is governed by the central 2 × 2 block of the matrix (17). The eigenstates of this block are
|χ 1 = [|−, 1 + i|+, 0 ]/ √ 2, |χ 2 = [|−, 1 − i|+, 0 ]/ √ 2,
corresponding to the eigen energiesĒ − E c andĒ + E c , respectively. These two excited eigenstates thus correspond to a maximum entanglement of charge qubit and resonator states, induced by the capacitive coupling between them. The time evolution of |ψ(t) is given by
|ψ(t) = 1 √ 2i e −i(Ē−Ec)t/h |χ 1 − e −i(Ē+Ec)t/h |χ 2 .(18)
We deduce that the state |ψ(t) oscillates coherently between |−, 1 and |+, 0 . In fact, these so-called quantum Rabi oscillations can be interpreted as the spontaneous emission and re-absorption of one excitation quantum by the resonator. An interesting quantity is the probability P 1 (t) to find the harmonic oscillator in the state |1 after a certain time t. This probability shows Rabi oscillations as a function of t with frequency 2E c /h,
P 1 (t) = | 1, −|ψ(t) | 2 = 1 2 [1 − cos(2E c t/h)].(19)
Since these Rabi oscillations are characteristic for the entanglement realized in the system, their measurement would provide direct evidence of the presence of the entangled states |χ 1 and |χ 2 . We will discuss the feasibility of such a measurement in the next Section.
V. DISCUSSION
For the numerical estimates presented below we will consider parameters related to a typical aluminium superconducting circuit 4,6 . For the Josephson charge qubit we have chosen the following characteristics: E j = 26.1µeV, E C,j = 70µeV, ∆ = 240µeV. As for the resonator, we take L r = 90pH and E C,r = 12neV, as a resulthω r = 26.1µeV. Finally, the coupling capacitance is chosen to be of the same order of C j , C c = 0.5fF, yielding E c = 256neV. Note that the coupling energy is indeed much smaller than the Josephson energy, which in turn is equal to the excitation energy of the resonator.
Using the above paramaters, we have plotted P 1 (t), Eq. (19), as a function of time in Fig. 2. We clearly see the Rabi oscillations with perdiodicity T Rabi ≈8 ns.
In order to be able to observe these oscillations, we need to satisfy various conditions. First of all, in order to avoid the presence of quasiparticles, the temperature must be much lower than the gap ∆. Secondly, it is necessary to have a decoherence time which is longer than T Rabi . In our system, the decoherence time will be the shorter of the lifetime τ r of the excitated state of the resonator and the decoherence time τ qubit of the charge qubit. A Q-factor of about 500 is quite realistic for a superconducting LC resonator. This yields a lifetime τ r > 10 ns. As for the qubit, the experiment by Nakamura 6 has indicated that τ qubit > 2ns. This lower bound is essentially the decoherence time associated with the coupling to the measuring probe. In principle, this time can be improved upon by modifications of the experimental set-up. Theoretical estimates 8 show that a time τ qubit ∼ 100 ns should be feasable. Finally, a measurement of the number of excitations should be performed on the resonator. This can be done, e.g., along the lines of Ref. 17 , where the discrete, oscillator-like energy levels of an underdamped Josephson junction were measured.
FIG. 2 .
2Probability P 1 (t) to find the system, prepared in the state |+, 0 at t = 0, in the state |−, 1 after a time t.
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| [] |
[
"The Quantized Monte Carlo method for solving radiative transport equations",
"The Quantized Monte Carlo method for solving radiative transport equations"
] | [
"Laetitia Laguzet ",
"Gabriel Turinici ",
"\nCEA-DAM-DIF\nF-91297ArpajonFrance\n",
"\nCEREMADE\nUniversité Paris -Dauphine -PSL\n75016ParisFRANCE\n"
] | [
"CEA-DAM-DIF\nF-91297ArpajonFrance",
"CEREMADE\nUniversité Paris -Dauphine -PSL\n75016ParisFRANCE"
] | [] | We introduce the Quantized Monte Carlo method to solve thermal radiative transport equations with possibly several collision regimes, ranging from few collisions to massive number of collisions per time unit. For each particle in a given simulation cell, the proposed method advances the time by replacing many collisions with sampling directly from the escape distribution of the particle. In order to perform the sampling, for each triplet of parameters (opacity, remaining time, initial position in the cell) on a parameter grid, the escape distribution is precomputed offline and only the quantiles are retained. The online computation samples only from this quantized version by choosing a parameter triplet on the grid (close to actual particle's parameters) and returning at random one quantile from the precomputed set of quantiles for that parameter. We first check numerically that the escape laws depend smoothly on the parameters and then implement the procedure on a benchmark with good results. | null | [
"https://export.arxiv.org/pdf/2301.11068v1.pdf"
] | 256,274,937 | 2301.11068 | 10d3565797c9f4046f73717d8e0de5a37026928c |
The Quantized Monte Carlo method for solving radiative transport equations
26 Jan 2023
Laetitia Laguzet
Gabriel Turinici
CEA-DAM-DIF
F-91297ArpajonFrance
CEREMADE
Université Paris -Dauphine -PSL
75016ParisFRANCE
The Quantized Monte Carlo method for solving radiative transport equations
26 Jan 2023(Dated: January 27, 2023)1
We introduce the Quantized Monte Carlo method to solve thermal radiative transport equations with possibly several collision regimes, ranging from few collisions to massive number of collisions per time unit. For each particle in a given simulation cell, the proposed method advances the time by replacing many collisions with sampling directly from the escape distribution of the particle. In order to perform the sampling, for each triplet of parameters (opacity, remaining time, initial position in the cell) on a parameter grid, the escape distribution is precomputed offline and only the quantiles are retained. The online computation samples only from this quantized version by choosing a parameter triplet on the grid (close to actual particle's parameters) and returning at random one quantile from the precomputed set of quantiles for that parameter. We first check numerically that the escape laws depend smoothly on the parameters and then implement the procedure on a benchmark with good results.
I. INTRODUCTION AND OUTLINE
The time dependent thermal radiative transport equations couple a transport equation with a internal matter density evolution equation. Simulating this dynamics is extremely time consuming and we are interested in the stochastic (Monte Carlo) approaches and more precisely in the situation involving a large range of opacities ; in such cases the particles used in the Monte Carlo simulation will undergo a wide range of behaviors : on one hand long-time rectilinear propagation interrupted by rare scattering events and on the other hand high intensity scattering with negligible overall displacement; but all other intermediate regimes are also present. The two extreme regimes can either be simulated directly or with good quality approximations and the corresponding works have been documented in the literature (see section II). But treating all regimes simultaneously has been a challenge and our contribution introduces a unified method to tackle this circumstance.
To this end we exploit a hidden smoothness in these models which is situated at the level of the statistics of the escape laws of a particle from a given domain.
The outline of the paper is the following : we present in section II some motivating comments and a presentation of the state of the art; then we describe in section III our method based on an offline-online approach that exploits the quantiles of the escape laws from a domain. The assumptions of the method are checked numerically in section IV A and then the method is tested * [email protected] † [email protected]; https://turinici.com on a benchmark with good results in sections IV B. Concluding remarks are presented in section V.
II. MOTIVATION AND STATE OF THE ART
We present briefly the principles of the Monte Carlo method used to solve the transport equation (1) and the problem of the diffusion limit in a general setting. We then present the state of the art of the methods that treat this high collisions regime. Consider the integro-differential transport equation of the form :
1 c ∂ t u(t, x, ω) + ω · ∇u(t, x, ω) + (σ a (t, x) + σ s (t, x))u(t, x, ω) = σ s (t, x) u (t, x),(1)with time t ∈ R + , position x ∈ D ⊂ R d (d ≥ 1 is the dimension), ω ⊂ S d (unit sphere in R d ) the angle of propagation and u (t, x) = S d u(t,x,ω )dω S d 1·dω
the angular average of u on S d .
The absorption opacity σ a and the scattering opacity σ s are (known) functions depending on the spatial discretization. To solve this equation we focus on the approaches described in [1] which interpret (1) as a time-evolving probability density and simulate the underlying stochastic process.
When σ s (t, x) → ∞ we are in the "diffusion limit" and the cost of the Monte Carlo is prohibitive [2] : each particle undergoes a high number of collisions with the mean time between two collisions being O 1 σs . But the asymptotic analysis [3] shows that (1) converges towards the diffusion limit equation :
1 c ∂ t u (t, x) = ∇ · 1 3σ s ∇ u (t, x) .(2)
The equation (1) appears in particular when solving radiative transfer equations where an isotropic scattering term is necessarily added by the Implicit Monte Carlo linearization method [4] in order to artificially represent the phenomena of absorption and re-emission. Another approach to avoid artificial scattering is proposed in [5] but the problem remains unchanged when important physical scattering terms are present. In the context of radiative transfer, several methods have been proposed exploiting the limit regime (2).
The Random Walk (RW) methods [6,7] exploit the fact that the trajectories of the particles are close to those of a Brownian motion: in an optically thick medium they replace (a part of ) the trajectory by a single diffusion step in the largest sphere contained in the mesh. The Random
Walk methods have the advantage to activate everywhere on the domain, and are easily applied to 3-dimensional problems as well as multi-group problems. Their use in a production context remains limited by their strong dependence on mesh size (the smaller the mesh size, the smaller the sphere where the method will be applied) and the loss of precision introduced by the use of the diffusion limit for transient regimes.
Initially called Implicit Monte Carlo Diffusion, the Discrete Diffusion Monte Carlo (DDMC) method [2,8,9] splits the domain into two regions: one optically thick region solved by a Monte
Carlo method using a diffusion equation and another part treated by the IMC method. The numerical simulation in the optically thick region uses a linearization similar to the IMC method.
A new type of particle is then introduced to solve the diffusion equation. The advantage of this method is that it does not have any net flux to consider between the diffusion and transport regions When the coefficient σ s (t, x) in (1) is large, the classical Monte Carlo method uses Markov particles that undergo an important number of scattering events. The randomness of the scattering part dominates and after a certain time the state of the particle follows a probability law; in this case the RW approximation is justified. However, there are always intermediary regimes when the number of collisions is big enough to slow down the computation but not high enough to justify the use of the diffusion approximation.
A new Monte Carlo method that is efficient regardless of the value of σ s and that does not reduce the accuracy of the solution is still a challenge. Ideally the method should not be sensitive to the mesh used (i.e. robust to the change in value of σ s and not limited to simple spatial domain e.g., a sphere); and it needs to be valid regardless of the value of σ s (or that activates according to criteria independent on a choice of spatial areas such as methods of RW type).
Our approach, called the Quantized Monte Carlo method, is to not use the diffusion limit approximation but to work with an approximation of the probability law of the exact solution of the escape time, position and direction from the spacial cell.
III. THE QUANTIZED MONTE CARLO METHOD
We will consider d = 1 in all this section and work on a segment (eventually divided in several sub-intervals). To ease notations we will also use σ instead of the scattering opacity σ s .
A. Toy model illustration
We recall here a simple example used later in the numerical tests in section IV B and that will be useful to describe the Quantized Monte Carlo method below.
Consider a 1D particle in the segment [x min , x max ] situated at the initial time t = 0 at position
x = x init with angle a ∈ {−1, 1}.
The total remaining simulation time is t max ; in the general simulation t max equals the overall time step ∆t decremented by any previous time increments for this particle (for instance when the particle traverses several cells during the same ∆t).
The exact evolution of the particle is the following: rectilinear movement in direction a for a time τ (exponential random variable of mean 1/σ) then a collision takes place. This collision changes the angle uniformly at random to a new value a ∈ {−1, 1}. Then the process repeats until either boundary is reached :
x = x min or x = x max or t = t max .
We are interested precisely in this escape place (one of the extremities of the segment or of the time domain) and the escape angle. This is a random variable whose distribution will be denoted
E(σ, , t max ) where = (x init − x min )/(x max − x min )
is the relative initial position of the particle.
The probability law E(σ, , t max ) has the support in
(x min , t), t ∈ [0, t max ] ∪ (x, t max ), x ∈ [x min , x max ] ∪ (x max , t), t ∈ [0, t max ] × {−1, 1}.(3)
Note that, although the distribution seems to be 3 dimensional, conditional on knowing the escape side, only one dimension is essential, for instance escaping through x min / x max implies that the angle is −1 / 1, otherwise it is random −1/1. An illustration is given in figure 1.
(x min , 0) (x max , 0) (x min , t max ) (x max , t max ) (x init , 0)
B. The method
The section II highlighted the difficulty of dealing with the diffusion limit of the equation (1) and the limitations of existing Monte Carlo methods. We propose a new Monte Carlo method, inspired by algorithms such as Random Walk, that works with the probability laws E(σ, , t max ) of escape from a cell and is based on vector quantization techniques [12].
1. We define grids of representative values of the main parameters concerned ; for instance in 1D, we employ a grid G sc for σ values (in practice a log-uniform grid from 7.5 × 10 −3 cm −1 to 9.0 × 10 6 cm −1 ), a grid G time for simulation time values t max (uniform grid from 400f s to 40000f s) and a grid G ini for relative initial position in the cell from 0% to 100% relative to left segment end. Each grid G sc , G time , G ini has 100 points. We denote |G| the size of a grid G.
2. An offline computation is done once and for all (independent of the final simulation) in order to obtain an approximation of the joint distribution (escape time, escape point, escape direction) E(σ, , t max ) as a probability distribution. For each point in G sc × G time × G ini we compute and store the quantiles of the law. This approximation is valid beyond the framework of the diffusion limit, in particular it does not use any analytical form. In practice we perform 1500 simulations for each point in G sc × G time × G ini but extract only J quantiles from the whole distribution (cf. previous remarks on the fact that distribution is essentially one dimensional) [13]. The quantiles are minimizers of the Huber-energy distance to the target and correspond to the optimal quantization of the measure, cf. [14, prop. 21] and [15, prop.
and 4]
; for J points the optimal quantiles chosen are j+1/2 J , j = 0, ..., J − 1. This part of simulation is highly parallelizable. The results are stored as a |G sc |×|G time |×|G ini |×J array of escape points x or t together with the 3 positive numbers (summing up to 1) indicating the probability of escape through each side; for us J = 100, the number of points is 100 3 × 103 requiring ∼ 800M b of storage.
3. During the online simulation, each time that a particle of parameters (σ, , t max ) needs to be advanced to its next escape point, a set of parameter values σ g , g , t g max from the 3D-grid G sc × G time × G ini is chosen (see below for details) and a random quantile from the stored distribution E(σ g , g , t g max ) is selected and returned to the user. The particle is advanced with the corresponding space/time increments prescribed by the escape quantile returned. The grid point σ g , g , t g max is chosen by identifying, for each of the parameters σ, , t max the 2 closest values of the grid : σ ∈ [σ k 1 , σ k 1 +1 ], t max ∈ [t k 2 max , t k 2 +1 max ], ∈ [ k 3 , k 3 +1 ] ; then we select one of them at random with probabilities depending on the relative distance between the actual parameters and the grid points, for instance σ g = σ k 1 with probability
(σ k 1 +1 − σ)/(σ k 1 +1 − σ k 1 ).
Such an approach does not raise questions of validity of the diffusion limit or of the calculation of the escape time from the spheres (which resort to partial differential equations with assumptions and boundary conditions sometimes difficult to tackle cf. [16,17]).
The method is called "quantized" because we always sample from a pre-defined list of quantiles.
In practice this dimension of quantization is not any more surprising than, e.g. space discretization of the mesh and if enough quantiles are considered the contribution to the overall error is negligible. The foundations of the method are well established (see [12] for general information on the mathematical objects and [14] more specifically tailored to our applications).
In order for the Quantized Monte Carlo method to work conveniently, one needs to ensure that the distribution E(σ, , t max ) is close to the mixing of the closest distributions E(σ g , g , t g max ) on the grids. This, at its turn, depends on the smoothness of the mapping (σ, , t max ) → E(σ, , t max ) that we investigate in the following. More precisely, we plot in figure 2 several histograms corresponding to different typical parameter values encountered in the numerical tests in section IV B.
As expected, the laws vary slowly with the parameters. For instance, in practice we noted that a grid of values for σ spaced log-uniform by about 25% increase from one point to another gives very satisfactory results. We test the method on the propagation of a Marshak-type wave in an opaque medium (see [18,19] for details) which is considered a good benchmark for difficult multi-regime computations.
We assume an ideal gas equation under the gray approximation. The Monte Carlo method used here is based on the Fleck and Cummings [4] linearization. We use a model with two temperatures (radiative and matter) : except mention of the contrary, the term temperature (noted T matter ) will indicate the matter temperature. This is a 1D benchmark in rod geometry (like the S N method [20] with N = 2) with symmetry conditions on the top and bottom edges of the mesh. The values and units used are specified in the table I. We then solve the system of equations (4) for t ∈ [t n , t n+1 [ where I + (t, x) = I(t, x, µ = 1) and I − (t, x) = I(t, x, µ = 1) and f n the Fleck factor : We analyze the wave profile at 1ns, 5ns and 10ns using a time step of ∆t = 4 × 10 −11 s. To do this, we perform a run for the classical Monte Carlo method and a run with our method with 50 cells and N obj = 200 (target number of particles by cell); we employ the local regularization method in [21,22] and compare the wave intensity to check for physical consistency.
1 c ∂ t I + + ∂ x I + + σf n I + = σ n f n acT 4 matter (t n ) 2 + σ n (1 − f n ) 1 2 (I + + I − ) 1 c ∂ t I − − ∂ x I − + σ n f n I − = σ n f n acT 4 matter (t n ) 2 + σ n (1 − f ) n 1 2 (I + + I − ) C V ∂ t T matter = σ n f n (acT 4 matter (t n ) − 2π(I + + I − ))(4)
The Quantized Monte Carlo is tested with a temperature dependent opacity given by the formula : σ = ρ × d × T −3 matter cm −1 [22]. The value used is computed at each iteration by the Fleck linearization method. Note that the Fleck factor induces a scattering term also depending on the matter temperature. This case illustrates the behavior of the method in a circumstance where the scattering values belong to different regimes. The results are presented in figure 3. The comparison with reference results shows good physical agreement, independent of the collision regime : moreover the number of events per particle is substantially reduced (by a factor 1000, cf. right axis in the right plot of figure 3), together with the computation time. Moreover, we notice that the computation time is no longer strictly proportional to the number of events as for the IMC classic method, which indicates that with this new method, the trajectography is no longer the limiting phase in the computation time, but the treatment carried out between each tracking phase (emission and regulation of the particles for example) becomes important (the time increases with the number of particles remaining at the end of the iteration).
V. CONCLUSION
We introduce the Quantized Monte Carlo method to solve a computationally intensive multiregime thermal radiative transport equation within an unifying framework. The method is independent on any random walk assumptions to treat the high collision regime and relies on a offline computation followed by online sampling from a database. We check empirically that the smoothness assumptions underlying the method are, for the applications considered, of satisfactory quality; we next test the approach on a 1D benchmark and obtain physically coherent results while improving the computational time. This opens the perspective of future work on more complicated geometries and higher dimensional settings.
(
the flux is carried by the particles) and the particles can go from one region to another (by a conversion) and, more importantly, can change the cell (having different σ a and σ s values) with no particular treatment. The introduction of a new type of particles to treat the diffusion region allows easy treatment of the interface between the transport and diffusion regions. Contrary to the RW methods, the efficiency of these methods is not dependent on the mesh; however their use is still restricted by the loss in precision introduced by the diffusion approximation when particles change the region.Hybrid approaches[10,11] solve the diffusion equation analytically in some spatial areas and use the IMC approach in others. Both methods are coupled by boundary conditions. The hybrid methods use an analytical resolution of the scattering equation when certain criteria are met (delimited areas or according to the frequency group). The use of these methods remains limited by the coupling between the analytical resolution of the diffusion equation and the Monte Carlo method solving the transport equation which is delicate as well as the choice of criteria (e.g. the definition of areas where the diffusion approximation can be used).
FIG. 1 .
1An illustration of the escape dynamics of a particle starting at x init and undergoing collisions after Exp(σ) time (exponential random variable of average 1/σ). The particle can escape through any of the domain's frontiers: either because it escapes the spatial domain (dotted trajectory) or because the time is up (dashed trajectory). The random events accumulate into a probability law denoted E(σ, , t max ) with support on the boundaries of the time-space domain (together with a escape angle direction attribute).
IP
(exit by xmax) =0.0 FIG. 2. We plot the escape times for a particle from a time-space domain [x min , x max ] × [0, t max ]. The probabilities of escape are given in the title of each plot. We take x min = 0.0, x max = 0.01, initial direction +1, t max = 4000f s, speed 3.0 × 10 −5 (speed of light in fs/cm), x init = 0.005 and change the σ parameter (in cm −1 ). The plots in the columns 1 to 3 correspond to the escape distributions for σ = 0.75 (first line), σ = 1 (second line) and σ = 1.25 (third line); the plots in columns 4 to 6 corresponds to σ = 7.5 (first line) σ = 10 (second line) and σ = 12.5 (third line). In these 3 latter column the collisions are too many and the point does no significantly move i.e., it only escapes because the time is consumed.
erg.cm −2 .s −1 a 7.56 × 10 −15 erg.cm −3 .K −4 ∆t 4 × 10 −11 s d 1.56 × 10 23 K 3 .g −1 .cm 2 ρ 3 g.cm −3 c 3 × 10 10 cm.s −1 T matter K T matter (0, ·) 11604 K C V 8.6177 × 10 7 erg.g −1 .K −1 T matter (·, left border) 11604000 K TABLE I. Values and units used in the numerical simulation of the propagation of a Marshak-type wave in an opaque medium.
FIG. 3 .
3Results of the simulation described in IV B (multi-regime physics, temperature dependent opacity). The number of events per particle for the Quantized Monte Carlo is reduced with respect to the reference while keeping the physical properties of the solution. Left image : temperature profile for the times 1ns, 5ns and 10ns. Right image dashed lines, right axis : mean number of particle events per iteration for the classical Monte Carlo trajectory compared with the quantized simulation; Right image solid lines, left axis : execution time per iteration for the two procedures. All plots refer to the same simulations.
ACKNOWLEDGMENTS L.L. and G.T. acknowledge the support from their institutions.
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one can sample this law using this diffusion approximation. In practice we use a very conservative approach by replacing, for σ large. When σ is large enough to ensure that the diffusion approximation is valid. several collisions with one collision provided that the diffusion approximation ensures that the probability to escape is less than 10 −6 . Note that this is only a way to compute faster the exact law but the Quantized Monte Carlo does not depend on this choice, any sampler of the exact escape law will doWhen σ is large enough to ensure that the diffusion approximation is valid, one can sample this law using this diffusion approximation. In practice we use a very conservative approach by replacing, for σ large, several collisions with one collision provided that the diffusion approximation ensures that the probability to escape is less than 10 −6 . Note that this is only a way to compute faster the exact law but the Quantized Monte Carlo does not depend on this choice, any sampler of the exact escape law will do.
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| [] |
[
"Aerosols optical properties in Titan's Detached Haze Layer before the equinox",
"Aerosols optical properties in Titan's Detached Haze Layer before the equinox"
] | [
"Benoît Seignovert \nGSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France\n",
"Pascal Rannou \nGSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France\n",
"Panayotis Lavvas \nGSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France\n",
"Thibaud Cours \nGSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France\n",
"Robert A West \nGSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France\n\nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n"
] | [
"GSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France",
"GSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France",
"GSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France",
"GSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France",
"GSMA\nUniversité de Reims Champagne-Ardenne\n7331-GSMA, 51687ReimsUMR, France",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA"
] | [] | UV observations with Cassini ISS Narrow Angle Camera of Titan's detached haze is an excellent tool to probe its aerosols content without being affected by the gas or the multiple scattering. Unfortunately, its low extent in altitude requires a high resolution calibration and limits the number of images available in the Cassini dataset. However, we show that it is possible to extract on each profile the local maximum of intensity of this layer and confirm its stability at 500 ± 8 km during the 2005-2007 period for all latitudes lower than 45 • N. Using the fractal aggregate scattering model ofTomasko et al. (2008)and a single scattering radiative transfer model, it is possible to derive the optical properties required to explain the observations made at different phase angles. Our results indicates that the aerosols have at least ten monomers of 60 nm radius, while the typical tangential column number density is about 2 · 10 10 agg.m −2 . Moreover, we demonstrate that these properties are constant within the error bars in the southern hemisphere of Titan over the observed time period. In the northern hemisphere, the size of the aerosols tend to decrease relatively to the southern hemisphere and are associated with a higher tangential opacity. However, the lower number of observations available in this region due to the orbital constraints is a limiting factor in the accuracy of these results. Assuming a fixed homogeneous content we notice that the tangential opacity can fluctuate up to a factor 3 among the observations at the equator. These variations could be linked with short scale temporal and/or longitudinal events changing the local density of the layer. | 10.1016/j.icarus.2017.03.026 | [
"https://export.arxiv.org/pdf/1704.00842v1.pdf"
] | 119,431,198 | 1704.00842 | 76f5bf87c322bc7f9a1573b94359b58f6ceb79f7 |
Aerosols optical properties in Titan's Detached Haze Layer before the equinox
Benoît Seignovert
GSMA
Université de Reims Champagne-Ardenne
7331-GSMA, 51687ReimsUMR, France
Pascal Rannou
GSMA
Université de Reims Champagne-Ardenne
7331-GSMA, 51687ReimsUMR, France
Panayotis Lavvas
GSMA
Université de Reims Champagne-Ardenne
7331-GSMA, 51687ReimsUMR, France
Thibaud Cours
GSMA
Université de Reims Champagne-Ardenne
7331-GSMA, 51687ReimsUMR, France
Robert A West
GSMA
Université de Reims Champagne-Ardenne
7331-GSMA, 51687ReimsUMR, France
Jet Propulsion Laboratory
California Institute of Technology
91109PasadenaCAUSA
Aerosols optical properties in Titan's Detached Haze Layer before the equinox
10.1016/j.icarus.2017.03.026TitanAtmosphereAerosols
UV observations with Cassini ISS Narrow Angle Camera of Titan's detached haze is an excellent tool to probe its aerosols content without being affected by the gas or the multiple scattering. Unfortunately, its low extent in altitude requires a high resolution calibration and limits the number of images available in the Cassini dataset. However, we show that it is possible to extract on each profile the local maximum of intensity of this layer and confirm its stability at 500 ± 8 km during the 2005-2007 period for all latitudes lower than 45 • N. Using the fractal aggregate scattering model ofTomasko et al. (2008)and a single scattering radiative transfer model, it is possible to derive the optical properties required to explain the observations made at different phase angles. Our results indicates that the aerosols have at least ten monomers of 60 nm radius, while the typical tangential column number density is about 2 · 10 10 agg.m −2 . Moreover, we demonstrate that these properties are constant within the error bars in the southern hemisphere of Titan over the observed time period. In the northern hemisphere, the size of the aerosols tend to decrease relatively to the southern hemisphere and are associated with a higher tangential opacity. However, the lower number of observations available in this region due to the orbital constraints is a limiting factor in the accuracy of these results. Assuming a fixed homogeneous content we notice that the tangential opacity can fluctuate up to a factor 3 among the observations at the equator. These variations could be linked with short scale temporal and/or longitudinal events changing the local density of the layer.
Introduction
Since the beginning of exploration of giant planets in the 80's, Titan, the largest moon of Saturn, always focused a lot of attention on its unique thick haze atmosphere. Voyager 1 and 2 flybys, revealed the existence of a complex stratified succession of haze layers above Titan's main haze layer (Smith et al., 1981(Smith et al., , 1982. One of these layers, the Detached Haze Layer (DHL), presented a large horizontal extent at 350 km and could be seen all along the limb surrounding Titan main haze between 90 • S up to 45 • N before merging with the north polar hood. Based on Voyager 2 radial intensity scans at high phase angles, Rages and Pollack (1983) were able to retrieve the vertical distribution of scattering particles and reveal an important depletion of the aerosol particle density below this layer. Toon et al. (1992) proposed the first explanation by an interaction of ascending winds with the vertical structure of the haze, yielding a secondary layer above the main layer. The aerosols are initially produced in a single production zone in the main haze then raised by the dynamics and stored inside the detached haze layer. On the other hand, Chassefière and Cabane (1995) presented a completely different formation scenario based on the photochemistry of polyacetylenes occurring at high altitudes producing fluffy aggregates trapped inside the detached haze layer. Finally Rannou et al. (2002Rannou et al. ( , 2004 proposed with a global circulation model (GCM) that the detached haze is an natural outcome of the interaction between the atmosphere dynamics and the haze microphysics. Therefore, the detached haze is submitted to a seasonal cycle driven by dynamics.
The arrival of the Cassini spacecraft in the Saturnian System in 2004, was a unique opportunity to investigate the persistence of the detached haze layer over time. Porco et al. (2005) confirmed its presence 24 years after its discovery at an altitude near 500 km. This new location was first explained by Lavvas et al. (2009) as a secondary layer under the limit of the Voyager camera sensitivity. Their mechanism based on the sedimentation and coagulation of particles of mono-dispersed spheres was able to produce a detached haze at 500 km altitude independently of the dynamical transport. However, the systematic survey made by the repetitive flybys of Titan by Cassini showed that its altitude at the equator remains constant at 500 km between 2005 and 2007, decreases to 480 km at the end of 2007 and dropped suddenly at 380 km in 2009, around the northern spring equinox, before disappearing in 2012 (West et al., 2011). Finally, recent 3D GCMs (Lebonnois et al., 2012;Larson et al., 2015), more sophisticated than the previous 2D GCM, still producing the detached haze, give a description of the complete annual cycle of the detached haze layer and predict a reappearance in early 2015.
Our purpose is to characterize the physical properties of the detached haze layer before the equinox, while it was stable. We analyzed ISS images of the detached haze layer at different phase angles at UV wavelengths where the detached layer can be easily seen. Then we interpret the observations using a single-scattering radiative transfer model (Tomasko et al., 2008) with fractal aggregate particles to constrain the aerosol optical properties based on the monomer radius, the number of monomers per aggregate and tangential column number density. Finally we derive the local temporal/longitudinal variability of the detached layer during the 2005-2007 period.
Observations
To determine the properties of the aerosols in the detached haze layer, we made a survey over the ISS images taken with the Narrow Angle Camera (NAC) before its first decrease in altitude at the end of 2007 (West et al., 2011). We limit our analysis to the CL1-UV3 filter (λ=338 nm) to get the best contrast between the detached haze layer and the main haze (Fig. 1a) and to minimize the multiple scattering. Assuming lambertian scattering and the albedo observations by Karkoschka (1994Karkoschka ( , 1998, we estimate that the multiple scattering is always less than 5% for UV wavelengths. We select images taken far enough from Titan to get the best latitude coverage on the limb and to get an accurate georeference calibration of data, but also close enough to get the highest pixel scale to probe properly the detached haze layer (<10 km). We also restrict our analysis to a short period of time between 2005 to 2007 to limit the temporal variability of the detached haze layer (1 Titan year = 29 Earth years). Among the acquired images only 15 (Tab. 1) satisfied the above criteria, covering a large range of phase angle (from 14 • to 155 • ).
The radiance factor on the ISS images (I/F ) is calibrated using the CISSCAL routines (West et al., 2010) and a Poisson Maximum-a-posteriori (PMAP) deconvolution is applied to improve the signal/noise ratio. Then the detached haze layer is automatically located at the maximum of contrast. Considered to be stable during this period (West et al., 2011), we use its altitude as a proxy to locate very precisely the center of Titan (Fig. 1b) by fitting an ellipse through these points. Therefore we are able to extract the I/F profile all along the limb of Titan according to the geographical coordinates and the illumination conditions during the acquisition.
The vertical I/F profile is sampled in latitude using 5 • bins (Fig. 2a). The uncertainty of the intensity on each pixel represents the photon noise (between 2 to 5%) and the flat field uncertainty (1%). To account for the ob- served variability of the profile inside the latitude bin and altitude ranges, we sum-up all pixel uncertainties with a moving average box of 1.5 times the pixel scale (i.e with an overlap of 50%) and containing at least 30 pixels (up to 150). Assuming a stochastic uncertainty distribution, the sum of these uncertainties is considered Gaussian and the value at 0.16 and 0.84 on the normalized cumulative uncertainty are equivalent to a Gaussian 1σ error (i.e. covering 68% of the integral). This uncertainty distribution provides average and variance values of the spatial variability. This method allows us to smooth the vertical profile and remove the outliers. Finally the local maximum I/F of the detached haze layer is extracted for each profile at all latitudes in the illuminated limb of Titan.
This process is applied to the whole image dataset. Figure 2b represents the distribution of the local maximum I/F of the detached haze layer as a function of phase angle. The variability of the local maximum I/F of the detached haze layer is resampled using bins of 5 • phase angle range in order to put similar weight on the different images at low and high phase angles.
Method
As mention before, we focus our study on the local maximum I/F of the detached haze layer. We consider that above the detached haze layer, the atmosphere of Titan is optically thin and the incoming flux from the Sun is not significantly attenuated down to the detached haze layer (i.e. the opacity along the incoming ray τ 0 ext 1 in the illuminated limb). We also neglect the multiple scattering inside and below the detached haze layer (evaluated at maximum to 5% at 338 nm). Then, in the limit of the single scattering approximation and the optically thin layers, the output flux can be simply described as:
I/F = ω · P (θ) 4 · [1 − exp (−N los · σ ext )](1)
with ω the single scattering albedo, P (θ) the phase function at the scattering angle θ (corresponding to 180 •phase angle of the observation), σ ext the extinction crosssection of the aerosols, and N los the tangential column number density (i.e. the amount of aggregates along the line of sight).
We assume that the detached haze layer is composed of fractal aerosols composed of single-size spherical primary (a) Intensity I/F vertical profile measured from ISS NAC in CL1-UV3 filter combination on N1551888681 1 close to the geographic equator of Titan (0 to 5 • N at -55 • from the photometric equator). Each pixel is considered with its own photon noise uncertainty (grey circles). The profile is smoothed with a moving average box (dashed black box) of 1.5 times the pixel scale of the image (e.g. 10.4 km here). The intensity spread of the profile (black lines) corresponds to the normalized Gaussian cumulative uncertainty at 1σ. (b) Collection of local maxima I/F of the detached haze layer (dots) close to the geographic equator of Titan (0 to 5 • N) over the different phase angles. Data used in the model (black squares) were integrated with a 5 • bin using Gaussian uncertainty integration with a 1σ Gaussian equivalent to keep the spatial and temporal spread of the data. For each latitude only the local maximum I/F on the detached haze layer and its spread is used in the rest of the study. particles called monomers (West and Smith, 1991;Cabane et al., 1993). Based on a tholin composition, we use the optical index n=1.64 and k=0.17 (Khare et al., 1984) for the ISS CL1-UV3 filter (λ=338 nm). Then, assuming a fractal dimension (D f ) of 2.0 and a maximum number of monomers per aggregate N ≤ 5, 000 (Tomasko et al., 2009), the optical properties (ω, P (θ) and σ ext ) of theses aerosols can be calculated using the model developed by Tomasko et al. (2008). Therefore, the aggregate itself can be described by only two parameters: R m the monomer radius and N the number monomers per aggregate. The bulk radius (R v ) is defined with the relation:
R v = 3 √ N · R m(2)
Between 2005 and 2007, we assume that the detached haze is stable and the aerosol content of the detached haze layer, i.e. the number of aggregates and their optical properties remain temporally constant (West et al., 2011) and also longitudinally stable due to mixing by zonal winds (Larson et al., 2015). Then we calculate a synthetic I/F synt with a set of 3 independent parameters: R m , N and N los , that we compare with the observed I/F obs as a function of phase angle for the different latitudes. This comparison is made with a χ 2 as a merit function (Press et al., 1992):
χ 2 (lat) = φ I/F φ,lat obs − I/F φ,lat synt (R m , N, N los ) ∆I/F φ,lat obs 2 (3)
with φ the phase angle, and ∆I/F φ,lat obs the uncertainty associated to observation I/F φ,lat obs . In practice, the mapping of the χ 2 is performed with a 2 nm step between 10 and 70 nm for the monomer radius (R m ), and 10 points per decade between 1 and 5,000 for the number of monomers per aggregates (N ). The upper limit on N ≤ 5, 000 correspond to typical aggregates observed close to the surface (Tomasko et al., 2009). We use a monodispersed distribution of particles to increase the speed of the calculations. The tangential opacity (τ ext = N los × σ ext ) is directly linked to the scaling factor between the phase function and the intensity and is fitted with a least squares minimization for all pairs of (R m , N ).
Results
The altitude of the local maximum I/F of the detached haze layer for the complete dataset (Fig. 3) presents a high stability at 500±8 km. Thus we confirm that the detached haze layer is a persistent feature, not only at the equator (West et al., 2011), but at all latitudes below 45 • N during the 2005-2007 period. The values for the latitudes higher than 45 • N are affected by the polar hood observed at this period (Griffith et al., 2006).
For each latitude bin, we compute the synthetic I/F for all pairs of (R m , N ) and we keep the best fit on the tangential opacity. All the solutions explored in the bin close to the geographic equator of Titan (0 to 5 • N) are represented on Figure 4a. Based on the Press et al. (1992) description of the confidence limits on estimated model parameters (chapter 15.6), we color code the ∆χ 2 = χ 2 − χ 2 min for a confidence level p = 68.3%. Thus all fits with ∆χ 2 < 3.53 (i.e. with 3 degree of freedom) match the observations within their error bars ( Fig. 4b and 4c). For this latitude (0 to 5 • N), the χ 2 min is achieved at R m = 60 nm, N = 159 and N los = 3.7 · 10 10 agg.m −2 with a value of 1.7. Then we consider each parameter independently (i.e. with 1 degree of freedom), so we select all the pairs of (R m , N ) for which ∆χ 2 < 1.00. At this latitude minima and maxima obtained are R m,min = 54 nm, R m,max = 64 nm, N min = 13 and N max = 5, 000 (upper limit tested). For these parameters, the size of the monomers (R m ) is constrained by the backscattering part of the phase function with the observations at low phase angle (Figure 4b). On the other hand, the determination of the number of monomers per aggregate (N ) is mainly driven by the shape of the forward peak ( Figure 4c). Unfortunately, observations at very high phase angles (> 160 • ) were not available in the CL1-UV3 filter during this period, and do not allow us to constrain the upper limit of N . Then, the bulk size of the aggregate (R v ) is poorly constrained between 150 nm to 1.1 µm.
The complete dataset of the best fits for each latitude bin is shown in Figure 5. We notice that the fits are very similar for all latitudes lower than 10 • N. At higher latitudes, the shape of the phase function changes and presents an increase in the backscattered direction (i.e. at low phase angle). This effect is considered to be an artifact due to the lower number of images available at these latitudes.
The parameter values derived for the different latitude bins are summarized in figure 6. Each point represents the average value inside the area covered by ∆χ 2 < 1.00 and the error bars correspond to the extreme values with (c) Same outputs but for different N with Rm fixed at 60 nm. In this case, only the forward peak is changed. Unfortunately, observations at very high phase angles were not available and do not provide an upper limit on N (e.g. N = 159 and N = 5, 000 are identical up to 170 • ). We also present on the same plot samples of the two local minima observed in (a). These parameters fit some observations but are not consistent with the others.
the same confidence level. The latitudinal variation of these parameters can be split into two main areas: below and above 10 • N. Below 10 • N, the monomer radius and the tangential opacity can be considered constant with R m = 60 ± 3 nm and τ ext = 0.078 ± 0.004. The number of monomers per aggregate is larger than ten monomers without upper constraint. We use N = 266 as a latitude mean (in the log-space). All the latitude means below 10 • N are summarized in Table 2. The optical properties derived from the Tomasko et al. (2008) model with these aggregates are listed in Table 3. Above 10 • N, we find a decrease in the aerosols size associated with an increase of the tangential opacity, i.e. smaller and more numerous particles or smaller particles with a higher extinction. However theses results inside the sparse dataset region need to be taken with caution due to the lower number of observation available in the 2005-2007 period as we mention before. For each latitude, we derive the tangential column number density by dividing the tangential opacity with the extinction cross-section calculated with the Tomasko et al. (2008) model. This value is also constant below 10 • N with N los = 1.9 ± 0.3 · 10 10 agg.m −2 but provide no consistent information above 10 • N. Finally, the slope of the I/F vertical profile (Fig. 2a) above the detached haze layer provide a measure of the aerosol scale height. The latitude mean value of this scale height (H = 35 km) can be used to estimate the geometric attenuation factor of the incoming optical depth in the
θ = 0 • - 185.8 - θ = 1 • - 178.1 - θ = 2 • - 157.2 - θ = 4 • - 98.9 - θ = 6 • - 52.6 - θ = 8 • - 28.4 - θ = 10 • - 17.4 - θ = 15 • - 8.0 - θ = 20 • - 4.7 - θ = 30 • - 2.2 - θ = 40 • - 1.1 - θ = 50 • - 0.58 - θ = 60 • - 0.36 - θ = 80 • - 0.21 - θ = 100 • - 0.144 - θ = 120 • - 0.117 - θ = 140 • - 0.112 - θ = 160 • - 0.115 - θ = 180 • - 0.117 -
detached haze layer (z = 500 km):
τ 0 ext ≈ τ ext · H 2π · (R T + z) · H = 3 · 10 −3 1 (4)
confirming that the incoming flux from the Sun is not significantly attenuated down to the detached haze layer. Moreover, this scale height can also be used as a proxy to derive the aggregate local number density (n):
n ≈ N los 2π · (R T + z) · H(5)
At 500 km, for latitudes lower than 10 • N, we estimate the local number density inside the detached haze layer at n ≈ 2 · 10 −2 agg.cm −3 . Then, assuming that the aerosols material has a density of 1 g.cm 3 and a fractal dimension of 2.0, we derive an estimation of the mass flux Latitudinal retrieval of (a) monomer radius, (b) number of monomers per aggregate, (c) tangential opacity, and (d) tangential column number density derived from the χ 2 exploration with a confidence level ∆χ 2 < 1.00. The circle and squares represents the local mean value (calculated in log-space for N and N los ). The solid lines and the drak gray boxes correspond to the mean and the standard deviation of these mean values between 90 • S and 10 • N where the dataset is most constraining. Latitudes in the light gray area are excluded due to their sparse dataset (winter in the northern hemisphere, cf. Tab. 1). The upper error-bars on N at 5,000 monomers per aggregate correspond to typical aggregates observed close to the surface (Tomasko et al., 2009). of ∼ 10 −14 g.cm −2 .s −1 for these aerosols (Lavvas et al., 2010). Above 10 • N the data are not consistent.
Discussion
During this analysis, we have decided to keep the fractal dimension (D f ) of the aerosols fixed at 2.0 owing to the Tomasko et al. (2008) model. Other models, like Rannou et al. (1997) provide a semi-empirical model of absorption and scattering by isotropic fractal aggregates of spheres for higher fractal dimension. However, after investigation we notice that the distribution of pairs of monomers (Rannou et al., 1997, equation A3) does not take into account the cumulative number of dimers (i.e. monomer touching each other). When we compare outputs with T-matrix cases provided by Tomasko et al. (2008), we notice a significant difference in the tail of the phase function in the backscattering direction at short wavelength. Therefore, without any additional constraints from the observations, we decided to discard this model and keep the fractal dimension fixed at 2.0.
We also considered the sensitivity of our retrievals on the wavelengths sampled by the instrument. The CL1-UV3 filter on the ISS Narrow Angle Camera is centered at 338 nm with a bandwidth of 68 nm. We find a difference smaller than 5% in the I/F synt calculation trough the transmission filter compared with the single central wavelength model presented in this study.
The 60 ± 3 nm for the monomer radius of the aerosols retrieved in this analysis is larger than the 40 ± 10 nm monomer observed in polarization by DISR (Tomasko et al., 2009) in the main haze at lower altitudes (<150 km). However, our value is consistent with the 60 nm derived with polarimetry (West and Smith, 1991) and the 66 nm (Rannou et al., 1997) retrieved at higher altitude (> 350 km) from Voyager observations. On the other hand, due to the poor determination of the number of monomers per aggregates, it is not possible to provide any The black dots correspond to the analysis made here and we linearly interpolate for intermittent times. We observe a temporal variability of a factor of ∼3 at the equator (see text for detail). We also added the sparse dataset output as a reference, however these latitudes must be taken with caution due to their poor determination of the (Rm,N ) parameters. (b) Tangential column number density (N los ) retrieved for each image as a function of phase angle. The mean over the phase angle and the standard deviation are represented by the solid line and the dark gray box, respectively. As expected, this parameter is independent of the phase angle where the dataset is enough constraining. The open squares represent the values retrieved on the sparse dataset (excluded from the mean).
constraints on the size distribution of the aerosols. Narrow and broad log-normal distributions where tested (with a scale parameter σ 0 = 0.3 and 0.5) but did not provide significantly different results compared to the mono-dispersed case. Moreover, we made a simple comparison with UVIS observations (Koskinen et al., 2011). The comparison between these instruments can help to constrain the aerosol distribution including the small, none scattering aerosols in the detached haze layer (Cours et al., 2011). The extinction tangential opacity retrieved at the local maximum around 500 km is about 0.75 at 190 nm (T41 and T53 flybys in 2008). We extend our results at this wavelength using the Khare et al. (1984) indices and we find a scattering tangential opacity between 0.07 and 0.11. As expected the tangential opacity found with ISS is smaller than the one found with UVIS which include small none scattering aerosols.
The fractal aggregates found in this study are composed of more than ten monomers of 60 nm. As expected from the GCM of Titan this type of aggregates needs long timescales to be formed and will be homogeneously distributed in longitude by the zonal wind (Rannou et al., 2004;Larson et al., 2015). As we showed, during the 2005-2007 period the detached haze layer is also very stable with latitude from the south pole up to the 10 • N with a possible decrease in particle size at higher latitudes where the dataset is less constraining. Finally, we use the optical properties derived from this analysis latitude by latitude to get an estimation of the temporal/longitudinal variability of the tangential opacity of the detached haze layer (Fig. 7a). The global variability is low at 0.08 ± 0.1 but we notice two patterns around the equator on the images taken in March 2006 and December 2006 (N1521213736 1 and N1546223487 1). The variability on the tangential opacity is about a factor 3 between these two structures (0.04 to 0.12). Both share the same acquisition geometry, a similar phase angle of observation (68 and 66 • ), a same local time (10h28 and 13h28) and a same position on the orbit (202 • and 254 • to aphelion), only the acquisition time and the longitude (21 • and 100 • W) are significantly distinct between these two images. Theses variations can be attributed to fluctuations of the tangential column number density. For now, its origin is unknown, but our best guess would be the propagation of waves. However, the large gaps between flyby acquisitions do not allow us to derive any temporal constraint. We also checked that our retrievals on the tangential column number density are not biased by the phase angle sampling. When we exclude from the statistics the observations corresponding to the sparser portions of the dataset, then the retrievals have the same spread at low, middle and high phase angle, and do not present any bias (Fig. 7b).
Conclusions
The analysis of Cassini UV images of Titan taken between 2005 and 2007 enable us to verify the stability of the detached haze layer at 500 ± 8 km for all latitudes lower than 45 • N during this period. Comparing the scattered intensity of the detached haze at its maximum for different phase angles with the synthetic intensity based on the Tomasko et al. (2008) model, we were able to constrain the optical properties of aerosols in the detached haze layer. We find that theses aerosols can have at least ten monomers of 60 nm radius for latitudes lower than 10 • N. The typical tangential column number density was constrained at 2 · 10 10 agg.m −2 in the same latitude range. At higher latitudes, we have shown that the size of the aggregates tends to decrease but the lower number of images available in this region due to observation limitations did not allow us to derive better constrains. Then, we used these aggregates (averaged between 90 • S and 10 • N) as a homogeneous content to get the local temporal/longitudinal variability of the detached layer during the 2005-2007 period and notice that the tangential opacity can fluctuate up to a factor 3. For now the cause of theses perturbations of the local density are still unknown but they seem to be linked to short scale temporal and/or longitudinal events. Finally, we stress the fact that fractal aerosols can have a fractal dimension higher than 2.0 but the current models are not able to investigate this property without more additional constraints. Observations at different wavelengths and with polarization filters could be used to provide these additional constraints in the future.
Fig. 1 .
1(a) Overexposed and contrast-enhanced ISS NAC N1551888681 1 image taken on 2007/03/06 at an observation phase angle of 143 • in CL1-UV3 filters (338 nm). The detached haze layer can be seen on the limb surrounding Titan's main haze. (b) Same image inverted. The small black dots represents the detected position of local maxima associated to the detached haze layer fitted by an ellipse (magenta). The center of the fit provides the location of Titan's center (i.e. the sub-spacecraft point (SC) in blue at 31 • E -17 • S). The orange point is given as a reference for the Spice Kernels (SK) predicted center. An offset of 55 pixels (430 km) can be noticed. The red lines are the geographic coordinates. The sub-solar point is located on the other side of Titan at 127 • W -13 • S and the yellow line represents the location of the terminator on the ground. The green cross behind Titan represents the photometric frame with a photometric equator (PE) at 79 • W -49 • S. The limb profiles are average in I/F in 5 • bins in latitude on the illuminated side of Titan.
Fig. 2 .
2Fig. 2. (a) Intensity I/F vertical profile measured from ISS NAC in CL1-UV3 filter combination on N1551888681 1 close to the geographic equator of Titan (0 to 5 • N at -55 • from the photometric equator). Each pixel is considered with its own photon noise uncertainty (grey circles). The profile is smoothed with a moving average box (dashed black box) of 1.5 times the pixel scale of the image (e.g. 10.4 km here). The intensity spread of the profile (black lines) corresponds to the normalized Gaussian cumulative uncertainty at 1σ. (b) Collection of local maxima I/F of the detached haze layer (dots) close to the geographic equator of Titan (0 to 5 • N) over the different phase angles. Data used in the model (black squares) were integrated with a 5 • bin using Gaussian uncertainty integration with a 1σ Gaussian equivalent to keep the spatial and temporal spread of the data. For each latitude only the local maximum I/F on the detached haze layer and its spread is used in the rest of the study.
Fig. 3 .
3Altitude map at the local maxima I/F on the detached haze layer as a function of the time and latitude. The black dots represent the profiles analyzed.
Fig. 4 .
4(a) (Rm, N ) pairs explored for latitudes close to Titan's geographic equator (0 to 5 • N). The confidence limits ∆χ 2 = χ 2 − χ 2 min are color coded with respect to χ 2 min =1.7 at Rm=60 nm and N =125. The black dash lines represent the aggregate bulk radius (Rv). (b) Evolution of I/Fsynt for different Rm with N fixed at 125 monomers per aggregate. The main changes are observed in the backscattering direction.
Fig. 5 .
5Best fits of I/F sorted by 5 • latitude bins between 90 • S and 50 • N. For each panel, the filled circles with associated error bars represent the observed I/F , and the black solid line corresponds to the best fit on Rm, N , and N los . The two gray solid lines represent the best fits in the extreme cases of minimum and maximum observed I/F . We can note that the fit is less consistent on the last two rows (> 10 • N) when the number of observations decreases.
Fig. 6 .
6Fig. 6. Latitudinal retrieval of (a) monomer radius, (b) number of monomers per aggregate, (c) tangential opacity, and (d) tangential column number density derived from the χ 2 exploration with a confidence level ∆χ 2 < 1.00. The circle and squares represents the local mean value (calculated in log-space for N and N los ). The solid lines and the drak gray boxes correspond to the mean and the standard deviation of these mean values between 90 • S and 10 • N where the dataset is most constraining. Latitudes in the light gray area are excluded due to their sparse dataset (winter in the northern hemisphere, cf. Tab. 1). The upper error-bars on N at 5,000 monomers per aggregate correspond to typical aggregates observed close to the surface (Tomasko et al., 2009).
Fig. 7 .
7(a) Temporal/latitudinal map of tangential opacity (τext) in the detached haze layer.
Table 1
1Dataset of 15 ISS NAC images used in this study, taken with CL1-UV3 filter combination (338 nm). The phase angle, the limb latitude coverage, the latitude of the photometric equator and the pixel scale are also listed below.ISS ID
Date
Phase
Coverage
Lat.
Photo.
Pixel
scale
(km)
Table 2
2Latitudinal average below 10 • N of the aerosol properties in the detached haze layer.Parameter
Symbol
Value
Std
Unit
Monomer radius
Rm
60
3
nm
Number of monomer per
aggregate
N
266
-
-
Aggregate bulk radius
Rv
0.4
-
µm
Tangential opacity
τext
0.078
0.004
-
Tangential column number
density
N los
1.9
0.3
×10 10
m −2
Table 3
3Aerosol optical properties derived from the table 2 and Tomasko et al. (2008) model (i.e. λ=338 nm, Rm=60 nm, N =266, Rv = 0.4 µm, D f =2.0).Parameter
Symbol
Value
Unit
Scattering cross-section
σsct
2.9e-12
m 2
Extinction cross-section
σext
4.2e-12
m 2
Absorption cross-section
σ abs
1.3e-12
m 2
Single-scattering albedo
ω
0.69
-
Phase function:
AcknowledgmentsThis work was supported by the French ministry of public research. The authors also thank the Programme National de Planétologie (PNP), the Agence National de la Recherche (ANR project "Apostic" No. 11BS65002, France) and the Centre National d'Étude Spatial (CNES) for their financial support. R.A. West would like to acknowledge the Region Champagne-Ardenne for its support (program "expertise de chercheurs invités" AO 2015).
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| [] |
[
"Phase-dependent Andreev molecules and superconducting gap closing in coherently coupled Josephson junctions",
"Phase-dependent Andreev molecules and superconducting gap closing in coherently coupled Josephson junctions"
] | [
"Sadashige Matsuo [email protected]@riken.jp \nCenter for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan\n",
"Takaya Imoto \nCenter for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan\n\nDepartment of Physics\nTokyo University of Science\n162-8601TokyoJapan\n",
"Tomohiro Yokoyama \nDepartment of Materials Engineering Science\nOsaka University\n560-8531OsakaJapan\n",
"Yosuke Sato \nCenter for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan\n",
"Tyler Lindemann \nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA\n\nDepartment of Physics and Astronomy\nPurdue University\n47907West LafayetteIndianaUSA\n",
"Sergei Gronin \nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA\n",
"Geoffrey C Gardner \nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA\n",
"Sho Nakosai \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Yukio Tanaka \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Michael J Manfra \nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA\n\nDepartment of Physics and Astronomy\nPurdue University\n47907West LafayetteIndianaUSA\n\nSchool of Materials Engineering\nPurdue University\n47907West LafayetteIndianaUSA\n\nElmore Family School of Electrical and Computer Engineering\nPurdue University\n47907West LafayetteIndianaUSA\n",
"Seigo Tarucha \nCenter for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan\n\nRIKEN Center for Quantum Computing\nRIKEN\n351-0198SaitamaJapan\n"
] | [
"Center for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan",
"Center for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan",
"Department of Physics\nTokyo University of Science\n162-8601TokyoJapan",
"Department of Materials Engineering Science\nOsaka University\n560-8531OsakaJapan",
"Center for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan",
"Birck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA",
"Department of Physics and Astronomy\nPurdue University\n47907West LafayetteIndianaUSA",
"Birck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA",
"Birck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Birck Nanotechnology Center\nPurdue University\n47907West LafayetteIndianaUSA",
"Department of Physics and Astronomy\nPurdue University\n47907West LafayetteIndianaUSA",
"School of Materials Engineering\nPurdue University\n47907West LafayetteIndianaUSA",
"Elmore Family School of Electrical and Computer Engineering\nPurdue University\n47907West LafayetteIndianaUSA",
"Center for Emergent Matter Science\nRIKEN\n351-0198SaitamaJapan",
"RIKEN Center for Quantum Computing\nRIKEN\n351-0198SaitamaJapan"
] | [] | The Josephson junction (JJ) is an essential element of superconducting (SC) devices for both fundamental and applied physics. The short-range coherent coupling of two adjacent JJs forms the Andreev molecule states (AMSs), which will provide a new ingredient to engineer the SC transport in JJs and control the Andreev qubits. However, no experimental evidence of the AMSs in the coupled JJs has been reported. Here we provide the tunnel spectroscopic results of electrically controllable two planar JJs sharing one SC electrode. We discover that the coupled JJ results are highly modulated from the single JJ results, due to formation of the phase-dependent AMSs, meaning that the two JJs are coherently coupled. In addition, the superconducting gap closing due to the AMS formation is observed. Our results would help in understanding the microscopic mechanism of the coherent coupling and promoting the AMS physics to apply for research of the topological superconductivity and quantum information technology. | null | [
"https://export.arxiv.org/pdf/2303.10540v1.pdf"
] | 257,632,292 | 2303.10540 | 64c3c60822926b74ca2bc754ff29852bf81d2445 |
Phase-dependent Andreev molecules and superconducting gap closing in coherently coupled Josephson junctions
Sadashige Matsuo [email protected]@riken.jp
Center for Emergent Matter Science
RIKEN
351-0198SaitamaJapan
Takaya Imoto
Center for Emergent Matter Science
RIKEN
351-0198SaitamaJapan
Department of Physics
Tokyo University of Science
162-8601TokyoJapan
Tomohiro Yokoyama
Department of Materials Engineering Science
Osaka University
560-8531OsakaJapan
Yosuke Sato
Center for Emergent Matter Science
RIKEN
351-0198SaitamaJapan
Tyler Lindemann
Birck Nanotechnology Center
Purdue University
47907West LafayetteIndianaUSA
Department of Physics and Astronomy
Purdue University
47907West LafayetteIndianaUSA
Sergei Gronin
Birck Nanotechnology Center
Purdue University
47907West LafayetteIndianaUSA
Geoffrey C Gardner
Birck Nanotechnology Center
Purdue University
47907West LafayetteIndianaUSA
Sho Nakosai
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Yukio Tanaka
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Michael J Manfra
Birck Nanotechnology Center
Purdue University
47907West LafayetteIndianaUSA
Department of Physics and Astronomy
Purdue University
47907West LafayetteIndianaUSA
School of Materials Engineering
Purdue University
47907West LafayetteIndianaUSA
Elmore Family School of Electrical and Computer Engineering
Purdue University
47907West LafayetteIndianaUSA
Seigo Tarucha
Center for Emergent Matter Science
RIKEN
351-0198SaitamaJapan
RIKEN Center for Quantum Computing
RIKEN
351-0198SaitamaJapan
Phase-dependent Andreev molecules and superconducting gap closing in coherently coupled Josephson junctions
The Josephson junction (JJ) is an essential element of superconducting (SC) devices for both fundamental and applied physics. The short-range coherent coupling of two adjacent JJs forms the Andreev molecule states (AMSs), which will provide a new ingredient to engineer the SC transport in JJs and control the Andreev qubits. However, no experimental evidence of the AMSs in the coupled JJs has been reported. Here we provide the tunnel spectroscopic results of electrically controllable two planar JJs sharing one SC electrode. We discover that the coupled JJ results are highly modulated from the single JJ results, due to formation of the phase-dependent AMSs, meaning that the two JJs are coherently coupled. In addition, the superconducting gap closing due to the AMS formation is observed. Our results would help in understanding the microscopic mechanism of the coherent coupling and promoting the AMS physics to apply for research of the topological superconductivity and quantum information technology.
The Josephson junction (JJ) is a representative superconducting (SC) device consisting of two weakly-linked superconductors through insulators or normal conductors [1]. The JJs have been studied to engineer quantum effects in solid-state devices, enabling to realize SC quantum computing and highly sensitive magnetic sensors. The recent development of JJs on the superconductorsemiconductor heterostructures has provided platforms for more exotic physics such as Andreev (spin) qubits [2][3][4][5][6], SC qubits with gate tunability [7,8] or Majorana zero modes (MZMs) [9][10][11][12]. From these aspects, engineering of coherent coupling between two JJs is an essential ingredient to explore novel SC phenomena, establish new control methods of JJs, and manage the qubit operation.
Recently a concept of short-range coherent coupling between two JJs has been proposed as Andreev molecule states (AMSs) [13][14][15]. In a single JJ consisting of two superconductors and a semiconductor, electrons are confined by the Andreev reflection at interfaces, forming the Andreev bound states (ABSs) [16][17][18]. For example, in the case of two adjacent JJs sharing one SC electrode, the ABS wavefunctions in the different JJs penetrate to the shared SC and overlap, which produces the coherently coupled wavefunctions called as AMSs. This is an analogy of the molecular orbital states formed by the coherent coupling of two atomic states. The recent experimental efforts on the coherently coupled JJs have revealed nonlocal SC transport assigned to the AMSs such as the nonlocal Josephson effect [19]. Additionally, the AMS physics in the coupled JJs may provide novel insights on the SC transport intermediated by the Cooper pair splitting in a parallel double quantum dot or double nanowire JJ, which can be regarded as the two JJs sharing two SC electrodes [20][21][22][23]. In order to understand the microscopic mechanisms of such SC transport for the novel SC device functionality and engineer operation and integration of the Andreev qubits using the coupled JJs [24], observation of the AMSs in the coupled JJs is indispensable. However, experimental evidence of the AMS formation in the coupled JJs is lacking although there are experimental reports of the AMS signatures formed in SC junctions other than JJs [25][26][27].
Additionally, the AMSs in the coupled JJ structures contribute to and develop the topological physics in the JJs. In the case of conventional two-terminal JJs, the Kramers degeneracy, which demands the energy levels of half-integer spin systems, is doubly degenerate in the presence of time-reversal symmetry even with the spin-rotation symmetry broken by the spin-orbit interactions (SOI), leads to a small amount of SC gap easily caused. For example, such SC gap in the short ballistic JJs closes only when the junction transmission is unity [16]. Lifting this Kramers degeneracy by breaking the timereversal symmetry in the JJs with SOI leads to various exotic SC phenomena including the realization of MZMs. Multiterminal JJ structures which have more than three or more SC electrodes contacted on a single normal metal [25,[28][29][30][31][32][33][34][35][36][37][38] enable to break the time-reversal symmetry only by controlling the phase differences with no strong magnetic fields. As a signature of the Kramers degeneracy lifting in the multiterminal JJs, the SC gap closing only by the phase control is predicted [28,37,38] and related experimental results have been reported [39]. Observation of this SC gap closing is an important step to realize and control more exotic SC phenomena predicted in the multiterminal JJs including Weyl fermion physics [29,31,40,41]. In the multiterminal JJ case, the ABSs are formed in the single normal metal, but in the coupled JJ case, the AMSs are not restricted to the single normal metal and spread over the two JJs. Therefore, the coupled JJ structure enables to design more diverse physics of AMSs hosted in various combinations of different JJs.
Here we experimentally study the Andreev spectrum in single and coupled JJs by tunnel spectroscopy [11,42] to elucidate the phase-dependent AMSs. For this sake, we fabricate an SC device of two JJs named JJL and JJR sharing an SC electrode from a high-quality InAs quantum well covered with the epitaxially grown thin Aluminum layer [43][44][45]. The scanning electron microscopic image and the schematic image of the coupled JJ device are shown in Figs. 1(a) and (b), respectively. The separation between the two JJs corresponding to the width of the shared SC electrode is designed as 150 nm which is sufficiently shorter than the coherence length of Aluminum (Al). The junction length and width are 80 nm and 1.9 μm, respectively. The two JJs are each respectively embedded in a SC loop which encloses the same area, to induce the same phase difference in JJL and JJR. With definition of L and R of the phase differences on JJL and JJR as L = s − l and R = r − s , respectively, the out-of-plane magnetic field B changes the phase differences with L = R . Here l , s , and r represent the phases of the left, shared, and right SC electrodes, respectively. The main concept of this design is to compare the spectroscopic results in the single JJ and the coupled JJ cases using the same device but by electrically independently controlling the two JJs. For this sake, we have fabricated the gate electrodes as highlighted in yellow in Fig. 1 (a). The gate electrodes on JJL and JJR are used to control the planar JJs with the gate voltages of gL and gR , respectively. Additionally, three gate electrodes are prepared to electrically form the quantum point contacts on the edges of the planar JJs (see supplementary Note 2). The gate voltages g1 , g2 , and g3 are applied on the electrodes as depicted in Fig. 1(b). We name the point contact formed on the edge of JJL (JJR) as QPCL (QPCR). When performing the spectroscopy of JJL, we pinch off the QPCR and detect the tunnel current through the QPCL as the schematic circuit diagram in Fig. 1(b). We measure this device at 10 mK of the base temperature in our dilution refrigerator. For the tunnel spectroscopy, we apply a D.C. bias voltage V with a small oscillation component (5 μ ) and measure the tunnel currents through QPCL and QPCR by Lock-in amplifiers. Then we obtain L and R of differential conductance through the QPCL and QPCR, respectively.
First, we perform the tunnel spectroscopy of the single JJs. For the sake, we measure L with JJR pinched off ( gL , gR )= (0 V, -6 V). In the measurement, QPCR is pinched off also and QPCL is tuned to allow L ∼ 0.10 2 /ℎ. Then the obtained L as a function of V and B is indicated in Fig. 2(a). As clearly seen, the subgap structure emerges inside the Aluminum SC gap energy (~0.18 meV). The observed SC gap inside the Aluminum gap energy is modulated periodically as a function of B. The period is around 0.126 mT consistent with the expected period evaluated from the loop area (0.172 mT). This periodic modulation has been reported in previous studies [11,18,42] and assigned to the phase modulation of ABSs in the single ballistic JJ [16,42]. Therefore, the oscillation period of the gap is equivalent to 2π and B producing the maximum (minimum) gap corresponds to L = 0 (mod 2 ) ( L = π (mod π) ). It is noted that JJR is pinched off so that R is not considered here (see supplementary note 3 and 4, and Figs. S3 and S5). In our results, the maximum (minimum) induced gap is around 0.1 meV (0.08 meV). The induced SC gap is defined as the ABSs with no momentum along the junction and the ABSs with finite momentum fill the states outer of the gap. These results imply that our method of tunnel spectroscopy using QPCL correctly detects the Andreev spectrum in JJL. We note that the jump of the data at = 0.066 mT occurs because of the charge jump around QPCL.
Then, we move to the characterization of JJR with ( gL , gR ) = (-6 V, 0 V). We measure R as a function of V and B with QPCL and JJL pinched off. The results are shown in Fig. 2(b). As well as the JJL results, the periodic oscillation of the SC gap is observed, which is the ABS oscillation of the single JJR as a function of R . In the JJR results, the maximum (minimum) induced gap is around 0.12 meV (0.075 meV). Comparing the JJR result with the JJL result, the oscillation periods in both results are consistent, which reflects the two SC loops have the same loop area. Therefore, the results imply that our device works correctly to detect the ABSs of both the single planar JJs with the phase biased by the out-of-plane magnetic field. Next, we explore the tunnel spectroscopy of JJL with JJR on ( gL , gR ) = (0 V, 0 V). Here two JJs are turned on, then both of L and R evolve as a function of B with L = R assured by the same SC gap oscillation period of the single JJL and JJR results. The obtained spectroscopic result ( L as a function of B and V) is shown in Fig. 2(c). Compared with the single JJ result in Fig. 2(a), the Andreev spectrum is drastically modulated. This drastic change is invoked by turning on JJR, namely turning on the coherent coupling between two JJs. Therefore, the observed spectroscopic result and the change from the single JJ results are assigned to the formation of AMSs in the coupled JJs. Here a significant feature that cannot be found in the single JJ results is observed. In the coupled JJ case, the SC gap becomes maximal where the gap in the single JJL result in Fig. 2(a) produces the minimum. This means that L = R = 0, π (mod 2π) give the maximal SC gap in Fig. 2(c) and the minimum SC gap in Fig. 2(c) is realized in 0 < L = R < . When the coherent coupling between JJL and JJR modulates the JJL Andreev spectrum, the JJR spectrum should also be modulated as well. Then we confirm the JJR spectrum by the tunnel spectroscopy of JJR with ( gL , gR )= (0 V, 0 V). The obtained R as a function of B and V is shown in Fig. 2(d). As explicitly found, the same feature as in the JJL results appears. The consistency obtained between the JJL result in Fig. 2(c) and the JJR result in Fig. 2(d) assures that the Andreev spectrum modulation is induced by the coherent coupling between the two planar JJs. We note that these features are reproducible (see supplementary note 1 and Figs. S1 and S2). Particularly, both results exhibit the same dependence on the phase differences, which means that the coherent coupling maintains the phase coherence. In the coupled JJ devices, realization of topological SC states only by the phase control is theoretically predicted [46]. In the prediction, the topological SC state is formed and the MZMs appear between the two SC gap closing points as seen in Fig. 3(c). For this realization, it is essential that the quantum well holds different Fermi velocities for the spin-up and spin-down states. In our device, this might be induced by mixing of the subbands formed by the confinement [47][48][49]. When the velocity difference is small, a space of the two SC gap closing points could be tiny in the numerical calculation and then the SC gap closing would be observed to emerge at a single point (see supplementary notes 7 and 8 and Figs. S7 and S8). In this scenario, our finding of the SC gap closing might be related to the topological SC states in the coupled JJ systems.
A similar SC gap closing is theoretically predicted in multiterminal JJs [28,37] and the related experimental results have been reported [39]. In theory, it is revealed that the SC gap closing is allowed when the normal metal of the multiterminal JJs holds the strong SOIs and the three SC phases encircle the origin, meaning /2 ≤ L = R ≤ 3 /2 in our definition. This reflects that the SC gap closing occurs when the Kramers degeneracy is lifted by breaking the time-reversal symmetry in the JJs with the broken spin-rotation symmetry. It is noted that the SC gap closing (lifting the Kramers degeneracy)
in JJs is one of the necessary ingredients to realize the exotic SC phenomena originating from the single Fermion states such as Majorana zero modes, Weyl singularity, and Andreev spin qubits [29,31,39]. In the multiterminal JJs, the time-reversal symmetry can be broken only by controlling the SC phases. Therefore, this SC gap closing is an important signature of breaking the Kramers degeneracy in the junctions holding the SOIs. This resembles our situation in that the AMSs consist of the ABSs in the different JJs coupled by the multipair processes of the double elastic cotunneling or double crossed Andreev reflection [13] and the normal metal of InAs quantum well holds the strong SOI. Therefore, we can conclude that the observed SC gap closing indicates that the Kramers degeneracy in the Andreev spectrum including the SOI can be lifted by controlling the phase differences in the coupled JJ devices. This leads to the use of the coupled JJs for the SC phenomena predicted in the multiterminal JJs [29,31,39] and design more exotic phenomena using the coupled JJs in which the respective JJs consists of the different normal metals.
Subsequently, we explore the gate voltage dependence of the AMSs and the SC gap closing.
Especially, we focus on the nonlocal gate control effect on the AMS results. Figure 4(a) shows R as a function of B and V at gL =-1, -2, -3, -4, and -5 V with gR = 0 V. It is clear that the AMS features disappear and the ABSs in the single JJR emerge for gL ≤-3 V. Then we fix B=-0.03 mT at the SC gap closing and sweep gL as shown in Fig. 4(b). The gap-closing behavior appears robust for gL from 0 to − 2 V and gapped for gL ≲ −2 V. With Fig. 4(a), this indicates that JJL is pinched off around gL =-2 V and the SC gap closing disappears when the AMSs disappear. This is supported by the magnetic field dependence of R measured as a function of gL shown in Fig. 4(c). The two bright vertical lines are observed in the gate voltage range of -2 V < gL ≤ 0 V. This indicates the SC gap closing of JJR occurs in -2 V < gL ≤ 0 V and almost at the same magnetic fields in the nonlocal gate control. Similar behavior is also observed in the nonlocal gate control of the JJL (see supplementary note 5 and Figs. S4 and S5). Furthermore, we confirm that the SC gap closing is robust against the control of gR as long as the AMSs are formed (see supplementary note 6 and Fig. S6).
These results reveal that the AMSs are formed when both JJL and JJR exist and are coherently coupled with each other even if their carrier densities are different. In addition, the SC gap closing is robust against the control of the carrier densities of JJL and JJR. On the other hand, the numerical calculation does not always produce the SC gap closing in the spectrum. This difference may be related to a hidden physical mechanism in the coherently coupled JJs or the finite energy resolution of the tunnel spectroscopy of our setup. Further studies are demanded. We note that similar spectroscopic results of a three-terminal JJ on an InAs quantum well have been reported on arXiv [50] while preparing this manuscript.
As a summary, we perform the tunnel spectroscopy of the electrically controllable planar JJs embedded in the two SC loops. We find that the Andreev spectrum of single JJs indicates the periodic ABS oscillation as expected in ballistic JJs while the coupled JJs exhibit the strongly modulated structures holding the SC gap closing. The modulated Andreev spectrum is discovered in both of the two JJs and reproduced by the theoretical calculation. From these results, it is concluded that the AMSs in the coupled planar JJs are detected. Furthermore, it is discovered that the SC gap closing resembles the theoretical prediction in multi-terminal JJs. This study will contribute to elucidating the microscopic mechanisms of the AMS formation and developing quantum control and integration of the Andreev qubits. Additionally, our results suggest that the coupled planar JJs will provide a new platform to engineer Majorana zero modes and Weyl singularities realized by the multiple phase differences.
Method
Sample growth
The wafer structure has been grown on a semi-insulating InP substrate by molecular beam epitaxy.
Device Fabrication
In the fabrication process, we have performed wet etching of the unnecessary epitaxial Aluminum with a type-D aluminum etchant to form JJs and SC loops. Then, we have grown a 30 nm-thick aluminum oxide layer through atomic layer deposition and fabricated a separate gate electrode on each junction with 5 nm-Ti and 20 nm-Au. Fig. 1 : A device for tunnel spectroscopy of the coupled JJs (a) A scanning electron microscopic image of the coupled planar JJ device. The blue and yellow regions represent the SC electrodes and the gate electrodes, respectively. Two JJs named JJL and JJR are coupled through a shared SC electrode and each is embedded in an SC loop. (b) A schematic image of the device. Tunnel currents through QPCL and QPCR are measured, and from the result, the differential conductance is calculated for each to implement the tunnel spectroscopy of JJL and JJR. The AMS features appear for gL ≥ -2 V but disappear for gL < -2 V. This suggests that the JJL is pinched off at gL ∼ -2 V. (b) R as a function of V and gL at gR = 0 V and B= -0.03 mT. B= -0.03 mT corresponds to the SC gap closing point. Then the result indicates no gap structure for gL ≥ -2 V while the SC gap grows as gL becomes smaller than -2 V. This indicates that the SC gap closing is robust as long as the AMS is formed. (c) R as a function of B and gL at V=0 mV. The two vertical bright lines represent B producing the SC gap closing. The lines disappear around gL ∼ -2 V, supporting that the SC gap closing is robust as long as the AMS is formed.
Figures
To evaluate the minimum SC gap energies in the coherent coupling case, we exhibit the line profiles at = 0.230, 0.273, and 0.295 mT, in Figs.2(a)-(d) in red, green, and blue in Figs. 2(e)-(h), respectively. Note that the SC gap changes from the maximum to the minimum in this B field range in the single JJ, while from the maximum to the minimum and then to the maximum in the coupled JJs. Figs. 2(e) and (f) indicate the line profiles of the single JJL and JJR results in Figs. 2(a) and (b), respectively. As seen in Figs. 2(e) and (f), the Andreev spectrum is always gapped. This holds at any B field. On the other hand, in Figs. 2(g) and (h), the green line profile does not touch the = 0 line. Therefore, the SC gap looks closed in our setup resolution (~0.01 mV). Note that the same result is obtained for each SC gap minimum. To reveal that the modulated Andreev spectrum reflects the AMS properties owing to the coherent coupling through the shared SC electrode, we perform numerical calculations on the tight-binding model (see supplementary note 7 and Fig. S7). Figures 3 (a) and (b) indicate the numerical calculation results without and with Rashba SOI, respectively. The numerical calculation results explain the properties of the experimentally observed spectrum. For example, the SC gap becomes minimal between 0 < L = R < π and then local maximal at L = R = 0, π. This behavior is caused by a coherent coupling between the ABSs in two JJs, which means the formation of the AMSs. Furthermore, Fig. 3 (c) is an enlarged view around the minimal SC gap in Fig. 3(b), indicating the SC gap is closed in the AMS spectrum owing to the SOI. Figure 3 (d) exhibits the energy of the positive lowest Andreev state as a function of L and R . At the mean free path being 217 nm roughly consistent with that in our InAs quantum well, we obtain the gap closing in the AMS spectrum in 19% of samples in the simulation (see supplementary note 8 and Fig. S8).
Fig. 2 :
2Tunnel spectroscopy results of the single and coupled JJs (a) Tunnel spectroscopic result of the single JJL as a function of B. The SC gap oscillates to B, assigned to the feature of ABS expected in short JJs. (b) Tunnel spectroscopic result of the single JJR. Almost the same features as in the JJL result are found. The periodicity of the ABS oscillation is the same as that in (a) because the two loops hold the same area. The white dashed lines indicate the B points giving the minimal SC gap. (c) Tunnel spectroscopic result of the JJL coupled with JJR. The result is drastically modulated from the single JJ case in (a). Especially, the SC gap becomes minimal away from L = R = π where the gap becomes minimal in the single JJ cases. (d) Tunnel spectroscopic result of the JJR coupled with JJL. The same features observed in (c) were acquired. The consistency between (c) and (d) assures that the AMSs are constructed due to the coherent coupling of the two JJs. (e) L vs. V at B=0.230, 0.273, and 0.295 mT, respectively measured for the single JJL. (f) R vs. V measured for the single JJR. (g) L vs. V measured for the coupled JJL. The SC gap is closed at B= 0.273 mT while the other two curves hold the SC gap. (h) R vs. V measured for the coupled JJL. As with the coupled JJL case, the SC gap is closed at B= 0.273 mT while the other two curves hold the SC gap.
Fig. 3 :Fig. 4 :
34The numerical calculation results indicating the AMS formation and SC gap closing AMS spectrum along L = R without SOI (a) and with SOI (b). Owing to the SOI, the gap closing is observed. (c) Enlarged view of the gap closing in (b). (d) The energy of the positive lowest Andreev state ( 1 /Δ) as a function of L and R . The white line indicates the condition of L = R corresponding to the experimental situation. Nonlocal gate control of the AMSs detected in the coupled JJR (a) R as a function of B and V at several gL .
GaAs are introduced to help passivate the wafer surface where the Al film is removed and to make the sample more compatible with the Al etchant, which does not attack GaAs. The two-dimensional electron gas (2DEG) is accumulated in the InAs quantum well.The stack materials from the bottom to top are a 100 nm In0.52Al0.48As buffer, a 5 period 2.5 nm
In0.53Ga0.47As/2.5 nm In0.52Al0.48As superlattice, a 1μm thick metamorphic graded buffer stepped from
In0.52Al0.48As to In0.84Al0.16As, a 33 nm graded In0.84Al0.16As to In0.81Al0.19As layer, a 25 nm
In0.81Al0.19As layer, a 4 nm In0.81Ga0.19As lower barrier, a 5 nm InAs quantum well, a 10 nm
In0.81Ga0.19As top barrier, two monolayers of GaAs and finally an 8.7 nm layer of epitaxial Al. The top
Al layer has been grown in the same chamber without breaking the vacuum. The two monolayers of
Competing interest statementThe authors declare no competing interests.Supplementary InformationSupplementary notes 1-8 and supplementary figures S1-S8
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| [] |
[
"Rate dependence of damage formation in metallic-intermetallic Mg-Al-Ca composites",
"Rate dependence of damage formation in metallic-intermetallic Mg-Al-Ca composites"
] | [
"Setareh Medghalchi \nInstitute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany\n",
"Muhammad Zubair \nInstitute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany\n\nDepartment of Metallurgical & Materials Engineering\nFaculty of Chemical\nMetallurgical & Polymer Engineering\nUniversity of Engineering & Technology (UET) Lahore\nPakistan\n",
"Ehsan Karimi \nInstitute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany\n",
"Stefanie Sandlöbes-Haut \nInstitute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany\n",
"Ulrich Kerzel \nData Science and Artificial Intelligence in Materials and Geoscience\nFakultät für Georessourcen und Materialtechnik\nRWTH Aachen University\nAachenGermany\n",
"Sandra Korte-Kerzel \nInstitute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany\n"
] | [
"Institute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany",
"Institute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany",
"Department of Metallurgical & Materials Engineering\nFaculty of Chemical\nMetallurgical & Polymer Engineering\nUniversity of Engineering & Technology (UET) Lahore\nPakistan",
"Institute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany",
"Institute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany",
"Data Science and Artificial Intelligence in Materials and Geoscience\nFakultät für Georessourcen und Materialtechnik\nRWTH Aachen University\nAachenGermany",
"Institute for Physical Metallurgy and Materials Physics\nRWTH Aachen University\nKopernikusstr. 1452074AachenGermany"
] | [] | We study a cast Mg-4.65Al-2.82Ca alloy with a microstructure containing α-Mg matrix reinforced with a C36 Laves phase skeleton. Such ternary alloys are targeted for elevated temperature applications in automotive engines since they possess excellent creep properties. However, in application, the alloy may be subjected to a wide range of strain rates and in material development, accelerated testing is often of essence. It is therefore crucial to understand the effect of such rate variations. Here, we focus on their impact on damage formation. Due to the locally highly variable skeleton forming the reinforcement in this alloy, we employ an analysis based on high resolution panoramic imaging by scanning electron microscopy coupled with automated damage analysis by deep learning-based object detection and classification convolutional neural network algorithm (YOLOV5). We find that with decreasing strain rate the dominant damage mechanism for a given strain level changes: at a strain rate of 5•10 -4 /s the evolution of microcracks in the C36 Laves phase governs damage formation. However , when the strain rate is decreased to 5•10 -6 /s, interface decohesion at the α-Mg/Laves phase interfaces becomes equally important. We also observe a change in crack orientation indicating an increasing influence of plastic co-deformation of the α-Mg matrix and Laves phase. We attribute this transition in leading damage mechanism to thermally activated processes at the interface. | null | [
"https://export.arxiv.org/pdf/2303.10477v1.pdf"
] | 257,632,467 | 2303.10477 | 33b8a556d38133e8639c64ea9c1ba877a2aea0ff |
Rate dependence of damage formation in metallic-intermetallic Mg-Al-Ca composites
Setareh Medghalchi
Institute for Physical Metallurgy and Materials Physics
RWTH Aachen University
Kopernikusstr. 1452074AachenGermany
Muhammad Zubair
Institute for Physical Metallurgy and Materials Physics
RWTH Aachen University
Kopernikusstr. 1452074AachenGermany
Department of Metallurgical & Materials Engineering
Faculty of Chemical
Metallurgical & Polymer Engineering
University of Engineering & Technology (UET) Lahore
Pakistan
Ehsan Karimi
Institute for Physical Metallurgy and Materials Physics
RWTH Aachen University
Kopernikusstr. 1452074AachenGermany
Stefanie Sandlöbes-Haut
Institute for Physical Metallurgy and Materials Physics
RWTH Aachen University
Kopernikusstr. 1452074AachenGermany
Ulrich Kerzel
Data Science and Artificial Intelligence in Materials and Geoscience
Fakultät für Georessourcen und Materialtechnik
RWTH Aachen University
AachenGermany
Sandra Korte-Kerzel
Institute for Physical Metallurgy and Materials Physics
RWTH Aachen University
Kopernikusstr. 1452074AachenGermany
Rate dependence of damage formation in metallic-intermetallic Mg-Al-Ca composites
*Corresponding authorsMg alloySEMmicrocracksdamagethermal activationconvolutional neural networkdeep learning
We study a cast Mg-4.65Al-2.82Ca alloy with a microstructure containing α-Mg matrix reinforced with a C36 Laves phase skeleton. Such ternary alloys are targeted for elevated temperature applications in automotive engines since they possess excellent creep properties. However, in application, the alloy may be subjected to a wide range of strain rates and in material development, accelerated testing is often of essence. It is therefore crucial to understand the effect of such rate variations. Here, we focus on their impact on damage formation. Due to the locally highly variable skeleton forming the reinforcement in this alloy, we employ an analysis based on high resolution panoramic imaging by scanning electron microscopy coupled with automated damage analysis by deep learning-based object detection and classification convolutional neural network algorithm (YOLOV5). We find that with decreasing strain rate the dominant damage mechanism for a given strain level changes: at a strain rate of 5•10 -4 /s the evolution of microcracks in the C36 Laves phase governs damage formation. However , when the strain rate is decreased to 5•10 -6 /s, interface decohesion at the α-Mg/Laves phase interfaces becomes equally important. We also observe a change in crack orientation indicating an increasing influence of plastic co-deformation of the α-Mg matrix and Laves phase. We attribute this transition in leading damage mechanism to thermally activated processes at the interface.
Introduction
Many high-performance alloys consist of two or more phases to combine and improve their respective properties in a microstructural composite. In lightweight magnesium alloys, cast microstructure with an intermetallic skeleton have been shown to lead to superior creep resistance, particularly at elevated temperature [1][2][3]. On the other hand, the presence of intermetallics naturally leads to decreased tensile elongation as the present Mg17Al12 and Ca(Mg,Al)2 Laves phases are hard and brittle, particularly below approximately two thirds of their melting temperature [4][5][6][7][8][9][10][11]. As both low and elevated temperature regimes, that is room temperature and temperature around and above 150 °C, are of interest in application, the underlying deformation and damage mechanisms and their thermal activation need to be understood. In particular, the competition between brittle fracture and plasticity, enabled by dislocation glide or diffusional processes, may be expected to govern any macroscopic transitions in behaviour with temperature. In this work, we address this by investigating the signatures of the dominant co-deformation mechanisms as a function of strain and rate in a Mg-Ca(Mg,Al)2 metallicintermetallic alloy by means of micromechanical testing and damage classification using artificial intelligence.
Laves phases with the general formula Ca(Mg,Al)2 precipitate in the Mg-Al-Ca ternary alloy system once the Al and Ca alloying content exceed the solubility limit [12]. Laves phases have an AB2 stoichiometry and are known for their high temperature strength, which is always accompanied with extreme brittleness at low temperatures [13][14][15]. However, the Laves phases usually show plasticity at high temperatures (above the ductile-brittle transition temperature) or at small scales [5,6,9,13,[15][16][17][18]. In conventional, cast Mg-Al-Ca alloys, they are usually present as thin interconnected struts with a thickness of the order of one to a few micrometre [19][20][21]. Their presence as interconnected reinforcement imparts good creep properties [1-3, 19, 22, 23], thermal stability [24], and adequate strength [25] to Mg-Al-Ca alloys. These alloys are thus intended for elevated temperature structural applications like automotive powertrains [26].
In Mg-Al-Ca alloys, the Mg and Laves phases have significantly different mechanical properties, which results in heterogenous deformation [8,27,28]. Under tensile loading or during indentation, cracks form in the Laves phase or at α-Mg/Laves phase interfaces [3,8,20,27,29,30]. In addition to cracking, plastic deformation has also been observed in the Laves phases surrounded by α-Mg matrix [8,16,28,31]. The co-deformation behaviour of α-Mg and Laves phases can be substantially affected by the strain rate, in addition to other factors like orientation relationships and temperature of deformation [28].
Strain rate and temperature are known to considerably affect the deformation and fracture behaviour of materials [32]. The flow stress or hardness decreases with decreasing strain rate and vice versa owing to thermally activated plastic deformation mechanisms and changing balance in their competition with fracture as temperature increases.
In this work, we investigate the effect of strain rate on the elevated temperature (≈170 °C) deformation behaviour of an Mg-4.65Al-2.82Ca alloy. As a consequence of deformation at different strain rates, any changes in prevalent deformation mechanism will lead to different amounts and characteristics of damage sites that are introduced in the microstructure during deformation. The Mg-Al-Ca metalintermetallic composites possess several critical microstructural length scales from the thickness of the intermetallic struts (~1 µm) over the strut spacing in the skeleton (10s of µm) to the grain size (100s of µm). These naturally expand into all three dimensions in this cast alloy, but three-dimensional characterisation is normally limited in the sense that interrogated volume and voxel size and therefore resolution scale. We therefore choose cross-sectional analysis after deformation by scanning electron microcopy (SEM), which is able to cover a large area of the order of 1 mm² at sufficiently high resolution to encompass all important length scales for this alloy. By using deep learning to assist in the analysis of the many damage sites induced across a large cross-sectional area after deformation, we consider the formed damage statistically in terms of its major characteristics or classes of damage observed.
Deep learning methods based on convolutional neural networks serve as a tool to learn spatial hierarchies of information about the content of image automatically and adoptively through backpropagation [33]. Here, we use two image analysis tasks: feature detection, which is the task of localising all instances of a specific class in an image [34], and classification of the objects in an image [35], which is the task of assigning features of an image to specific classes [36]. In the case of this work, damage sites in panoramic micrographs are detected and classified with the objective of damage analysis. In this context, deformation induced damage sites can be detected and subsequently classified with respect to their appearance, as shown by Kusche et. al and Medghalchi et. al. for damage sites in dual phase steel [37,38].
In this work, we explore the prevalent mechanisms of co-deformation and their dependence on strain and rate in a Mg-Ca(Mg,Al)2 metallic-intermetallic composite microstructure. To this end, we use micromechanical testing and scanning electron microscopy coupled with automated image analysis and damage classification to identify and quantify the dominant damage mechanisms of brittle failure in the intermetallic and interfacial decohesion at the internal interfaces. These insights are essential to guide future material design strategies dedicated to achieving a damage tolerant microstructure with tailored strength, creep resistance and elongation to failure for a given application.
Experimental Methods
Sample synthesis, deformation, and imaging
A protective atmosphere of argon was used to melt the raw materials in a steel crucible using a vacuum induction melting system. Mg-4.65Al-2.82Ca (wt.% alloy) was then solidified in a copper mould as described in [39].
Dog bone shaped specimens with a 10 mm gauge length for tensile deformation were cut by spark erosion. The grinding and polishing procedure for the samples was the same as discussed comprehensively in [12]. Samples were deformed to 3, 5 and 7% global strain at a temperature of 170 °C and strain rate of 5 x 10 -4 /s. One sample was also deformed at 5 x 10 -6 /s to the intermediate strain of 5% to study the effect of strain rate on damage formation.
Post deformation microscopic analysis was done using secondary electron (SE), back-scattered electron (BSE) panoramic imaging, and electron back-scatter diffraction (EBSD) in scanning electron microscopes (SEM, FEI Helios 600i and Zeiss LEO1530). Panoramic imaging was done to cover maximum microstructural region without losing resolution or image quality. For example, the microstructural region presented in Figure 9 (a) is comprised of 255 individual images, each with 50 µm (1024 px) of horizontal field width. An acceleration voltage ranging from 8-10 kV was used for this purpose. Images were stitched using image composite editor [40]. EBSD was performed at an accelerating voltage of 20kV.
Automatic damage analysis method
Object detection and classification model
Yolo (You Only Look Once) as one of the popular object detection and classification algorithms has gained lots of attention in a wide range of applications such as object detection and image segmentation [41]. In contrast to two-stage detector methods, in which region proposal and classification tasks are sequentially performed, Yolov5 is a one-stage detector methods, in which region proposal and classification are solved simultaneously [42].This algorithm is relatively easier to implement and can be trained on the entire image without the need to divide the image or filter the objects of interest [43].
Its fifth version (YoloV5), which is the most recent one at the time of doing this research, has overperformed other variants such as fast R-CNN in terms of speed and accuracy. Here, we use YoloV5s, which has one of the fastest learning rates among all other options with 37.2 validation mAP (mean average precision) @ [0.5:0.95] on COCO API dataset [41,44] [45].
We use Yolov5s to detect and classify the damage sites in the microstructure of the Mg-Al-Ca alloy across panoramic SEM images. For further damage analysis, we used the Hough transformation to extract the inclination of cracks in the Laves phase.
Definition of damage classes
The deformed microstructure is composed of Mg matrix with the Ca(Mg,Al)2 Laves phase, and also contains deformation features such as slip lines, twins and cracks or interfacial decohesion sites. The cracks evolve in the Laves phase, while interface decohesion occurs at the α-Mg/Laves phase interfaces and pores of different shape also form at inclusions. Thus, depending on location and shape of the damage sites, we distinguish three different types in the microstructure (
Data preparation
As the starting step, a dataset of 452 SEM images collected by both high-resolution scanning electron microscopes with varying sizes of 1024 x 768 and 3072 x 2048 pixels and varying resolutions of 16 and 48 nm/pixel were introduced to the LabelIing software [46]. The three damage site classes, Laves phase cracks, interface decohesion and inclusion, were annotated using the bounding box defining tool by drawing a rectangular box around the damage site and specifying its type. The annotation was performed by a single person after the appearance of each class was defined and agreed between several scientists. As the output, an annotated label file with .xml format is generated and assigned for each SEM image containing the information with respect to coordinates and size of the bounding boxes of the damage sites in the annotated images.
Subsequently, the annotated images together with the respective labels were uploaded to the Roboflow [47] website where the annotation files were converted to YoloV5 format. Thereafter, the data were Data augmentation serves as a tool to increase the network robustness against different variations and in turn, enhances the robustness of network to classify the objects (here damage sites) obtained from different environments [38] , [48]. The common augmentation method used in the context of YoloV5 is mostly Mosaic data augmentation and colour space augmentation [48]. In Mosaic data augmentation, four training images are mixed. In colour space augmentation, the hue and saturation of the images are varied. In addition, other geometrical augmentation methods like flipping, rotation, translation, scaling which exist in the default settings of the method and do not change the definition of the three classes.
Network Training
The training of the network was initiated from scratch, particularly without utilizing pre-existing weights that had been previously trained on an alternate dataset such as COCO [45] (a method known as transfer learning [49]). The rationale behind this decision relates to the higher evaluation metrics, particularly mAPs, which is a comprehensive single indicator summarizing precision, recall and intersection over the union (IoU), when compared to transfer learning methods implemented on the same dataset. This approach has been adopted to ensure a more rigorous and accurate evaluation of the network's performance.
In the YoloV5, the optimal weight is automatically saved based on the best fitness function. We utilised the standard saved weight of the network to perform prediction on our data. By using the best saved weight, we were able to detect and classify damage sites across the previously unseen SEM images, and extract the relevant quantities of each type. In order to assess the performance of the trained network, a number of metrics were employed and are described in detail below.
Precision is calculated as the ratio between the number of Positive objects in the image (e.g Laves crack here) correctly classified, to the total number of Positive objects (either correctly or incorrectly classified). Equation 1 provides the corresponding formula for calculating the precision [50]. The evaluation of object detection and classification algorithms relies fundamentally on both precision and recall metrics. These metrics are interdependent and exhibit a trade-off relationship. Increasing recall, for example, by generating a greater number of predictions, would result in a corresponding decrease in precision, which refers to the accuracy of prediction. To effectively balance these two metrics, it is necessary to aggregate them into a single metric, which is described in the following. where B is the area of the predicted bounding box and B GT is the area of ground truth bounding box [52]. where n is the number of the classes, and APk is the average precision of each class [53].
Evaluation of crack inclination
To study the dependence of the inclination of the formed Laves phase cracks relative to the loading direction, we designed a method based on the application of the Hough transformation. In computer vision, the Hough transform is a widely recognized technique used for the detection of object instances with specific class of shape like lines, circles, or ellipses inside the image [54]. The detection process is conducted through convolutional operations to extract the object boundaries in the image. The collected information can then be used to accumulate Hough votes to finally find the instances of the specific shapes in the image [55].
Our procedure is illustrated in Figure 4. First, the damage sites classified as Laves phase cracks were isolated and cropped to a size of 100 x 100 pixels around the centroid of the object to encompass a single damage site as well as some surrounding area. Sites where more than one crack were contained within the cropped window were omitted. For precise recognition of the crack boundaries, its edges were slightly thickened by means of erosion morphology correction [56] (function parameters: 3*3
kernel size under 1 iteration). Then the edges of the crack were detected by Canny edge detection [57]. connected to the "sure edge" or "non edge" [58] . In the Hough transformation ρ is the distance of each line from the coordination origin and vote is the value of the accumulator [58]. The higher vote here, the higher is the weight which conveys information about the strength of the supporting evidence of the lines [59]. All the packages were implemented using OpenCV library [58]. Finally, the inclination angles of the Laves cracks were calculated in terms of the angle of the isolated parallel lines of the crack edges with respect to the horizontal, along which the tensile stress was applied.
Technical Configuration
All computation including training and validation of the network, calculation of numerical results, and detection and classification of damage sites in new images were performed on central high performance computing facilities with a GPU node providing "Nvidia Tesla V100" and 16 GB of memory and 1TB
storage. Using this system configuration, training YOLOv5 took around 22 hours. Laves phase crack detection and angle calculation were performed on a workstation with an AMD FX (tm)-8350 eightcore processor and 2 TB of memory.
Results
Microstructural features of the deformed Mg-4.65Al-2.82Ca alloy
The deformed microstructure of the Mg-465Al-2.82Ca alloy is presented in Figure 5and EDS presented in previous work [39]. The brighter phase is the C36 Ca(Mg,Al)2 Laves phase. A detailed characterisation of this phase and the Mg-C36 interface has already been presented in reference [39].
Deformation in the α-Mg matrix is dominated by basal slip ( Figure 5) and extension twinning ( Figure 6). Small cracks in the Laves phase tend to nucleate at the points where basal slip lines in α-Mg phase intersect the Laves phase (depicted by red arrows in Figure 5 b). In some instances, slip is also Examples of the formation of extension twins in the α-Mg phase are given in Figure 6. The three variants of extension twins within one α-Mg grain are highlighted in Figure 6 a. There is more cracking in Laves phases in α-Mg grains containing significant deformation features, such as slip lines and deformation twins ( Figure 6 a-b). On the other hand, grains which do not show these deformation features appear to contain fewer cracks in the Laves phase. (Figure 6 c). The cracks highlighted by red arrows (Figure 6 This shows that the type of damage initiated in the Mg-4.65Al-2.82Ca alloy is rate dependent.
By this selective and manual analysis, we therefore find that the type of damage and inclination of Laves phase cracks depends on the local microstructural and deformation conditions. To investigate this further, a statistical analysis is needed, which is provided in the following using a combination of artificial intelligence and classical image analysis.
Figure 6: SE images of 5% deformed Mg-4.65Al-2.82Ca sample at a strain rate of 5•10 -4 /s, (a) three variants of extension twins visible in same α-Mg grain, (b) slip lines in α-Mg phase, c) α-Mg grain bounded by blue lines shows no evidence of slip or twinning, (d-e) cracks in the Laves phase as depicted by red arrows and slip lines by blue arrows. The orientation of cracks in C36 phase is same as that of slip traces in C36
phase. (f) microstructure of sample deformed to 5% strain at a strain rate of 5•10 -6 /s
Damage analysis using deep learning
Network training
The history of network training in terms of precision, recall and mean average precision at two IoU thresholds of 0.5 and 0.5 to 0.95 are depicted in the graphs in Figure 7. The highest mean average precision (mAP) at the intersection over union in the range of 0:5 to 0.95 reaches 70%.
The batch size and number of epochs were 4 and 2500 respectively. Batch sizes of 2 and 8 were also explored, however, the smaller batch size slowed down the training process and the higher one resulted in a reduction of the generalizability of the model and also increased the memory usage. The number of the epochs was fixed to 2500 as a compromise between increasing trend of network learning potential and overfitting. predominantly cracks running through the Laves phase struts (see Figure 8). The frequency of red squares and arrows (depicting cracking in Laves phase) is much higher than the green boxes and arrows (enclosing cracks at the α-Mg/Laves phase interfaces).
Figure 8. (a) and (b): BSE panorama from a sample surface deformed upto 5 % global strain at a strain rate of 5•10 -4 /s and temperature of 170 °C. Magnified images of the microstructural regions enclosed by the blue, orange and yellow rectangles are presented in (b), (c), and (d), respectively. Green boxes highlight interface decohesion site at α-Mg/Laves phase interfaces, red corresponds to cracks in the Laves phase, while blue boxes represent the inherent defects in the material visible on sample surface, such as inclusions and other pores.
Much more interface decohesion, as highlighted by green squares and arrows, was observed in samples deformed at a strain rate of 5•10 -6 /s (compare Figure 8 and Figure 9). Although Laves phase cracking is also visible in the sample deformed at the lower strain rate of 5•10 -6 /s, the ratio between Laves phase cracking and interface decohesion is lower compared to the sample deformed at higher strain rate. The quantitative data for all experiments is presented in Figure 10. As expected, the total number of damage sites per area for a given strain rate varied directly with strain
Discussion
Microstructural and mechanical heterogeneity in the Mg-Al-Ca alloy
During straining, deformation therefore usually initiates in the α-Mg phase [8,28] and then it either extends into the Laves phase (Figure 5c and Figure 6d,e) or creates enough local stress to initiate cracks in the Laves phase (Figure 5b and Figure 6). This observation is consistent with previous experimental and computational work [8,28] and result from the differences in mechanical properties and crystal structure between α-Mg and Laves phases [27]. In its as-cast form, the Mg-4.65Al-2.82Ca alloy has a microstructure consisting of α-Mg phase and C36 Laves phase. The α-Mg phase in Mg-Al-Ca alloys is a solid solution primarily of Al in Mg [3,8,12,27,28,60]. This phase predominantly deforms via basal slip and deformation twinning [8,12,27,28]. In pure Mg, these two deformation mechanisms have the lowest critical resolved shear stresses (CRSS) of < 1 MPa and < 10 MPa respectively at the macroscopic scale [61][62][63][64]. In contrast, the CRSS for prismatic and pyramidal slip are of the order of 40 MPa [65][66][67]. Laves phases on the other hand demonstrate much higher hardness and strength but extreme brittleness at low temperatures [13][14][15]. However, these phases show plasticity at small scales even at room temperature, for example, in micropillar compression [6,16,68,69]. The CRSS values for basal, prismatic, and pyramidal slip values in the C14 (CaMg 2, structurally quite close to the C36 Laves phase) phase were found to be of the order of 0.5 GPa [6]. These values decrease with temperature but they remain well above the CRSS of Mg even at high temperature of the order of 250 °C [5]. Similarly, the CRSS values observed for {111}(11 � 0) of C15 (Ca Al2) Laves phase was reported to be nearly ten times to that observed in α-Mg phase [16].
In addition to this generally heterogeneous deformation resulting from the mechanical contrast, we also observed that cracking in the Laves phase is more concentrated in those α-Mg grains that exhibited a higher amount of basal slip and extension twinning (see panoramic images in Figure 1 and Figure 2). This is again in agreement with earlier work on a similar alloy, where it was shown that cracks initiate in the Laves phase either at places where slip lines or twins in the α-Mg phase intersect with the Laves phase [27]. The behaviour of the alloy investigated here is therefore in good agreement with previous reports in the literature. However, the effect of strain and in particular rate has not been investigated explicitly to our knowledge, although a transition from cracking to plastic co-deformation has been predicted by atomistic simulations for an α-Mg-CaMg2 C14 Laves interface [28].
Quantitative damage analysis by deep learning
We used Yolo5s as a single-stage target recognition algorithm. This method proved simple and more effective compared to other deep learning object detection and classification methods like R-CNN, which is more complex and computationally intensive. Owing to its single-shot approach, it is well suited for the detection of small objects, like fine damage sites in our case [44].
The classification performance of the trained network is visualised in Figure 12 for two randomly selected unseen sets of image data using confusion matrices in which actual damage type (from manually labelled ground truth data) and predicted damage type (by the network) are compared. The relatively high scores show that the classification is quite reliable, in particular for the classification of Laves cracking, while the classification of interface decohesion and inclusion sites is more variable.
This may be improved by providing more labelled training data, further image augmentation or by including image artefacts or further damage types as additional classification options [38].
Strain dependence of damage
The number of deformation induced damage sites formed in the Mg-4.65Al-2.82 Ca alloy increased with strain (Figure 10 a). This is as expected as higher strains will result in more dislocation pileups at α-Mg/Laves phase interfaces and thus more interface decohesion and cracking at and in the intermetallic phase [8]. In polycrystalline Ti-6Al with low interstitial content, Huang et al. [70] have shown that the geometric incompatibility and lack of slip transfer across grain boundaries are mainly responsible for nucleation of microcracks at these boundaries. Significant stress concentrations at grain boundaries can alternatively be released by slip transfer across boundaries, resulting in more homogenous plasticity in the sample [70,71]. However, in the material investigated here, the dominant interface is a phase boundary rather than grain boundary and the crystal structure of the C36 Laves phase [72] is very different from that of α-Mg phase in spite of both possessing a hexagonal unit cell. Moreover, a dominant orientation relationship found for such alloys places the basal plane of the C36 Laves phase at an approximately perpendicular angle to the basal plane of the α-Mg phase [8,28]. The different crystal structure and lattice parameters [72] of C36 Laves phase, together with a much higher CRSS [5,6] for basal slip, as compared to the Mg phase, and unfavourable orientation relationship [8,28] with
Mg therefore severely restricts basal to basal slip transfer into the C36 phase. Slip could possibly transfer when non basal slip is initiated in regions close to α-Mg/C36 Laves phase interfaces as a result of the high and multiaxial stresses induced by a basal dislocation pileup [8].
Consistent alignment of slip traces indicating slip transfer at high angles between the slip planes has indeed been found here (blue arrows in Figure 5 and Figure 6). However, cracks are observed much more commonly, and we therefore could not yet investigate the conditions for slip transfer statistically using the employed neural networks. There may also be a transition from slip in the C36 phase to crack opening along the slip plane, as has been observed in micropillars of similarly complex intermetallic phases [73]. We consider this in more detail below as part of the rate dependence of damage formation, as decohesion of slip planes within the Laves phase would be expected to depend on the extent of slip transmission at the interface and dislocation motion in the intermetallic, both of which are expected to be thermally activated and therefore rate dependent.
Rate dependence of damage
The variation of strain rate resulted in a more varied effect on the formation of damage sites. The total number of sites per area for the same strain of 5% reduced to nearly half as the strain rate was decreased by two orders of magnitude, from 5•10 -4 /s to 5•10 -6 /s (Figure 10 b). Moreover, the type of damage changed from predominantly Laves phase cracking to a combination of interface decohesion and Laves phase cracking Figure 11. This change in dominant damage mechanism is likely due to thermally activated phenomena at the interfaces.
In Figure 13, the angle of the Laves phase cracks relative to the loading direction is shown for both strain rates. The angles were calculated for all sites detected and classified by the neural network with sufficient crack spacing to apply the Hogh transformation as described above and benefits from the reliable classification of the Laves phase cracks( Figure 12). The results indicate that there are in fact likely three aspects to consider: in addition to (1) brittle cracking of the Laves phase driven by normal stresses and (2) interface decohesion, (3) Laves phase fracture following slip in the Laves phase or at least introduction of a critical dislocation density near the interface in the Laves phase may occur. Only the latter may be expected to lead to the change in dominant angle away from the perpendicular orientation at 90°, as found for the higher strain rate ( Figure 13). In contrast, at the lower rate, the distribution does not show a clear trend and if the data are interpreted to contain a maximum, then it is at a much lower angle. Additional studies on the errors associated with angle distribution measurements relating to, for example, orientation relationships and texture, are needed to investigate whether a preferential angle does exist and , if so, how it is determined. Purely brittle failure should be independent of thermal activation in the absence of thermally activated mechanisms promoting critical defect formation or pronounced changes in the crystal's stiffness and decohesion energy. We therefore consider here the formation of perpendicular cracks as not thermally activated as a first estimate and discuss in the following how thermal activation may affect the formation of both interface decohesion and non-perpendicular cracks.
Cracks at angles significantly away from 90° to the loading axis are likely to result in large number not simply from local variations of the macroscopic stress field due to the skeleton, but mainly from the formation of stress concentrations, crack nuclei and crystal planes with low decohesion energy in the Laves phase. The first will result directly from the intersection of slip bands and twins in the metallic matric with the intermetallic, whereas the deformation induced initiation of crack nuclei and dislocations in the Laves phase that may lower decohesion energy, will form subsequently at stress concentrations. In combined experimental and computational work on a comparable system, using a similar alloy in micromechanical testing and TEM experiments and a Mg/CaMg 2 composite in correlated atomistic simulations, evidence of plastic co-deformation between Mg and Laves phase, interface sliding and fracture has also been observed, consistent with our experimental findings, as shown for example in Figure 5. In case of slip transfer for plastic co-deformation, the active planes are very limited in both phases with <a> dislocations moving on the basal plane of Mg and the basal and prismatic planes of the Laves phase. The dominant orientation relationships identified for other Mg-Al-Ca alloys are based on parallel [22] and near perpendicular basal plane orientations [28,74], with the latter observed in alloys with very similar processing and microstructure to those investigated here. This implies a strong change in slip plane orientation where co-deformation takes place, which is thought to be enabled by the formation of non-basal dislocations in the Mg matrix as a result of the stress field of the pile-up of basal dislocations at the interface [8]. A likely case of this kind of plastic co-deformation is pictured in Figure 5c. Plastic deformation in the Laves phase is strongly localised with large strain accommodated on individual parallel slip planes. This is typical and in the structurally related µ-phase has been shown to coincide with decohesion of planes on which plasticity led to a high dislocation density [5,6,68,73,75]. The latter gives a route from plastic co-deformation to the observation of cracks along slip planes in the Laves phase. Atomistic simulations considering Mg/Laves interfaces suggest that the slip transfer may be thermally activated [8,28] and therefore expected to become more frequent as the temperature is increased or the rate is lowered. Similarly, the motion of dislocations in the Laves phase after slip transfer or nucleation from a stress concentration may be thermally activated, although the extent of this thermal activation at up to 170 °C may be very limited, as the hardness of the related CaMg 2 C14 Laves phase has been found to be constant between room temperature and 250 °C [5] or drop only a little up to 60% of its melting temperature (at 320 °C) [9]. Only limited data exist on individual slip systems from microcompression at elevated temperature [5].
Possible mechanisms that give rise to damage classified here as interface decohesion have also been considered previously in experiments and computational studies [8,28]. In the case of interfacial sliding, it was shown in atomistic simulations that dislocations that pile up against the intermetallic are absorbed into the interface [28]. In nanomechanical testing, a clear signature of thermally activated deformation was observed, however, as both sliding and slip transfer were found to occur undeath indentations [8], a direct correlation with either mechanism was not possible by indentation.
The approach presented here now adds valuable additional insights. It allows us to study the prevalence of the individual mechanisms and how their competition is affected by applied strain and strain rate. In particular, it has revealed a transition in active mechanism and, within the observed class of cracks in the Laves phases, important clues also towards changing mechanisms of plastic co-deformation and their impact on crack formation. These insights were found to be consistent with previous experimental and computational studies on very similar alloys and related strengthening Laves phases or idealised by directly relevant phase boundaries. In future experiments, additional parameters may now be considered, especially deformation temperature, the morphology of the intermetallic skeleton and the local orientation relationships. The intermetallic strut size, volume fraction and present Laves phase can be controlled by alloy composition, cooling rate after casting and subsequent annealing [74]. Concurrent orientation imaging by EBSD across large areas at sufficiently low kV to successfully index the small Laves phase volumes, will rely heavily on appropriate metallographic preparation and indexation method [12,76]. Digital image correlation and surface topography analysis for (quasi) in-situ experiments may additionally help to reveal the role of interfacial sliding both in and normal to the surface plane and it may be possible to integrate this data into a workflow with automated damage type analysis.
Conclusions
We investigated the changes in deformation induced damage density and type with strain and strain rate at 170 °C in a Mg-4.65Al-2.82Ca alloy consisting of an interconnected C36 Laves phase skeleton embedded in an α-Mg matrix. The main conclusions from this work are
• at higher rate (5•10 -6 /s), damage formation is dominated by cracking in the Laves phase (≈88% of all deformation induced damage sites),
• at lower rate (5•10 -4 /s), a transition towards interface decohesion (≈46% vs 54% Laves phase cracking) is observed
• the orientation of the formed cracks in the Laves phase changes from predominant cracking perpendicular to the loading axis at the higher rate to much more random crack inclination, which we associate with beginning plastic co-deformation across the interfaces.
• In the two phase microstructure spanning microstructural length scales from single to hundreds of µm, these insights were made possible based on high resolution data from an area greater than 4mm 2 . The use of deep learning for damage detection and classification as well as subsequent image analysis proved successful as well as essential in analysing this large image dataset.
Future work may build on these insights and methods to further unravel the role of thermal activation at interface boundaries and the role of plasticity in the reinforcing Laves phase, which may facilitate greater ductility of the alloy by plastic co-deformation on the one hand or lead to increased microscopic damage formation on the other.
Acknowledgement
The authors are grateful for the financial support received from the Deutsche Forschungsgemeinschaft
References
Figure 1 )
1: (1) Laves phase cracks are linear shaped groups of black pixels lying on the white Laves phases, (2) interface decohesion at the α-Mg/Laves phase interfaces are black islands lying at the boundary of the white Laves phase and the grey matrix of the Mg, and (3) inclusions are normally randomly shaped and sized with either black, grey, or white pixels lying in the matrix. The pores presumably formed during casting are also placed in the latter category. The Laves phase cracks and interface decohesion sites are of particular interest here as they are deformation induced and not intrinsically dependent on casting or melt conditions.
Figure 1 .
1Illustration of the three types of the damage sites considered in the Mg-Al-Ca microstructure.
randomly split into the training and test set with 80-20% proportion respectively (362 training and 90 validation). The size of the bounding box, containing the whole area of the damage site, varied depending on the size of the damage sites in the microstructure. Out of 1082 annotated damage sites in the images of the training dataset, 856, 452 and 143 were annotation as associated with Laves cracks, interface decohesion and inclusions, respectively. The distributions of the different classes depending on the position in the microstructure and more details about the data distribution in the sense of dimensions and coordinates of the bounding boxes are mapped in Figure 2. As can be perceived from the training data distribution maps, there is no specific concentration of the positions of the damage sites within the image, which in turn would introduce no bias to the localization of the damage sites in the model training process and indicates that the selected area is large enough to represent the locally changing microstructure and its variable damage distribution.
Figure 2 .
2a) The number of the annotation of each class, b) the relative position of the bounding box on an SEM image c) the normalized dimensions of the bounding boxes (the highly occupied coordinates and dimensions are denoted with different colour).
Recall is calculated as the ratio between the number of Positive objects in the image correctly classified, to the number of Positive objects correctly classified and Negative objects incorrectly classified. It is basically a measure of how well the model finds all the relevant cases in the images (
Figure 3
3Average precision (AP) is a parameter calculated by averaging all the precision values across various recall values under different thresholds. The precision is calculated for different classes and, depending on the number of the classes to be detected and classified, several precision-recall relations are built. The threshold utilised is the Intersection Over Union (IoU), which expresses the overlap between the area under the ground truth bounding boxes and the predicted bounding boxes over the whole areas under both bounding boxes (
Figure 3 .
3Illustration of the intersection over union on example case of Laves phase crack. Mean average precision (mAP) is the mean value of all AP values across all classes. mAP is the most robust way of evaluating the method. It is mostly used with [email protected] and [email protected]:0.95, with the numbers referring to the range of the IoU. mAP is expressed as
Finally
, the edge lines of Laves phase cracks were isolated by means of a Hough line transformation. Due to varying brightness and contrast condition of the images, the parameters which influence the voting accumulation, such as intensity threshold values in the canny edge detection as well as ρaccuracy and vote values of the Hough transformation were dynamically adjusted within the iteration for each image until it captured the lines with the highest number of the votes as the representation of the longest edge line of the crack. In the canny edge detection, the intensity threshold determines the limit to define an edge line as a line with respect to its connectivity, such that if the line portions are
Figure 4 .
4Illustration of Laves phase crack angle calculation. a) SEM image of the crack cropped from the panorama, b) application of erosion morphology correction, c) Canny edge detection, d) angle calculation.
Figure 6 .
6The darker phase is the α-Mg phase and has the composition 98.8Mg-1.2Al (at. %) as determined by STEM-
transmitted from α-Mg to the Laves phase (Figure 5 c).
Figure 5 .
5SE images of 5% deformed Mg-4.65Al-2.82Ca sample, (a) panoramic image of large microstructural area, (b) magnified image of the area enclosed by white rectangle in (a), and (c) magnified portion of the area covered by blue rectangle in (b). Loading direction is horizontal for all images. The basal slip trace is represented by green line in (b).
d and e) have the same orientation as those of the slip lines highlighted by blue arrows in the C36 Laves phase. This indicates that the cracks may have appeared in the Laves phase after plastic deformation through dislocation slip. Moreover, as the strain rate during deformation decreases, other deformation features, such as α-Mg/Laves phase interface decohesion, begin to appear more frequently (Figure 6 f).
Figure 7 .
7Evaluation metrics during training history of the network3.2.2 Rate dependence of damage formationWith decreasing strain rates from 5•10 -4 /s to 5•10 -6 /s, the relative fraction of the observed damage mechanisms was found to change significantly. Both samples presented inFigure 8and
Figure 9
9170 °C to 5% global strain. The sample deformed at a higher strain rate (5•10 -4 /s) exhibits
Figure 9 .
9(a): SE panorama from a sample surface deformed upto 5 % global strain at a strain rate of 5•10 -6 /s and temperature of 170 °C. Magnified images of the microstructural regions enclosed by the blue, red and yellow rectangles are presented in (b), (c), and (d). Green boxes highlight interface decohesion sites at α-Mg/Laves phase interfaces, red indicates the cracks in the Laves phase, while blue boxes represent inclusions and other pores.
Figure 10 .
10Data calculated using AI from the Mg-4.65Al-2.82Ca deformed at a temperature of 170 °C. (a) Total number of damage sites per area in the alloy when subjected to different strain levels at the same rate, (b) change in total number of damage sites per area for the two different strain rates .
Figure 10 a
10. Higher strain resulted in an increased density of damage sites. At all three strain levels and the higher strain rate of 5•10 -4 /s, the fraction of Laves cracks exceeded that of interface decohesion sites by a factor of two or more. However, the nucleation of damage sites reduced significantly as the strain rate was reduced from 5•10 -4 /s to 5•10 -6 /s for the same global strain of 5%. Further, the relative fraction of the two deformation induced damage types changed with strain rate( Figure 11). Interface decohesion is a significant mode of damage at a lower strain rate (5•10 -6 /s), accounting for approximately half the total number of damage sites, while at high strain rate (5•10 -4 /s), cracks in the Laves phase dominate.
Figure 11 .
11Change in Laves phase crack and interface decohesion fraction as a function of strain rate in the Mg-4.65Al-2.82Ca alloy deformed at 170 °C to 5% global strain.
Figure 12 .
12Confusion matrices of classification associated with 2 randomly selected datasets.
Figure 13 .
13Angle of damage sites nucleated in the Laves phase relative to the loading direction at the two different strain rates. Sites perpendicular to the loading direction are represented by an angle of 90°, while those parallel to the loading direction are represented by an angle of 0°.
(
DFG) as part of Collaborative Research Center CRC 1394 -Structural and Chemical Atomic Complexity: From Defect Phase Diagrams to Material Properties (project ID 409476157) and Collaborative Research Center TRR 188 -Damage controlled forming (project ID 278868966). This work is also funded by the state of North Rhine-Westphalia as part of the NHR Program. Calculations were performed with computing resources granted by RWTH Aachen University under project rwth0535.
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| [] |
[
"LOCAL REGION-LEARNING MODULES FOR POINT CLOUD CLASSIFICATION A PREPRINT",
"LOCAL REGION-LEARNING MODULES FOR POINT CLOUD CLASSIFICATION A PREPRINT"
] | [
"Kaya Turgut [email protected] \nDepartment of Electrical-Electronics Engineering\nDepartment of Electrical-Electronics Engineering Eskisehir\nEskisehir Osmangazi University Eskisehir\nTurkey\n",
"Helin Dutagaci [email protected] \nOsmangazi University Eskisehir\nTurkey\n"
] | [
"Department of Electrical-Electronics Engineering\nDepartment of Electrical-Electronics Engineering Eskisehir\nEskisehir Osmangazi University Eskisehir\nTurkey",
"Osmangazi University Eskisehir\nTurkey"
] | [] | Data organization via forming local regions is an integral part of deep learning networks that process 3D point clouds in a hierarchical manner. At each level, the point cloud is sampled to extract representative points and these points are used to be centers of local regions. The organization of local regions is of considerable importance since it determines the location and size of the receptive field at a particular layer of feature aggregation. In this paper, we present two local region-learning modules: Center Shift Module to infer the appropriate shift for each center point, and Radius Update Module to alter the radius of each local region. The parameters of the modules are learned through optimizing the loss associated with the particular task within an end-to-end network. We present alternatives for these modules through various ways of modeling the interactions of the features and locations of 3D points in the point cloud. We integrated both modules independently and together to the PointNet++ object classification architecture, and demonstrated that the modules contributed to a significant increase in classification accuracy for the ScanObjectNN data set. | 10.48550/arxiv.2303.17338 | [
"https://export.arxiv.org/pdf/2303.17338v1.pdf"
] | 257,834,173 | 2303.17338 | 8b0f96c6ac38b8f02aa82d7fdc8799e3babc9150 |
LOCAL REGION-LEARNING MODULES FOR POINT CLOUD CLASSIFICATION A PREPRINT
March 31, 2023
Kaya Turgut [email protected]
Department of Electrical-Electronics Engineering
Department of Electrical-Electronics Engineering Eskisehir
Eskisehir Osmangazi University Eskisehir
Turkey
Helin Dutagaci [email protected]
Osmangazi University Eskisehir
Turkey
LOCAL REGION-LEARNING MODULES FOR POINT CLOUD CLASSIFICATION A PREPRINT
March 31, 2023point cloud · deep learning · self-attention · adaptive region update
Data organization via forming local regions is an integral part of deep learning networks that process 3D point clouds in a hierarchical manner. At each level, the point cloud is sampled to extract representative points and these points are used to be centers of local regions. The organization of local regions is of considerable importance since it determines the location and size of the receptive field at a particular layer of feature aggregation. In this paper, we present two local region-learning modules: Center Shift Module to infer the appropriate shift for each center point, and Radius Update Module to alter the radius of each local region. The parameters of the modules are learned through optimizing the loss associated with the particular task within an end-to-end network. We present alternatives for these modules through various ways of modeling the interactions of the features and locations of 3D points in the point cloud. We integrated both modules independently and together to the PointNet++ object classification architecture, and demonstrated that the modules contributed to a significant increase in classification accuracy for the ScanObjectNN data set.
Introduction
Analysis and semantic interpretation of 3D point clouds became a popular research topic due to the widespread use of 3D sensors for many public, commercial and scientific activities. Many 3D sensors provide output in the form of point clouds. In order to process these 3D point clouds for various tasks, point-based deep neural network architectures have been developed [1][2][3], following the unrivaled success of deep learning methods in pattern recognition and computer vision.
One of the main challenges such 3D point-based deep learning approaches face is the lack of inherent indexing structure that defines the spatial relationships between data points. A point cloud is an unordered set of points, each independently represented by their Cartesian coordinates. As opposed to the grid-based indexing, which provides information on spatial relationships between pixels in images, the spatial relationships between points in the point cloud have to be explicitly specified.
In order to define spatial relationships within the point cloud, some deep learning architectures involve organization of the point data in a hierarchical manner [4][5][6]. At each organizational layer, the point cloud is subsampled to obtain representative points (sampling), then a local region around each representative point is defined (grouping) for aggregation of point features within this region. Similar to the spirit of Convolutional Neural Networks (CNN), the number of representative points is reduced at each layer and their corresponding receptive fields are widened. While the features of numerous points in the first layers encode local surface information, the features of fewer points in subsequent layers represent more of the global structure.
arXiv:2303.17338v1 [cs.CV] 30 Mar 2023
The organization of the point cloud in such hierarchical manner depends on two design choices: 1) Determination of the representative points, and 2) The size of the neighborhood defined by the representative points at each layer. In PointNet++ [4], which is one of the first 3D point-based deep learning approaches, the representative points are selected using Farthest Point Sampling (FPS), and the neighborhood size is fixed to a layer-specific radius value for all representative points. In order for the network reconsider these choices in a task-dependent manner, we propose two local region-learning modules: 1) Center Shift Module (CSM), and 2) Radius Update Module (RUM).
Farthest point sampling, random sampling, and grid sampling are widely used techniques to define representative points that cover the point cloud. We propose the employment of the Center Shift Module to infer the appropriate replacement of each representative point using the local information surrounding it. Similarly, Radius Update Module, as opposed to using a layer-specific fixed radius for grouping around all center points, alters the radius (i.e. the size of the receptive field) for each center point. CSM and RUM learn to infer the replacement of the points and the radii of their receptive fields, respectively, through training with loss functions specified for a particular task; object classification in our study.
Our main contribution is the development of two novel modules that learn to determine the location and size of local regions through training to maximize the classification accuracy of point cloud objects. The two modules can be integrated into any hierarchical point-based deep learning network, where, at each layer, the points are sampled into representative points and grouped within spherical receptive fields around these points. For each module, we propose alternative approaches for inferring the effective locality of each point. These variations correspond to the manner of point interactions within the point cloud. We especially drew inspiration from attention-based approaches [7][8][9][10] to encode these interactions.
Related Work
In this section, we discuss methods that form local regions on point clouds, alternative to common practices such as FPS, random sampling, ball query and KNN search, for deep learning architectures. The methods are mainly divided into two categories: 1) Methods that determine the representative points, i.e., centers of local regions, and 2) methods that group the local points around center points. We discuss these lines of work in the following subsections.
Determining Centers of Local Regions
The heuristic method FPS has been widely used to determine the center points of local regions. Since the points obtained by FPS are distributed uniformly in the Euclidean space, full coverage of the point cloud is provided. However, the selected points do not necessarily correspond to the distinctive parts of objects. To address this problem, two approaches had been proposed: Pre-processing of the point cloud and processing within the end-to-end deep learning framework dedicated to the particular task.
In the first approach, centers of local regions are determined independently from the main network that performs classification or segmentation of the point cloud. In [11], a self-organized map (SOM) is trained to produce a twodimensional representation of the input point cloud in terms of m × m nodes. Initially, a fixed number of nodes are selected randomly and the final locations of these nodes are determined via unsupervised competitive learning. Zhang and Jin [12] proposed an adaptive clustering method to prepare the original point cloud for the deep learning network. They determined the number of clustering centers according to the change of the within-cluster sum of squared errors (SSE). Sampling centers are obtained through forcing the mutual distance between initial cluster centers as far apart as possible. In [13], partitioning the point cloud into geometrically homogeneous regions in the form of a superpoint graph is proposed. In Grid-GCN [14], a Coverage-Aware Grid Query (CAGQ) module is introduced. This module consists of a center voxels sampling framework, and a node points querying algorithm to pick K points around each center point.
The parameters of the data organization methods described above are not optimized jointly with the parameters of the task network. In the second type of approach, parameters of the procedure that determine the center points are learned through an end-to-end deep learning architecture. In [15], a sampling network called S-NET is proposed. During training, S-NET is connected to a pre-trained and fixed task network, and the parameters of S-NET are learned through minimization of the task's loss and a sampling loss. However, in the inference stage, the sampled points need to be matched with the original point cloud. Lang et al. [16] extended S-Net [15] to integrate the matching step into the learning process by using soft projection. In [17], Gumbel Subset Sampling (GSS) is proposed to select representative points within the end-to-end task-dependent network. In CP-Net [18], Critical Points Layer (CPL) is introduced to keep representative points, according to a point's level of contribution to the global max-pooling. PointASNL [19] is proposed to reduce the effect of noise-induced points. The coordinates and attributes of the center points, which are initially sampled by the FPS algorithm, are updated using a self-attention approach. In SK-Net [20], unlike SO-Net [11], the nodes are jointly optimized with regression over extracted features by the end-to-end framework. In PointDAN [21], designed for domain adaption, center points are determined initially by FPS and the shift amounts of the center points are learned by a sub-network. Lin et al. [22] introduced a density-adaptive deep learning framework called DA-Net. Initial center points are learned in relation to the aggregated feature in a global manner and then, they are shifted with a density adaptive sampling module to ease the effect of noise.
Grouping Local Points
Determination of adaptive receptive fields based on geometric similarities within the point cloud is a challenging problem. Similar to the determination of sampling points, adaptive adjusting of the grouping process can be done independently or learned within the task network. As an example for the first approach, Sheshappanavar and Kambhamettu [23] proposed a statistical method based on the distribution of local points. The receptive field is considered as an ellipsoid and its size and orientation are calculated using the eigenvalues and eigenvectors of covariance analysis within the defined initial local region.
There are also works that modify the receptive fields around the center points within the task network. Inspired by dilated convolution, Qiu et al. [24] proposed the Adaptive Dilated Point Grouping module. The dilation factor for nearest neighborhood algorithm is learned by a network over local regions determined with maximum dilation factor. In another work [25], instead of learning the dilation factor, local regions are constituted by a nearest neighbor search algorithm with a dilation factor. The adaptive feature of regions are extracted over local points through learned weight parameters. The attention search module proposed in [26] uses an adaptive combination of spatial and feature distances over nearest neighbours. The weights on spatial and feature distances are learned by an attention mechanism and thus the receptive field is altered implicitly for nearest search algorithm.
In order to jointly define representative points and their local neighborhoods through learned parameters, the Spatial Location Feature Transform Function was proposed in [27]. The dimension of the resulting features of the transform is constrained to the desired number of center points. The channels of the features of the point cloud are sorted independently in descending order and the point index at the top of each sorted channel is selected as a center point. The following K points at each sorted channel constituted the local region around the corresponding center point.
In our work, we propose modules that provide adaptive updates for both the locations of center points and the sizes of the receptive fields around them. These modules can operate within any end-to-end network necessitating point sampling and grouping. The parameters that infer the local regions are learned through minimization of a combination of the task-dependent loss function and a loss function that constraints the amount of alterations on center points and radii. We designed most variations of our Center Shift Module to be heavily influenced by attention-based modeling of point interactions. Also, the proposed Radius Update Module is different from the above mentioned approaches in that it directly increases or decreases the radius of the receptive field for each center point depending on point feature interactions and spatial organization of points in the locality.
Method
In 2D CNNs, at each layer, the size of the image is reduced by pooling while the effective region of each convolution is increased. This approach is mimicked for point clouds through sampling and grouping operations. In Fig. 1, a representative layer for a hierarchically-organized 3D point-based deep network is given together with the alterations provided by CSM and RUM. At each layer L, N L new representative points are sampled from the N L−1 representative points of the previous layer L − 1, with N L < N L−1 . Previous representative points are grouped around new representative points, which can also be referred to as "center points".
Let the set of representative points corresponding to layer L − 1 be denoted by
P L−1 = {p L−1 1 , p L−1 2 , ..., p L−1 N L−1 }, p L−1 i ∈ R 3 ,
and their corresponding point features be in the set
F L−1 = {f L−1 1 , f L−1 2 , ..., f L−1 N L−1 }, f L−1 i ∈ R D L−1 . Through FPS, a subset of P L−1 is formed as C L = {c L 1 , c L 2 , ..., c L N L }, c L j ∈ R 3 , with corresponding features G L = {g L 1 , g L 2 , ..., g L N L }, g L j ∈ R D L−1 , where G L ⊂ F L−1 . Let R L j = {p j,1 , p j,2 , .
.., p j,K }, with p j,k ∈ P L−1 denote the set of K points randomly selected from the spherical region, centered at c L j , with a specific radius. Without the proposed Center Shift Module (CSM) and Radius Update Module (RUM), a hierarchical deep learning architecture aggregates the features of these K points into the region specific feature f L j ∈ R D L , attached to c L j , to be forwarded to the next layer. The procedure is repeated in the next layer with P L = C L , and the corresponding feature set F L .
With the integration of CSM and RUM, K points in R L j are incorporated into the task of updating the locations of sampled center points and the radii of the groupings through examining point interactions. After each update, R L j is Figure 1: Integration of CSM and RUM modules into a hierarchical point-based deep learning framework.
also updated accordingly. As seen in Fig. 1, once initial center points c L j are obtained through FPS algorithm, the task of CSM is to infer the shift vector ∆c L j for each center point so that the center point is updated as:
c L j = c L j + ∆c L j(1)
RUM is responsible of determining ∆r L j for each center point. The receptive field of the jth center point is thus updated as:r
L j = r L + ∆r L j (2)
We suggest a number of variants for the design of both CSM and RUM. The details of these variants are given in the following subsections. Unless necessary, the superscript L, indicating the layer is dropped from the notation for the sake of simplicity.
Center shift module
The objective of CSM is to move the center points sampled by FPS algorithm using local and/or global organization and interactions of other points and their features. Through various ways of encoding these interactions, we implemented five variations of CSM.
CSM-I
CSM-I is similar to the geometric guiding shift learning proposed by Qin et al. [21] in that the edges between the point to be shifted and its neighbors are weighted by a transformation of the feature relations. Our contribution is to update the feature of the center point with an attention-based mechanism, and determine the edge weights through the difference between the features of the surrounding points and the aggregated feature vector of the center point.
Given an initial center point, c j with point feature vector g j , let R j be the local region centered at c j . The radius of the spherical region R j is set to be 2r L . K points are randomly selected from this region defining the neighboring points p j,k with point feature vectors f j,k . We define the displacement vector affected by point p j,k on c j as:
∆E j,k = γ(ĝ j − f j,k ) (c j − p j,k ), k = 1, ..., K(3)
where represents the Hadamard product, andĝ j is the updated feature vector of the center point. The displacement vector ∆E j,k ∈ R 3 is a weighted version of the relative location vector between the center point, c j and the neighbor point, p j,k . The weights are determined by the 3D output of γ, which is an MLP network with two hidden layers, operating on the difference between the updated feature vector of the center point,ĝ j and the feature vector, f j,k of the neighbor point. ReLU is used as the activation function of the first hidden layer, while tanh is preferred for the second hidden layer to limit the size of the displacement vector. The final displacement vector ∆c j is obtained through averaging the displacement vectors affected by K neighbor points:
∆c j = 1 K K k=1 ∆E j,k(4)
Notice that, instead of using the feature vector, g j , transmitted from the previous layer, we use an updated version of it, g j , for assessing the center point's similarity to the feature vectors of the neighbor points:
g j = g j + g sa j(5)
The vector g sa j is an attention-based aggregation of the feature vectors of the neighbor points:
g sa j = φ K k=1 a j,k ψ(f j,k ) (6) a j,k = ρ β(g j )ϕ(f j,k ) T d qk (7)
where ρ is the softmax function, and φ is an MLP with ReLU activation function. In this attention-based feature aggregation scheme [7], β(g j ) corresponds to the query vector, which is obtained through a linear transformation of g j . The linear transformations of the feature vectors of the neighbor points, ϕ(f j,k ) and ψ(f j,k ) serve as key and value vectors, respectively. d qk is the dimension of the query and key vectors. The transformations applied to the feature vectors of the center point and neighbor points to obtain the query, key, and value vectors reduce their original dimension to half. The weights a j,k are determined through the dot product of the query vector β(g j ) and the key vectors ϕ(f j,k ). In other words, the similarity between the feature vectors of the center point and the neighbor points are reflected to the weights a j,k , which determine the contribution of the neighbor points to the aggregation. g sa j is the output of φ, which transforms the weighted sum of the value vectors and increases the dimension of the feature vector back to the original dimension of g j .
The transformations involved in CSM-I are embedded in the functions γ, φ, ϕ, β, and ψ are learned through training. This scheme allows the output displacement vector ∆c j be inferred through CSM, whose parameters are adjusted through minimization of a loss function designated for a particular task.
CSM-II
The main difference between CSM-I and CSM-II is in the determination of the weights a j,k used in Eq. 6. In CSM-II variant, for updating the feature vector of the center point, positional relationships, δ jk , between query and key vectors are integrated into the computation of a j,k :
a j,k = ρ (ϑ (concat [δ jk , d (β(g j ), ϕ(f j,k ))]))(8)
where δ jk is a transformed version of the relative position vector, c j −p j,k through a linear transformation Θ : R 3 → R 3 :
δ jk = Θ(c j − p j,k )(9)
ρ in Eq. 8 is the softmax function, and ϑ is a two-layer network with a ReLU activation function in between the layers. d(., .) defines the relation between the query and key vectors, β(g j ) and ϕ(f j,k ). We considered various alternatives for the similarity measures of these query and key vectors, d (β(g j ), ϕ(f j,k )), including subtraction (sub), summation (sum), concatenation (cat), and dot product (dot). This similarity is then integrated with positional relationship δ jk via concatenation.
The rest of the operation of CSM-II is the same as CSM-I. The final displacement vector ∆c j is obtained using Eqs. 3 to 6.
CSM-III
In CSM-III, for determining the shift vector of the center point, relationships with close center points are incorporated in addition to the interactions between the particular center point and the points p j,k in its vicinity. The final displacement vector ∆c j is obtained with Eqs. 3 and 4 as in CSM-I. However, the updated feature vectorĝ j of the center point is computed as:ĝ
j = g sa j + g saC j(10)
where g sa j is obtained through the attention-based aggregation defined by Eqs. 6 and 7. g saC j refers to the attention-based aggregation among the updated feature vectors belonging to the U closest center points of the particular center point c j . The feature vectors of the closest U center points g j,u , with u = 1, ...U , are updated as:
g j,u = g j,u + g sa j,u(11)
Then, g saC j is computed as:
g saC j =φ U u=1 b uψ (ḡ j,u )(12)with b u = ρ β (ḡ j )φ(ḡ j,u ) T d qk(13)
where ρ is the softmax function, andφ is a nonlinear transformation network consisting of an MLP with ReLU activation function. Here,β(ḡ j ) is the query vector. The linear transformations of the updated feature vectors of the U center points,φ(ḡ j,u ) andψ(ḡ j,u ) correspond to key and value vectors, respectively. With CSM-I and CSM-II, we only consider point interactions within the locality of radius 2r L of each center point to shift it. With CSM-III, the spatial extent of point interactions is enlarged. The relative locations and feature similarities of other center points also contribute to the shift of a particular center point.
CSM-IV
In CSM-IV, feature vectors of all center points c j , j = 1, .., N L are first updated using Eq. 10. While computing g saC j component in Eq. 12, all center points contribute rather than the nearest U center points. Consequently, global interactions between a center point and all other center points covering the object are encoded in g saC j . Likewise, the relative position vectors between c j and all other center points contribute to the shift. The displacement vector affected by another center point c l on c j is defined as:
∆E(j, l) = θ(ĝ j −ĝ l )(c j − c l )(14)
where θ is a nonlinear transformation function that takes the difference of center points features as input.
The final shift vector is the average of ∆E(j, l) over all center points:
∆c j = 1 N L N L l=1 ∆E(j, l)(15)
Among all versions of the Center Shift Module, CSM-IV considers point relationships most globally. It shifts sampled center locations by taking into account all center points covering the entire object.
CSM-V
CSM-V is fundamentally different from the other versions in that pairwise point interactions within R j are employed.
For each center point, c j weighted pairwise distances within its neighborhood R j are formulated as
∆E j (k, l) = θ(f j,k − f j,l )(p j,k − p j,l )(16)
The relative location vector between point p j,k and p j,l is weighted by the 3 × 3 output of the network θ, which is a two-layered network with a ReLU activation function in between the layers. The pairwise feature differences are reduced to 64 and 9 dimensions in the first and second layer, respectively. The final displacement vector ∆c j is calculated as
∆c j = max k=1,...,K ∆G j,k(17)
with
∆G j,k = 1 K K l=1 ∆E j (k, l)(18)
Another important distinction of CSM-V from the first variations is that feature update through attention-based aggregation is not employed prior to calculation of feature interactions.
Radius update module
RUM is responsible to update the radius of the neighborhood of each center point c j . The convention is setting the radius r L constant for all center points at a particular layer. We propose to update the radius for each center point by ∆r j produced by RUM, through Eq. 2. In this way, the size of the receptive field is defined in an adaptive way for each center point.
For the sake of simplicity, the superscript L is dropped. The points from the previous layer within 2r distance of a center point c j are gathered randomly. Let the feature vectors of these points be denoted as f j,s , s = 1, .., S, where S is the number of points in the neighborhood of radius 2r. Let the feature vector of the center point be denoted as g j . We propose two variations of RUM for inferring ∆r j from these points.
RUM-I
First, the difference between the feature vectors of the center point and each neighbor point is transformed through the nonlinear function ζ:
e j,s = ζ(g j − f j,s )(19)
where ζ is an MLP with ReLU activation function. The spherical region of radius 2r surrounding c j is partitioned into T concentric spheres, B j,t , each with radius t T 2r. The number of points in B j,t is denoted as S j,t . Transformed feature differences of the points in each region is aggregated through either averaging
R j,t = 1 S j,t s∈Bj,t e j,s(20)
or maxpooling
R j,t = max s∈Bj,t e j,s(21) .
In the experiments, we considered and compared both alternatives for feature aggregation as RUM-I (cum) for Eq. 20 and RUM-I (max) for Eq. 21. The aggregated features R j,t are arranged in a matrixR j following the order t = 1, ..., T (i.e., from smaller to larger spheres). SinceR j is invariant to permutation of point indices, we apply an MLP with activation function tanh to produce a scalar value, ∆r j , which is the output of RUM-I. The activation function tanh is used to limit the magnitude of ∆r j .
RUM-II
In RUM-II, the neighborhood of each center point c j is organized by the same manner as in RUM-I. The features aggregated for each concentric sphere t are updated aŝ
R j,t = R j,t + R sa j,t(22)
To obtain R sa j,t , an attention-based transformation is applied to the features R j,t :
R sa j,t = T v=1 a t,vψ (R j,v )(23)
with weights a t,v computed as
a t,v = ρ β (R j,t )φ(R j,v ) T d qk(24)
Here, ρ is the softmax function,β(R j,t ) is a linear transformation of R j,t , and corresponds to the query vector. The linear transformations of R j,v , v = 1, ..., T are designated as the keyφ(R j,v ) and valueψ(R j,v ) vectors, with dimension d qk .
For RUM-II, these linear transformations do not involve dimension reduction.
As in RUM-I, the featuresR j,t are arranged to form the matrixR j , and ∆r j is computed by an MLP with tanh activation function. For both RUM-I and RUM-II, we set T = 4 in our experiments.
Loss function
In 3D point-based classification architectures, usually, cross-entropy is used as the loss function. We introduced additional terms, L L csm , and L L rum , to the loss function in order to limit the shift generated by CSM and the radius update introduced by RUM: (25) where L ce is the cross-entropy function and α 1 and α 2 are constant weights, which are set to 0.01 in our experiments. L ce ensures that the network learns the parameters of both feature extraction layers and CSM and RUM modules via maximizing the classification performance on the training data. With L L csm , we introduce a regularization term that penalizes large shifts of the center points computed at layer L. L L csm is composed of two terms:
L = L ce + α 1 L L L csm + α 2 L L L rumL L csm = L L f it + L L range (26) L L f it = 1 N L N L j=1 ĉ L j −p L j (27) L L range = 1 N L N L j=1 max(0, ∆c L j − r L )(28)
wherep L j is the closest point to the updated center pointĉ L j from the set P L−1 . L L f it forces the updated center pointŝ c L j to remain close to the object surface, while L L range penalizes the excess in the magnitude of the shift vector ∆c L j in relation to r L , the initial radius of local regions at a particular layer L.
The other term, L L rum , limits the magnitude of the radius update ∆r L j :
L L rum = 1 N L N L j=1 |min(0, r L + ∆r L j )| + max(0, ∆r L j − r L )(29)
In this way, ∆r L j is encouraged to remain in the interval [−r L , r L ].
Integration to PointNet++
The proposed local region-learning modules, CSM and RUM can be integrated into any hierarchical deep learning framework consisting of sampling and grouping stages, where group centers and group radii are needed to be determined. The classification network architecture of PointNet++ [4], shown in Fig. 2a, is selected to demonstrate the effectiveness of the proposed modules. The global descriptors representing each object are extracted by passing its point cloud representation through three Set Abstraction (SA) layers. The category scores are obtained by applying fully connected layers to the global descriptor.
At each SA layer, the center points c L j of N L local regions are determined with the FPS algorithm. CSM and RUM can be integrated to either or both of the first two layers as shown in Fig. 2b. If CSM is "ON" at a particular layer, the center points are shifted by ∆c L j as in Eq. 1, otherwise they are not altered. If RUM is "OFF", a local region with radius r L , which is kept constant for layer L, is established around each center point. If RUM is "ON", then the radius is updated using Eq. 2, and a local region of radius r L j is formed around center point c L j . Grouping is performed by randomly selecting K points in the local regions. After the features of the K points are mapped to higher dimensions independently with MLP layers, the features of the local region centers are calculated by taking the maximum among the feature channels of grouped points.
The third abstraction layer is responsible for computing the global descriptor for the entire point cloud of the object. At this layer, a single center point is defined, with its "local region" covering all the center points conveyed from the previous layer.
Results
We conducted various experiments to observe the effect of CSM and RUM on the performance of PointNet++. The experiments were performed on the real-world classification data set ScanObjectNN [28]. First, variations of only CSM or only RUM were integrated to the set abstraction layers of PointNet++. The modules were either integrated to the first layer, or to the second, or to both. In the final experiments, various combinations of versions of CSM and RUM were then inserted together to the second layer of PointNet++.
Data Set
The ScanObjectNN [28] data set consists of real-world point cloud object data collected from the public 3D indoor scene data sets SceneNN [29] and ScanNet [30]. 700 unique scenes of SceneNN and ScanNet mesh indoor scans were selected, and from these scenes, 2902 objects were cropped and manually filtered. These objects are categorized into 15 common categories (bag, bed, bin, box, cabinet, chair, desk, display, door, pillow, shelf, sink, sofa, table, and toilet). Each point cloud is represented with local and global coordinates, surface normals, color attributes, and semantic labels. Different variants of each object have been generated to add levels of difficulty to explore the robustness of classification algorithms. We experimented with two variants referred to as OBJ_ONLY and OBJ_BG. The first variant, OBJ_ONLY, includes objects neatly delineated from the background. The second variant, OBJ_BG, corresponds to an axis-aligned bounding box around the object, and contains parts of the background and nearby objects as well.
We used the training and test split and the parameters as specified in [28]. For all experiments, each point cloud was randomly sampled to 1024 points, centered at zero, and scaled to fit in the unit sphere. As input features, only the local coordinates (x, y, z) of the points were used.
Experimental results
Three types of experiments were performed for each variation of CSM and RUM: 1) the module is applied only to the first layer (1 st ), 2) the module is applied only to the second layer (2 nd ), and 3) the moudle is applied to both the first and second layers (1 st -2 nd ). The results are compared in terms of overall accuracy (Acc), which is equal to the ratio of the number of correctly classified objects to the total number of objects in the test set. We also give the mean of the accuracy (M Acc), which corresponds to the class-based accuracy values averaged over all categories. Table 1 gives the contribution of variations of CSM to the classification performance of PointNet++ on the ScanObjectNN data sets OBJ_ONLY and OBJ_BG. In the first row, the classification accuracy of PointNet++ is provided as reported in [28]. All variations of CSM, except for CSM-IV applied only to the first layer, and CSM-V applied to both layers, increased the performance of PointNet++. However, for most of the variations, applying CSM only to the second layer resulted in a higher increase in performance than applying it to only the first layer or to both. At the very local level, keeping the center points selected via FPS gives close results to moving them with CSM. As the region size grows in the second layer, semantic content aggregated at center points gains prominence for classification. CSM module displaces these center points in accord with their contribution to the minimization of the task loss. We provide visual results of the effect of CSM for sample objects belonging to various categories from the OBJ_ONLY set in Fig. 3. Here, CSM-III was applied only to the second layer of PointNet++. In the first row for each object, center points sampled by FPS algorithm are given from front and side views. In the second row, center points shifted by CSM are shown. We can observe that CSM leaves some portions of the object with relatively few center points, while populating other parts with more representative points. We can view this tendency as the network's response to emphasize certain regions that are salient within a category. For example, as seen in Fig. 3, the center points of objects in the chair, table, and bed categories at the flat portions were kept in the same plane, however points at legs were widely displaced to counter intra-class variability. Legs are the portions that cause the most intra-class shape variation in categories such as chair and table. The salient part is the flat portion and its relative size with respect to the other parts.
For all variations of CSM, application to the second layer increases the classification performance of PointNet++. For OBJ_ONLY data set, the highest increase in M Acc is +4.64%, achieved by CSM-III. The increase in classification accuracy with integration of CSM is more pronounced for the OBJ_BG data, reaching +6.17 of M Acc with CSM-II (sum). Two examples from the OBJ_BG data are given in Fig. 4 to observe the displacements of center points affected by CSM-II (sum). The main difference of the chair object in Fig. 4a as compared to the chair object in Fig. 3a is the presence of the ground points, which do not belong to the object but to the background. However, the ground points still provide context for the distinction of the chair category. Here, again, we observe that the non-salient chair points at legs that show greater within class variability tend to move to populate the salient portions such as the seat, and the ground in the case of OBJ_BG. The points of the back of the chair moved widely. We conjecture that these points tend to be distributed to even out the variability of the back of the chairs in the training data set. We observe a similar effect for the display object shown in 4b, where the center points remained on the plane, however the distribution on the plane is altered to fit a certain aspect ratio. We can also notice that the center points behind the display are shifted towards the display to suppress the contribution of the points belonging to the irrelevant background. Table 2 gives the increase in the classification performance of PointNet++ by integrating variations of RUM. For OBJ_ONLY data, applying RUM to the first layer did not contribute to an increase in performance; on the contrary, caused a drop with most variations. However, similar to the case with CSM, applying RUM only to the second layer resulted in higher increase in performance than applying it to both layers. The strategy of keeping the region size fixed for extracting low level surface features at the first layer while varying region size in the second layer where partial semantic information comes into prominence proves to be effective.
Examples of local regions affected by RUM are given in Fig. 5 for sample objects from OBJ_ONLY data set. The first column shows sample regions where the radii are reduced by RUM-I-(max). The points excluded from the updated regions are marked with red color. We can observe that the regions shrink to fit a semantic part of the object while excluding the points not belonging to the part. For example, the radius of the region enclosing the leg of the chair is reduced to exlude points from the seat and base (Fig. 5a). Similarly, a region on the top of the desk is shrank to be confined to the planar region (Fig. 5b). In the second column, the opposite effect is demonstrated, where the radii are increased by RUM to include more points (marked with green) from the same semantic part of an object. Examples are the back of the chair and the side of the desk. The last column shows instances where the update is zero; hence the particular region did not change. Figure 5: Updated region radii for sample objects from OBJ_ONLY through RUM-I-(max) applied only to the second layer. The objects belong to the chair(a) and desk(b) categories. Black points represent the centers of the updated local regions. The red points are the points excluded from the region when the radius is decreased by RUM. The green points indicate the points included to the region when the radius is increased. Points in magenta remained in the region.
Among all RUM variations applied to the second layer only, RUM-I (max) yielded the largest increase in M Acc as +4.03% for the OBJ_ONLY data. For the OBJ_BG data, RUM-II (cum) resulted in the largest increase as +6.41%. As in the case with CSM, RUM boosts the classification accuracy for OBJ_BG data more than the OBJ_ONLY data.
In Fig. 6, the effect of the RUM is demonstrated for sample objects from OBJ_BG data set. We can observe a number of tendencies of the RUM while processing the noisy data, additional to the category-sensitive region adapting. One is to isolate salient object parts both from the object itself and also from irrelevant background. Examples are the excluded background points from the bin object in Fig. 6a and the region shrinking to the back and seat of the sofa in Fig. 6b. Another tendency of RUM is to isolate background regions from the object points as can be seen at the updated regions of display and toilet objects in Fig. 6c and 6d. In the cases where RUM increased the region size for a center point (the second column of Fig. 6) more points making the region distinct and salient for an object category are included in the region. Guided by the results obtained by integrating CSM and RUM variations separately to PointNet++ architecture, we opted to apply them together to only to the second layer. We report in Table 3 the results of the experiments conducted by combining variations of CSM and RUM. The combination providing the highest increase in performance is CSM-I and RUM-I (max) for both OBJ_ONLY and OBJ_BG data. M Acc increased by 5.25% for OBJ_ONLY data, and 6.75% for OBJ_BG, surpassing the increase obtained by integrating CSM or RUM separately.
Conclusion
In this work, we proposed variations of two modules, Central Shift Module and Radius Update Module to learn to infer the positions and radii of local regions organized in hierarchical 3D point-based deep learning architectures. We integrated variations of CSM and RUM to the classification network of PointNet++, and demonstrated their effectiveness on the publicly available ScanObjectNN data set. We observed that applying all variations of these modules to the
Figure 2 :
2The integration of proposed modules to PointNet++ framework: a-) The network for the classification problem. r L for L = 1 (first layer) is set to be 0.2. r L for L = 2 (second layer) is 0.4. b-) adaptive local region inference framework.
Figure 3 :
3Shifted local center points for sample objects from OBJ_ONLY. The objects belong to the chair(a), table(b), and bed(c) categories. CSM-III is integrated only to the second layer of PointNet++.
Figure 4 :
4Shifted local center points for sample objects from OBJ_BG. The objects belong to the chair(a) and display(b) categories. CSM-II-(sum) is integrated only to the second layer of PointNet++.
Figure 6 :
6The visual result of updated radius through RUM-II-(cum) applied only to the second layer. The objects belong to bin(a), display(b), sofa(c), and toilet(d) categories.
Table 1 :
1Results on ScanObjectNN data set with CSM . +1.73 84.34 +2.24 86.23 +3.93 83.36 +3.46 85.54 +1.21 84.00 +1.90 87.26 +4.96 84.27 +4.37 CSM-III 85.89 +1.56 83.99 +1.89 87.78 +5.48 85.26 +5.36 88.81 +4.48 86.74 +4.64 87.09 +4.79 84.90 +5.00 85.54 +1.21 83.43 +1.33 85.71 +3.41 82.94 +3.04Method
OBJ_ONLY
OBJ_BG
1 st 2 nd
Acc
+/-
M Acc
+/-
Acc
+/-
M Acc
+/-
PointNet++
84.33
-
82.1
-
82.3
-
79.9
-
CSM-I
85.71 +1.38 83.62 +1.52 86.75 +4.45 83.57 +3.67
87.44 +3.11 84.89 +2.79 86.75 +4.45 85.15 +5.25
85.03 +0.70 82.59 +0.49 86.06 +3.76 83.08 +3.18
CSM-II (sub)
85.20 +0.87 83.84 +1.74 86.06 +3.76 81.90 +2.00
87.26 +2.93 85.31 +3.21 86.75 +4.45 84.61 +4.71
84.38 +0.05 83.36 +1.26 86.58 +4.28 82.75 +2.85
CSM-II (sum)
86.06 +1.73 84.92 +2.82 87.09 +4.79 83.96 +4.06
86.92 +2.59 85.32 +3.22 88.47 +6.17 86.07 +6.17
86.75 +2.42 85.23 +3.13 87.09 +4.79 85.13 +5.23
CSM-II (cat)
84.51 +0.18 81.98 +0.12 86.06 +3.76 82.94 +3.04
85.37 +1.04 83.93 +1.83 86.06 +3.76 83.60 +3.70
86.23 +1.90 83.28 +1.18 86.58 +4.28 83.31 +3.41
CSM-II (dot)
85.03 +0.70 81.93
-0.17 85.37 +3.07 82.28 +2.38
86.23 +1.90 85.46 +3.36 86.92 +4.62 85.13 +5.23
87.44 +3.11 86.02 +3.92 85.89 +3.59 82.88 +2.98
CSM-II (hadamard)
85.20 +0.87 83.81 +1.71 86.58 +4.28 84.27 +4.37
86.06 CSM-IV
84.34 +0.01 81.99
-0.11 86.92 +4.62 84.56 +4.66
86.06 +1.73 84.35 +2.25 87.61 +5.31 83.97 +4.07
84.85 +0.52 83.66 +1.56 86.40 +4.10 83.98 +4.08
CSM-V
85.37 +1.04 83.15 +1.05 86.92 +4.62 82.94 +3.04
87.09 +2.76 84.68 +2.58 88.47 +6.17 85.11 +5.21
83.99 -0.34
81.19
-0.91 85.03 +2.73 81.86 +1.96
Table 2 :
2Results on ScanObjectNN data set with module RU M . +1.69 81.60 +1.70 87.09 +2.76 85.45 +3.35 86.92 +4.62 85.17 +5.27 86.23 +1.90 84.30 +2.20 85.20 +2.90 82.17 +2.27 +3.24 81.88 +1.98 87.95 +3.62 85.66 +3.56 87.44 +5.14 85.81 +5.91 84.85 +0.52 82.81 +0.71 86.40 +4.10 84.00 +4.10 RUM-II (cum) 83.99 -0.34 82.22 +0.12 85.37 +3.07 82.49 +2.59 87.44 +3.11 86.13 +4.03 88.30 +6.00 86.31 +6.41 85.54 +1.21 83.58 +1.48 86.40 +4.10 83.89 +3.99 RUM-II (max) 84.68 +0.35 81.83 -0.27 85.71 +3.41 81.86 +1.96 85.71 +1.38 83.58 +1.48 87.44 +5.14 85.87 +5.97 86.58 +2.25 84.87 +2.77 87.78 +5.48 85.34 +5.44Method
OBJ_ONLY
OBJ_BG
1 st 2 nd
Acc
+/-
M Acc
+/-
Acc
+/-
M Acc
+/-
PointNet++
84.33
-
82.1
-
82.3
-
79.9
-
RUM-I (cum)
82.79 -1.54
80.52
-1.58 83.99 RUM-I (max)
83.48 -0.85
81.40
-0.7
85.54
Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.second layer, where semantic regions are formed, increased the classification performance significantly. Integrating CSM and RUM separately only to the second layer resulted in 6.17% and 6.41% increase in M Acc, respectively, for OBJ_BG data of the ScanObjectNN data set, while combining both modules yielded a performance increase of 6.75%. These results demonstrate that the proposed tools for learning the location and size of the local regions, hence the receptive fields, within the task network through minimization of the task loss are effective at improving data organization for 3D point-based deep networks.
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| [] |
[
"ARTICLE TYPE On the True Nature of the Contact Binary CRTS J192848.7-404555",
"ARTICLE TYPE On the True Nature of the Contact Binary CRTS J192848.7-404555"
] | [
"Surjit S Wadhwa \nSchool of Science\nWestern Sydney University\nNSWAustralia\n",
"| Ain \nSchool of Science\nWestern Sydney University\nNSWAustralia\n",
"Y Dehorta \nSchool of Science\nWestern Sydney University\nNSWAustralia\n",
"Miroslav Filipović \nSchool of Science\nWestern Sydney University\nNSWAustralia\n",
"| Nick \nSchool of Science\nWestern Sydney University\nNSWAustralia\n",
"F H Tothill \nSchool of Science\nWestern Sydney University\nNSWAustralia\n"
] | [
"School of Science\nWestern Sydney University\nNSWAustralia",
"School of Science\nWestern Sydney University\nNSWAustralia",
"School of Science\nWestern Sydney University\nNSWAustralia",
"School of Science\nWestern Sydney University\nNSWAustralia",
"School of Science\nWestern Sydney University\nNSWAustralia",
"School of Science\nWestern Sydney University\nNSWAustralia"
] | [] | CRTS J192848.7-404555 was recognised as a potential contact binary merger candidate on the basis of survey photometry analysis. We have carried out follow up ground based photometry of the system and show that at the recorded coordinates for the system there are two stars approximately 3 seconds of arc apart. Our analysis shows that the fainter of the two stars is the actual variable while the slightly brighter star is of fixed brightness. In addition we show that the reported survey photometry is the result of both stars being treated as a single light source with resultant erroneous light curve solution. The true nature of CRTS J192848.7-404555 shows it to be a low mass contact binary system with a high mass ratio of 0.425, high amplitude of 0.69 magnitude and shallow 24% contact. The system does not have features of orbital instability and is not a potential merger progenitor. | 10.1002/asna.20220066 | [
"https://export.arxiv.org/pdf/2212.08209v1.pdf"
] | 254,823,499 | 2212.08209 | e5e9ad5a0808b7be21ad614b969ffd2e40c6322e |
ARTICLE TYPE On the True Nature of the Contact Binary CRTS J192848.7-404555
Surjit S Wadhwa
School of Science
Western Sydney University
NSWAustralia
| Ain
School of Science
Western Sydney University
NSWAustralia
Y Dehorta
School of Science
Western Sydney University
NSWAustralia
Miroslav Filipović
School of Science
Western Sydney University
NSWAustralia
| Nick
School of Science
Western Sydney University
NSWAustralia
F H Tothill
School of Science
Western Sydney University
NSWAustralia
ARTICLE TYPE On the True Nature of the Contact Binary CRTS J192848.7-404555
Received: 00 Month 0000 Revised: 00 Month 0000 Accepted: 00 Month 0000Correspondence *Surjit Wadhwa, Email:Contact BinaryLight Curve SolutionPhotometry
CRTS J192848.7-404555 was recognised as a potential contact binary merger candidate on the basis of survey photometry analysis. We have carried out follow up ground based photometry of the system and show that at the recorded coordinates for the system there are two stars approximately 3 seconds of arc apart. Our analysis shows that the fainter of the two stars is the actual variable while the slightly brighter star is of fixed brightness. In addition we show that the reported survey photometry is the result of both stars being treated as a single light source with resultant erroneous light curve solution. The true nature of CRTS J192848.7-404555 shows it to be a low mass contact binary system with a high mass ratio of 0.425, high amplitude of 0.69 magnitude and shallow 24% contact. The system does not have features of orbital instability and is not a potential merger progenitor.
(Drake et al., 2017)
with a period of 0.320028d, brightest V band magnitude of 12.04 and V band amplitude of 0.27 magnitudes. Catalina magnitudes were converted to V band as per The Catalina Surveys Data Release 2 notes available online at http://nesssi.cacr.caltech.edu/DataRelease/ (accessed 10/10/2022). Other surveys such as the All Sky Automated Survey -Super Nova (ASAS-SN) (Jayasinghe et al., 2020;Shappee et al., 2014), SuperWASP (Thiemann, Norton, Dickinson, McMaster, & Kolb, 2021) and All sky Automated Survey (Pojmanski, 2002) have all reported similar basic parameters. The light curves from these surveys are illustrated in Figure 1.
The SuperWasp photometry of C1928 was modelled by and indicated a very low mass ratio ( = 0.08), low inclination ( = 67 • ) system in marginal contact. As part of our ongoing project to identify and confirm low mass ratio contact binary systems, in this study we present the results and analysis of ground based photometry of C1928 and show that the survey photometry is blended from a nearby star approximately 3 seconds of arc roughly west of the contact binary system. The contact binary system is significantly fainter than reported with a much larger amplitude and significantly higher mass ratio. As would be expected physical parameters such as masses, radii and separation of the components are also different to those derived from analysis of the survey photometry.
OBSERVATIONS
C1928 was observed with the 0.4m telescope network of the Las Cumbres Observatory (LCO) over 4 days in June 2022. Full cycle Bessel V band photometry was acquired with 366 observations in total with typical exposure time of 45 seconds. In addition concurrent 50 Bessel B band observations (exposure time 60 seconds) during times of minima were acquired to document the (B-V) colour of the system. Review of the initial FIGURE 1 Light curves of C1928 from 4 sky surveys as labelled. All show a bright contact binary system light curve with amplitude near 0.3. There is slight variation in the actual magnitude values is due to the different band pass for each survey. The ASAS and ASAS-SN surveys use standard V-Band filters, Catalina survey data is unfiltered while the SuperWASP survey employs a broadband filter from 400 to 700 nm V band images ( Figure 2) clearly shows two stars approximately 3 seconds of arc apart at the recorded location of C1928. As most survey data is obtained either with a telephoto lens or small aperture telescopes, we postulated there maybe some that there was a potential for mis-identification and or blending of the survey light curves and the true nature of C1928 may differ from that described based on published survey photometry. The main aim of this study was to identify which of the two stars at the reported location was the variable and once identified to fully elucidate the nature of the likely contact binary system.
We performed aperture photometry on both stars using 2MASS-19284698-4046286 (V = 13.29, B = 14.27) as the comparison star and 2MASS-19285265-4046355 as the check star. We used the AAVSO Photometric All-Sky Survey (Henden, Levine, Terrell, & Welch, 2015) calibrations for the comparison star magnitudes. The data was folded using the revised period (see below) and normalised to the brightest magnitude. The V band light curve of the slightly brighter (upper star in Figure 2) is essentially constant with V magnitude of 12.76. There is possibly some minor variability when compared to FIGURE 3 Light curves of the two stars at the location of C1928. Top curve (black) illustrates the light variation when both stars are taken as a single light source. The middle (purple) curve is that of the bright star while the bottom (green) curve is of the fainter star. The yellow curve represents the check star the check star however it is likely to be less than 0.1 magnitude in amplitude. We could not find any short term periodicity. The V band light curve of the fainter (lower star in Figure 2) shows a classical contact binary light curve however the amplitude, unlike survey photometry, is considerably larger at 0.69 (V: 12.86 -13.55) magnitude. In other respects the light curve has similar shape to the survey light curves. To confirm our suspicion that the survey light curves were the result of full contamination by the non variable star we performed photometry with both stars acting as a single light source. The resulting light curve is very similar to the survey light curves with a maximum brightness V magnitude of 12.08 and amplitude of 0.29 magnitudes. All three light curves and the check star curve are illustrated in Figure 3.
Based on the times of minima observed we revised the orbital elements as follows:
= 2459753.121103(±298) + 0.3200310(±233)
The B-V value was the same at both the primary and secondary eclipses at 0.77 which is quite different to 0.65 reported with the ASAS-SN survey. We note that the B-V value of the pair taken as a single light source is 0.67, similar to the value reported by ASAS-SN again confirming that the survey photometry data includes both stars.
LIGHT CURVE ANALYSIS
Given the presence of complete eclipses, light curve analysis without radial velocity data is feasible and should yield accurate results for the geometric parameters (Terrell & Wilson, 2005). We used the 2009 version of the Wilson-Devenney code to analyse the V band photometry data. As there is no appreciable O'Connell effect only unspotted solutions were modelled. The accepted grid/q search method was used to obtain the mass ratio ( ). The effective temperature of the primary ( 1 ) is usually fixed during the search procedure. As the recent EDR 3 separates the two stars (Anders et al., 2022;Gaia Collaboration et al., 2022) we estimated 1 as 5900K. As usual gravity darkening coefficients were fixed as ( 1 = 2 = 0.32), bolometric albedo was fixed at ( 1 = 2 = 0.5) and simple reflection treatment was applied. Limb darkening coefficients were interpolated from van Hamme (1993). The mass ratio search grid along the observed and fitted light curves are illustrated in Figure 4 and the light curve solution summarised in Table 1.
The system is observed edge on ( = 90 0 ) and found to have a slightly hotter secondary ( 2 = 6148 ) with a mass ratio of 0.425 and shallow fillout of 24%. The light curve solution is very different to the solution based on survey data which suggested a very low mass ratio of 0.08 with low inclination and marginal contact.
PHYSICAL PROPERTIES
Apart from the mass ratio the next most critical parameter required to estimate other physical characteristics is the mass of the primary. It is well established that the primary component of a contact binary systems follow in general a main sequence profile (Yildiz & Doğan, 2013). We estimate the mass of the primary ( 1 ) as the mean of combined distance and color based calibrations.
As there is a risk of contamination with published color magnitudes of the system we first determined the distance and reddening corrected ( − ) 0 value of the system as follows: Firstly, we determined the line of sight reddening at infinity ( − ) ∞ centered on the system coordinates using the Schlafly & Finkbeiner (2011) dust maps. The value was then scaled to the distance (461.4±10 ) (Anders et al., 2022) ( − ) using the equation (Bilir et al., 2008):
( − ) = ( − ) ∞ 1 − − | | ℎ(1)
In the equation is the distance, is the galactic latitude of the system and ℎ is the galactic scale height, taken as ℎ = FIGURE 4 Observed (green) and fitted (black) light curves are illustrated on the left while the mass ratio search grid on the right. The search grid residuals were normalised to the minimum value and only the portion near the minimum residual is illustrated for clarity 125 as per Bilir et al. (2008). The distance and reddening corrected ( − ) 0 was determined as 0.70.
As the mission does separate the two stars we also make use of the color ( − ) = 0.94 corrected to ( − ) 0 = 0.82 using the line of extinction of ( − ) ∞ = 0.13 and ( − ) = 0.12. In addition to the two color estimates we estimated the luminosity of the primary based on the distance and our photometry data. The eclipses are of near equal depth thus the apparent magnitude at mid eclipse (13.53) represents the apparent magnitude of the primary component. Given the distance and the distance corrected line of sight extinction determined above we calculate the absolute magnitude of the primary ( 1 ) as 5.42 ± 0.05. From the difference in peak and eclipse brightness we estimate the absolute magnitude of the secondary ( 2 ) as 5.59 ± 0.05. We used the April 2022 updated calibration tables of Pecaut & Mamajek (2013) for low mass (0.6 ⊙ < < 1.5 ⊙ ) stars along with ( − ) 0 , ( − ) 0 and 1 to interpolate three different values for the mass of the primary as 0.95 ⊙ , 1.01 ⊙ and 0.91 ⊙ respectively. We adopt the mean of these as the mass of the primary ( 1 = 0.96 ± 0.01 ⊙ ). We use the error based on the distance estimate as this was of the highest order. From the mass of the primary we estimate the mass of the secondary ( 2 ) as 0.41 ± 0.02 ⊙ . Other physical parameters such as the radii of the components ( 1,2 ), the separation ( ) and density ( 1,2 ) can be estimated from the fractional radii of the components (from light curve solution), mass ratio and Keplers' third law as previously described (Wadhwa, de Horta, Filipović, & Totohill, 2022). Errors were propagated from the errors in the mass of the components by far the greatest contributors to overall error. Physical parameters are summarised in Table 1. Wadhwa et al. (2021) introduced simplified quadratic relations linking the mass of the primary of contact binary systems with the mass ratio at which orbital instability is likely. Based on those relations one can estimate a narrow range of the mass ratio at or below which orbital instability is likely. when analysing the survey photometry of C1928 estimated the mass of the primary based on the main sequence calibration of the J-H magnitude as 1.14 ⊙ yielding an instability mass ratio range of 0.082 -0.093. As noted above their analysis of the survey photometry suggested a mass ratio of 0.08 for C1928 and they concluded that the system was likely a merger candidate. The current analysis suggests that the primary is likely significantly smaller with a mass of 0.96 ⊙ with a resulting instability mass ratio range of 0.107 -0.126. Our light curve solution indicates a mass ratio considerably higher thus suggesting that the system is likely quite stable and unlikely to be a merger candidate. Similarly, the estimated separation of the system at 2.18 ⊙ is significantly higher than the predicted instability separation range of 1.72 ⊙ − 1.81 ⊙ based on the formulae described in (Wadhwa et al., 2021).
ENERGY TRANSFER AND ANGULAR MOMENTUM LOSS
Energy Transfer and Density
The atypical nature of the secondary component of contact binary systems has long been recognized (Struve, 1948). In particular, the secondary is brighter and larger than the main sequence counterparts. Lucy (1968) suggested that the discrepancy in the brightness and size of the secondaries was due to the transfer of energy from the primary to the secondary within the common envelope. A number of authors have explored the relationship between the mass ratio and luminosity and the transfer of energy between the components (Csizmadia & Klagyivik, 2004;Wang, 1994) with general agreement that energy transfer is a function of both the mass ratio and luminosity ratio. Csizmadia & Klagyivik (2004) introduced the energy transfer parameter defined as:
= 1, 1,(2)
They showed that the transfer parameter can be estimated as (3) and the transfer of the luminosity from the primary to the secondary can be estimated as
Δ = (1 − ) 1 (4)
In the equations is the mass ratio, 1,2 the temperatures of the primary and secondary components, 1 is luminosity of the primary in solar units and Δ the transferred luminosity in solar units.
indicates Zero Age Main Sequence. We estimate the luminosity transfer from the primary to the secondary to be in the order of 0.22 ⊙ ± 0.05 which is quite significant given the intrinsic luminosity of the primary is in the order of 0.66 ⊙ .
The effect of energy transfer on the radius of the secondary was explored by Jiang, Han, Jiang, & Li (2009). They showed that higher the energy transfer rate the greater the impact on the radius of the secondary. As expected given the high luminosity transfer in the system the estimated radius of the secondary 2 = 0.73 ⊙ is nearly double the calibrated main sequence star radius.
Yang & Liu (2001) suggested that the over-luminosity of the secondary is related to the relative higher density of the secondary. The density (in −3 ) of the components can be expressed as a function of the period, radii and mass ratio (Mochnacki, 1981) and it is easy to show that the difference in densities of the components can be expressed as:
Δ = 0.0189 3 2 (1 + ) 2 − 0.0189 3 1 (1 + ) 2(5)
where is the current mass ratio, 1,2 relative radii of the primary and secondary and is the period in days. Δ for C1928 was estimated as 0.017 ± 0.002 −3 indicating near equal density despite vastly different masses and radii.
Angular Momentum Loss
Most contact binary systems have periods of less than 1 day with short period cutoff at approximately 0.22 days. Variation in period has been suggested as a marker of orbital instability, however, variation is commonly observed with decreasing and increasing variations observed equally (Liu, Qian, & Xiong, 2018). Numerous mechanisms such as light time travel effects, apsidal motion, magnetic activity cycles, mass transfer or loss can result in either shortening or lengthening of the period. Long term period decrease however is usually due to angular momentum loss (Liu et al., 2018). C1928 does not have historical high cadence observations to mount a meaningful period study however it is possible to estimate the current potential period decrease due to angular momentum loss (AML).
Bradstreet & Guinan (1994) deduced a theoretical constraint on the rate of AML as per the following equation: ≈ 1.1 × 10 −8 −1 (1 + ) 2 × ( 1 + 2 ) −5∕3 2 × ( 1 4 1 + 2 4 2 ) −7∕3 × 86400 (6) where 1,2 are masses of the primary and secondary in solar units, is the mass ratio, is the period in days, 1,2 are the radii of the primary and secondary components, and is the gyration radius of the primary. The resultant rate change in the period is in seconds yr −1 .
We interpolated the value of the gyration radius ( ) for low mass rotating and tidally distorted stars as described in (Wadhwa et al., 2021). We estimate AML for the system as −0.01 . −1 . If AML was the only source of period change then it would be unlikely any change would have been detected during the 30 or so years of survey observations.
SUMMARY AND CONCLUSIONS
Analysis of survey photometry in the detection of interesting, particularly low mass ratio contact binary systems has received considerable attention recently (Christopoulou et al., 2022;Devarapalli et al., 2020;. Although it is clear that in most cases survey photometric data is suitable and yields acceptable light curve solutions, careful follow-up evaluation of any interesting system is essential as demonstrated by the present study. The current study shows that at least 4 sets of survey data namely, ASAS, ASAS-SN, SuperWASP and CRTS sampled a double star as a single system yielding erroneous estimates of brightness and amplitude of the light curve. When combined with the reported colour estimates, also shown to be erroneous, have previously yielded physical parameter estimations for the system that are far removed from the values presented in this study. Although survey photometric data offers a vast pool data available for analysis and there are numerous examples in the literature of favourable comparisons of survey and dedicated observations yielding comparable results Zheng, Li, & Xia, 2021) care must be taken avoid potential sources of error such as observation scatter and blending. Blending is potentially a common problem given the low resolution of many surveys such as ASAS which has an approximate resolution of 23" (Pojmanski, 2002), ASAS-SN 17"(Shappee et al., 2014), CRTS 9" (Drake et al., 2017) and SuperWASP 60" (Thiemann et al., 2021). Other examples of survey data yielding results not confirmed by other methods include CN Hyi (Özkardeş, Erdem, & Bakış, 2009) and V1187 Her (Caton et al., 2019). In both cases the survey analysis overestimated the mass ratio compared to spectroscopic mass ratio in the case of CN Hyi and dedicated ground based observations in the case of V1187 Her.
In the case of C1928 survey data analysis suggested that the system was of extreme low mass ratio and likely merger candidate. The system actually is a high mass ratio system with a large amplitude. The geometric parameters such as the mass ratio, separation and fill-out are far removed from values indicating orbital instability and we conclude that the system is quite stable and not a potential merger candidate. In other respects the system is similar to other contact binary systems having a secondary that is considerably brighter and larger than the corresponding main sequence counterpart of the same mass. Similarly, the density of the secondary is relatively high and near equal to that of the primary.As the temperature of the secondary is higher than that of the primary, technically the system would be classified as a W-Type. As noted by Csizmadia & Klagyivik (2004) W-type systems are more commonly seen where the mass ratio is greater than 0.35 and as such C1928 is similar to other stable systems.
The present study serves as a cautionary tale as to the reliance on survey photometry for formal light curve analysis and absolute parameter determination. As part of our long term project to formally observe survey identified low mass ratio contact binary systems we suspect we will find other such examples.
54.539) (= ASASSN-V J192848.87-404554.0, 1SWASP J192848.72-404554.6, ASAS J192849-4045.9) was recognised as a contact binary system by the Catalina Real-time Transient Survey (CRTS)
FIGURE 2
2At the location of C1928 there are clearly two stars one slightly brighter.
ACKNOWLEDGMENTSThis work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos .esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos .esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
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| [] |
[
"FDH knockout and TsFDH transformation led to enhanced growth rate of Escherichia coli",
"FDH knockout and TsFDH transformation led to enhanced growth rate of Escherichia coli"
] | [
"Roya Razavipour [email protected] \nDepartment of Biology, Science and Research Branch IAU\n1477893855TehranIran\n",
"† ",
"Saman Hosseini Ashtiani \nDepartment of Biochemistry and Biophysics and Science for Life Laboratory\nStockholm University\n106 91StockholmSweden\n",
"Abbas Akhavan Sepahy \nDepartment of Microbiology\nFaculty of Science\nDepartment of Genetics, Medical School\nNorth Branch IAU\n1651153311TehranIran\n\nFaculty of Industrial and Environmental Biotechnology\[email protected] 5. Department of Systems biotechnology\nTehran University of Medical Sciences\n1416753955TehranIran\n\nfor Genetic Engineering and Biotechnology (NIGEB)\nNational Institute\n1497716316TehranIran\n",
"Mohammad Hossein Modarressi ",
"Arne Elofsson \nDepartment of Biochemistry and Biophysics and Science for Life Laboratory\nStockholm University\n106 91StockholmSweden\n",
"Bijan Bambai [email protected] "
] | [
"Department of Biology, Science and Research Branch IAU\n1477893855TehranIran",
"Department of Biochemistry and Biophysics and Science for Life Laboratory\nStockholm University\n106 91StockholmSweden",
"Department of Microbiology\nFaculty of Science\nDepartment of Genetics, Medical School\nNorth Branch IAU\n1651153311TehranIran",
"Faculty of Industrial and Environmental Biotechnology\[email protected] 5. Department of Systems biotechnology\nTehran University of Medical Sciences\n1416753955TehranIran",
"for Genetic Engineering and Biotechnology (NIGEB)\nNational Institute\n1497716316TehranIran",
"Department of Biochemistry and Biophysics and Science for Life Laboratory\nStockholm University\n106 91StockholmSweden"
] | [] | Background:Increased Atmospheric CO2 to over 400 ppm has prompted global climate irregularities. Reducing the released CO2 from biotechnological processes could remediate these phenomena. In this study, we sought to reduce the released CO2 into the atmosphere from bacterial growth by reducing formic acid conversion into CO2. Since E. coli is the biotechnological workhorse and its higher growth rate is desirable, another goal was to monitor the bacterial biomass after the metabolic engineering.Results:The biochemical conversion of formic acid to CO2 is a key reaction. Therefore, we compared the growth of control strains K12 and BL21, alongside two strains (in which two different genes coding two formate dehydrogenase (FDH) subunits were deleted) in complex and simple media. Our observations demonstrated that the knockout bacteria significantly grew more efficiently than the controls in both media. TsFDH, an FDH with moderately more catalytic efficiency, in contrast to other known FDHs for converting CO2 to formate, increased the growth of both knockouts compared with the controls and the knockouts without TsFDH. This difference was more accentuated in M9+Glycerol. Through a transcriptomics-level in silico analysis of the knockout genes, RNA-seq-based correlation outcome revealed that the genes negatively correlated with the target genes (knockout genes) belong to tRNA-related pathways.Conclusion:Observing higher cell biomass for the knockout and transformed strains at equal concentrations of carbon source in both media indicates possible underlying mechanisms leading to reduced carbon leakage and increased carbon assimilation, which need more detailed investigations. These results may also provide a phenotypic-level clue for the inconsistency of predictions in previous metabolic models that declared glycerol as a suitable carbon source for the growth of E. coli but failed to achieve it in practice. Gene expression correlations and pathway analysis outcomes suggested possible over-expression of the genes involved in tRNA processing and charging pathways. | 10.21203/rs.3.rs-1146279/v1 | [
"https://export.arxiv.org/pdf/2110.01045v3.pdf"
] | 238,259,707 | 2110.01045 | 8e6bc440df81c24b6bf33acf8613b7a05ec566da |
FDH knockout and TsFDH transformation led to enhanced growth rate of Escherichia coli
Roya Razavipour [email protected]
Department of Biology, Science and Research Branch IAU
1477893855TehranIran
†
Saman Hosseini Ashtiani
Department of Biochemistry and Biophysics and Science for Life Laboratory
Stockholm University
106 91StockholmSweden
Abbas Akhavan Sepahy
Department of Microbiology
Faculty of Science
Department of Genetics, Medical School
North Branch IAU
1651153311TehranIran
Faculty of Industrial and Environmental Biotechnology
[email protected] 5. Department of Systems biotechnology
Tehran University of Medical Sciences
1416753955TehranIran
for Genetic Engineering and Biotechnology (NIGEB)
National Institute
1497716316TehranIran
Mohammad Hossein Modarressi
Arne Elofsson
Department of Biochemistry and Biophysics and Science for Life Laboratory
Stockholm University
106 91StockholmSweden
Bijan Bambai [email protected]
FDH knockout and TsFDH transformation led to enhanced growth rate of Escherichia coli
1 * Corresponding Authors: Bijan Bambai, [email protected] Tel: +982144787320 and Saman Hosseini Ashtiani, [email protected], ORCID ID: 0000-0003-2381-3410; Tel.: +46-762623644 † Have equally contributed to the manuscript as the first author 2
Background:Increased Atmospheric CO2 to over 400 ppm has prompted global climate irregularities. Reducing the released CO2 from biotechnological processes could remediate these phenomena. In this study, we sought to reduce the released CO2 into the atmosphere from bacterial growth by reducing formic acid conversion into CO2. Since E. coli is the biotechnological workhorse and its higher growth rate is desirable, another goal was to monitor the bacterial biomass after the metabolic engineering.Results:The biochemical conversion of formic acid to CO2 is a key reaction. Therefore, we compared the growth of control strains K12 and BL21, alongside two strains (in which two different genes coding two formate dehydrogenase (FDH) subunits were deleted) in complex and simple media. Our observations demonstrated that the knockout bacteria significantly grew more efficiently than the controls in both media. TsFDH, an FDH with moderately more catalytic efficiency, in contrast to other known FDHs for converting CO2 to formate, increased the growth of both knockouts compared with the controls and the knockouts without TsFDH. This difference was more accentuated in M9+Glycerol. Through a transcriptomics-level in silico analysis of the knockout genes, RNA-seq-based correlation outcome revealed that the genes negatively correlated with the target genes (knockout genes) belong to tRNA-related pathways.Conclusion:Observing higher cell biomass for the knockout and transformed strains at equal concentrations of carbon source in both media indicates possible underlying mechanisms leading to reduced carbon leakage and increased carbon assimilation, which need more detailed investigations. These results may also provide a phenotypic-level clue for the inconsistency of predictions in previous metabolic models that declared glycerol as a suitable carbon source for the growth of E. coli but failed to achieve it in practice. Gene expression correlations and pathway analysis outcomes suggested possible over-expression of the genes involved in tRNA processing and charging pathways.
Introduction
CO2 is easily formed by the oxidation of organic molecules during respiration in living organisms or combustion in regular mechanical engines. This molecule is thermodynamically stable with a low chemical activity. Today, the atmospheric CO2 concentration is approaching alarming levels (from 300 ppm to 417 ppm in about 50 years). This has led to elevated frequencies of extreme climate conditions like drought, flooding, wild fire and tropical storms in different regions of the world (1). The development of innovative methods for reducing the released CO2 into the atmosphere and assimilating CO2 into organic matter is in demand more than ever (2).
Respiration is a more economical process for extracting chemical energy from organic matter compared with anaerobic fermentation. There is still an undesired side effect of respiration, i.e., the release of CO2. Engineering bacterial strains involved in biotechnological processes with the aim of reducing carbon dioxide release into the atmosphere or even fixing the atmospheric CO2 into biomass has environmental and economic advantages. There have been a number of efforts to reduce the carbon dioxide release during biomass production by metabolic engineering (3). The central pathways and cycles of metabolism are the first targets for manipulating enzymes responsible for critical biochemical reactions or regulatory proteins controlling the expression of certain enzymes to reduce CO2 release (4).
One of the interesting candidates for reducing the amount of released CO2 is formate dehydrogenase (FDH). Theoretically, FDHs are enzymes capable of reversible conversion of CO2 to formate, which is the simplest organic acid (5). However, the major drawback of the biotechnological application of FDHs is the fact that the majority of these enzymes favor the oxidation of formate to produce CO2 under physiological conditions (6). There are three known FDHs in E. coli genome, namely, FdhH, FdhN, and FdhO. The newly identified pressure induced FDH (FHL) is another identified FDH in E. coli. In search of an "ideal" FDH we chose to express FDH from Thiobacillus sp. KNK65MA (TsFDH) as an enzyme with high catalytic efficiency (7). The crystal structure of active enzyme (PDB: 3WR5) shows a homo-tetramer of 406 amino acidlong polypeptide. There are 5 extra residues at N-terminal of the recombinant protein, compared with the sequence in UniProt (8) (accession code: Q76EB7).
In this study we sought to monitor the growth of E. coli fdhD and fdhF knockout strains (JW3866 and JW4040, respectively) with and without TsFDH, as well as control strains, i.e., K12 and BL21. For consistency's sake, we will refer to knockout strains JW3866 and JW4040 as ∆fdhD and ∆fdhF, respectively. Both knockout strains demonstrated growth advantage in LB as well as M9 + glycerol media compared with the control E. coli strains with wild type FDH. Our observations demonstrate a clear growth rate advantage in knockouts expressing the recombinant enzyme (TsFDH) compared with knockouts and controls without TsFDH, particularly in M9+Glycerol medium the difference was higher. In order to perform an in silico study related to our observations, we opted for examining the transcriptomic-level correlations between the target (knockout) genes and the rest of the genes in E. coli. The correlation analysis based on an independent E. coli. RNA-seq gene expression profile data set followed by pathway analysis disclosed that all the genes significantly anti-correlated with both target genes belong to tRNA charging or tRNA processing pathways.
Materials and Methods
Escherichia coli Strains, Plasmids and Media
All E. coli strains and media used in this study are presented in Table 1. Escherichia coli BL21(DE3) was used for the expression of the recombinant FDH from Thiobacillus sp. KNK65MA (TsFDH). Two FDH knockout strains, ∆fdhD and ∆fdhF were purchased from Keio Collection. (10) and the published articles were reviewed and compared in different bacteria (Table S1). This approach revealed some interesting FDHs with relatively better kinetic parameters.
Although, the results obtained by TsFDH might be interesting, we assume there are still some FDHs that deserve attention for replacing the indigenous FDHs of E. coli to improve the growth efficiency. Our mentioned assumption is based on the ambiguity of assay conditions for some of the reported FDHs and the lack of a gold standard for the kinetics comparisons. Scanning the kinetic parameters for a desired FDH suggested Thiobacillus sp. KNK65MA FDH (7).
Amino acid and nucleotide sequences of Thiobacilusis sp KNK65MA formate dehydrogenase were obtained from UniProt (accession # Q76EB7). cDNA of TsFDH was synthetized in pET21a by ZistEghtesadMad based on reference sequence (Q76EB7). Two knockout strains of K12 Escherichia coli, ∆fdhD and ∆fdhF, with the deletion of fdhD and fdhF genes, respectively, were purchased from Dharmacon. The stocks of the Knockout E. coli strains were cultured on LB broth and M9+Glycerol media followed by incubation at 37°C for 24 hours (11).
Strains K12 and BL21 were used as control to compare the growth rates. All strains were cultured at the same time under the same conditions on LB broth media at 37 °C and 200 rpm. Competent cells of the BL21, E. coli ∆fdhF, and E. coli ∆fdhD were prepared as previously mentioned (11). pET21, a plasmid containing a fusion gene to express format dehydrogenase of Thiobacillus sp. KNK65MA (pET+TsFDH), was transformed in competent BL21cells, E. coli ∆fdhD and E. coli ∆fdhF on LB Agar with Amp (100 mg / ml) followed by incubation overnight at 37°C. The colonies containing plasmid were selected and cultured on a 10 ml LB broth with Amp as a primary culture and were incubated at 37°C, 200 rpm for 24 hours. Then the culture was carried out in 200 ml of the LB broth containing Amp (100 mg / ml) and they were incubated at 37°C and at 200 rpm for 24 hours. Also, the bacteria K12, BL21, E. coli ∆fdhD and E. coli ∆fdhF lacking the plasmid were simultaneously cultivated and incubated on LB broth and M9-Glycerol + 50 µg/ml Kanamycin under identical conditions with plasmid-containing strains.
Media and culture conditions
M9 medium + glycerol containing 30 µg kanamycin was used for measuring bacterial growth with and without pET+TsFDH. The same media with ampicillin were also used. In all BL21 samples containing pET+TsFDH, IPTG (0.5 mM final concentration) was added to the medium. LB media were purchased from Merck.
Growth measurements
Bacteria were grown in batch cultures at 37 o C in shaker incubator in 50 mL flasks. 1000 µL samples were taken in triplicate at indicated time intervals and the absorbance was measured at 600nm. During incubation, plasmid-free bacteria and plasmid-containing ones were sampled at different times, namely 0h, 2h, 4h, 6h, 8h, 10h, 12h and 24hr (Table S2). To determine the growth rates of bacteria at above time intervals optical absorption was measured using a spectrophotometer at 600 nm wavelength,.
In silico analysis
Data preparation
To perform a transcriptomic-level study related to our observations, we searched for an independent E. coli expression dataset which could reflect the maximum possible transcriptional variations so that we would be able to achieve significant correlations between as many genes as possible. Moreover, the number of genes involved in the gene expression profile was important to calculate as many correlations as possible. With this aspiration, we fetched an E. coli RNA-seq dataset comprising 152 RNA-seq count samples under 34 different growth conditions (GEO accession GSE94117). These samples were taken from both exponential and stationary phases. One unique aspect of this highly pertinent dataset is the fact that it is sampled under 34 different growth conditions leading to a wider range of differentially expressed genes thanks to different metabolic needs (12). Subsequently, correlation analysis, PCA and pathway analysis were applied to this data set.
Data preprocessing
Using Python version 3.6.1, 152 samples of RNA-seq count files were merge. The counts were converted into count per million (CPM) and were log2 transformed. The resulting data were zscore transformed per gene across all samples. Quality control was performed as sample-level box plots before and after data preprocessing ( Fig. S1 and S2).
Correlation analysis
The Spearman rank-order correlation coefficient (Spearman's ρ), being a nonparametric measure, examines the monotonic relationship between the ordinal values of the variables. Contrary to the Pearson correlation, the Spearman's rank correlation is not based on the assumption that the variables are normally distributed. Spearman correlation coefficient spans between -1 and +1 with 0 indicating no correlation. Correlation coefficients of -1 or +1 imply perfect monotonic relationship. Using the spearman function from the sub-package scipy.stats (13) the correlations between each of the two target genes (knockout genes) and the rest of the genes were calculated. The correlated genes were chosen for further analysis, all of which with FDR-adjusted p-values < 0.01 and |Spearman's ρ| > 0.4.
Principal Component Analysis (PCA)
PCA as a dimensionality reduction technique was used to compare the gene expression profile dispersion of the bacteria based on the variations of their genes' expression levels. "pca" function from mixOmics R package was used for this purpose (14).
Pathway analysis
BioCyc (15) database of microbial genomes and metabolic pathways was used to find the pathways each of the correlated genes are assigned to.
Results
Experimental results
Growth measures of bacterial cells with or without plasmid on LB and M9+Glycerol are presented in Fig. 1 and 2, respectively. Samples were taken within 24 hours at different time intervals. On LB media the preferential growth dynamics of knockout strains with or without TsFDH over control strains were recognizable more clearly after eight hours post inoculation up to 24 hours. In samples grown on M9+Glycerol, we observed a pronounced shift in the growth divergence of FDH knockout strains with or without TsFDH from control strains during earlier time intervals up to 24 hours compared with LB medium. On M9+Glycerol, BL21 with TsFDH showed higher growth up to eight hours compared with the standard BL21. Moreover, on M9+Glycerol, both knockouts with TsFDH showed relatively higer growth rates from 12 hours onward compared with the respective growth rates on LB. SDS-PAGE analysis of the strains confirms the expression of TsFDH under the experimental conditions (Fig. S3).
Growth comparisons in LB
Figure 2. Growth rate comparison among bacteria on M9+Glycerol medium at OD 600 nm at different time points. At earlier hours, unlike on LB medium, on M9+Glycerol the growth rates of the bacterial species with TsFDH are vividly higher than those without TsFDH except for K12. Particularly from 12 hours onward, the difference between the knockouts with and without TsFDH is relatively more accentuated compared with the growth rates on LB at the same time points.
In silico analysis results correlation analysis:
All correlations with each of the knockout genes were calculated (Table S3 and S4) and the ones with FDR-adjusted p-values < 0.01 were chosen (Table S5 and S6).
PCA analysis:
According to the PCA results (Fig. S4a), it is postulated that the top anti-correlated genes for both knockouts are closely associated with one another. As a comparison, the PCA plot was also generated using all the genes, indicating that the other genes show more expression divergence. Moreover, the second PCA plot (Fig. S4b) could be an indication that the dataset, being based on different growth media, reflects a wide range of expression levels for different genes in each data point, which is critical for reflecting the correlations between the fluctuating gene expression levels.
Growth comparisons in M9+Glycerol
Metabolic pathways
Since the knockout of FDH main subunits is synonymous to absolute down regulation of the FDH gene in knockout strains, we initially focused on the significantly negatively correlated genes, which may reveal the genes that undergo up regulation accordingly. Using BioCyc database of microbial genomes and metabolic pathways, all the significantly negatively correlated genes (with FDR adjusted p-values < 0.01) were shown to be involved in tRNA charging pathway and tRNA processing pathway (PWY0-1479). To evaluate whether the mentioned pathways, are more frequent among the negatively or positively correlated gene, all the positively correlated genes assigned to the same pathways, i.e., tRNA charging and tRNA processing pathways, were selected for comparison. All the pathways other than the two mentioned pathways for both negatively and positively correlated genes were categorized as background pathways and were named "All other pathways". For better visualization of the distribution of all positive and negative correlation coefficients for both mentioned pathways the corresponding density plots were used (Fig 3. and 4.). The comparisons of the percentages of the number of correlated genes in each pathway category are given for both fdhD and fdhF genes ( Table 2).
Discussions
Increasing the growth efficacy of industrially important microorganisms is a novel goal in biotechnological applications. One of the strategies to boost the bacterial growth rate is to reduce the organic carbon leak, i.e., the release of CO2 as one of the main end products in the respiration process. Different E. coli, strains are the workhorse for the production of some well-known biopharmaceuticals, like G-CSF, Romiplostim and Asparaginase. Therefore, E. coli is a suitable model microorganism and developing a strain of E. coli with higher growth rate is in demand particularly in Biotechnology.
Since the Kelvin cycle is the mainstream of CO2 fixation pathway in plants, algae and cyanobacteria, most engineering efforts are directed towards the Kelvin cycle for converting CO2 into valuable materials. Heterotrophic microorganisms generally do not assimilate CO2 through the central metabolism (16). Over the past decade, there has been great success in the production of CO2 derivatives, which have the potential to be used as fuel and valuable chemicals by bacteria (17). Previously, other researchers have approached this challenge by defining fermentation conditions and controlling aeration rates (18) or with genetically overexpressing ArcA transcription factor (19).
Here, we introduce a new approach by targeting one of the main enzymes responsible for converting organic formate into inorganic wasteful CO2, i.e., FDH. There are three known FDHs in E. coli, namely, respiratory FDH, anaerobically expressed FDH and newly identified pressure induced FDH (FHL). FDHF is the cytosolic form, while FDHN and FDHO are membrane bound, with FDHN responsible for nitrogen cycle and FDHO active in sulfur metabolism (20). All these enzymes prefer the oxidation of formate into CO2 under physiological conditions. Scanning BRENDA for FDHs with tendency towards the production of formate from CO2 revealed that there are few candidate FDHs with relatively higher formate production (CO2 reduction) catalytic efficiency. A comprehensive search of E. coli's metabolism using the Regulon DB (21) showed that the formate dehydrogenase enzymes were successfully expressed in recombinant form in E. coli (22). According to a study (16), E. coli's FDHs have a strong tendency for regenerating CO2 from formate. Among the studied formate dehydrogenases, TsFDH has potential advantages as a biocatalyst in the field of CO2 reduction (7). We hypothesized that perhaps the E. coli strains harboring TsFDH could lead to developing bacterial strains for biotechnological applications with higher biomass to carbon source ratio owing to the possible lowering of carbon dioxide leakage. This possible solution has so far been remained out of sight and to the best of our knowledge not tried yet.
We expressed the recombinant fdhD gene from Thiobacillus in Escherichia coli strain K12 and BL21. In order to better compare the role of TsFDH in growth efficacy on similar growth conditions (media), we also transformed two FDH knockout strains from K12 and BL21 with pET+TsFDH. Significant growth rate differences were observed between the knockout strains and the recombinant knockouts containing TsFDH on either growth medium. On the other hand, both knockouts with TsFDH showed vividly higher growth rate starting from 8 hours all the way to 24 hours on both media post induction. Additionally, the growth advantage of the control strains (K12 and BL21) with TsFDH lasted up to 12 hours on both media. A possible reason for the less increase in growth rate of BL21 cells transformed with TsFDH plasmid compared with the transformed knockouts is the presence of the original FDH in these strains. BL21 also contains IPTG as a gene expression inducer thanks to having a chromosomally encoded bacteriophage T7 RNA polymerase (T7 RNAP), which may be responsible for some metabolic leakage leading to less growth compared with K12-derived knockouts and transformed knockouts (23). The original K12 cells and mutants derived from this cell lack the complementary T7 RNA-polymerase.
In this study, we showed that removing either subunit of the wild type formate dehydrogenase gene from E. coli and its replacement with the formate dehydrogenase gene from Thiobacillus Sp. KNK65MA can increase the growth rate of E. coli cells. In a previous study by Palsson et al., the whole-cell in silico model of E. coli metabolic network predicted that glycerol should be a preferred carbon source over glucose. However, the experimental findings were not consistent with the mentioned predictions. They indicated the adaptive evolution phenomenon for the bacteria to go from sub-optimal to the predicted optimal growth rate on glycerol (24)(25)(26). Concerning our results, the replacement of the native FDH with TsFDH might lead to potential metabolic rewirings leading to an increased glycerol efficiency as a carbon source. Our outcomes may suggest an initial clue to start more mechanistic metabolic investigations such as flux balance analysis and CO2
leakage measurements to address these discrepancies between the in silico predictions and the experimental outcomes. Considering our in silico outcomes, we hypothesised that the omission of the original E. coli FDH may ultimately lead to the increased expression of some genes playing roles in tRNA charging and processing pathways. These in silico findings could be considered as a gene expression-level study related to our experimental observations; nonetheless, more detailed investigations are necessary to verify the hypotheses arising from our results.
Conclusions
The main problems with studying most FDHs published so far are protein instability, sensitivity to oxygen and the low conversion rate. The FDH of this study (TsFDH) shows some biochemical advantages over the previously studied FDHs such as Candida bolidini's FDH including higher turnover number and insensitivity to the environmental oxygen (17,27,28).
We showed that both knockouts and transformation with TsFDH leads to an overall increased growth in all strains. In the initial incubation periods, the growth rates were approximately equal between transformed and non-transformed knockouts on LB medium, while in 8-24h the growth rates of the transformed and knockout bacteria were much higher than those of controls in both media, which implies the negative effect of wild type FDH gene on the E. coli growth rate. In other words, there would be a higher growth rate by eliminating the wild type FDH chains.
The increased growth rate of the transformed knockouts on both media might be a consequence of decreased CO2 leakage due to less formate oxidation by TsFDH compared with the wild type FDH. One plausible hypothesis regarding the relatively more accentuated growth of the transformed knockouts on glycerol medium could be the preferred utilization of glycerol after knockout and transformation besides the probable decrease of CO2 leakage due to the above-mentioned conjecture. To evaluate these hypotheses, further studies are necessary such as CO2 absorption/emission measurements and flux balance analysis (FBA). These observations could also be a starting point for more sophisticated molecular-level studies on TsFDH contribution to the growth efficiency of E. coli on glycerol as the carbon source.
List of abbreviations
Figure 1 .
1Growth rate comparison among bacteria on LB medium at OD 600 nm at different time points. At earlier time points there is no considerable difference between the bacterial species with and without TsFDH. From eight hours onward, there is a significant difference between all bacterial species with and without TsFDH.
Figure 3 .
3Correlations distribution for fdhF. The distribution of all genes' correlations with fdhF for each pathway. The gold density plot represents the probability density of achieving either tRNA charging or tRNA processing for the corresponding correlation coefficients on the x axis. The gray density plot represents the probability density of getting any pathway other than the two mentioned ones for the corresponding correlation coefficients on the x axis.
All Genes Correlations with fdhF by Pathway
Figure 4 .
4Correlations distribution for fdhD. The distribution of all genes' correlations with fdhD for each pathway. The gold density plot represents the probability density of achieving either tRNA charging or tRNA processing for the corresponding correlation coefficients on the x axis. The gray density plot indicates the probability density of getting any pathway other than the two mentioned ones for the corresponding correlation coefficients on the x axis.
Table 1 .
1Bacterial strains and media. All bacterial strains and plasmids used in this study and their characteristics as well as the sources they were obtained. //www.genome.jp/pathway/map01200+C00058). Using the results from KEGG pathway search for all the carbon fixation reactions, the contributing FDHs were identified. The kinetic parameters including Kcat and Km of FDHs (EC: 1.17.1.9) for formate formation were obtained from Brenda enzyme data bankStrains and plasmids
Related characteristics
Source
Strains
E. coli K12
E. coli BL21(DE3)
E. coli JW3866
E. coli JW4040
Plasmids
pET-21α
pET-21α-TsFDH
Wild type
[lon] ompT gal (λ DE3) [dcm] ∆hsdS
K12 ∆fdhD
K12 ∆fdhF
Ap R , T7 promoter, lac operator
pET-21α, containing TsFDH gene from Thiobacilusis sp
KNK65MA
NIGEB stocks
Invitrogen
Dharmacon
Dharmacon
Novagen
This research
M9 medium with glycerol as carbon source and LB medium as a complex medium all containing
30 µg kanamycin were used for measuring bacterial growth. For BL21 with pET+TsFDH, the
same media with ampicillin were used. In all samples containing pET+TsFDH, IPTG (0.5 mM
final concentration) was added to the medium. The metabolic reactions consuming or producing
formate
(map01200
and
C00058)
were
obtained
from
KEGG
(9)
(https:
Table 2 .
2Correlations with fdhF and fdhD. Comparison of the number of positively and negatively correlated genes assigned to the unique set of pathways.Knockout gene
Pathway
Negatively
correlated genes
Positively
correlated genes
fdhF
t-RNA charging or
processing
96%
1%
All other pathways
4%
99%
fdhD
t-RNA charging or
processing
100%
0.8%
All other pathways
0%
99.2%
PCA: Principal Component Analysis; FDH: Formate Dehydrogenase; TsFDH: Thiobacillus sp. KNK65MA; DE3: Escherichia coli BL21; CPM: Count Per Million Ethics approval and consent to participate -Not applicable.Declarations
Consent for publication
-Not applicable.
AcknowledgementsWe acknowledge the fund by EU-ITN project ProteinFactory (MSCA-ITN-2014-ETN-642836) and the Swedish Research Council (Grant 2016-03798). We thank Payam Emami, Rui Benfeitas and Paulo Czarnewski at the National Bioinformatics Infrastructure Sweden (NBIS) at SciLifeLab for their fruitful discussions and help with the RNA-seq data analyses. We also thank Roghaieh Ghaderi Ternik for her help.Authors' contributionsRR and BB conducted experiments. AAS and MHM helped with the design of experiments. SHA conceived and executed the bioinformatics analysis sections. AE helped with the bioinformatics analyses and visualizations. BB, RR and SHA wrote the manuscript. All authors read and approved the manuscript.Funding-International Cooperation for Applied Research Development (ICARD) grant to BB.Availability of data and materials -All the data supporting the conclusions of this paper are included in the context of the paper and the additional files.Competing interests -Not applicable.
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| [] |
[
"QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS",
"QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS",
"QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS",
"QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS"
] | [
"Gregory Natanson correspondence:[email protected] \nAI-Solutions Silver Spring MD 20904\nU.S.A\n",
"Gregory Natanson correspondence:[email protected] \nAI-Solutions Silver Spring MD 20904\nU.S.A\n"
] | [
"AI-Solutions Silver Spring MD 20904\nU.S.A",
"AI-Solutions Silver Spring MD 20904\nU.S.A"
] | [
"Acta Polytechnica",
"Acta Polytechnica"
] | The paper advances Odake and Sasaki's idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including 'juxtaposed' pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states. | 10.14311/ap.2022.62.0100 | [
"https://arxiv.org/pdf/2110.13913v1.pdf"
] | 239,998,736 | 2110.13913 | 3094ee52d1ecf7542a9d38fdb1727ab56d594c20 |
QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS
2022
Gregory Natanson correspondence:[email protected]
AI-Solutions Silver Spring MD 20904
U.S.A
QUANTIZATION OF RATIONALLY DEFORMED MORSE POTENTIALS BY WRONSKIAN TRANSFORMS OF ROMANOVSKI-BESSEL POLYNOMIALS
Acta Polytechnica
621202210.14311/AP.2022.62.0100Translationally form-invariant Sturm-Liouville equationgeneralized Bessel polynomialsRomanovski-Bessel polynomialsrational Darboux-Crum transformationspolynomial Wronskians
The paper advances Odake and Sasaki's idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including 'juxtaposed' pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states.
Introduction
In recent publication [1] Alhaidari pointed to a new form of 'quasi-rational' [2] solutions (q-RSs) of the Schrödinger equation with the Morse potential in terms of generalized Bessel polynomials [3][4][5][6], instead of using the conventional q-RSs composed of weighted Laguerre polynomials [7][8][9][10][11]; though, to be more accurate, the possibility to quantize the Morse potential by Romanovski-Bessel (R-Bessel) polynomials [12,13] has been already recognized by Quesne [14], with reference to Cotfas' papers [15,16] (see also [17]). It should be also emphasized that Odake and Sasaki in their in-depth study [18,19] on rational Darboux-Crum [20,21] transforms (RDCT s) of translationally shape-invariant (TSI) potentials did implicitly express eigenfunctions of the Morse potential in terms of R-Bessel polynomials with degree-independent indexes as a substitute for commonly used classical Laguerre polynomials [22]. (Though the Bochner-type differential equation for generalized Bessel polynomials was also listed in Table 1 in [7] on the line linked to the Morse potential the authors used the conventional representation for eigenfunctions [22] to construct rationally deformed Morse potentials.)
The remarkable feature of the new rational realization for the Morse oscillator is that the resultant rational canonical Sturm-Liouville equation (RCSLE) can be converted by an energy-independent gauge transformation to the Bochner-type eigenequation with a linear coefficient function of the first derivative independent of degrees of sought-for polynomial solutions. Using terminology of our recent study [23] on translationally form-invariant (TFI) CSLEs this implies that the given RCSLE belongs to TFI Group A and we should give full credit to Odake and Sasaki [18,19] who initially came up with this breakthrough idea to treat the Morse oscillator as a rational TSI potential of Group A.
Keeping in mind that the TFI equation under consideration has only two basic solutions the net of its RDCT s is uniquely specified by a single series of Maya diagrams [24] and therefore any rationally deformed Morse potential can be re-expressed in terms the Wronskian of generalized Bessel polynomials with a common degree-independent index, as it has been done in [19] though in slightly different terms. The novel representation of seed eigenfunctions [19] is in a sharp contrast with their conventional representation in terms of classical Laguerre polynomials with degree-dependent indexes [10,11].
The main purpose of this work is to present new simplified expressions for eigenfunctions of the Schrödinger equation with a rationally deformed Morse potential by re-writing them in terms of finite exceptional orthogonal polynomial (EOP) sequences formed by Wronskian transforms of R-Bessel polynomials.
TFI Sturm-Liouville equations
Liouville-Darboux transformations
Let ϕ τ [ξ; Q] be a solution of the generic CSLE
d 2 dξ 2 + I 0 [ξ; Q] + ε τ (Q)ρ[ξ] ϕ τ [ξ; Q] = 0(1)
at an energy ε τ (Q), where the index τ specifies the factorization function (FF) in question. In the problems of our current interest I 0 [ξ; Q] is a rational function of ξ termed 'reference polynomial fraction' (RefPF). We prefer to keep this notation in the general case when I 0 [ξ; Q] is an arbitrarily chosen real function of ξ also dependent on some parameters Q. We will replace Q by ⇀ a , b after restricting the analysis solely to TFI CSLEs. In [23] we identified four families of RefPFs associated with rational TSI potentials termed 'Jacobi', 'Laguerre', 'Routh' and 'Bessel' (or J Ref, L Ref, RRef, and BRef for briefness) so the corresponding q-RSs are composed of polynomials (with degree-dependent indexes in general) from one of four conventional differential polynomial systems (DPSs) [25,26]. The density function ρ[ξ] plays a crucial role in our analysis because, as indicated by Eq. (4) below, it determines the change of variable converting CSLE (1) to the Schrödinger equation [27,28].
It was Rudjak and Zakhariev [29] who extended the intertwining technique [30] from the Schrödinger equation to the CSLE. Here we however use a slightly different definition of the socalled [31,32] 'generalized' Darboux transformations introducing them via the requirement that the function
* ϕ τ [ξ; Q] ∝ ρ −1/2 [ξ]/ϕ τ [ξ; Q](2)
is a solution of the transformed CSLE at the same energy ε τ (Q), i.e.,
d 2 dξ 2 + I 0 [ξ; Q | τ ] + ε τ (Q)ρ[ξ] * ϕ τ [ξ; Q] = 0.(3)
Rudjak and Zakhariev's reciprocal formula (2) thus plays a crucial role in our approach to the theory of TFI CSLEs.
Since various authors give the term 'generalized Darboux transformation' completely different meanings it seems preferable to refer to these operations as 'Liouville-Darboux' transformations keeping in mind that they can be performed in three sequential steps: (1.) The Liouville transformation ξ(x):
ξ ′ (x) = ρ −1/2 [ξ(x)](4)
from the CSLE
d 2 dξ 2 + I 0 [ξ; Q] + ερ[ξ] Φ[ξ; Q; ε] = 0(5)
to the stationary 1D Schrödinger equation with the potential [27,28]
V [ξ(x); Q] = −ρ −1 [ξ(x)]I 0 [ξ(x); Q] − 1/2 {ξ, x}(6)
where {ξ, x} stands for the 'Schwarzian derivative';
(2.) the Darboux deformation of Liouville potential (6) using the FF
ψ τ (x; Q) = ρ 1/4 [ξ(x)]ϕ τ [ξ(x); Q];(7)d 2 dξ 2 + I 0 [ξ; ⇀ a , b] + ε ±,0 ( ⇀ a , b)ρ[ξ] ϕ ±,0 [ξ; ⇀ a , b] = 0 (9)
related via the following reciprocal formulas:
ϕ ±,0 [ξ; ⇀ a ± ⇀ 1 , b] = ρ −1/2 [ξ]/ϕ ±,0 [ξ; ⇀ a , b].(10)
It has been proven [23] that
I 0 [ξ; ⇀ a , b | ±, 0] = I 0 [ξ; ⇀ a ± ⇀ 1 , b] + E ±1 ( ⇀ a , b)ρ[ξ],(11)
where
E ±1 ( ⇀ a , b) ≡ ε ∓,0 ( ⇀ a ± ⇀ 1 , b) − ε ±,0 ( ⇀ a , b).(12)
The
V [ξ; ⇀ a , b | +, 0] = V [ξ; ⇀ a + ⇀ 1 , b] − E +1 ( ⇀ a , b) (13) or V [ξ; ⇀ a , b | −, 0] = V [ξ; ⇀ a − ⇀ 1 , b] − E −1 ( ⇀ a , b)(14)
depending on which basic solution ϕ +,0 [ξ;
⇀ a , b] or ϕ −,0 [ξ; ⇀ a , b]
represents the lowest energy eigenfunction. Note that the Russian word 'форма' used by Gendenshtein [34] has two meanings 'form' and 'shape'. The term 'form invariant' with reference to CSLEs was adopted by us from the English translation of Gendenshtein's joint paper with Kreve [35] while the commonly accepted term 'shape-invariance' is preserved for the corresponding Liouville potentials. The shift of the translational parameters ⇀ a by 1 thus retains the analytical form of the TFI CSLE while preserving the 'shape' of its Liouville potential. It is true that the Liouville transformation of the TFI CSLE results in a 'translationally shape-invariant (TSI) potential. However the Class of TFI SLEs is defined via (10) with no reference to the associated Schrödinger equation.
Equivalence theorem for Darboux-Crum transforms of a TFI CSLE with two basic solutions
It has been proven [23] that any TFI CSLE has at least two infinite sets of solutions
ϕ ±,m+1 [ξ; ⇀ a , b] = ρ −1/2 [ξ]W [ξ; ⇀ a ± ⇀ 1 , b | ∓, 0; ±, m]/ϕ ±,0 [ξ; ⇀ a ± ⇀ 1 , b],(15)
where
W [ξ; ⇀ a , b | ±, m; ∓, m ′ ] ≡ W ϕ ±,m [ξ; ⇀ a , b]ϕ ∓,m ′ [ξ; ⇀ a , b] .(16)
The cited 'raising' recurrence relations can be conveniently re-written as
f ±,m+1 [ξ; ⇀ a ± ⇀ 1 ; b] =ḟ ±,m [ξ; ⇀ a , b],(17)
where
f ±,m+1 [ξ; ⇀ a , b] ≡ ϕ ±,m [ξ; ⇀ a , b]/ϕ ∓,0 [ξ; ⇀ a , b](18)
and dot denotes the first derivative with respect to ξ.
The solutions ϕ ±,m [ξ;
⇀ a , b] also obey the´lovering´recurrence relations:
ϕ ±,m [ξ; ⇀ a , b] ≡ ρ −1/2 [ξ]w[ξ; a, b | ± . . .0, m]/ϕ ±,0 [ξ; ⇀ a , b] = = −E ±,m−1 ( ⇀ a ± ⇀ 1 , b)ϕ ±,m−1 [ξ; ⇀ a ± ⇀ 1 ; ⇀ b ] for m ≥ 1,(19)
where
E ±,m ( ⇀ a , b) ≡ ε ±,m ( ⇀ a , b) − ε ∓,0 ( ⇀ a , b).(20)
Solutions from both infinite sets can be then used as seed functions for Darboux-Crum transformations (DCTs) of the given TFI CSLE which results in an infinite net of solvable SLEs specified by a single series of Maya vol. 62 no. 1/2022
Quantization of rationally deformed Morse potentials. . . diagrams [24]. Following the arguments presented in [33] for rationally deformed TSI potentials we [23] proved that any CSLE in this net can obtained using only seed solutions of the same type. Let us parametrize a set of seed functions of the same type,
M (∆ 1→L ) = m 1 , . . . , m |δ1→L| ,(21)
by two partitions of an equal size L:
∆ 1→L ≡ δ 1→L ; δ ′ 1→L(22)
such that
m k = δ ′ 1 + k − 1 for 1 < k ≤ δ 1 ,(23)m |δ 1→l−1 |+1 = m |δ 1→l−1 | + δ ′ l + 1 = | ∆ 1→l−1 | +δ ′ l + 1 for 1 < l ≤ L,(24)m |δ 1→l−1 |+k = m |δ 1→l−1 |+1 + k − 1 for 1 < l ≤ L, 1 < k ≤ δ l ,(25)
One can easily verify that the largest element in partition (21) coincides with the sum of the partition lengths | δ 1→L | and | δ ′ 1→L |, i.e.,
m |δ 1→L | =| ∆ 1→L |≡| δ 1→L | + | δ ′ 1→L | .(26)
It has been proved in [23] that use of the conjugated set of seed solutions of opposite type,
∆ ′ L→1 ≡ δ ′ L→1 ; δ L→1 ≡ δ ′ L , δ ′ L−1 , . . . , δ ′ 1 ; δ L , δ L−1 , . . . , δ 1 ,(27)
results in an equivalent CSLE so the corresponding Liouville potential V [ξ;
⇀ a (δ) , b | ∓, M (∆ ′ L→1 )] computed at shifted values of the translational parameters, ⇀ a (δ) ≡ ⇀ a + δ ⇀ 1 ,(28)
where δ is a nonzero integer, differs from the Liouville potential V [ξ;
⇀ a , b | ±, M (∆ ′ 1→L )]
only by a zero-point energy.
In [23] we have derived the following relation between the Wronskians of two equivalent sets of seed solutions of the same type
w[ξ; ⇀ a , b | + . . .M (∆ 1→L )] ρ 1/4|δ 1→L |(|δ 1→L |−1) [ξ] L l=1 χ −δ l [ξ; ⇀ a (|∆ ′ l→1 |−δ l ) , b] ∝ w[ξ; ⇀ a (|∆ ′ L→1 |) , b | − . . .M (∆ ′ L→1 )] ρ 1/4|δ ′ L→1 |(|δ ′ L→1 |−1) [ξ] L l=1 χ δ l [ξ; ⇀ a (|∆ ′ l−1→1 |+δ ′ l ) , b] ,(29)
where
χ ∓|N | [ξ; ⇀ a, b] ≡ |N |−1 k=0 ϕ ±,0 [ξ; ⇀ a (±k) , b].(30)
For any CSLE from Group A the derived relation turns into the equivalence relations between the Wronskians of the corresponding seed polynomials discovered in the breakthrough paper by Odake and Sasaki [19]. We illuminate these relations in more details in subsection 3.4 below using Wronskians of generalized Bessel polynomials as an example.
If the given rational TSI potential has only a finite number of eigenfunctions then the set of seed functions +, m or −, m which starts from these eigenfunctions (−, m in case of our current interest) also contains infinitely many q-RSs vanishing at only one quantization end (virtual state wavefunctions in Odake and Sasaki's terms [18,19]), with the Gendenshtein (Scarf II) potential [34] as the sole exception (including its symmetric limit represented by the sech-squared potential well). The DCTs using nodeless q-RSs of the selected type results in a net of isospectral potentials. Therefore, except for the Gendenshtein potential, we don't need to include 'state-inserting' solutions ('pseudo-virtual state wave functions' in Odake and Sasaki's terms) into the given set of seed functions-a remarkable corollary of the 'extended' Krein-Adler theorem [11].
If the given partition ∆ 1→L is composed of alternating even and odd integers staring from an even integer δ ′ 1 then all the integers
δ ′ l = m |δ1→1|+1 − m |δ1→1| − 1 > 0 for any l < L(31)
must be also even which implies that the set of seed solutions ±, M (∆ ′ L→1 ) is composed of L segments of even lengths [11,19] or in other words is formed by 'juxtaposed' [36][37][38] pairs of seed solutions ±, m ′ , ±, m ′ + 1. Similarly if the set of seed solutions, ±, M (∆ ′ ) L→1 is formed by 'juxtaposed' pairs of seed solutions ±, m, ±, m+1 then the conjugated set is formed by seed solutions ∓, m ′ with only even gap lengths, again starting from an even number. We refer the reader to subsection 3.4 below for a scrupulous analysis of this issue in connection with juxtaposed pairs of eigenfunctions of the Schrödinger equation with the Morse potential in the BRef representation [19].
3.
Quantization of rationally deformed Morse potentials by Wronskian transforms of R-Bessel polynomials
Schrödinger equation with Morse potential in Bessel form
In this paper we focus solely on the TFI CSLE
d 2 dy 2 + I 0 [y; a] + ε ∞ ρ ⋄ [y] ∞ Φ[y; a; ε] = 0(32)
with the RefPF
I 0 [y; a] = 2ay −3 − y −4 + 1/4y −2(33)
and the density function
∞ ρ ⋄ [y] ≡ ∞ σ −1 [y] = y −2(34)
One can directly verify that CSLE (32) has a pair of 'basic' solutions
∞ ϕ ±,0 (y; a) = y 1±a e ±1/y (y > 0)(35)
at the energies
∞ ε ±,0 (a) = −(a ± 1/2) 2 .(36)
Examination of solutions (35) shows that they obey the following symmetry relations
∞ ϕ ±,0 [y; a + k] = y ±k ∞ ϕ ±,0 [y; a](37)
for any integer k and
∞ ϕ +,0 (y; a) ∞ ϕ −,0 (y; a) = y 2(38)
whereas the function
f ±,0 [ξ; ⇀ a , b] ≡ ϕ ∓,0 [ξ; ⇀ a , b]/ϕ ±,0 [ξ; ⇀ a , b](39)
takes form
∞ f ±,0 [ξ; a] ≡ y ∓2a e ∓2/y .(40)
We thus proved that the pair of basic solutions in question satisfy the TFI condition [23]
∞ ϕ ∓,0 [y; a ± 1] = ∞ ρ −1/2 ⋄ [y]/ ∞ ϕ ±,0 (y; a).(41)
One can directly verify that
∞ ε ∓,0 (a ± 1) = ∞ ε ±,0 (a)(42)
and thereby
∞ E ±1 (a) ≡ ∞ ε ∓,0 (a ± 1) − ∞ ε ±,0 (a) = 0 (43) so the symmetry condition [23] E ∓1 (a ± 1) = −E ±1 (a).(44)
trivially holds. The gauge transformations
∞ Φ[y; a; ε] = ε ϕ ± [y; a] ∞ F ± [y; a; ε](45)
convert CSLE (32) to a pair of Bochner-type eigenequations
y 2 d 2 dy 2 + ∞ τ ± [y; a] d dy + [ε − ∞ ε ±,0 (a)] ∞ F ± [y; a; ε] = 0,(46)
with
∞ τ ± [y; a] = 2(1 ± a)y ∓ 2.(47)
vol. 62 no. 1/2022
Quantization of rationally deformed Morse potentials. . .
We define generalized Bessel polynomials as
Y (α,β) n (y) ≡ Y (α) n (y/β),(48)
where the polynomial Y (α)
n (x) is given by (2) in [4] and thereby coincides with polynomial (9.13.1) in [6]
Y (α) n (x) ≡ y n (x; α).(49)
Note that Chihara's relation (4.3) in [5] is apparently based on Brafman's definition [39] for the polynomial y n (x; α, β) such that y n (x; α + 2, 2) = y n (x; α). Adding the second index to the conventional notation [4,5] allows us to avoid uncertainties in the definition of the variable used to differentiate a polynomial in the reflected argument, keeping in mind that
Y (α) n (−y) ≡ Y (α,−2) n (y).(50)
Eq. (37) for the Bessel DPS in [40] thus corresponds to the polynomials Y (α−2,β) n (y) in our terms. (We prefer to preserve symbol 'B' for their orthogonal subset composed of R-Bessel polynomials [12,13].) It is also worth mentioning that Alhaidari [1] introduced a slightly modified notation for generalized Bessel polynomials:
J a n (1/2y) ≡ Y (2a) n (y) = (2n + 2a) n (y/2) n 1 F 1 (−n; −2a − n; 2/y),(51)
with the Pochhammer symbol (a) n standing for the falling factorial. And indeed it would be possibly more convenient to use the parameter a as the polynomial index keeping in mind that the forward and backward shift relations change the polynomial index by 1. However we prefer to stick to the more conventional notation. The basic solution ∞ ϕ ±,0 [y; a] is thus nothing but a constant solution of eigenequation (46) converted back by gauge transformation (45). Similarly the reverse gauge transformation of each of the DPSs composed of polynomials Y (±2a,∓2) m (y) results in pairs of infinite sequences of q-RSs of CSLE (32):
∞ ϕ ±,m [y; a] = ∞ C ±,m (a) ∞ ϕ ±,0 [y; a]Y (±2a,∓2) m (y).(52)
The multiplier l C ±,m will be chosen below in such a way that q-RSs (52) satisfy recurrence relations (15). The crucial advantage of expressing q-RSs in terms of generalized Bessel polynomials, instead of Laguerre polynomials [7][8][9][10][11], is that the weight function ∞ ϕ ±,0 [y; a] in the right-hand side of (52) does not depend on the polynomial degree -the direct consequence of the fact that the given TFI CSLE belongs to Group A [18,19,23], in contrast with the conventional representation of eigenfunctions of the Schrödinger equation with the Morse potential in terms of classical Laguerre polynomials [22]. According to the general theory of Bochner-type eigenequations [41] differential equation (46) has a polynomial solution of degree m at
ε = ∞ ε ±,m (a) = ∞ ε ±,0 (a) − m[2(1 ± a) + m − 1],(53)
which, coupled with (36), gives
∞ ε ±,m (a) = −(m + 1/2 ± a) 2 .(54)
This brings us to the simplified version of the raising ladder relations [23] for the energies of q-RSs (15):
∞ ε ±,m+1 (a) = ∞ ε ±,m (a ± 1)(55)
with E ±1 (a) ≡ 0 .
To be historically accurate, it is worth mentioning that Cotfas' Eq. (10) in [16] with the leading coefficient
σ(s) = s 2 does list Al-Salam's [4] formula Y (α) n (y) = n! (−y/2) n L (−α−2n−1) n (2/y)(56)
for the generalized Bessel polynomials in terms of Laguerre polynomials in the reciprocal argument 2/y (though without mentioning the former polynomials by name). Actually Cotfas discusses only eigenfunctions of the corresponding Sturm-Liouville problem so the cited formula specifies R-Bessel polynomials expressed in terms of classical Laguerre polynomials in 2/y:
B (A) n (y) ≡ Y (−2A−1) n (y) = n! (−y/2) n L (2A−2n) n (2/y) for n < A,(57)
with Cotfas' parameter α standing for 1 − 2A here. The remarkable feature of this finite subsequence of generalized Bessel polynomials is that the polynomials in question are orthogonal on the positive semi-axis as prescribed by orthonormality relations (9.13.2) in [6]:
∞ 0 ∞ ρ ⋄ [y] ∞ ϕ 2 −,0 [y; A + 1/2]B (A) n (y)B (A) n (y)dy ≡ ∞ 0 y −2A−1 e −2/y B (A) n (y)B (A) n (y)dy = n! Γ(2A + 1 − n) 2A − 2n − 1 δ nñ .(58)
Making use of (39) we can represent backward shift relation (9.13.8) in [6] as d dy
î ∞ f +,0 [ ξ; a] Y (−2a,2) m (y) ó = 2 ∞ f +,0 [ξ; a + 1]Y (−2a−2,2) m+1 (y)(59)
so the functions
∞ f +,m [ξ; a] = ∞ C −,m (a) ∞ f ∞,0 [ξ; a]Y (−2a) m (y)(60)
satisfy raising relation (17) provided we choose
∞ C −,m+1 (a) = 2 ∞ C −,m (a − 1) ≡ 2 m+1(61)
keeping in mind that ∞ C −,0 (a) ≡ 1. Substituting (54) into (20) gives
∞ E −,m−1 (a − 1) = −m(m + 1 − 2a)(62)
so recurrence relation (19) can be re-written as
2 m yẎ (−2a,2) m (y) = m(m + 1 − 2a) ∞ ϕ −,m−1 [y; a − 1]/ ∞ ϕ −,0 [y; a].(63)
Combining (52), (61), and (37) with k = 1 brings us to 'forward shift operator' (9.13.6) in (6)
Y (−2a,2) m (y) = 0.5m(m + 1 − 2a)Y (2−2a,2) m−1 (y).(64)
To formulate the Sturm-Liouville problem of our interest it is worthy to convert CSLE (32) to its 'prime' [42] form at ∞ using the gauge transformation
∞ ̸ Ψ [y; a; ε] = y −1/2 ∞ Φ[y; a; ε](65)
and then to solve the resultant RSLE
The main advantage of converting CSLE (32) to its prime form with respect to the regular singular point at infinity comes from our observation [42] that the characteristic exponents for this singular end have opposite signs and therefore the corresponding principal Frobenius solution is unambiguously selected by the DBC. Prime RSLE (66) can be also re-written in the form of the 'algebraic' [42] Schrödinger equation
y d dy y d dy − y −2 + 2ay −1 + ε ∞ ̸ Ψ [y; a; ε] = 0.(68)
(As discussed in the following subsections this is the common remarkable feature of RCSLEs with density function (34) assuming that the singular point at infinity is regular.) Reformulating the given spectral problem in such a way allows us to take advantage of powerful theorems proven in [43] for
Note also that eigenfunctions (69) are orthogonal with the weight y −1 and that any solution normalizable with this weight must vanish at infinity.
The presented argumentation does not exclude existence of eigenfunctions with the number of nodes larger than N (a) − 1. To confirm that the problem in question is indeed exactly solvable one can simply take advantage of the conventional analysis of the Schrödinger equation with the Morse potential [22] in the L Ref representation.
The reader can argue that the problem must be exactly solvable since the Morse potential is TSI. However the author [44] has an issue with this assertion. Though the Gendenshtein's claim [34] concerning the exact solvability of shape-invariant potentials is most likely correct it has been never accurately proven to our knowledge. The catch is that Gendenshtein's arguments decreasing the translational parameter a one by one bring us to the Sturm-Liouville problem with | a |< 1/2 and then we still need to prove that the resultant SLE has no discrete energy spectrum.
The change of variable y(x) = e x converts BRef CSLE (32) into the Schrödinger equation with the Morse
potential ∞ V [y(x); a], where ∞ V [y; a] = −y 2 I 0 [y; a] + 1/4 (71) = −2ay −1 + y −2 .(72)
Comparing (72) with (1) in [10] shows that ∞ V [y(x); A + 1/2] = V A,1 (x) in Quesne's notation. According to the general theorem presented in [43] for singular SLEs solved under the DBCs any principal solution ∞ ̸ ψ −,m [y; a] near the singular end point y = 0 has nodes at the positive semi-axis iff it lies above the ground energy level. Examination of the inequality
∞ ε −,m (a) < ∞ ε −,0 (a)(73)
thus shows that the q-RS ∞ ̸ ψ −,m [y; a] with m ̸ = 0 preserves its sign on the positive semi-axis iff
m > 2a − 1 = 2A(74)
(cf. (12) in [10]). It will be proven in next subsection that one can use any combination of admissible q-RSs
∞ ̸ ψ −,Y (−2a,+2) m (y) = 2 −m (2m − 2a) mŶ (−2a,+2) m (y)(75)
where, in following [5], we use hut to indicate that the polynomial in question is written in its monic form. It is essential that the multiplier
(2m − 2a) m = m−1 l=0 (2m − 2a − l) = m l ′ =1 (m − 2a + l ′ )(76)
necessarily differs from 0 if either 2m − 2a < −1 (R-Bessel polynomials) or m = m > 2a − 1 (generalized Bessel polynomials with no positive zeros) so the polynomial degree is equal to m in both cases of our primary interest.
RDCT s of principal solutions near singular end points
Using an arbitrary set M p = m 1 , . . . , m p of seed functions ∞ ϕ ±,m k [y; a] of the same type (0 < m k < m k+1 for k = 1, . . . , p − 1) we can represent the corresponding RDCT of BRef CSLE (32) as
d 2 dy 2 + ∞ I 0 [y; a | ± . . .M p ] + εy −2 ∞ Φ[y; a; ε | ± . . .M p ] = 0,(77)
and the symbolic expression ld standing for the logarithmic derivative. When deriving (78) we also took into account that the so-called [42] 'universal correction'
∆I {ρ(y)} ≡ 0.5 » ρ(y) d dy ld ρ(y) ρ(y)(81)
in Schulze-Halberg's [45] generic formula for zero-energy free term of the transformed CSLE vanishes in the case of our current interest: ρ(y) = y −2 .
The common remarkable feature of Wronskians (80) for TFI CSLEs from group A (originally noticed by Odake and Sasaki [19] in their scrupulous study on RDCT s of the corresponding TSI potentials) is that each can be represented as the weighted polynomial Wronskian
∞ w[y; a | ± . . .M p ] = ∞ ϕ p ±,0 [y; a] ∞ W N M p [y; a | ± . . .M p ],(82)
where the Wronskian
∞ W N M p [y; a | ± . . .M p ] ≡ W Y (±2a,∓2) m1 (y), . . . , Y (±2a,∓2) mp (y)(83)
is a polynomial of degree
N M p =| M p | −0.5p(p − 1)(84)
(see (61) in [19]). When it seems appropriate we will drop the index specifying the degree of polynomial Wronskians in question. Substituting (82) into (78), coupled with (33) and (35), one finds
∞ I 0 [ y; a | ± . . .M p ] = 2(a ± p)y −3 − y −4 + 1/4y −2 + 2 y d dy Å y ld ∞ W[y; a | ± . . .M p ] ã .(85)
Each RCSLE under consideration can be alternatively obtained via sequential RDTs with the FFs
∞ Φ ±,mp [y; a | ± . . .Mp −1 ] = y p−1 ∞ w[y; a | ± . . .Mp] ∞ w[y; a | ± . . .Mp −1 ] (p = 1, . . . , p)(86)
so RefPFs (85) can be determined via the following sequence of recurrence relations
∞ I 0 [y; a | ± . . .M p ] = ∞ I 0 [y; a | ± . . .M p−1 ] + 2 y d dy Å y ld ∞ Φ ±,mp [y; a | ± . . .M p−1 ] ã(87)
(a natural extension of the renown Crum formulas [21] to the CSLEs).
For an arbitrary choice of the partition M p RefPF (85) generally has poles on the positive semi-axis and therefore RCSLE (77) cannot be quantized analytically. So let us choose a set M ± p = m ± 1 , . . . , m ± p of seed solutions of the sane type, ∞ ϕ ±,m k [y; a] (0 < m k = m ± k < m k+1 = m ± k+1 for k = 1, . . . , p − 1), in such a way that the seed function ∞ ϕ ±,m1 [y; a] and all Wronskians ∞ w[y; a | ± . . .M ± p ] forp = 2, . . . , p preserve their sign on the positive semi-axis. In particular Odake and Sasaki [19] and nearly the same time Gomez-Ullate et al [11] constructed the subnet of rationally deformed Morse potentials
∞ V [y; a | + . . .M + p ] = ∞ V [y; a] + y 2 ∞ I 0 [y; a] − ∞ I 0 [y; a | + . . .M + p ](88)
using seed solutions infinite at both quantization ends. In next subsection we will introduce another subnet of rationally deformed Morse potentials
∞ V [y; a | − . . .M _ p ] = ∞ V [y; a] + y 2 ∞ I 0 [y; a] − ∞ I 0 [y; a | − . . .M _ p ](89)
constructed by means of FFs vanishing at the origin. The subnet starts from the potential ∞ V [y; a | − . . .m] with a positive integer m > 2a − 1 -potential function (16) in [10] with A = a − 1/2, B = 1. Substituting (82) into (86) and also making use of (37) with k = p, shows that RCSLE (77) has an infinite set of q-RSs
∞ Φ ±,m [y; a | ± . . .M p ] = ∞ ϕ ±,0 [y; a ± p] ∞ W[y; a | ± . . .M p , m] ∞ W[y; a | ± . . .M p ] .(90)
Apparently q-RS (90) with the label '−' represents the principal solution approaching 0 as y δ−(M p ) e −1/y in the limit y → +0. On other hand q-RS (90) labelled by '+' infinitely grows as y δ+(M p ) e 1/y in this limit. In both cases Note that the last summand in sum (85) has a simple pole at y = 0 so an arbitrary principal solution of RCSLE (77) near its irregular singular point at y = 0 can be approximated as
ld ∞ Φ ±,∞ Φ 0 [y; a; ε | ± . . .M p ] ∝ y ∆±(a;M p ) e −1/y for y << 1,(94)
where ∆ ± (a; M p ) stands for a finite power exponent which particular value is non-essential for our discussion. Examination of the quasi-rational function
∞ Φ 0 [y; a; ε | ± . . .M p+1 ] = y W ∞ Φ ±,mp+1 [y; a | ± . . .M p ] , ∞ Φ 0 [y; a; ε | ± . . .M p ] ∞ Φ ±,mp+1 [y; a | ± . . .M p ] = y ∞Φ0 [y; a; ε | ± . . .M p ] − y ld ∞ Φ ±,mp+1 [y; a; | . . .M p ] ∞ Φ 0 [y; a; ε | ± . . .M p ](95)
representing the RDT of the principal solution of RCSLE (77) near its irregular singular point at y = 0 confirms that it is a principal solution of the transformed RCSLE near the singular point in question. Vice versa the quasi-rational function
y W * Φ ±,mp [y; a | ± . . .M p ] , ∞ Φ 0 [y; a; ε | ± . . .M p+1 ] * Φ ±,mp+1 [y; a | ± . . .M p ] = y ∞Φ0 [y; a; ε | ± . . .M p+1 ] − y ld * Φ ±,mp+1 [y; a | ± . . .M p+1 ] ∞ Φ 0 [y; a; ε | ± . . .M p+1 ](96)
representing the reverse RDT of the principal solution (95) is the principal solution of RCSLE (77) near its irregular singular point at y = 0.
To study a behavior of Frobenius solutions near a regular singular point of RCSLE (77) at infinity it is convenient to convert this equation to its 'prime' form [42] using the gauge transformation
∞ ̸ Ψ [y; a; ε ± . . .M p ] = y −1/2 ∞ Φ[y; a; ε | . . .M p ](97)
which gives
d dy y d dy − y −3 + 2(a ± 1)y −2 + 2 d dy (y ld ∞ W[y; a | ± . . .M p ] ) + εy −1 ∞ ̸ Ψ [y; a; ε | ± . . .M p] = 0(98)
As explained above the main advantage of this representation comes from the fact that the characteristic exponents of two Frobenius solutions of RSLE (98) near this singular end have opposite signs, with the principal Frobenius solution decaying as y − √ −ε when y → ∞. Again RSLE (98) is nothing but the 'algebraic' [42] form of the Schrödinger equation with the rationally deformed Morse potentials (88) or (89) accordingly -the common feature of RCSLEs with density function (34) as far as the given SLE has a regular singular point at infinity. Apparently
∞ ̸ Ψ [y; a; ε | ± . . .M p+1 ] ≡ y −1/2 ∞ Φ[y; a; ε | ± . . .M p+1 ] = y W ∞ ̸ Ψ [y; a; ε ±,mp+1 (a) | ± . . .M p ], ∞ ̸ Ψ [y; a; ε | ± . . .M p ] ∞ ̸ Ψ [y; a; ε ±,mp+1 (a) | ± . . .M p ](99)
Here we are only interested in cases when the FF appearing in the denominator of PF (99) is the non-principal Frobenius solution of RSLE (98) near the singular point at infinity so ] at the energy ε * (a) < 0. Applying the reverse RDT with the FF
∞ ̸ Ψ [y; a; ε | ± . . .M p+1 ] ≈ − [ √ −ε + » −ε ±,y 1/2 / ∞ Φ[y; a; ε ±,mp+1 (a) | ± . . .M ± p ] = ∞ ̸ Ψ −1 [y; a; ε ±,mp+1 (a) | ± . . .M ± p ](101)
to the new eigenfunction we would come to the solution which obeys the DBC at infinity:
W ∞ ̸ Ψ −1 [y; a; ε ±,mp+1 (a) | ± . . .M ± p ], ∞ ̸ Ψ [y; a; ε * (a) | ± . . .M ± p+1 ] ∞ ̸ Ψ −1 [y; a; ε ±,mp+1 (a) | ± . . .M ± p ] ≈ [ » −ε ±,mp+1 (a) − » −ε * (a)]y − √ −ε±,m p+1 (a)
for y >> 1
assuming that ε * (a) ̸ = ε ±,mp+1 (a). On other hand, the quasi-rational function on the left is related to principal solution (96) via gauge transformation (97) with ε = ε * (a) and therefore the solution in question would obey both DBCs which contradicts the assumption that ε * (a) is a new eigenvalue. The only exception corresponds to the case ε * (a) = ε ±,mp+1 (a), when the RDT with FF (100) insert the new bound energy state below the ground energy level of rationally deformed Morse potential (88) or (89) accordingly.
Isospectral family of rationally deformed Morse potentials with a regular spectrum
Let us prove that any set M
∞ ̸ Ψ [y; a; ε n (a) | − . . .m 1 ] ≡ ∞ ̸ Ψ −,n [y; a | − . . .m 1 ] = y −1/2 ∞ Φ −,n [y; a | − . . .m 1 ](105)
at the energies ∞ ε −,n (a) with n varying from 0 to N (a) − 1. Making use of (90) with p = 1 and m = n, they can be also re-written in the quasi-rational form
∞ ̸ Ψ −,n [y; a | − . . .m 1 ] = ∞ ̸ ψ −,0 [y; a − 1] ∞ W[y; a | − . . .m 1 , n] Y (−2a) m1 (y)(106)
Keeping in mind that the PF in the right-hand side of the latter expression is proportional to y n−1 for y >> 1 one can immediately confirm that eigenfunctions (105) vanish in the limit y → ∞ for any n < a − 1/2.
Subnet of rationally deformed Morse potentials quantized via Wronskians of R-Bessel polynomials
Another family of solvable RDCT s of CSLE (32) can be constructed using juxtaposed pairs of eigenfunctions ∞ ϕ −,n k [y; a], ∞ ϕ −,n k +1 [y; a] (0 < n k < n k+1 − 1 < N (a) for k = 1, . . . , J). The simplest double-step representative of this finite family of rationally deformed Morse potentials with n 1 = 1, J = 2 was constructed by Bagrov and Samsonov [38,48] in the late nineties based on the conventional L Ref representation of the Schrödinger equation with the Morse potential. The extensions of their works to an arbitrary number of juxtaposed pairs of eigenfunctions in both L Ref and BRef representations were performed more recently in [11] and [19] accordingly. For any TFI RCSLE from Group A one can by-pass an analysis of the pre-requisites for the Krein-Adler theorem [49,50] by taking advantage of the fact that the Wronskians of eigenfunctions are composed of weighted orthogonal polynomials with the common degree-independent weight and therefore the numbers of their positive zeros are controlled by the general Conjectures proven in [51] for Wronskians of positive definite orthogonal polynomials. In particular we conclude that any Wronskian formed by juxtaposed pairs of R-Bessel polynomials of non-zero degrees may not have positive zeros.
Let N 2J be a set of R-Bessel polynomials of degrees
N 2J = M (∆ ′ L→1 ) = n 1 : n 1 + 2j 1 − 1, n 2j1+1 : n 2j1+1 + 2j 2 − 1, . . . , n 2J−2j L +1 : n 2J (n 1 > 0, n 2J < N )(117)
with even δ ′ l = 2j l (l = 1, . . . , L).
Examination of the q-RS functions
∞ ̸ Ψ −,n [y; a | − . . .N 2J ] = y −1/2 ∞ Φ −,n [y; a | − . . .N 2J ] = ∞ ̸ ψ −,0 [y; a − 2J] ∞ W[y; a | − . . .N 2J , n] ∞ W[y; a | − . . .N 2J ] (n / ∈ N 2J )(119)
Note that the PF in the right-hand side of (119) is proportional to y n−2J for y >> 1 so each solution with n / ∈ N 2J < N (a) represents an eigenfunction of RSLE (120). Again these eigenfunctions must be orthogonal with the weight y −1 and therefore N (a) − 2J Wronskians ∞ W[y; a | − . . .N 2J , n] with n / ∈ N 2J < N (a) form a polynomial set orthogonal with the positive weight
∞ W [y; a | − . . .N 2J ] = ∞ ̸ ψ 2 −,0 [y; a − 2J] y ∞ W 2 [y; a | − . . .N 2J ](122)
If sequence (117) starts from n 1 = 1 then the finite EOP sequence in question lacks the first-degree polynomial.
Otherwise it always starts from a polynomial of non-zero degree
| N 2J | −J(2J + 1) > (n 1 − 1)(δ ′ 1 − 1) ≥ 1.(123)
In both cases the pre-requisites of the Bochner theorem are invalid as expected [46].
We refer the reader to Conjectures in [51] to verify that the number of zeros of each Wronskian in the constructed orthogonal polynomial set changes exactly by 1 even if a jump in the polynomial degree is larger than 1. However vol. 62 no. 1/2022
Quantization of rationally deformed Morse potentials. . . even if we take advantage of these elegant results we still need to prove that there are no additional eigenfunctions with a number of nodes larger than N (a) − 2J − 1. In contrast with the analysis presented in the previous section, this proof is complicated by the fact that the RDT at each odd step results in a non-solvable RSLE with singularities on the positive semi-axis. Luckily we deal with the TFI CSLE so its RDCT using juxtaposed pairs of eigenfunctions can be alternatively obtained via sequential RDTs with seed solutions from the second sequence +, m [11,19]. Namely, as already mentioned in the end of section 2 the conjugated partition
M + |δ 1→L | = M (∆ 1→L )(125)
is formed by alternating even and odd integers starting from an even integer δ ′ 1 . The reverse is also true: if the partition M
+ p = M ( p δ 1→Lp ; p δ ′ 1→Lp )(126)
is composed of alternating even and odd integers starting from an even integer δ ′ 1 then each segment of the conjugated partition
p N 2Jp = M ( p δ ′ Lp→1 ; p δ Lp→1 )(127)
must have an even length, with the largest element
m + | p δ ′ 1→Lp | =| p ∆ 1→Lp | −1 = m | p δ Lp →1 | ∈ p N 2Jp ,(128)
where p ∆ 1→Lp ≡ p δ 1→Lp ; p δ ′ 1→Lp . Making use of (37) one can verify that quasi-rational functions (30) can be decomposed as
∞ χ ∓N [y; a] = y 1/2N (N −1)∓N δ [ξ] ∞ ϕ N ±,0 [y; a (δ) ](129)
and therefore the denominators of the fractions in equivalence relations (29) take form
y −1/2|δ L |(|δ L |−1) L l=1 ∞ χ −δ l [y; a (|∆ ′ l→1 |−δ) ] = y Σ L ∞ ϕ |δ L | −,0 [y; a](130)
Note that decomposition (129) holds for any TFI CSLE of Group A provided that we replace y 2 for the leading coefficient l σ[y] of the corresponding counter-parts of differential eigenequations (46). This brings us to the equivalence relations for polynomial Wronskians discovered by Odake and Sasaki [19] in their pioneering analysis of TSI potentials from Group A. If a > 1/2 then, according to (128), the largest element of the partition p N 2Jp is smaller than a+ | p ∆ l→Lp | −1/2 and therefore the Wronskian in the right-hand side of (133) with ∆ ′ L→1 replaced for p ∆ ′
not true in general.) Re-formulating the given spectral problem in such a very specific way allowed us to take advantage of powerful theorems proven in [43] for zeros of principal solutions of SLEs solved under the DBCs at singular ends. We [42] also used this simplified version of the conventional spectral theory to prove that any RDT of a principal (non-principal) Frobenius solution near the regular singular point at ∞ is itself a principal (non-principal) Frobenius solution of the transformed RSLE. This assertion plays a crucial role in our proof of the exact solvability of the constructed DC net of isospectral rational potentials. It is commonly presumed that the Krein-Adler theorem [49,50] is applied to an arbitrary potential regardless its behavior near the singular end points. In [42] we examined this presumption more carefully for the Dirichlet problems of our interest again taking advantage of the theorems proven in [43] for zeros of juxtaposed eigenfunctions. However one can by-pass this analysis for any TFI RSLE from Group A keeping in mind that the Wronskians in questions are formed by orthogonal polynomials with degree-independent indexes and therefore the numbers of their positive zeros are controlled by the general Conjectures proven in [51]. In particular this implies that any Wronskian formed by juxtaposed pairs of R-Bessel polynomials of non-zero degrees may not have positive zeros.
∞
− y −3 + 2ay −2 + εy −1 ∞ ̸ Ψ [y; a; ε] ̸ Ψ [y; a; ε n ] = lim y→∞ ∞ ̸ Ψ [y; a; ε n ] = 0.
zeros of principal solutions of SLEs solved under the DBCs at singular ends.The eigenfunctions of RSLE (66) thus take form∞ ̸ ψ −,n [y; a] = y −1/2 ∞ ϕ −,n [y; a] = 2 n y 1/2−a e −1/y B (a−1/2) n (y) for n = 0, . . . , N (a).(69)One can then directly verify that each eigenfunction obeys the DBC at both singular ends. Since R-Bessel polynomials (57) form an orthogonal sequence the eigenfunction ∞ ̸ ψ −,n [y; a] must have exactly n nodes and therefore[43] the sequence of eigenfunctions (69) corresponds to ⌈A⌉ = N (a) + 1 lowest eigenvalues of RSLE (66) with N (a) = ⌊a − 1/2⌋ ≡ ⌊A⌋.
∞
p of seed solutions ϕ −,m k [y; a] (0 < m 1 < m k < m k+1 ≤ p) is admissible if the generalized Bessel polynomial Y (−2a) m k (y)does not have positive zeros so each seed function ∞ ϕ −,m k [y; a] preserves its sign on the positive semi-axis. According to (73), this is possible if m > 2a − 1 for any m ∈ M _ p . In other words we have to prove that polynomial Wronskian (83) does not have positive zeros if this is true for each polynomial Y (−2a) m k (y). This assertion is obviously trivial forp = 1. It also directly follows from the arguments presented in previous subsection that the RDT of BRef CSLE (32) with the FF ϕ −,m1 [y; a] preserves the discrete energy spectrum so the prime RSLE d dy y d dy + y ∞ I 0 [y; a | − . . .m 1 ] + (ε + 1/2)y −1 ∞ ̸ Ψ [y; a; ε | − . . .m 1 ̸ Ψ [y; a; ε n (a) | − . . .m 1 ] = lim y→∞ ∞ ̸ Ψ [y; a; ε n (a) | − . . .m 1
∞
χ −δ ′ l [y; a (|∆ ′ l→1 |−δ ′ ) ] = y Σ L ∞ ϕ |δ ′ L | −,0 [y; a (|∆ 1→L |) ](131)accordingly, where | ∆ 1→L |=| ∆ ′ come to the following equivalence theorem for the Wronskians of generalized Bessel polynomials ∞Ŵ [y; a | + . . .M (∆ 1→L )] = ∞Ŵ [y; a (|∆ 1→L |) | − . . .M (∆ ′ L→1 )].
is formed by juxtaposed pairs of R-Bessel polynomials. This confirms that none of the polynomial Wronskians∞Ŵ [y; a | + . . .M + p ] has zeros on the positive semi-axis and therefore each partition M + p specifies an admissible sequence of seed solutions ∞ ϕ +,m k [y; a] (m k ∈ M + p for k = 1, . . . , p). Based on the arguments presented in subsection 3.2 we thus assert that the RDTs in question may insert only one bound energy level at the energy ∞ ε +,mp+1 (a) which by definition lies below the ground energy level ∞ ε +,mp (a) of the Liouville potential ∞ V [a | M + p ]. On other hand all the existent energy levels remain unchanged. As the simplest example we can cite the partition 1, 2, . . . , 2J = M (2J, 1) = M † (1, 2J) for 2J ≤ ⌊A⌋
We call a CSLE 'translationally form-invariant' if it has two 'basic' solutions ϕ +,0 [ξ,(3.) reverse Liouville transformation from the Schrödinger equation to the new CSLE
d 2
dξ 2 + I 0 [ξ; Q | τ ] + ερ[ξ] Φ[ξ; Q; ε | τ ] = 0.
(8)
Obviously any TFI theorem proven for Liouville-Darboux transformations of CSLE (5) can be directly
applied to the resultant Liouville potential thus linking the new technique to the conventional Darboux-Crum
theory of TSI potentials [11, 19, 33].
2.2. Translational from-invariance of Sturm-Liouville equation
⇀
a , b] and ϕ −,0 [ξ,
⇀
a , b]
m [y; a] as seed functions to construct an exactly solvable RDCT of the BRef CSLE.According to (9.13.1) in [6]
<< 1, where we dropped subscript ∞ in the notation of the FF for the reverse RDT: * Φ ±,m [y; a | ± . . .M p ] ≡ y/ ∞ Φ ±,m [y; a | ±m [y; a | ±
. . .M p ] ≈ ∓y −2
(91)
and consequently
ld * Φ ±,m [y; a | ±
. . .M p ] ≡ ld y − ld ∞ Φ ±,m [y; a | ±
. . .M p ] ≈ ±y −2
(92)
for 0 < y . . .M p ] .
(93)
shows that they all represent principal solutions near the irregular singular point of the prime RSLE assuming again that the latter equation is solved under DBCs lim y→0 ∞ ̸ Ψ [y; a; ε n | − . . .N 2J ] = lim y→∞ ∞ ̸ Ψ [y; a; ε n | − . . .N 2J ] = 0.d
dy
y
d
dy
+ y ∞ I 0 [y; a | −
. . .N 2J ] + (ε + 1/2)y −1
∞ ̸ Ψ [y; a; ε | −
. . .N 2J ] = 0
(120)
The Liouville potentials in question can be thus expressed in terms of the admissible Wronskians ∞ W[y; a | − . . .N 2J ] as follows ∞ V [y; a | − . . .N 2J ] = ∞ V [y; a − 2J] − 2y d dy (y ld ∞ W[y; a | − . . .N 2J ]).
AcknowledgementsI am grateful to A. D. Alhaidari for bringing my attention to the alternative representation of eigenfunctions of the Schrödinger equation with the Morse potential in terms of R-Bessel polynomials with degree-independent indexes. This alteration helped me to fully comprehend Odake and Sasaki's suggestion to place the Morse oscillator into Group A of rational TSI potentials.Examination of q-RS (110) reveals that it vanishes at the origin and therefore represents a principal solution of prime SLE (103) near its irregular singular point. Since this solution lies below the lowest eigenvalue it must be nodeless[43]and therefore no Wronskian ∞ W[y; a | − . . .M_ p ] has positive zeros. All the q-RSsvanish at infinity for n < N (a) = ⌊A⌋ since the power exponent of the PF in the right-hand side of (111) is equal to n − p in the limit y → ∞. This confirms that the Direchlet problem for SLE (103) has exactly N (a) eigenfunctions defined via (111) with n < N (a). Since these eigenfunctions must be orthogonal[43]with the .If the Morse potential has at least 2 energy levels the sequence starts from a polynomial of degreein this case. The finite EOP sequence in question thus starts from a polynomial of at least second degree and therefore[46]does not obey the Bochner theorem[47].we can then explicitly express corresponding Liouville potential (89) in terms of the admissible WronskianAs mentioned in previous subsection this net of isospectral rational potentials starts from potential function(16)in[10]with A = a − 1/2 , B = 1, after the latter is expressed in terms of the variable y = e x .As a direct consequence of the equivalence theorem we find thatwhere the Wronskian on the right is formed by 2J sequential R-Bessel polynomials of non-zero degrees smaller than A and therefore may not have positive zeros for a > −1/2[51]. As initially proven in[7]and then illuminated in more details in[8]using the so-called 'Kienast-Lawton-Hahn's theorem'[52][53][54]the latter assertion holds for any positive J despite the fact that the seed functions ∞ ̸ ψ −,m [y; a (2J+1) ] have nodes on the positive semi-axis for a (2J+1) + 1/2 < m < 2A.Indeed, representing (56) asshows that the absolute value of the negative m-dependent Laguerre indexis larger than the polynomial degree and therefore the polynomial in question may not have zeros at negative values of its argument.Isospectral rational extensions of Krein-Adler SUSY partners of Morse potentialSince any RDCT of the Morse potentials using pairs of juxtaposed eigenfunctions N 2J keeps unchanged the ground-energy level a set of seedis an admissible set of seed polynomials specified in subsection 3.3. We can then use the same arguments as in subsection 3.3 to prove that any Liouville potentialhas exactly the same discrete energy spectrum as rationally deformed Morse potential (124) constructed by means of juxtaposed pairs of R-Bessel polynomials of non-zero degrees. Its eigenfunctions expressed in terms of the variable y = e x can be represented askeeping in mind that the corresponding prime RSLE is nothing but the Schrödinger equation re-written in its algebraic form.ConclusionsThe presented analysis illuminates the non-conventional approach[19]to the family of rationally deformed Morse potentials using seed solutions expressed in terms of Wronskians of generalized Bessel polynomials in the variable y = e x . As a new achievement compared with Odake and Saski's[19]study on RDCT s of the Morse potential (see also[11]where a similar analysis was performed within the conventional L Ref framework) we constructed a new RDC net of isospectral potentials by expressing them in terms of the logarithmic derivative of Wronskians of generalized Bessel polynomials with no positive zeros. The constructed isospectral family of rationally deformed Morse potentials represents a natural extension of the isospectral RDT s of the Morse potential discovered by Quesne[10]. An important element of our analysis often overlooked in the literature is the proof that the sequential RDTs in question do not insert new bound energy states. The widespread argumentation in support of this (usually taken-for-granted) presumption is based on the speculation that the theorems of the regular Sturm-Liouville theory[55]are automatically applied to singular SLEs. We can refer the reader to the scrupolous analysis performed in[43]for SLEs solved under the DBCs as an illustration that this is by no means a trivial issue. To be able to prove the aforementioned assertion we converted the given RCSLE to its prime form such that the characteristic exponents of Frobenius solutions for the regular singular point at ∞ have opposite signs and therefore the principal Frobenius solution near this singular end is unambiguously selected by the corresponding DBC. (In the particular case under consideration the prime RSLE accidently coincides with the Schrödinger equation re-written in the 'algebraic'[42]form but this is
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| [] |
[
"Opacity dependence of transverse flow, pre-equilibrium and applicability of hydrodynamics in heavy-ion collisions",
"Opacity dependence of transverse flow, pre-equilibrium and applicability of hydrodynamics in heavy-ion collisions"
] | [
"Victor E Ambrus \nInstitut für Theoretische Physik\nJohann Wolfgang Goethe-Universität\nMax-von-Laue-Strasse 1D-60438Frankfurt am MainGermany\n\nDepartment of Physics\nWest University of Timis\noara, Bd. Vasile Pârvan 4300223TimisoaraRomania\n",
"S Schlichting \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n",
"C Werthmann \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n\nIncubator of Scientific Excellence-Centre for Simulations of Superdense Fluids\nUniversity of Wroc law\npl. Maxa Borna 9, 50-204 Wroc lawPoland\n"
] | [
"Institut für Theoretische Physik\nJohann Wolfgang Goethe-Universität\nMax-von-Laue-Strasse 1D-60438Frankfurt am MainGermany",
"Department of Physics\nWest University of Timis\noara, Bd. Vasile Pârvan 4300223TimisoaraRomania",
"Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany",
"Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany",
"Incubator of Scientific Excellence-Centre for Simulations of Superdense Fluids\nUniversity of Wroc law\npl. Maxa Borna 9, 50-204 Wroc lawPoland"
] | [] | We evaluate the full opacity dependence of collective flow in high-energy heavy-ion collisions within a microscopic kinetic description based on the Boltzmann equation in the conformal relaxation time approximation. By comparing kinetic theory calculations to hydrodynamic and hybrid simulations for an average initial state, we point out shortcomings and inaccuracies of hydrodynamic models and present modified simulation setups to improve them. | 10.1103/physrevd.107.094013 | [
"https://export.arxiv.org/pdf/2211.14379v2.pdf"
] | 254,044,543 | 2211.14379 | cf7e08fa19fd688c4c976473f7cada08f808afee |
Opacity dependence of transverse flow, pre-equilibrium and applicability of hydrodynamics in heavy-ion collisions
Victor E Ambrus
Institut für Theoretische Physik
Johann Wolfgang Goethe-Universität
Max-von-Laue-Strasse 1D-60438Frankfurt am MainGermany
Department of Physics
West University of Timis
oara, Bd. Vasile Pârvan 4300223TimisoaraRomania
S Schlichting
Fakultät für Physik
Universität Bielefeld
D-33615BielefeldGermany
C Werthmann
Fakultät für Physik
Universität Bielefeld
D-33615BielefeldGermany
Incubator of Scientific Excellence-Centre for Simulations of Superdense Fluids
University of Wroc law
pl. Maxa Borna 9, 50-204 Wroc lawPoland
Opacity dependence of transverse flow, pre-equilibrium and applicability of hydrodynamics in heavy-ion collisions
(Dated: May 12, 2023)
We evaluate the full opacity dependence of collective flow in high-energy heavy-ion collisions within a microscopic kinetic description based on the Boltzmann equation in the conformal relaxation time approximation. By comparing kinetic theory calculations to hydrodynamic and hybrid simulations for an average initial state, we point out shortcomings and inaccuracies of hydrodynamic models and present modified simulation setups to improve them.
Relativistic heavy ion collisions have proven to be an important tool for probing the dynamical properties of QCD matter in and out of equilibrium. Many current efforts are concerned with using experimental data to assess the conditions under which a quark gluon plasma (QGP) forms in the collision, as well as its properties [1][2][3]. Since the QGP itself cannot be directly observed, its properties have to be inferred by studying suitable aspects of the data and comparing with model descriptions. Hydrodynamics has proven to be a powerful tool for simulating the QGP dynamics [4][5][6][7][8][9] and can accurately describe data for transverse flow, which is an important indicator of collective behaviour. Modern Bayesian inference frameworks based on simulations using hydrodynamics are able to provide significant constraints on the transport properties of the medium created in the collision [1,3,10].
However, the conditions for applicability of hydrodynamics to describe hadronic collisions is still an open question. It is doubtful whether it can be applied to small systems with a dilute medium and large local gradients. Certainly it cannot describe the far-fromequilibrium stage right after the collision. The system will quickly approach equilibrium and start behaving hydrodynamically, but the time scales of its applicability in realistic systems are yet unclear. The topics of applicability to small systems and the properties of the preequilibrium stage have been the focus of many recent endeavors, as described below.
In an effort to find clear distinctive features that indicate the presence or absence of a QGP, small systems have been extensively studied in experiment and have proven to feature non-vanishing transverse flow [11][12][13][14][15][16] and therefore display an onset of collective behaviour. There have been many efforts in simulating these systems in hydrodynamics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], which have produced reasonable results. However, in contrast to nucleus-nucleus collisions, such calculations are subject to much larger uncertainties, where in addition to the poorly constrained initial state geometry [26,32,33], one may question the theoretical justification for employing a hydrodynamic description for a system which features a very short lifetime and consists of very few degrees of freedom. Hence, alternative descriptions with a more sound motivation of their applicability have been put forward. For example, it has been studied whether initial state effects as described by the color glass condensate model could be the source of collective flow in small systems [34][35][36][37][38][39][40][41][42][43][44][45]. However, it turns out that these dynamics fail to describe the important systematics [46].
On the other front, significant progress has been made in pushing the theoretical understanding of the dynamics in the pre-equilibrium stage and the approach to hydrodynamic behaviour and eventually equilibrium in large and small systems [47][48][49] (see also [50,51] for recent reviews). Descriptions of Bjorken flow have been found to exhibit universal behaviour across different dynamical models and initial conditions [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67]. The farfrom-equilibrium behaviour depends on the setup, but the approach to equilibrium proceeds in the same way, by means of an attractor solution that has been studied extensively. This concept has since also been applied to systems with trivial and nontrivial transverse expansion [68][69][70][71]. Phenomenologically, it has been shown that the pre-equilibrium stage has a non-negligible influence on final state observables [71][72][73][74][75] and it is therefore crucial to employ realistic descriptions thereof.
An appropriate alternative dynamical model for small systems as well as pre-equilibrium is kinetic theory, which is a mesoscopic description of the phase space distribution of interacting particles and is therefore less constrained in its applicability to very dilute systems and far-from-equilibrium dynamics. Applications of kinetic theory to heavy ion collisions have been proposed already 30 years ago [76][77][78][79] and have been used in different model scenarios to various levels in complexity. Among others, this has lead to the simulation code Boltzmann Approach to Multi-Parton Scattering (BAMPS) [80,81]. Several efforts have succeeded in describing transverse dynamics and the buildup of transverse flow within this description [82][83][84][85][86][87][88][89][90][91][92], in some cases even with event-byevent simulations [39,49,93,94].
The success of relativistic hydrodynamics in describing experimental observables, demonstrated repeatedly dur-ing the past two decades [95], is heavily dependent on the various theoretical models that lead to such an effective description of strongly-interacting matter. Key ingredients include initial-state generators such as IP-Glasma [96] or MC-Glauber [97], QCD equation of state and realistic transport coefficients [98], hadronization models [99], as well as particle-based hadronic transport such as UrQMD [100]. All of these stages introduce sometimes unquantifiable uncertainties, while statistical approaches such as the Bayesian analysis can be used to pinpoint the most probable parameter-set values of each of these models [101].
In our previous work [71], we found that for final state observables related to transverse flow, results from purely hydrodynamic simulations are in disagreement with results from kinetic theory even at very large opacities due to differences between the dynamics in these two theories during the pre-equilibrium phase. Even though equilibration proceeds on arbitrarily short timescales for sufficiently large opacities, conversely the rate of change of observables in this period increases, such that it still has a tangible effect on their final values. We also examined how at early times, even an inhomogeneous system obeying boost invariance can be described locally by 0+1D Bjorken flow and used the corresponding universal attractor solution to predict the time evolution before the onset of transverse expansion. This also allowed us to describe the discrepancies between hydrodynamics and kinetic theory due to pre-equilibrium in quantitative detail and verify that the size of this effect matches with the described discrepancy of final state observables.
Motivated by these results, the aim of this paper and its companion paper [102] is to examine how in practice simulations of heavy ion collisions based on hydrodynamics can be brought into agreement with kinetic theory simulations. In the present paper, we perform an in-depth theoretical analysis of the non-equilibrium dynamics in different time evolution models based exclusively on mid-central collision events, while a broad phenomenological analysis inferring conclusions for the applicability of hydrodynamics in small systems is presented in Ref. [102].
The time evolution is modeled in a simplified description based on the relaxation time approximation (RTA) of conformal kinetic theory. In such a simplistic model, the ultrarelativistic equation of state = 3P = aT 4 can realistically describe the quark-gluon plasma only in the ultra-high temperature phase, when interactions become negligible [103]. Furthermore, the bulk viscous pressure vanishes identically for a conformal fluid, while Bayesian studies indicate that bulk viscosity can play a significant role on final-state observables [104]. Also, our conformal model gives a constant shear viscosity to entropy density ratio, η/s = const, which is a crude approximation for the expected temperature variation of this ratio [101,105]. Nevertheless, due to its simplistic evaluation of the collision kernel, the RTA has the clear advantage of being computationally cheaper over more realistic collision kernels (e.g. AMY [106]). Such kernels are typically too expensive to be implemented in deterministic solvers, such as the lattice Boltzmann approach that we employ in this paper [71,107,108]. Previous implementations of higher-dimensional dynamics (e.g. BAMPS [80,81]) therefore rely on a test particle algorithm and thus suffer from statistical noise. Furthermore, the firstand second-order transport coefficients computed for the RTA can be readily implemented in the relativistic hydrodynamics solver, allowing for a well-defined comparison between the two theories. Within this model, we perform an analysis of the circumstances under which hydrodynamics becomes applicable as a function of opacity and time, as determined by comparing results for a set of observables related to cooling and transverse flow to kinetic theory. Due to the above simplifications our simulation results cannot be expected to realistically describe experimental data, nevertheless we expect that our conclusions regarding the applicability of hydrodynamics also hold for more realistic models. One argument for this is that the low-momentum behaviour close to equilibrium -which is the relevant part for a comparison to hydrodynamics -should be qualitatively similar between all collision kernels. The model setup, initial conditions and the set of observables are introduced in Sec. II.
Apart from kinetic theory and hydrodynamics, in our work we also used other evolution models, which are discussed in Sec. III. We employed an expansion scheme of kinetic theory that linearizes in opacity and should agree with full kinetic theory in the limit of small interaction rates. We also employed KøMPøST [109,110] as an alternative to using a full kinetic theory simulation of the pre-equilibrium phase. Switching from this description to hydrodynamics for the equilibrated system in a hybrid simulation framework is one way to properly include pre-equilibrium dynamics. KøMPøST is an approximation of the dynamics of kinetic theory, which we were able to verify in quantitative detail by comparing to full kinetic theory, the results of which are presented in Sec. III E.
Before presenting our results, we first discuss in detail how pre-equilibrium is described in hydrodynamics and kinetic theory, pointing out the differences between the two theories. To this end, in Sec. IV we introduce the 1D Bjorken flow attractor solution. This description is valid locally also for early times in 3D simulations assuming boost invariance. We use it to make predictions of the pre-equilibrium behaviour in both evolution models, including a prediction of pre-flow. Based on our results for the differences of kinetic theory and hydrodynamics in this phase, we then introduce a scaling scheme for the initial condition of hydro that can counteract these differences. This scheme relies on a timescale separation of equilibration and the onset of transverse expansion.
In Sec. V, we discuss the time evolution of the system at three example opacities. On the basis of transverse profiles, we indicate how the picture changes from a closeto-free-streaming to an almost fully equilibrated system in kinetic theory. We compare the time evolution in kinetic theory and viscous hydrodynamics as well as in hybrid schemes. Within these hybrid schemes, the first part of the system's evolution is modelled using kinetic theory. Afterwards, we switch to hydrodynamics to model the remainder of the evolution. For sufficiently large opacities, our proposed scaling scheme indeed brings hydrodynamics into agreement with kinetic theory after pre-equilibrium. Based on the system's equilibration, we present a useful criterion for the applicability of hydrodynamics, which can be used to define the switching times for hybrid schemes. This criterion is reached at later evolution times for smaller opacities and in some cases is never fulfilled. We find that when switching sufficiently late, hybrid schemes are also in good agreement with kinetic theory. KøMPøST + viscous hydro simulations yield similar results as simulations with full kinetic theory + viscous hydrodynamics.
The range of applicability of the different schemes can best be assessed by studying the opacity dependence of final state observables. In Sec. VI, we compare first naive and scaled hydrodynamics to kinetic theory and establish 4πη/s 3 as the opacity range where the scaling scheme brings agreement. We then show results from the two hybrid simulation schemes, which can improve on scaled hydro results in the intermediate opacity range around 4πη/s ∼ 3.
In Sec. VII, we present our conclusions and give a brief outlook. Appendix A summarizes the details regarding the relativistic lattice Boltzmann solver that we employ for solving the kinetic equation. Appendices B 1 and B 2 provide further details on how the linearized results in opacity expansion were obtained, while in Appendices C and D we discuss some additional results for the time evolution of the system.
II. INITIAL STATE AND OBSERVABLES
We will describe the time evolution of the plasma created in a collision under the assumption of boost invariance in the longitudinal direction, when the phase-space distribution f ≡ f (x, p) of single particles depends only on the difference of the pseudorapidity y = artanh(p z /p t ) and the spacetime rapidity η = artanh(z/t). We also assume that at initial time τ 0 , the particles comprising the fluid have an isotropic distribution in transverse momentum p ⊥ and vanishing momentum along the longitudinal direction, or in other words, the longitudinal pressure P L measured in the local rest frame vanishes [111]. For the latter assumption to be valid in kinetic theory simulations, we choose the initialization time τ 0 to be small enough for the system to start from the early-time free-streaming attractor of kinetic theory [60]. Further assuming that the interparticle interactions can be modeled in the relaxation-time approximation (RTA), and describing only a reduced distribution function with no dependence on total momentum [71,88], the initial state is fully determined by the initial transverse energy density per unit rapidity, dE 0 ⊥ /dηd 2 x ⊥ . The detailed reduced distribution functions are given in Appendix A.
A. Initial state
We will use a realistic average initial condition for the 30−40% most central Pb-Pb collisions (see also our com-panion paper [102] for a comparison of hydrodynamization in different centrality classes). This initial condition was generated numerically on a transverse grid of size 512 × 512 in the following way. A saturation model based initial state generator was used to generate 8 × 10 6 events with aligned directions of the impact parameter, which were then divided into centrality classes. Then the pointwise average of all events in each centrality class was taken. We made sure that in the resulting event averages statistical fluctuations are sufficiently suppressed by checking that they feature no local peaks above an energy density level of 10 −6 times its maximum. More details on this event generation procedure can be found in [112].
Given this initial condition for dE 0 ⊥ /dηd 2 x ⊥ , the full initial state can be constructed according to the model assumptions. Enforcing at initial time τ 0 a vanishing longitudinal pressure P L and ignoring possible initial-state transverse-plane dynamics, the initial energy-momentum tensor is diagonal and has the following components:
T µν (τ 0 , x ⊥ ) = diag( 0 , 0 /2, 0 /2, 0) ,(1)
where the initial energy density 0 ≡ (τ 0 , x ⊥ ) is given by
(τ 0 , x ⊥ ) = 1 τ 0 dE 0 ⊥ dηd 2 x ⊥ .(2)
In order to characterize the initial energy distribution, we define the total transverse energy per rapidity dE 0 ⊥ /dη
dE 0 ⊥ dη = x ⊥ τ 0 0(3)
and effective radius R
R 2 dE 0 ⊥ dη = x ⊥ τ 0 0 x 2 ⊥ ,(4)
where x ⊥ ≡ d 2 x ⊥ , as well as the eccentricities n
n (τ ) = − x n ⊥ cos [n(φ x − Ψ n )] x n ⊥ ,(5)
where Ψ n are event plane angles and the energy density weighted average over transverse space is defined as
O (τ ) = x ⊥ O(τ, x ⊥ ) (τ, x ⊥ ) x ⊥ (τ, x ⊥ ) .(6)
Based on the definitions in Eqs. (3) and (4), we introduce the opacity of a system with shear viscosity to entropy density ratio η/s viâ
γ = 1 5η/s R πa dE 0 ⊥ dη 1/4 ,(7)
where a is related to the equation of state via
a = T 4 = π 2 ν eff 30 ,(8)
where T is the local temperature and ν eff = 42.25 represents the effective number of degrees of freedom of high temperature QCD [113,114]. The characteristic properties for the initial condition we use are summarized in Table I. As we use a fixed profile, the parameters R and dE 0 ⊥ /dη are also fixed and we varyγ by changing η/s. Hence, throughout this paper, whenever discussing opacity dependencies, we will characterize the opacity via the value of the shear viscosity to entropy density ratio η/s. Note, however, that these two quantities are inversely proportional.
B. Observables
We consider a set of observables which are measured as a function of time τ . Their final state values are taken at finite time, τ /R = 4. These observables are chosen such that they can be easily computed within the two frameworks considered in this paper, namely kinetic theory and hydrodynamics.
Specifically, we focus on observables that are derived from the energy-momentum tensor, which is the fundamental object of hydrodynamics and can be calculated in kinetic theory as
T µν = p µ p ν ,(9)
where angular brackets denote the microscopic average of an observable O with respect to the single-particle distribution function f :
O ≡ dP f O ,(10)
while dP = ν eff √ −gd 3 p/[(2π) 3 p 0 ] is the generallycovariant integration measure in momentum space.
We work in the Landau frame, where the local restframe energy density and flow velocity u µ are given as the timelike eigenvalue and eigenvector of the energymomentum tensor:
T µν u ν = T µν eq u ν = u µ ,(11)
where the energy-momentum tensor in thermal equilibrium reads
T µν eq = ( + P )u µ u ν − P g µν .(12)
In order to facilitate the comparison between kinetic theory and relativistic hydrodynamics, we use as a substitute for dE ⊥ /dη = τ x ⊥ p τ p ⊥ the integral of the transverse part of the trace of the energy-momentum tensor, tr ≡ T τ τ − τ 2 T ηη = T xx + T yy , computed as
dE tr dη = τ x ⊥ (T xx + T yy ),(13)
which is equal to the actual transverse energy per rapidity dE ⊥ /dη whenever the rapidity component of the particle momentum is negligible, p η 0. Similarly, instead of the flow harmonics v n , we will focus on the ellipticity of the energy flow ε p , defined in terms of the transverse components of the energy-momentum tensor as
ε p e 2iΨp = x ⊥ (T xx − T yy + 2iT xy ) x ⊥ (T xx + T yy ) ,(14)
where Ψ p is the symmetry plane angle of the elliptic flow ε p . In order to characterize the expansion rate in the transverse plane, we consider the energy-weighted average of the transverse four-velocity, defined as
u ⊥ = (u 2 x + u 2 y ) 1/2 .(15)
The local departure from equilibrium can be characterized in terms of the inverse Reynolds number,
Re −1 (τ, x ⊥ ) = 6π µν (τ, x ⊥ )π µν (τ, x ⊥ ) 2 (τ, x ⊥ ) 1/2 ,(16)
where π µν is defined as the non-equilibrium part of the energy-momentum tensor:
π µν = T µν − T µν eq .(17)
With the above normalization, Re −1 = 1 when T µν = diag( , /2, /2, 0), corresponding to the initial preequilibrium free-streaming limit. As a global measure of non-equilibrium effects in the system, we use the energyweighted average inverse Reynolds number Re −1 .
III. EVOLUTION MODELS
We want to compare the dynamics of several different time evolution frameworks, which first have to be introduced. In Sec. III A, we discuss the relativistic kinetic model based on the relaxation time approximation (RTA), which is solved using the relativistic lattice Boltzmann approach [107]. Sec. III B discusses an analytical approach aimed at approximating the solution of the kinetic theory model for small opacities. Sec. III C summarizes the equations of relativistic hydrodynamics, which are solved using the vHLLE code [115]. Finally, Sec. III D introduces the linear response framework KøMPøST [109,110], which was modified to include RTA Green's functions [116].
A. Kinetic Theory (RTA)
As the primary tool to investigate the time evolution of the initial configurations discussed in Sec. II A, we employ the relativistic Boltzmann equation in the Anderson-Witting relaxation time approximation (RTA) [117][118][119][120]:
p µ ∂ µ f = − p · u τ R (f − f eq ),(18)
where p µ = (p τ , p ⊥ , p η ) is the particle four-momentum of massless on-shell particles (p 2 = 0) and τ R = 5(η/s)/T is the relaxation time [121]. The prefactor is determined by the fact that in conformal RTA, the shear viscosity is given as η = 4τ R P/5 and the entropy density as s = 4P/T . For the remainder of this paper, we will consider that the specific shear viscosity η/s is constant. The restframe velocity u µ and energy density = aT 4 are determined according to Eqs. (9), (11). As τ R ∝ 1/T , the system obeys conformal symmetry, which simplifies its dynamics. Introducing the reference length scale ref = R and reference energy density
ref = 1 πR 3 (dE 0 ⊥ /dη), Eq. (18) can be non-dimensionalized as v µ∂ µ f = −v µ u µγ T (f − f eq ),(19)
where 1/4 . In this formulation of the equation, it becomes apparent that the time evolution of f parametrically depends only on the opacityγ introduced in Eq. (7). The equilibrium distribution appearing on the right-hand side of Eq. (18) can be identified as the Bose-Einstein distribution
v µ = p µ /p τ ,∂ µ = ref ∂ µ , T = T /T ref and T ref = ( ref /a)f eq = 1 exp (p · u(x)/T (x)) − 1 ;(20)
however, as pointed out in [88], the dynamics depend only on the fact that this distribution is isotropic in the local rest frame. The initial state corresponding to vanishing longitudinal pressure is modeled via
f (τ 0 , x ⊥ , p ⊥ , y − η) = (2π) 3 ν eff δ(y − η) τ 0 p ⊥ dN 0 d 2 x ⊥ d 2 p ⊥ dy ,(21)
where y − η = artanh(τ p η /p τ ). The initial particle distribution is assumed to be isotropic with respect to the azimuthal angle ϕ p = arctan(p y /p x ), being connected with the initial transverse-plane energy distribu-
tion dE 0 ⊥ /dηd 2 x ⊥ via dE 0 ⊥ dηd 2 x ⊥ = 2π ∞ 0 dp ⊥ p 2 ⊥ dN 0 d 2 x ⊥ d 2 p ⊥ dy .(22)
In this paper, we employ the relativistic lattice Boltzmann (RLB) method [122][123][124] to solve Eq. (19). The full details of the algorithm are given in Sec. IV.B of Ref. [71]. The key ideas and simulation parameters are summarized in Appendix A. In the following, we will refer to the numerical solution obtained using the lattice Boltzmann algorithm as described above as "kinetic theory".
B. Opacity expansion
For small systems, the dynamical behaviour is expected to be close to free-streaming, with only slight corrections coming from the small but finite number of interactions. In the limit of small opacity, we expand the solution of the Boltzmann equation in opacity up to linear order: f ≈ f (0) + f (1) . We follow the expansion scheme that was introduced in [82,83], which has recently also been used in other works examining small systems [84,86,87,89,92]. To zeroth order, there are no interactions, and the time evolution of the phase space distribution is computed in the free-streaming limit
p µ ∂ µ f (0) = 0 .(23)
Parametrizing the momentum space in terms of (p ⊥ , y), f (0) can be related to the distribution at initial time via
f (0) (τ, x ⊥ , η, p ⊥ , y) = f τ 0 , x ⊥ − v ⊥ t(τ, τ 0 , y − η), y − arcsinh τ τ 0 sinh(y − η) , p ⊥ , y ,(24)
where
t(τ, τ 0 , y − η) = τ cosh(y − η) − τ 2 0 + τ 2 sinh 2 (y − η).(25)
The linear order correction f (1) vanishes at initial time. Its time evolution is given by the scattering rates of the zeroth order solution,
p µ ∂ µ f (1) = C[f (0) ].(26)
Its explicit expression and properties are presented in Appendix B. As parametrically the collision kernel is proportional to the opacityγ, cf. Eq. (A3), we can indeed identify it as the expansion parameter in this scheme. To enable our scheme to deal with arbitrary input data, the linear order results have to be computed numerically. The computation requires performing a 6D integral, which in part can be done analytically. The details of the code for linear order results are explained in Appendix B.
C. Ideal and viscous hydrodynamics
Relativistic hydrodynamics [125] is an effective macroscopic description based on the conservation equations ∇ µ T µν = 0 for energy and momentum. After decomposing the energy-momentum tensor T µν according to Eqs. (17,12), the equations can be cast in the forṁ + ( + P )θ − π µν σ µν = 0,
( + P )u µ − ∇ µ P + ∆ µ λ ∂ ν π λν = 0,(27)
where θ = ∂ µ u µ is the expansion scalar and σ µν = ∇ µ u ν is the shear tensor, while A µν = ∆ µν αβ A αβ ,
∆ µν αβ = 1 2 (∆ µ α ∆ ν β + ∆ ν α ∆ µ β ) − 1 3 ∆ µν ∆ αβ and ∆ µν = g µν − u µ u ν .
Equation (27) provides only four evolution equations, governing the dynamics of and u µ , leaving the dissipative shear-stress π µν as defined in Eqs. (17,12) unspecified. In ideal hydrodynamics, π µν = 0 at all times, such that the system of equations in (27) becomes closed.
Modeling dissipative effects by means of the Navier-Stokes constitutive equation π µν π µν NS = 2ησ µν , where η is the shear viscosity, leads to parabolic equations which violate causality and are thus incompatible with special relativity [126,127]. In this paper, we will consider the Müller-Israel-Stewart-type theory of secondorder hydrodynamics [128,129], by which π µν evolves according to the following equation [120,130]:
τ ππ µν + π µν = 2ησ µν + 2τ π π µ λ ω ν λ − δ ππ π µν θ − τ ππ π λ µ σ ν λ + φ 7 π µ α π ν α ,(28)
where
ω µν = 1 2 [∇ µ u ν −∇ ν u µ ]
is the vorticity tensor. The relaxation time τ π , as well as the other coupling coefficients, represent second-order transport coefficients, the values of which are chosen for compatibility with RTA [131][132][133]:
η = 4 5 τ π P, δ ππ = 4τ π 3 , τ ππ = 10τ π 7 , φ 7 = 0,(29)
while τ π = τ R . Numerical solutions of Eqs. (27) and (28) reported in this paper are obtained using the open-source viscous HLLE (vHLLE) code [115], 1 which we modified to allow the implementation of the initial state considered in this paper (we employed vHLLE also in Ref. [71] for a similar application). Specifically, we employed the square simulation domain [−8R, 8R] × [−8R, 8R], which we discretized using 401 × 401 equidistant points. The simulations were performed until the final time τ f = 5R. The initial state was prepared using insight on the hydrodynamic attractor, as will be discussed in Sec. IV D. In the initial state, a background value of 10 −7 × R τ0 ref was added to the energy density to prevent free-streaming artifacts in the system outskirts. The time step δτ was chosen dynamically,
δτ (τ ) = min τ δτ τ M , R δτ R M ,(30)
where (δτ /τ ) M = 0.01 and (δτ /R) M = 10 −3 .
D. KøMPøST
The open-source simulation code KøMPøST [109] implements a linearized non-equilibrium time evolution of the energy-momentum tensor T µν based on the dynamics of a kinetic theory description. It has been developed as a practical tool for describing the early-time far-from-equilibrium dynamics of heavy ion collisions, where the system has not yet hydrodynamized and a nonequilibrium description is required. The original version of KøMPøST was based specifically on the effective kinetic theory for pure glue QCD [106]. To perform accurate comparisons with the other evolution models used in this paper, a modified version based on the dynamics of RTA was used. For this, we imported the RTA Green's functions calculated in [116]. This version of KøMPøST is available on Git.
KøMPøST evolves a given input initial state from an initial time τ 0 to a final time τ in a single propagation step. Conceptionally, the output is expected to describe a hydrodynamized system and can be used as input for a subsequent hydrodynamic evolution model. Since the computation of this step involves linearizations in perturbations around a local average value, KøMPøST has a limited range of applicability in the evolution time.
More specifically, in its default mode with energy perturbations, KøMPøST propagates the energy momentum tensor in the following way: the values at each point x in the final state are computed from the initial values of T µν at all causally connected points x in the initial state, meaning points that fulfill |x − x | < c(τ − τ 0 ). The energy-momentum tensor is divided into a spatial average of the causal past and perturbations around this average:
T µν x (τ 0 , x ) =T µν x (τ 0 ) + δT µν x (τ 0 , x ) ,
where the subscript x denotes the fact that the average depends on the position for which the causal past is considered. The average value is evolved according to the laws of Bjorken flow dynamics, assuming local homogeneity in the transverse plane and boost invariance, while the perturbations are propagated in a linear response scheme:
δT µν (τ, x) = d 2 x G µν αβ (x, x , τ, τ 0 ) × δT αβ x (τ 0 , x )T τ τ x (τ ) T τ τ x (τ 0 ) .(31)
The Green's functions G µν αβ (x, x , τ, τ ) have been computed in the respective underlying kinetic theory description and are included in KøMPøST.
Energy perturbations (δT µν ) can also be switched off, in which case KøMPøST propagates only the average energy-momentum tensor taken over the causal past, as discussed above. Some of the phenomenological implications of this mode are discussed below. For all other results in this paper, we employed the modified RTA-KøMPøST with energy perturbations.
E. Validation of KøMPøST
Before employing KøMPøST to describe preequilibrium, we first checked to what extent results from modified RTA-KøMPøST are in agreement with results from full kinetic theory in RTA for our specific initial condition. This comparison was done on the basis of the time evolution of the observables we examined in this paper but also for cross sections through profiles of the energy-momentum tensor after some evolution time. All KøMPøST results presented here were obtained using an initial time of τ 0 = 10 −6 R. Figure 1 shows a comparison of the time evolution of four different transverse space integrated observables at three different values of the shear viscousity, namely 4πη/s = 0.5, 2, 10. The results from KøMPøST are plotted with symbols "+" for the mode with and "×" for the mode without energy perturbations and are benchmarked for times up to τ = 0.5R against the results obtained using a full kinetic theory description, which are plotted with lines.
The decrease of transverse energy dE tr /dη is described very well in both modes. As without energy perturbations, the energy-momentum tensor is propagated as if there were no local gradients, it predicts zero transverse flow velocity u ⊥ and elliptic flow ε p . The mode with energy perturbations can describe the buildup of u ⊥ correctly. On the other hand, while giving nonzero results, it still vastly underestimates the buildup of anisotropic flow ε p . The inverse Reynolds number Re −1 is well described by both modes at early times, but results from the mode with energy perturbations deviate at very late times.
Generally, the comparison suggests that, for certain observables, KøMPøST results can be accurate way beyond the timeframe it was intended for, which is on the order of 0.1 fm. Other observables, in particular those related to anisotropies, are not described correctly.
In a further comparison of KøMPøST to full kinetic theory data, we also investigated profiles of certain components of T µν at fixed shear viscosity 4πη/s = 2 and three different fixed times τ = 0.1R, 0.3R and 0.5R. The same comparisons were also performed in the local rest frame with analogous quantities that are defined via the variables , u µ and π µν . Figure 2 illustrates our findings. This time, KøMPøST results are plotted with lines and full kinetic theory results with symbols.
The results confirm that for energy or energy flow observables like T τ τ , T τ y and T xx + T yy , KøMPøST works well even on a local level and in the outskirts of the system for all evolution times that we examined. The only part of the energy-momentum tensor for which KøMPøST results shows significant deviations are anisotropies in the shear stress, as measured by T xx − T yy . While this observable is still correctly reproduced in the central part of the system, it exhibits sizeable deviations of up to a factor of five at a radial distance of r R. These deviations also explain the errors in elliptic flow ε p .
IV. EARLY-TIME DYNAMICS OF DIFFERENT MODELS
Before the onset of transverse expansion, at times τ R, the system's dynamics is dominated by longitudinal expansion and the effects of transverse expan- (16)]. Plotted are results from KøMPøST (RTA) with (+ symbols) and without (× symbols) energy perturbations compared to full kinetic theory results (solid lines) at three different opacities 4πη/s = 0.5 (green), 2 (yellow) and 10 (blue). In the plot of transverse flow velocity, results at different opacities are shifted in value in order to be distinguishable.
sion can be neglected. Under these conditions, at each point in the transverse plane, the system evolves independently of the transverse neighbourhood and can locally be described by (0 + 1)D longitudinally boost-invariant Bjorken flow. In Bjorken flow, the trajectories of energy, pressure and stress for different initial conditions are known to rapidly converge to a common time evolution curve called the Bjorken flow attractor curve [59,60]. This means that at late times the system always evolves in the same way. If it is initialized on the attractor, then its entire time evolution is given by the attractor curve. We will describe the features of the attractor scaling solution for both the Müller-Israel-Stewart-type second-order hydrodynamics theory and for the conformal kinetic theory in RTA. In Sec. IV A, the quantities describing the attractor solutions are introduced. Sec.s IV B and IV C discuss how the pre-equilibrium evolution impacts the observables of interest, highlighting the possible discrepancies between RTA, viscous hydrodynamics and ideal hydrodynamics. Finally, in Sec. IV D, we discuss how viscous and ideal hydrodynamics can be brought in agreement with RTA at late times by scaling the initial conditions.
A. Bjorken attractor
The (0+1)D Bjorken flow can be described in terms of the Bjorken coordinates (τ, x, y, η), with respect to which the velocity becomes u µ ∂ µ = ∂ τ . The energy-momentum tensor takes the diagonal form
T µν = diag( , P T , P T , τ −2 P L ),(32)
where P T and P L are the transverse and longitudinal pressures, respectively. The shear-stress tensor also becomes diagonal,
π µν = diag 0, − 1 2 π d , − 1 2 π d , 1 τ 2 π d ,(33)-1.5 -1 -0.5 0 0.5 1 1.5 1.8τ ϵ -1.5 -1 -0.5 0 0.5 1 1.5 7u y ϵτ -1.5 -1 -0.5 0 0.5 1 1.5 4τ (π xx + π yy ) −20τ (π xx − π yy ) Position y/R τ ≈ 0.1R τ ≈ 0.3R τ ≈ 0.5R KøMPøST w/o pert.
Energy-momentum tensor The upper row shows, from left to right, the following components of the energy-momentum tensor: T τ τ (left), T τ y (middle), as well as T xx + T yy and T xx − T yy (right). The lower row shows analogous local rest-frame quantities, namely (left), u y (middle), as well as π xx + π yy and π yy − π xx (right). Notice the change in sign for the latter when compared to the upper panel. All observables were multiplied with τ and rescaled with a constant factor to adjust their magnitudes such that they can be plotted on the same total range of 80 GeV/fm 2 c.
where π d can be related to P T and P L via
P T = P − π d 2 , P L = P + π d ,(34)
such that π d = 2 3 (P L − P T ). The observables of interest for the following section are the inverse Reynolds number defined in Eq. (16), and the sum tr = T xx + T yy , which become
Re −1 = − 3π d , tr = 2 3 − π d = 3 (2 + Re −1 ). (35)
The evolution of the energy density is governed by the conservation equations ∇ µ T µν , where ∇ µ is the covariant derivative, which reduces to
τ ∂ ∂τ + 4 3 + π d = 0.(36)
In ideal hydrodynamics, π d = 0 and τ 4/3 (τ ) = τ 4/3 0 0 , where 0 is the energy density at initial time τ 0 .
In RTA, the dynamics of π d is governed directly by the Boltzmann equation. In viscous hydrodynamics, the evolution of π d can be found from Eq. (28) and reads:
τ ∂π d ∂τ + λ + 4πw 5 + 2πw 5 φ 7 π d π d + 16 45 = 0,(37)
wherew is the conformal parameter,
w = 5τ 4πτ π = τ T 4πη/s .(38)
In the above, s = ( + P )/T is the entropy density for an ultrarelativistic gas at vanishing chemical potential, while η = 4 5 τ π P , as shown in Eq. (29). In Eq. (37), we introduced the notation
λ = δ ππ τ π + τ ππ 3τ π ,(39)
which evaluates to 38/21 when using the values for the second-order transport coefficients given in Eq. (29). We note that in the original MIS theory, λ evaluates to 4/3, while the value 31/15 was advocated in Ref. [65] in order to mimic the early time attractor of RTA.
Equations (36) and (37) admit scaling solutions with respect to the conformal parameterw. To see this, we note that the time derivative ofw satisfies
τ dw dτ =w 2 3 − f π 4 ,(40)
where we defined
f π = π d .(41)
Using Eqs. (36), (37), and (40), f π can be shown to satisfỹ
w 2 3 − f π 4 df π dw + 16 45 + λ − 4 3 + 4πw 5 + 2πw 5 φ 7 f π − f π f π = 0,(42)
where φ 7 = 0 for consistency with RTA [see Eq. (29)]. Demanding that f π remains finite whenw → 0, its earlytime behavior in viscous hydro can be obtained as
f π (w 1) = f π;0 + f π;1w + f π;2w 2 + O(w 3 ),(43)
where
f π;0 = 1 2 λ − 4 3 − λ − 4 3 2 + 64 45 , f π;1 = 16π 25 f 2 π;0 (f π;0 − 4 15 ) 2 + 16 75 1 + 1 2 φ 7 f π;0 , f π;2 = 8π 15 f π;0 f π;1 f π;0 − 4 9 2 + 16 405 1 − 25f π;1 16π + f π;0 φ 7 .(44)
When λ = 38/21, we find f π;0 = 1 105 (25 − 3 √ 505) −0.404, which is different from the limit −1/3 in kinetic theory. 2 At large values ofw, f π (w) behaves like
f π (w 1) = − 4 9πw + O(w −2 ),(45)
which is the leading order gradient expansion [54] and therefore valid in both viscous hydrodynamics and in RTA. Due to the relations in Eq. (35), our observable Re −1 = −3f π also exhibits attractor behaviour. Its attractor curve is represented as a function ofw in the top panel of Figure 3. Its asymptotic forms at small and largẽ w can be found from Eqs. (43) and (45), respectively. We now turn to the energy equation, Eq. (36). On the attractor, when f π depends only onw, it is possible to write (cf. [59,67])
τ 4/3 (τ ) = τ 4/3 0 0 E(w 0 ) E(w),(46)
2 Expressing λ = f π;0 + 4 3 − 16
45f π;0
, it can be seen that f π;0 = −1/3 leads to λ = 31/15, as pointed out in Ref. [65]. Energy attractors E, E tr Scaling variablew = τ T /(4πη/s) where the scaling function E(w) satisfies
E RTA E hydro E tr RTA E tr hydro Largew Smallw ∼ 1 − 2 3πw ∼ 2 3 (Pre-equilibrium)(Hydrodynamization)w 2 3 − f π 4 dE dw + f π E = 0.(47)
Due to Eq. (35), tr also admits a scaling solution,
τ 4/3 tr (τ ) = τ 4/3 0 0 E(w 0 ) E tr (w), E tr (w) = 2 3 − f π (w) E(w).(48)
Forw 1, E(w) can be obtained as
E(w 1) = C −1 ∞w γ (1 + E 1w + . . . ),(49)
where the exponent γ and the correction E 1 are given by
γ = 12f π;0 3f π;0 − 8 , E 1 = − 32 3 f π;1 (f π;0 − 8 3 ) 2 .(50)
The constant C ∞ appearing in Eq. (49) is taken such that limw →∞ E(w) = 1, in which case E has the following late-time asymptotic behavior:
E(w 1) = 1 − 2 3πw .(51)
In the case of ideal hydrodynamics, obviously f π = 0 (such that f π;0 = γ = 0) and E(w) = C ∞ = 1. The functions E(w) and E tr (w) are shown in the bottom panel of Figure 3 for both viscous hydrodynamics and for kinetic theory. The normalization factor C ∞ can be obtained in each theory by computing the attractor curve [59]. For completeness, we list below the values of γ and C ∞ in the relevant theories:
RTA : γ = 4 9 , C ∞ 0.88, (52a) Visc. Hydro : γ = √ 505 − 13 18 , C ∞ 0.80, (52b) Ideal Hydro : γ = 0, C ∞ = 1. (52c)
Due to the normalization limw →∞ E(w) = 1, the quantities τ −2/3w and τ 4/3 can be rewritten as
τ −2/3w = (τ −2/3w ) ∞ E 1/4 (w),(53a)τ 4/3 = (τ 4/3 ) ∞ E(w),(53b)
where (τ −2/3w ) ∞ and (τ 4/3 ) ∞ represent the corresponding asymptotic, late-time hydrodynamic limits, satisfying
(τ −2/3w ) ∞ = (τ 4/3 ) 1/4 ∞ a 1/4 4πη/s , (τ 4/3 ) ∞ = τ 4/3 0 0 E(w 0 ) .(54)
Taking now the initial time such thatw 0 1, Eq. (49) can be used to obtain
(τ 4/3 ) ∞ C ∞ 4πη s a 1/4 γ × τ ( 4 3 −γ)/(1−γ/4) 0 0 1−γ/4 .(55)
Equation (55) tells us that the equilibration dynamics introduce a nontrivial relation between energy densities in equilibrium and in the initial state, as the dependence is nonlinear and the exponents depend on the model description, which was one of the main points of Ref. [59].
In the pre-equilibrium regime,w 1. Under the early-time approximation (49),w can be written in terms of (τ −2/3w ) ∞ as
w τ 2 3 /(1−γ/4) C −1/4 ∞ (τ −2/3w ) ∞ 1/(1−γ/4) ,(56)
which allows (w 1) to be obtained as
(w 1) τ (γ− 4 3 )/(1−γ/4) × C −1 ∞ 4πη s a 1/4 −γ (τ 4/3 ) ∞ 1/(1−γ/4) .(57)
Substituting the expression (55) for (τ 4/3 ) ∞ manifestly shows that τ ( 4 3 −γ)/(1−γ/4) becomes independent of τ as τ → 0:
(w 1) τ 0 τ ( 4 3 −γ)/(1− γ 4 ) 0 .(58)
B. Pre-equilibrium evolution
We now consider a system which is no longer homogeneous in the transverse plane, such that the energy density becomes a function of both τ and x ⊥ , ≡ (τ, x ⊥ ). At early times τ R we can neglect transverse dynamics and describe the dynamics locally by Bjorken flow (we will discuss early-time transverse expansion effects on the build-up of flow in the Sec. IV C). Under this approximation, at each point x ⊥ of the transverse plane, we can assume that (τ, x ⊥ ) follows an evolution along the attractor curve according to the local value of the conformal variable,w ≡w(τ, x ⊥ ). Moreover, we consider thatw 0 (x ⊥ ) 1 throughout the system, such that the full pre-equilibrium evolution is captured during the system's evolution.
Neglecting the dynamics in the transverse plane, such that T xx = T yy = 1 2 tr , dE tr /dη defined in Eq. (13) can be written as where Eq. (35) was employed to replace tr and f π = −Re −1 /3. Using now Eqs. (53b) and (55) to replace , we arrive at
dE tr dη = τ x ⊥ 2 3 − f π ,(59)dE tr dη = τ −1/3 a 4πη s γ x ⊥ τ ( 4 3 −γ)/(1−γ/4) 0 0 a 1− γ 4 × 2 3 − f π C ∞ E. (60)
The above equation can be employed to estimate the evolution of dE tr /dη due solely to longitudinal expansion over the whole range of τ . At a fixed value of τ , the conformal parameterw spans the interval 0 (reached at infinitely large distances from the system's center of mass) up to the valuew max corresponding to the maximum value of the temperature. For sufficiently small values of τ ,w max 1 and Eqs. (43), (58) can be used to approximate f π and , leading to
dE tr dη τ 0 τ 1 3 (1−9γ/4)/(1−γ/4) dE 0 tr dη .(61)
The above relation shows that in RTA (γ = 4/9), dE tr /dη remains constant during pre-equilibrium. Conversely, in viscous hydrodynamics, γ > 4/9 and consequently dE tr /dη increases with time. As expected, in ideal hydrodynamics, dE tr /dη decreases as τ −1/3 . In the limitw 1, f π ∼w −1 and E 1, as shown in Eqs. (45) and (51), such that τ 4/3 can be approximated by (τ 4/3 ) ∞ by virtue of Eq. (53b). Using Eq. (55), dE tr /dη reduces to dE tr dη
2τ −1/3 3 C ∞ 4πη s a 1/4 γ τ 4 3 −γ 0 x ⊥ 1−γ/4 0 .(62)
The above equation shows that at late times, dE tr /dη decrease as τ −1/3 . The amount of energy available at a given time τ depends explicitly on the dynamical theory (ideal and viscous hydrodynamics, RTA). We now consider another important effect arising due to the pre-equilibrium evolution, namely inhomogeneous cooling. During pre-equilibrium, neighbouring points in the transverse plane undergo cooling at differing rates according to their local attractors. As pointed out in Refs. [59,71], the characteristics of the inhomogeneities in the transverse plane change during pre-equilibrium, as can be seen by looking at the eccentricity n , defined as
n = − x ⊥ x n ⊥ cos[n(φ x − Ψ n )] x ⊥ x n ⊥(63)
Whenw 1, Eq. (58) can be employed to show that n (τ ) n (τ 0 ) and the eccentricities n remain constant during pre-equilibrium. Whenw 1, n is modified to
n − x ⊥ x n ⊥ 1−γ/4 0 cos[n(φ x − Ψ n )] x ⊥ x n ⊥ 1−γ/4 0 .(64)
The above relation shows that inhomogeneous cooling leads to modifications of all eccentricities of the initial profile, except in the case of ideal hydrodynamics (γ = 0). The effects of the different behaviour for global and inhomogeneous cooling in different model descriptions are illustrated in Figure 4. It shows the pre-equilibrium evolution of the energy density profile multiplied by the Bjorken time, τ , for an example event in the 30 − 40% centrality class of Pb-Pb collisions in kinetic theory and viscous hydrodynamics with either the same or a scaled initial condition. At very early times, this quantity is constant in kinetic theory, but later it decreases slightly due to equilibration. Meanwhile, in hydrodynamics it increases first before transitioning to a decreasing trend. The speed of these transitions in both cases depends on the local temperature, meaning that e.g. the peak values will start decreasing earlier than the values in the outskirts of the system, i.e. the system cools inhomogeneously. After equilibration, the time evolution will uniformly follow the same power law in both models, but the differences due to the different pre-equilibrium evolution will persist. But the knowledge of the local attractor scaling behaviour allows to anticipate the differences between kinetic theory and hydrodynamics and apply a corresponding local scaling prescription to the initial condition of hydro. It then initially takes smaller values than in kinetic theory but dynamically approaches it during pre-equilibrium and reaches agreement after equilibration. This initialization scheme is explained in more detail in Sec IV D.
C. Pre-flow estimation
We now estimate the buildup of flow during the preequilibrium evolution, which we quantify via the observable u ⊥ defined in Eq. (15). The basis of our analysis is to consider that the transverse dynamics represent a small perturbation on top of the purely-longitudinal dynamics discussed in Secs. IV A and IV B, which we consider to be dominant. The idea of this calculation is similar to the one presented in Ref. [134].
At early times τ R, when the transverse flow is negligible, we can write
T µν = T µν B + δT µν , where T µν B = diag( B , P B;T , P B;T , τ −2 P B;L )
is the background (Bjorken) solution of the local, equivalent (0 + 1)D system (we also consider that at initial time,w 0 1 throughout the transverse plane). Further assuming that δT µν is small and imposing the Landau frame condition, T µ ν u ν = u µ , we write u µ = u µ B + δu µ and = B + δ and find
δ = δT τ τ , δu i = δT τ i B + P B;T ,(65)
while δu τ = 0 as required by u µ B δu µ = 0. Thus, the flow buildup can be estimated from the build-up of δT τ i .
We can now derive a dynamical equation for T τ i via the conservation equations ∇ µ T µν = 0, which in a general coordinate system reads
∇ µ T µν = ∂ µ T µν + Γ µ λµ T λν + Γ ν λµ T µλ ,(66)
where Γ λ µν = 1 2 g λρ (∂ ν g ρµ + ∂ µ g ρν − ∂ ρ g µν ) are the Christoffel symbols. In the Bjorken coordinate system (τ, x, y, η), the only non-vanishing Christoffel symbols are Γ τ ηη = τ and Γ η τ η = Γ η ητ = τ −1 , such that the equation
for ν = i becomes 1 τ ∂(τ T τ i ) ∂τ + ∂ j T ij = 0 .(67)
Splitting the energy-momentum tensor into a local Bjorken flow part and a small perturbation as discussed above, we find:
1 τ ∂(τ δT τ i ) ∂τ + ∂ i P B;T + ∂ j δT ij = 0 .(68)
Noting that δT ij represents a higher-order correction, the leading-order contribution to δT τ i can be obtained by solving
∂(τ δT τ i ) ∂τ −τ ∂ i P B;T .(69)
In the above, P B;T evolves according to the local Bjorken attractor, such that P B;T B 1
3 − 1 2 f π;B . Using Eq. (53b) to replace B , the spatial gradient of P B;T can be obtained as:
∂ i P T P T = ∂ i (τ 4/3 ) ∞ (τ 4/3 ) ∞ + E E − 1 2 f π 1 3 − 1 2 f π ∂ iw ,(70)
where the prime denotes differentiation with respect tõ w. Here and henceforth, we will drop the B subscript for brevity, keeping in mind that all instances of P T , , f π and the corresponding conformal variablew are evaluated according to the background (0 + 1)D Bjorken attractor.
Since (τ 4/3 ) ∞ depends on the transverse coordinates only through the initial profile [see Eq. (55)], the first term on the right-hand side of the above relation evaluates in the limitw 0 1 to
∂ i (τ 4/3 ) ∞ (τ 4/3 ) ∞ = 1 − γ 4 ∂ i 0 0 .(71)
The gradient ofw appearing in Eq. (70) can be written in terms of that of (τ −2/3w ) ∞ starting from Eq. (53a),
∂ iw w = 1 −w E 4E −1 ∂ i (τ −2/3w ) ∞ (τ −2/3w ) ∞ = 1 4 1 − γ 4 1 −w E 4E −1 ∂ i 0 0 ,(72)
where the equality on the second line is established using the relations (54) and (71). Substituting Eqs. (71) and (72) into Eq. (70) gives
∂ i P T P T = 1 − γ/4 1 −w E 4E 1 −w f π 8 3 − 4f π ∂ i 0 0 .(73)
Substituting Eq. (73) in Eq. (69) and integrating with respect to τ , we arrive at
δT τ i = − 1 τ 1 − γ 4 ∂ i 0 0 τ τ0 dτ 1 3 − 1 2 f π −w 8 f π 1 −w 4E E τ .(74)
Considering now thatw 1 throughout the system, we can use Eqs. (49), (43) and (58) to approxi-
mate f π f π;0 = −(2γ/3)/(1 − γ/4), E C −1 ∞w γ and = (τ 0 /τ ) 2−α 0 , where α = (γ +4/3)/[2(1−γ/4)]
, which reduces to α = 2/3, 1 and 1.071 in ideal hydro, RTA and viscous hydro, respectively. To leading order, we find
τ 2−α δT τ i = − τ 2 1 − τ 0 τ α ∂ i (τ 2−α 0 0 ),(75)
which allows the macroscopic velocity to be estimated as
δu i (w 1) − 3τ 8 1 − γ 4 1 − τ 0 τ α ∂ i 0 0 .(76)
As expected, the flow velocity is driven by the gradients of the initial energy density profile. In addition, when τ τ 0 , δu i exhibits a linear increase with τ , independently of the value of γ. The prefactor governing the overall amplitude of δu i is however γ-dependent. We can now estimate the early-time evolution of u ⊥ , defined in Eq. (15), as follows:
u ⊥ ,early 3τ 8 1 − γ 4 1 − τ 0 τ α × x ⊥ 0 −1 x ⊥ |∇ ⊥ 0 |,(77)
where
|∇ ⊥ 0 | = [(∂ x 0 ) 2 + (∂ y 0 ) 2 ] 1/2 .
In general, the time dependence of the integrand in Eq. (74) is too complicated to integrate analytically. But it again takes a simple form in the Bjorken flow equilibrium stage, where τ 4/3 P T 1 3 (τ 4/3 ) ∞ . At late times, when the duration of pre-equilibrium is small compared to the elapsed time, its contribution in the time integration is negligible and δT τ i and δu i asymptote to
δT τ i (w 1) − 1 2τ 1/3 1 − τ 0 τ 2/3 ∂ i (τ 4/3 ) ∞ , (78a) δu i (w 1) − 3τ 8 1 − τ 0 τ 2/3 ∂ i (τ 4/3 ) ∞ (τ 4/3 ) ∞ , (78b) such that u ⊥ becomes u ⊥ ,late 3τ 8 1 − τ 0 τ 2/3 x ⊥ |∇ ⊥ 1−γ/4 0 | x ⊥ 1−γ/4 0 .(79)
Note that the above equation was derived under the assumption that δu i is small and thus holds only when the system hydrodynamizes before transverse expansion sets in.
The right-hand side of Eqs. (77) and (79) can be evaluated numerically for the 30 − 40% centrality profile that we are considering in this paper. The results for the different theories (kinetic theory, ideal hydrodynamics and viscous hydrodynamics) are shown in Table II. Here, we contrast the "naive" and "scaled" initial conditions for hydrodynamics, which will be discussed in detail in the following subsection. In the early-time regime, it can be seen that kinetic theory leads to more flow than viscous hydrodynamics (2% and 1% more for the naive and scaled initialization, respectively), while ideal hydrodynamics leads to more flow than kinetic theory (13% and 7% more for the naive and scaled initializations, respectively). In the late-time limit, both ideal and viscous hydrodynamics are brought in agreement with kinetic theory when the scaled initialization is employed. In the case of the naive initialization, ideal hydrodynamics gives about 5% more flow, while viscous hydrodynamics underestimates the flow by less than 1%.
D. Setting initial conditions
From the discussion in the previous subsection, it becomes clear that the pre-equilibrium evolution of the fluid depends on the theory employed to describe it. We take as the "correct" evolution that described by kinetic theory, when dE tr /dη remains constant during the freestreaming stage of pre-equilibrium. This can be seen by setting γ = 4/9 in Eq. (61). Since in viscous hydrodynamics, γ 0.526 > 4/9, dE tr /dη will actually increase during pre-equilibrium, thus leading for the same initial energy profile to an unphysically higher transverse plane energy at late times, as illustrated in Figure 4. Similarly, the change in eccentricity due to the pre-equilibrium evolution will be different compared to kinetic theory. We will now discuss how these phenomena specifically affect the pre-equilibrium evolution of our initial state as given in Sec. II A and how they are counteracted by locally scaling the initial condition. We will then give the quantitative details of the scaling prescription. Figure 5 illustrates the size of the effect on transverse energy dE tr /dη in the top panel and ellipticity 2 in the bottom panel. In naive hydrodynamics using the same initial condition for the energy density as kinetic theory and initial pressure determined by the hydrodynamic attractor, dE tr /dη rises to a value which is about 1.5 times larger than in kinetic theory at the onset of equilibration and will remain in disagreement throughout the rest of the evolution. The dashed lines show predictions of the behaviour in the local Bjorken flow scaling approximation according to Eq. (60). In our proposed scheme the initial value of dE tr /dη is scaled down in such a way that it dynamically reaches agreement with kinetic theory. Similarly, we find that the ellipticity decreases in both kinetic theory and in hydro, but more so in the lat- ter case. This means that in naive hydro the eccentricity will have a smaller value at the onset of the buildup of transverse flow than in kinetic theory, which will result in smaller final values of elliptic flow. With the scaling scheme, the initial ellipticity is scaled up in hydrodynamics and will come into agreement with kinetic theory after equilibration.
As the local scaling factor for the hydrodynamic initial condition is computed in the local Bjorken flow approximation, it assumes that the system will fully equilibrate before the onset of transverse expansion. How well this works in practice will be discussed in Sec. V B.
We now move to the quantitative analysis of the preequilibrium behaviour in the two hydro schemes. In the first one, dubbed "naive hydrodynamics", we will impose the same energy density 0 at initial time τ 0 as in kinetic theory. We first note that the RTA initial conditions given in Eq. (1) are not compatible with the hydrodynamic attractor.
Indeed, noting the relations P T = ( 1 3 − fπ 2 ) and P L =
( 1 3 + f π ), the early-time expression for T µν reads
T µν 0 = 0 × diag 1, 1 3 − f π 2 , 1 3 − f π 2 , 1 3τ 2 + f π τ 2 ,(80)
where f π ≡ f π (w 0 ) depends on the local value of the conformal variable,w 0 ≡w 0 (x ⊥ ) = τ 0 T 0 (x ⊥ )/(4πη/s), with T 0 (x ⊥ ) = [ 0 (x ⊥ )/a] 1/4 . In order to evaluate f π (w 0 ), we follow Ref. [135] and employ a simple Padé approximation interpolating between thew 1 andw 1 limits given in Eqs. (43) and (45):
f π (w) c 0 + c 1w d 0 + d 1w + d 2w 2 ,(81)
where the coefficients c 0 , c 1 , d 0 , d 1 , and d 2 are computed to ensure second order accuracy at smallw and first order accuracy at largew:
d 0 = 4f π;1 9π − f 2 π;0 , d 1 = f π;0 f π;1 − 4f π;2 9π , d 2 = f π;0 f π;2 − f 2 π;1 , c 0 = d 0 f π;0 , c 1 = − 4d 2 9π .(82)
The coefficients f π;0 , f π;1 , and f π;2 are given in Eq. (44).
In the limitw → 0, when f π → f π;0 = − 2γ 3 /(1 − γ/4), Eq. (80) reduces to
T µν 0 = 0 1 − γ/4 diag 1 − γ 4 , 1 3 + γ 4 , 1 3 + γ 4 , 1 3τ 2 − 3γ 4τ 2 ,(83)
This explains why at initial time the naive hydro curve in Figure 5 starts above the kinetic theory one. Acknowledging that viscous hydrodynamics does not capture correctly the pre-equilibrium evolution of the fluid, we propose to change the initialization of hydrodynamics in such a way that the energy density locally agrees with the kinetic theory prediction at late times. In principle, this works only when the pre-equilibrium evolution ends before the onset of transverse expansion. Taking a and η/s to be identical in the two theories and demanding that they both reach the same (τ 4/3 ) ∞ value when τ → ∞, Eq. (55) shows that the local modification of the initial energy density in hydrodynamics (denoted 0,γ ) is
0,γ = 4πη/s τ 0 a 1/4 1 2 − 9γ 8 C RTA ∞ C γ ∞ 9/8 0,RTA 8/9 1−γ/4 ,(85)
where the specific shear viscosity η/s is considered to have the same value in viscous hydrodynamics and in kinetic theory. Using the above energy profile in Eqs. (62), (64) and (79) shows that after pre-equilibrium (i.e., at largew), dE tr /dη, the eccentricities n and the average flow velocity u ⊥ will reach the corresponding RTA limits, irrespective of the value of γ. We note, however, that the pre-equilibrium behaviour of all of the above observables will still be different from that in RTA.
Before ending this section, we emphasize that the rescaling of the initial conditions shown in Eq. (85) is not only possible, but also mandatory for ideal hydrodynamics simulations, when γ = 0 and C ∞ = 1. While when applying the scaling procedure to viscous hydrodynamics, η/s was considered as an invariant physical parameter, in ideal hydrodynamics (when η = 0), this is no longer the case. Instead, the factor η/s helps rescale the initial energy density such that at late times, τ 4/3 obtained in ideal hydrodynamics would match the one in a hypothetical RTA system with that given value of η/s. The agreement between ideal hydro and RTA can be expected of course only in the limit η/s → 0. Specifically, Eq. (85)
When comparing the ideal hydro result to kinetic theory calculations, we employ the above formula with 4πη/s = 1, and for dE tr /dη we rescale the final results with (4πη/s) 4/9 according to Eq. (86) when comparing to kinetic theory at other values of 4πη/s.
V. SPACE-TIME EVOLUTION AT DIFFERENT OPACITIES AND IN DIFFERENT SETUPS
The different behaviour of hydrodynamics compared to kinetic theory in pre-equilibrium can best be assessed via the time dependence of the studied observables. This also allows to study the behaviour during different stages of the collision. In Sec. V A, we discuss the time evolution of transverse profiles of the system in kinetic theory. Sec. V B compares the time evolution of the tracked observables in kinetic theory and scaled viscous hydrodynamics. These are then used as the basis for a discussion of the time evolution in hybrid simulation schemes in Sec. V C.
A. Evolution of transverse profiles in kinetic theory
We now want to discuss the system's time evolution at different opacities resolved in transverse space. This is illustrated in Figure 6 via heat map plots of the time scaled local rest frame energy density τ together with a vector plot of the transverse components of the flow velocity u µ at three different values of the shear viscosity, 4πη/s = 0.5, 3, 10, which are representative of the regimes of hydrodynamic behaviour, close-to-freestreaming behaviour and the intermediate transitioning regime. The time evolution of these profiles is sampled at three different times, τ = 0.1R, 1R and 2R, which mark the beginning, peak and end of the buildup of elliptic flow ε p , as will be discussed in Section V B. At the earliest time, τ = 0.1R, transverse dynamics have not had a large effect yet: flow velocities are negligible and the main geometric properties of the profile remain unchanged. The only obvious difference is the overall scale. At smaller η/s, the system starts cooling sooner, performing more work against the longitudinal expansion, resulting in significantly smaller energy densities when compared to larger η/s. τ = 1R marks the characteristic time where transverse expansion effects become significant. Here, we see the profile taking on a more circular shape. We also see significant flow velocities, which rise in magnitude with the distance from the center. For smaller shear viscosity η/s, meaning larger interaction rates, the system tends to lump together more, resulting in a smaller spatial extent and smaller flow velocities compared to larger η/s. At the largest selected time, τ = 2R, the interaction rate in the system has significantly decreased due to the dilution caused by the transverse expansion. Over time, the dynamics will approach a free-streaming expansion in all directions. It is apparent in all three cases that the system has expanded mainly in the directions of larger gradients in the initial state. For small shear viscosity η/s, the system's energy density is still peaked in the center due to stronger collective behaviour. On the other hand, at large η/s, the system evolution is closer to a free-streaming propagation of the initial state, resulting in two high-density areas at distances r ≈ τ from the center. Though the difference is barely visible, the builtup flow velocities are larger for larger η/s. We can discern additional spatially resolved information on the opacity dependence of the system's evolution by also comparing profile plots of the anisotropic stress, T xx − T yy , which are presented in Figure 7. Per definition in Eq. (14), the transverse integral of this quantity is proportional to elliptic flow ε p , which builds up more at smaller values of η/s. Note that the symmetry-plane phase factor takes the value e 2iΨp = −1 in this case, such that a negative integral results in positive ε p . The plots show that the transverse plane separates into re- gions with different sign of the anisotropic stress. The behaviour in the outskirts is dictated by transverse expansion, resulting in positive values in ±x-direction and negative values in ±y-direction. The buildup of elliptic flow seems to proceed mainly via the positive parts decaying more than negative ones. At small opacities in the right panel, particles propagate with few interactions. Due to the initial almond shape, most of the particles in the center propagate in ±x-direction, resulting in a larger T xx than T yy . At large opacities in the left panel, the system is hydrodynamized and the anisotropic stress comes mostly from the direction of flow. Since the flow components u x and u y are zero in the center of the system, the anisotropic stress vanishes there.
B. Time evolution of observables in kinetic theory and hydrodynamics
We will now examine the time evolution of certain characteristic transverse space integrated observables in both kinetic theory and the scaled hydrodynamics scheme that was proposed in Sec. IV D as a countermeasure to the unphysical pre-equilibrium behaviour of hydrodynamics discussed in Sec. IV B. This will provide additional insights into the system's behaviour but also reveal how well the scaled hydro scheme works at different opacities. Figure 8 shows comparisons of the time evolution of transverse energy dE tr /dη, elliptic flow ε p , average transverse flow velocity u ⊥ and average inverse Reynolds number Re −1 in both models at three different opacities. Since we are using a fixed initial profile, we plot ε p without normalization to the initial state eccentricity 2 . As an illustration of the motivation for the scaling scheme in hydrodynamics, for dE tr /dη and Re −1 we also compare with the time evolution in the absence of transverse expansion, where we describe the system as a collection of local Bjorken flows.
The time evolution of transverse energy dE tr /dη closely follows results from Bjorken flow scaling at early times, as predicted in Sec. IV B. In Bjorken flow scaling, this observable starts out being constant in the freestreaming period of kinetic theory, while in hydrodynamics, it follows a positive power law, cf. Eq. (61). From there, in both cases the time evolution smoothly transitions to a late time equilibrium power law dE tr /dη ∼ τ −1/3 . The timescale of this transition depends on the opacity and is smaller at smaller η/s. In RTA 3 , it scales as τ eq ∼ (η/s) 4/3 [71]. By construction, results from scaled hydrodynamics agree with kinetic theory results in the late time limit of Bjorken flow scaling. The time evolution in full simulations follows this behaviour up to times τ ∼ R, when effects of transverse expansion become significant. The rapid dilution due to transverse expansion decreases interaction rates and causes dE tr /dη to approach a constant value. For large opacities like 4πη/s = 0.5, the Bjorken flow equilibrium where both models agree sets in long before transverse expansion and even afterwards the results will stay in agreement. Intermediate opacities around 4πη/s = 3 mark the transition region where results for dE tr /dη from both models just barely come into agreement before approaching a constant value. At small opacities like in the case of 4πη/s = 10, the onset of transverse expansion interrupts the Bjorken flow scaling period before the two model descriptions have come into agreement. The residual discrepancy then persists throughout the evolution of the system and leads to a mismatch of final state observables,which becomes worse as η/s is increased.
The second line of Figure 8 shows the time evolution of the elliptic flow coefficient ε p . Again, like in the case of dE tr /dη, because of the decrease of interaction rates due to the dilution caused by transverse expansion, ε p reaches a late-time plateau. Thus, at all opacities, ε p builds up in a timeframe of τ 2R. Contributions from early times are negligible, such that effectively the buildup starts at τ 0.1R. As indicated in the log-log insets, the kinetic theory curves exhibit at early times an approximate power-law increase, ε p ∝ τ 8/3 . In contrast, the scaled hydro curve for ε p first dips to negative values. For 4πη/s = 0.5, when equilibration is achieved before the onset of transverse expansion, the scaled hydro curve merges into the RTA one as implied by the discussion in Sec. IV D. At small opacity (4πη/s = 10), the merging process is interrupted by transverse expansion. The scaled hydro result for ε p is in perfect agreement with kinetic theory at large opacities and stays in good agreement at intermediate opacities. Due to a smaller overall interaction rate, the ε p -response decreases with decreasing opacity. For small opacities, a negative trend in the early time behaviour of hydrodynamics causes discrepancies with kinetic theory. This trend will become dominant at even smaller opacities, resulting in negative values of the late time plateaus.
As discussed in Sec. IV C, at early times, u ⊥ builds up linearly with the elapsed time ∆τ = τ − τ 0 in kinetic theory. For finite initialization time τ 0 , the detailed behaviour in hydrodynamics is slightly different, but almost indistinguishable from linearity in ∆τ . Hence, we plot the ratio u ⊥ ∆τ /R and indicate the early time limit using horizontal dashed blue lines. The plots confirm that there are slight differences in the early time be-haviour of the flow velocities in hydrodynamics and kinetic theory, however they come into agreement on similar timescales as dE tr /dη. This is partly owing to the fact that early time contributions to the total u ⊥ are negligible. u ⊥ enters a period of superlinear rise during transverse expansion. While this period ends earlier at larger opacities due to dilution of the system and transition to free-streaming, the total rise of u ⊥ ∆τ /R is nevertheless larger. Comparing hydrodynamic results to kinetic theory results, the late time free-streaming does not seem to be accurately reproduced, as hydrodynamics underestimates u ⊥ . Problems in the late time behaviour are less relevant for the other observables we discuss, as they tend to plateau at late times. This late time discrepancy between hydrodynamics and kinetic theory is thus a phenomenon that mainly affects u ⊥ among the observables that were tracked in this work, and is not related to pre-equilibrium.
Finally, we look at the time evolution of the average inverse Reynolds number, which is a measure of the size of non-equilibrium effects in the system. We normalized this in such a way that in RTA, its initial value is equal to one (note that on the hydro attractor, Re −1 ∼ 1.212 when τ 0 → 0). Like for dE tr /dη, the two model descriptions will come into agreement in the late time limit of Bjorken flow scaling, on timescales that are larger for smaller opacities. Due to equilibration, in this period Re −1 experiences a phase of rapid decay towards 0, as expected since Re −1 measures non-equilibrium effects. The effect of transverse expansion on this observable is not straightforwardly understood, except for the fact that due to the additional dilution, Re −1 must be larger in full simulations than in Bjorken flow scaling. For large opacities, transverse expansion seems to only slow down the approach to equilibrium. However, at intermediate opacities we see a significant rise in Re −1 . We also computed results for the limit of vanishing opacity. Here, the inverse Reynolds number remains constant at early times, but later increases due to transverse expansion, e.g. at τ = 4R to a value of Re −1 (τ = 4R) = 1.322. However, an increase due to transverse expansion cannot be the only late time effect, as we can see from the results at 4πη/s = 10, where the trend of this quantity changes multiple times. It first departs from the local Bjorken flow prediction at τ /R ∼ 0.3, but later the curve returns to decreasing at a rate comparable to that during the Bjorken flow stage. At late times, the behaviour transitions to a rise in the inverse Reynolds number. Despite this peculiar behaviour, our numerical results indicate that Re −1 reaches a minimum value that is larger for smaller opacities. For very small opacities, it will not drop significantly below its initial value of 1 before starting to rise.
For a more detailed examination of the opacity dependence of the time evolution in kinetic theory ranging from very small (4πη/s = 1000) to very large opacities (4πη/s = 0.01), please see Appendix C.
After examining the time evolution of these observables and establishing some understanding about the implications of their buildup, we now want to invert this logic. As the change in these observables carries information on the state of the system, e.g. the progress of its equilibration or the onset of transverse expansion, we want to track the first times when these observables reach a specific milestone of their time evolution as a function of opacity. Figure 9 shows plots of kinetic theory results for these curves for five different milestone criteria. Specifically, we tracked when the average transverse flow velocity reaches a value of 0.1 as a criterion for the onset of transverse expansion, the time when the elliptic flow response builds up to 5% of its maximum value at the given opacity as a criterion for the beginning of the buildup of flow, and the time when the average inverse Reynolds number reaches values of 0.4, 0.6 and 0.8, which tells us to what degree hydrodynamization has progressed. As it turns out, the curve for the flow velocity criterion is almost perfectly flat at a value of τ c ≈ 0.15R, meaning that the early time buildup of u ⊥ is mostly independent of the opacity. The elliptic flow criterion is met at similar times as the flow velocity criterion at large opacities, but at slightly later times τ c ≈ 0.3R for small opacities. Despite the general timeframe of ε p -buildup being independent of opacity, it seems to start slightly earlier at larger opacities. The most interesting criterion curves are those for the average inverse Reynolds number. The system's adherence to early time Bjorken flow scaling leads to a power law behaviour τ c ∝ (η/s) 4/3 for all three of these curves at large opacities. The curves deviate from this power law when the criterion is not reached before transverse expansion sets in at times τ ∼ R. For small opacities, the criteria are never met, as the average inverse Reynolds number reaches a minimum value larger than the criterion value, as already stated in the discussion of Figure 8. The behaviour of dE tr /dη resembles that of dE ⊥ /dη, which we already discussed in our previous paper [71]. Similarly to Re −1 , it follows Bjorken flow scaling at early times, resulting in a similar power law behaviour.
C. Time evolution in hybrid schemes
Another way to alleviate discrepancies due to the behaviour of hydrodynamics in the pre-equilibrium phase as discussed in Sec. IV B is to model the time evolution via a hybrid scheme, switching from a kinetic theory based description at early times to hydrodynamics at later times, i.e. initializing the hydrodynamic simulation with the energy-momentum tensor computed from the kinetic theory based time evolution. This requires to fix a criterion for when to switch descriptions.
As we argue that hydrodynamics becomes viable only after some timescale related to equilibration, we also expect the accuracy of hybrid scheme results to depend on the switching times. Due to the opacity dependence of equilibration, it might be beneficial to choose switching times as a function of opacity. Hence we tested both a hybrid scheme with fixed switching times at two different times τ = 0.4 fm and τ = 1 fm, which are in the range of switching times typically used in phenomenological descriptions, and with dynamically determined switching times.
In order to tie this definition to the phenomenon of equilibration, we determine the dynamical switching times on the basis of the decrease of the average inverse Reynolds number Re −1 , i.e. we switch as soon as this quantity first reaches a specific value. Specifically, we chose the values Re −1 = 0.8, 0.6 and 0.4 (sometimes we will consider switching also when Re −1 drops below 0.2). In the case of a transversally homogeneous system, Figure 3 shows that these values for the inverse Reynolds number correspond to various degrees of hydrodynamization of the system. Specifically, Re −1 = 0.8 (w 0.2) corresponds to the onset of hydrodynamization. When Re −1 = 0.6 (w 0.6), the system significantly progressed through the hydrodynamization process, while when Re −1 = 0.4 (w 1), the system has hydrodynamized and the kinetic theory and hydrodynamics attractor curves are almost merged. Due to the relation (38) betweenw and the Bjorken time τ , the characteristic times τ c when Re −1 drops below a certain threshold increase with 4πη/s (see Sec. V B for a detailed discus- The results are illustrated by the time evolution of transverse energy dE tr /dη, elliptic flow ε p and average transverse flow velocity u ⊥ compared for different choices of the switching times, as plotted in Figure 10 at three different opacities. The early time evolution was computed with the RLB method of simulating kinetic theory. The plots also compare to results from a pure kinetic theory simulation as well as from our scaled viscous hydro scheme. Here we plot all results including the ones for elliptic flow ε p on a logarithmic scale of the time axis so that the different switching times are discernible. The ε p plots also feature an inset plot on log-log scale. It can be seen that the curves corresponding to the hybrid setups tend to detach from the RTA curve towards lower values of ε p . Since in viscous hydro, the equilibration process leads to a decrease of spatial eccentricity 2 (see lower panel of Figure 5), the hybrid simulations with early switching times will lead to lower late-time values of ε p (see the discussion in the next section).
∼ ( η / s ) 4 / 3 Characteristic timescale τ c /R Shear viscosity 4πη/s Opacityγ ε p (τ ) = 0.05ε p ⟨u ⊥ ⟩ ϵ = 0.1 ⟨Re −1 ⟩ ϵ = 0.8 ⟨Re −1 ⟩ ϵ = 0.6 ⟨Re −1 ⟩ ϵ = 0.4
At a small shear viscosity of 4πη/s = 0.5, all switching schemes yield accurate results for all three observables. Since the equilibration timescale is small for this opacity, the system has equilibrated by the time we switch descriptions such that kinetic theory and hydrodynamics are in agreement. The Re −1 -based criteria are fulfilled early on in the system's evolution such that the dynamically chosen switching times are significantly smaller than the fixed ones. However, when comparing results from pure kinetic theory or viscous hydrodynamics, they are within the timeframe where both descriptions are in acceptable agreement. The only curve where a devia-tion from kinetic theory is clearly visible is the one for Re −1 = 0.8, where hydrodynamization has only partly progressed before this criterion is fulfilled.
The results at 4πη/s = 3 now show that it is indeed necessary to give the choice of switching times some thought, as here we see a significant increase in accuracy of results for dE tr /dη and u ⊥ with later switching times. For this opacity, the dynamically chosen switching times are on a similar scale as the fixed ones. We also see that the nature of any discrepancies with pure kinetic theory results is the same as in the case of hydrodynamics. As soon as we switch, the curves of these observables follow a trajectory that is qualitatively similar to the pure hydrodynamics result, meaning that dE tr /dη is increased, while u ⊥ and ε p are decreased relative to the kinetic theory result.
The strength of the dynamically chosen switching times is well displayed for results at 4πη/s = 10. In this case, the system is still far from hydrodynamized at the two fixed switching times, leading to sizeable inaccuracies in the corresponding hybrid scheme results for all three observables, but especially for dE tr /dη. As Re −1 does not drop low enough, two of the three criteria for the dynamical switching were not reached. However, the result for switching at the largest of the three values of Re −1 retains a similar level of accuracy as at smaller shear viscosity and is a significant improvement to fixed time switching results.
Overall, we find that while switching at fixed time is conceptionally straightforward and always possible, the accuracy of this scheme strongly depends on the opacity and results at small opacity show large deviations from Table II).
full kinetic theory. On the other hand, switching based on Re −1 is not always possible because this quantity does not drop to the desired values at small opacities, but whenever it is possible, the accuracy of the result can be estimated beforehand and depends only little on opacity. In other words, the dynamical definition yields the earliest possible switching time for a desired accuracy, and whenever Re −1 does not drop enough for it to be determined, hydrodynamics is not viable in the first place.
Finally, we also tested hybrid schemes with the same switching times but with an early time evolution computed in KøMPøST. We found that due to its limited range of applicability, some of the later switching times could not be viably reached with this description. But whenever we were able to obtain results, they were in good agreement with the results from the previously discussed scheme, except for some systematic errors in ε p and u ⊥ . These results are presented in more detail in Appendix D, together with analogous time dependence plots to those presented in Figure 10.
VI. OPACITY DEPENDENCE OF FINAL STATE OBSERVABLES IN DIFFERENT TIME EVOLUTION MODELS
The previous section's comparison of the time evolution in different models has provided insights into the nature of different sources of discrepancies and at what opacities to expect them. For a detailed opacity-resolved analysis, it is convenient to study the dependence of final-state observables on a wide range of opacity, from the free-streaming regime to ideal fluid behaviour. In Sec. VI A, we present opacity dependencies in kinetic theory, naive viscous hydrodynamics and scaled viscous hydrodynamics. Section VI B discusses results for hybrid simulation schemes.
A. Scaled and naive hydrodynamics compared to kinetic theory
First, we assess the performance of scaled hydrodynamics as described in Sec. IV D when compared to a common naive initialization scheme of hydrodynamics, where the simulation is started at a time τ 0 where hydrodynamization is likely to have set in, with the same initial condition for τ 0 (τ 0 , x ⊥ ) as we are using for kinetic theory simulations initialized in the early time free-streaming limit. Fig 11 shows the opacity dependences of transverse energy dE tr /dη, elliptic flow ε p and average transverse flow velocity u ⊥ in kinetic theory, scaled hydrodynamics and naive hydrodynamics initialized on the hydrodynamic attractor at two different times τ 0 = 0.4 fm and τ 0 = 1 fm, which are in the range of values typically used in phenomenological descriptions.
For all three observables, the kinetic theory results smoothly interpolate between limiting cases of small and large opacities. For dE tr /dη and ε p , we compare at small opacities to results from the linear order opacity expansion that is introduced in Sec. III B. Results from full kinetic theory are in excellent agreement with these approximations for 4πη/s 20. In the case of u ⊥ , we present results for the free-streaming limitγ → 0, to which the full kinetic theory results converge at small opacities.
On the other end of the opacity spectrum, the results from both kinetic theory and scaled viscous hydrodynamics converge to those of scaled ideal hydrodynamics in the limit η/s → 0. Even though η/s = 0 by definition in ideal hydrodynamics, we represent the scaled ideal hydro results as a function of 4πη/s in the equivalent RTA simulation [see discussion around Eq. (86)], leading to a power-law dependence dE tr /dη ∝ (4πη/s) 4/9 , which is confirmed by the scaled viscous hydrodynamics and kinetic theory results (this result was derived from earlytime Bjorken scaling in [71]). The curves for ε p and u ⊥ converge at large opacities to the ideal hydrodynamics limit that was obtained with a scaled initial condition. This is not a priori obvious but rather an achievement of 15)] for kinetic theory (black), scaled hydro (purple) and naive hydro at two different initialization times τ0 = 0.4 fm (brown) and 1 fm (yellow). Also plotted are the small opacity limits of an opacity-linearized result (blue) in the top two plots, the free-streaming result (blue, dashed) in the bottom plot as well as the opacity-scaled ideal hydrodynamics results (grey, dashed). The latter follows a (η/s) 4/9 scaling law for dEtr/dη as per the initialization scheme in Eq. (86). The ideal hydro results are 611 GeV · (4πη/s) 4/9 for dEtr/dη, 0.244 for εp and 3.01 for u ⊥ . The red shaded region shows the realistic values for QCD according to Bayesian estimates. our proposed scheme. Ideal hydrodynamics is the large opacity limit of kinetic theory only after hydrodynamization. At any finite opacity, kinetic theory simulations feature a pre-equilibrium period which is absent in ideal hydro. In this period, the ellipticity 2 decreases in kinetic theory, such that with the same initial condition, it would result in a smaller elliptic flow ε p than in ideal hydro. The agreement is only reached after scaling the hydro initial condition as discussed in Sec. IV D.
Comparing now to hydrodynamic results, for all three obserables, the large opacity limits of scaled hydrodynamics and kinetic theory are in excellent agreement. Going to small opacities, all observables are underestimated in hydro, as will be further discussed in the following. Agreement holds for 4πη/s 3.
On the other hand, for naive hydrodynamics initialized at τ 0 = 0.4 fm and 1.0 fm, the opacity dependence curves show qualitatively similar behaviour to kinetic theory but remain in quantitative disagreement for all opacities. This is obvious in the case of dE tr /dη, which is significantly overestimated. We find that the large opacity power law is not captured. There are different reasons for this in the two limiting cases of large and small opacity. For small opacities 4πη/s 10, despite the initialization time being large, it is still smaller than the equilibration timescale and the simulation will partly undergo a pre-equilibrium phase. As we have seen, in the hydrodynamic description of this phase dE tr /dη increases before the onset of transverse expansion, while staying constant in kinetic theory, so it is overestimated in hydro. For the smaller initialization time τ 0 = 0.4 fm, the system is in pre-equilibrium for a longer time compared to τ 0 = 1 fm. This is why results for τ 0 = 0.4 fm yield a larger final value of dE tr /dη at small opacities. On the other hand, for large opacities 4πη/s 3, the system would have been in equilibrium for a significant amount of time if it had been initialized at an earlier time. In the equilibrated phase before transverse expansion, dE tr /dη drops proportionally to τ −1/3 . The larger the initialization time, the more of this period is cut out of the simulation, resulting in a larger final value. This is why the curve for initialization at τ 0 = 1 fm is above the one for τ 0 = 0.4 fm at large opacities, resulting in a crossing of the two at intermediate opacities 4πη/s ∼ 5. The equilibration timescale becomes smaller and smaller at larger and larger opacities, meaning that for fixed initialization time more and more of the τ −1/3 -scaling period is cut out. This is why the large opacity power law is not reproduced in naive hydrodynamics.
These problems are cured in scaled hydrodynamics. It is initialized at very early times, so no part of the time evolution is lost. The discrepancies due to hydrodynamic pre-equilibrium behaviour are cured via scaling the initial energy density as discussed in Sec. IV D such that agreement with kinetic theory is reached only after equilibration. However, for small opacities 4πη/s 3, the underlying assumption of a timescale separation of equilibration and transverse expansion no longer holds. In this case, scaled hydrodynamics underestimates dE tr /dη, as transverse expansion interrupts its approach to kinetic theory behaviour before agreement is reached.
Of the three presented observables, ε p in naive hydrodynamics shows the weakest deviations from kinetic theory . This is in alignment with our expectations, as we know that hydro has been very successful in phenomenological descriptions of anisotropic flow. The reasons might be that ε p builds up on timescales that are fully captured by simulations at initialization times of ∼ 1 fm and depends very little on the overall energy scale. But certainly, this level of agreement was not to be expected a priori and should be regarded as a coincidence. The influence of the initialization time is as follows. At small opacities 4πη/s 10, a part of the early time negative trend in hydrodynamics is cut out, resulting in larger results for later initialization times. For large opacities 4πη/s 1, ε p already has positive contributions at early times which might be cut out, resulting in smaller final values for later initialization times. But very early initialization times cannot bring hydro into agreement with kinetic theory. As discussed in Sec. IV B, hydrodynamics initialized at very early times exhibits a larger decrease of the eccentricity during preequilibrium, resulting in lower final values of ε p than in kinetic theory. However, the scaling procedure counteracts this phenomenon by increasing the eccentricity in the initial state of hydrodynamic simulations, such that scaled hydrodynamics is in perfect agreement with kinetic theory at large opacities 4πη/s 3. For small opacities 4πη/s 10, on the other hand, due to the early initialization scaled hydrodynamics features a longer period of the aforementioned early time negative buildup of ε p , resulting in final values which are lower than in the case of the naive hydro initializations discussed above.
The flow velocity results from naive hydrodynamics again show two effects. One of them is straightforward: as this observable rises monotonically with time, for larger initialization times, there is less time for u ⊥ to build up, resulting in an underestimate. This effect is cured in scaled hydrodynamics due to the early initialization. At small opacities 4πη/s 10, we see an additional decrease of hydrodynamic results compared to kinetic theory due to its inability to describe the latetime free-streaming behaviour. This is an effect that both hydro schemes (based on naive and scaled initial conditions) have in common.
B. Hybrid simulations
As described in Sec. V C, another way to bring hydrodynamic results into agreement with kinetic theory is to use hybrid schemes switching from a kinetic theory based early time description to hydrodynamics at later times. We tried switching both at fixed times as well as at the first times equilibration has proceeded to a given extent, which we quantified by the inverse Reynolds number dropping to a specific value. We also tested two different model descriptions for early times: full kinetic theory and KøMPøST. As described in the previous section, the time evolution curves of all observables instantaneously change behaviour when the models are switched, such that switching too early will be affected by the inac-curate description of pre-equilibrium in hydrodynamics. We now want to quantitatively assess the accuracy of various switching schemes as a function of opacity.
We first discuss results for the opacity dependence in hybrid simulations with Re −1 -based switching, which are plotted in Figure 12. For early switching times on the timescale of equilibration, hybrid results may reflect the inaccurate pre-equilibrium behaviour in hydrodynamics. Of course, in this case, there is no scaling of the initial condition to counteract this behaviour. However, this also means that these schemes do not suffer from discrepancies due to an incomplete approach of a scaled initial condition to kinetic theory behaviour before the onset of transverse expansion, and therefore tend to be more accurate than scaled hydrodynamics at intermediate opacities, i.e. for 4πη/s ∼ 3. However, results plotted with smaller crosses and dashed lines were obtained in simulations with switching times larger than 0.5R, so in this case it is questionable whether these schemes could be considered hybrid results, as the crucial parts of the time evolution were actually described in kinetic theory.
Going into more detail, hybrid results typically overestimate dE tr /dη because of the hydrodynamic preequilibrium increase after switching. ε p is underestimated, however, the hydrodynamic negative early time trend is alleviated, such that results from kinetic theory + viscous hydro are typically larger than scaled hydro results. Hybrid results show a consistent underestimation of u ⊥ , but on a relative scale this error is negligible. This could be due to hydrodynamic flow velocities typically being smaller than those in kinetic theory at early times, causing a dip in u ⊥ relative to kinetic theory after switching.
Comparing kinetic theory + viscous hydrodynamics in the left column of the figure to KøMPøST + viscous hydrodynamics in the right column, one obvious difference is that in the latter, some of the results for smaller opacities are missing, because there the Re −1 -based switching times were too late to be reached with KøMPøST. 4 Where it does work, it produces almost the same results for dE tr /dη as kinetic theory. The underestimation of u ⊥ is slightly more severe in KøMPøST. It does seem to have a systematic component on top of the one related to switching early. But the total deviation is still negligible. The largest difference is seen in ε p , which is not built up at all in KøMPøST simulations, thus there is a significantly larger underestimation at smaller opacities, where a larger part of the time evolution is described in KøMPøST.
Next, we shift our attention to results from hybrid schemes at fixed switching times τ s = 0.4 fm and τ s = 1 fm, which are presented in Figure 13. As expected from the discussion of the time evolution in Sec. V C, again kinetic theory + viscous hydrodynamics yields perfectly accurate results at large opacities 4πη/s 1 and improves on scaled hydrodynamics at intermediate opacities 4πη/s ∼ 3, but less so than for dynamically chosen switching times. The upshot is that hybrid schemes with fixed switching times are applicable for arbitrarily small opacities. However, here the results for the three tracked observables show similar problems to those obtained in naive hydrodynamics simulations discussed earlier in this section. Due to incomplete equilibration at early switching times, dE tr /dη increases after switching. ε p suffers from the early time negative trend in hydrodynamics, but slightly less than scaled hydrodynamics. u ⊥ is again only slightly underestimated in hybrid schemes when compared to scaled hydrodynamics due to the different pre-equilibrium. This is an improvement over naive hydrodynamics, as instead of starting at late times with no flow velocity, the early time buildup is described in kinetic theory. Both schemes suffer equally from the inability of hydrodynamics to describe flow velocities in the late time free-streaming limit.
Also for fixed switching times, KøMPøST + viscous hydrodynamics results for dE tr /dη and u ⊥ are in good agreement with those obtained in kinetic theory + viscous hydrodynamics simulations. We again see the effect of the absence of ε p -buildup in KøMPøST. Since we do not increase the duration of time evolution in KøMPøST, the effect is not larger at small opacities 4πη/s 10. In fact, here we see agreement with results from kinetic theory + viscous hydrodynamics, as there is no significant buildup of ε p at early times. However, at large opacities 4πη/s 5, this buildup starts earlier, which is why KøMPøST + viscous hydrodynamics results underestimate the final values in these cases.
VII. CONCLUSIONS
In this work, we examined hydrodynamic and kinetic theory simulations of hadronic collisions. Within a simplified model setup based on RTA and using a fixed initial profile that was obtained as an average of events in the 30 − 40% centrality class of Pb-Pb collisions, we scanned the dynamical behaviour on the whole range in interaction rates as parametrized by the opacityγ defined in Eq. (7), which for our fixed profile is inversely proportional to shear viscosity,γ = 11.3/(4πη/s). This study was based on results for the transverse energy dE tr /dη, elliptic flow ε p , radial flow u ⊥ and shear stress as measured via the inverse Reynolds number Re −1 . At small opacities 4πη/s 20, kinetic theory agrees with results from a linearization in opacity. Here, the system is too dilute for hydrodynamics to be applicable, which was confirmed quantitatively in Sec. VI A: the time evolution of transverse energy, radial flow and shear stress is in significant disagreement in hydrodynamic simulations compared to kinetic theory. For large opacities 4πη/s 0.1, in the limit of high interaction rates, kinetic theory is expected to converge to hydrodynamics. Our results confirm that the two descriptions are in agreement after pre-equilibrium. Going down to intermediate scription of kinetic theory approaches a free-streaming behaviour. In both of these regimes, hydrodynamic results are in quantitative disagreement with kinetic theory, which can be seen at the level of final state observables, as discussed in Sec. IV B. Omitting the preequilibrium period or naively employing hydrodynamics 15)] in hybrid kinetic theory + viscous hydro (solid lines) and KøMPøST + viscous hydro simulations (dashed lines) when switching at fixed times τ = 0.4 fm (light red) and τ = 1 fm (dark red). The results are compared to kinetic theory (black), scaled hydro (purple) and the small opacity limits of an opacity-linearized result (blue) in the top two plots, the free-streaming result (blue, dashed) in the bottom plot, as well as to the large opacity limit of scaled ideal hydro (grey, dashed), which scales as (η/s) 4/9 in the top plot. The red shaded region shows the realistic values for QCD according to Bayesian estimates. The bottom part of the plot shows the ratios of all results to those from kinetic theory.
to describe it will yield inaccurate results. On the other hand, at late times where interactions die out, these observables no longer build up and approach constant values, such that hydrodynamic descriptions yield similar results to kinetic theory. However, the late time freestreaming stage does have an effect on radial flow, which is underestimated in hydrodynamics.
We examined two different modified setups of hydrodynamic simulations that can alleviate problems with the pre-equilibrium evolution. The first setup follows the idea of an early initialization of hydrodynamics with a scaled initial condition relative to kinetic theory to counteract the differences in the pre-equilibrium evolution. These differences are predicted locally based on insights from Bjorken flow, which is accurate in systems with a timescale separation of equilibration and the onset of transverse flow. By construction, this setup yields accurate results at large opacities 4πη/s 3, but fails at smaller opacities, where equilibration takes longer and is interrupted by transverse expansion. The results obtained in this setup are presented in Sec. V B and VI A.
The second setup is a hybrid simulation switching from kinetic theory based descriptions at early times to hydrodynamics for later times. In these schemes, as described in Sec. V C, we saw an immediate change of the time evolution behaviour at the moment that we switch the dynamical descriptions. Thus, the accuracy of hybrid simulations depends on the extent to which the kinetic theory and hydrodynamic descriptions of the system's time evolution have come into agreement by the time of the switch. This approach to agreement between the two descriptions is what we call hydrodynamization. We found that this criterion can in practice be quantified via the inverse Reynolds number. Figure 3 shows that the system is partly hydrodynamized when Re −1 = 0.8, significantly hydrodynamized when Re −1 = 0.6 and has reached almost perfect agreement of the two descriptions at Re −1 = 0.4. The accuracy of hybrid simulations when switching at a fixed value of Re −1 can be estimated beforehand and is almost independent of the opacity. As detailed in Sec. VI B, results from simulations with late switching times are accurate at high opacities 4πη/s 1 and can slightly improve on our first setup at intermediate opacities 4πη/s ∼ 3. At small opacities 4πη/s 20, Re −1 does not drop below 0.8, meaning the system does not equilibrate enough for hydrodynamics to become applicable at any point during the system's evolution.
For the early time kinetic theory description in hybrid models, we used both full kinetic theory and the compact KøMPøST code. The latter uses a linearization scheme in perturbations around local homogeneity to propagate the energy-momentum tensor according to the Boltzmann equation under the relaxation time approximation (the original version [109,110] is based on the QCD effective kinetic theory [106]). We first tested the performance of KøMPøST as detailed in Sec. III E and found that it can accurately reproduce full kinetic theory results for transverse energy, radial flow and isotropic shear stress, but due to the linearization it significantly underestimates elliptic flow. It is by construction limited to times on the order of 0.5R. In hybrid simulations with switching times in this regime, transverse energy and radial flow results reach similar accuracy as when employing full kinetic theory. However, the underestimation of elliptic flow causes discrepancies when the switching time is non-negligible compared to the timescale of transverse expansion. These shortcomings have already been reported in the original KøMPøST paper [136] and will require further investigation in the future.
This work provides the baseline for analyses of hadronic collisions in frameworks based on the microscopic dynamics of kinetic theory. It is part of a series of recent efforts [49,71,88,89,94] to push the practical applicability of these dynamics in theoretical simulations. One important goal that has yet to be reached is to improve the codes that implement them in order to be able to also run event-by-event simulations. At the moment, the tool that is closest to fulfilling this goal is KøMPøST, which we confirmed to function properly for its intended use, but it is confined to the pre-equilibrium phase of heavy-ion collisions.
Broadly speaking, our results confirm that in principle hydrodynamics is the proper tool for describing mid-central collisions, if and only if pre-equilibrium is described correctly. Issues with this phase in hydrodynamic descriptions can be alleviated by changing the interpretation of the initial state in the way discussed in Sec. IV D. As alluded to in Sec. IV A as well as in previous works [65,137,138], appropriate changes to the evolution equations might achieve similar improvements. If such changes are not incorporated, we discussed in Sec. VI that hydrodynamic results can be in significant disagreement with kinetic theory. We also refer the interested reader to our companion paper [102], where we extract a more general criterion for the applicability of hydrodynamics and infer phenomenological conclusions for the description of the space-time dynamics of highenergy collisions.
ACKNOWLEDGMENTS
We thank P. Aasha, N. Borghini The first step in applying the relativistic lattice Boltzmann (RLB) method is the parametrization of the momentum space, which we perform using two sets of coordinates, namely the spherical (subscript s) and freestreaming (subscript fs) coordinates:
(p s , v z;s ) = p τ , τ p η p τ ,(A1a)(p fs , v z;fs ) = p s ∆ s , τ v z;s τ 0 ∆ s ,(A1b)
where ∆ s = [1 + ( τ 2 [139]. The azimuthal coordinate ϕ p = arctan(p y /p x ) is employed in both parametrizations.
τ 2 0 − 1)v 2 z;s ] 1/2
Due to the particularly simple nature of RTA, the dynamics of the observables introduced in Sec. II are fully described by the reduced distribution F * ( * ∈ {s, fs}), obtained from the phase-space distribution f via
F * = ν eff πR 2 τ 0 (2π) 3 dE 0 ⊥ dη −1 ∞ 0 dp * p 3 * f .(A2)
Using the non-dimensionalization conventions introduced around Eq. (19), the non-dimensional function F s ≡ F s (τ ,x T , ϕ p , v z;s ) satisfies
∂ τ + 1 − v 2 z;s v ⊥ ·∇ ⊥ + 1 + v 2 z;s τ F s − 1 τ ∂[v z;s (1 − v 2 z;s )F s ] ∂v z;s = −γv µ u µT (F s − F eq s ) , (A3) while F fs ≡ F fs (τ ,x T , ϕ p , v z;fs ) obeys ∂ τ + 1 ∆ fs 1 − v 2 z;fs v ⊥ ·∇ ⊥ F fs = −γv µ u µT (F fs − F eq fs ) , (A4) with ∆ fs = [1 − (1 −τ 2 0 τ 2 )v 2
z;fs ] 1/2 . The equilibrium functions F eq * are given by where
F eq s = ∆ 4 fs F eq fs =τ 0˜ 4π(v µ u µ ) 4 ,(A5)v µ u µ = γ 1 − 1 − v 2 z;s v ⊥ · β = γ 1 − 1 ∆ fs 1 − v 2 z;fs v ⊥ · β ,(A6)
with γ = u τ ≡ 1/ 1 − β 2 being the local Lorentz factor. In the above, β = β(cos ϕ u , sin ϕ u ) and v ⊥ = (cos ϕ p , sin ϕ p ) are transverse-plane vectors.
Vanishing longitudinal pressure and azimuthal momentum isotropy imply the following initial state for the reduced distributions F * :
F * (τ 0 ,x ⊥ , ϕ p , v z; * ) = δ(v z; * ) 2πτ 0˜ 0 (x ⊥ )(A7)
and depends only on the initial transverse energy distribution dE 0 ⊥ /dηd 2 x ⊥ = τ 0 0 [see Eq. (2)]. Note that at τ = τ 0 , ∆ s = ∆ fs = 1 and (p fs , v z;fs ) = (p s , v z;s ).
Due to the singular nature of the Dirac delta function δ(v z ), Eq. (A7) cannot be achieved exactly with our numerical approach. We instead employ the Romatschke-Strickland distribution with anisotropy parameter ξ 0 ,
f RS = exp p τ Λ 0 1 + ξ 0 v 2 z − 1 −1 ,(A8)
where ξ 0 = 0 corresponds to the isotropic Bose-Einstein distribution, while ξ 0 → ∞ is required in order to achieve Eq. (A7). The parameter Λ 0 ≡ Λ 0 (x ⊥ ) represents an energy scale satisfying
Λ 0 = 2 1/4 T 0 arctan √ ξ 0 √ ξ 0 + 1 1 + ξ 0 −1/4 ,(A9)
reducing to the initial temperature T 0 when ξ 0 = 0. Thus, the system is initialized according to
F 0;s = F 0;fs =τ 0˜ 0 2π (1+ξ 0 v 2 z ) 2 arctan √ ξ 0 √ ξ 0 + 1 1 + ξ 0 −1 . (A10)
We now summarize the characteristics and parameters of our RLB solver. The advection operator v ⊥ · ∇ ⊥ is implemented using the upwind-biased finite-difference fifth-order weighted essentially non-oscillatory (WENO) scheme [125,140] (see Ref. [107] for details). The spatial domain consists of a square box of size 16R centered on the system's center of mass and is discretized equidistantly using S 2 cells. Periodic boundary conditions are employed at the domain edges. When initializing the system, a background value th = 10 −10 × R τ0 ref is added to the energy density to avoid numerical underflow.
The time stepping is performed by solving the equatioñ ∂ τ F * = L[τ , F * (τ )] using the third-order Runge-Kutta scheme [125,141,142], as described in Ref. [107]. The time step δτ is chosen dynamically as
δτ (τ ) = min τ δτ τ M , max ⊥ (τ R ) 2 , δτ M ,(A11)
where max ⊥ (τ R ) represents the maximum value of τ R (τ ,x ⊥ ) taken over the entire flow domain at timẽ τ = τ /R, while the values of (δτ /τ ) M and δτ M are shown in Table III. The discretization of δϕ p is done equidistantly using Q ϕ points (the employed values of Q ϕ are summarized in Table III).
The v z; * degree of freedom is discretized using Q z values. When employing the spherical coordinate v z;s , these points are chosen according to the Gauss-Legendre quadrature rules as the roots of the Legendre polynomial of order Q z , i.e., P Qz (v z;j ) = 0. When v z;fs is employed, the discretization is performed equidistantly at the level of the parameter χ = artanh(Av z;fs ), namely χ j = ( 2j−1 Qz − 1)artanhA. In this paper, we take 1 − A = 10 −6 (see Sec. IV.B of Ref. [71] for more details).
As shown in Table III, the (s) and (fs) approaches are employed when 4πη/s ≤ 5 and 4πη/s ≥ 10, respectively. Employing the (s) approach at larger values of 4πη/s requires increasing Q z , otherwise the time evolution leads to energy-momentum tensor configurations which are incompatible with the Landau frame. Using Q z = 200 gives reliable results for 4πη/s 10. Because the computation of the force term involving ∂ vz F is quadratic with respect to Q z (see Sec. III.E of Ref. [107] for details), the (s) strategy becomes inefficient when Q z 200.
Conversely, the (fs) approach requires larger Q z as τ f /τ 0 is increased (we ran all simulations up to τ f = 5R). Since in the (fs) approach, the computational time scales linearly with Q z , we employed Q z = 1000. With our choice of parameters, this limits the lower value of τ 0 to 10 −3 R, which is insufficient to correctly capture the early-time dynamics of the system when 4πη/s 1.
Finally, the choice of ξ 0 in preparing the initial state depends on the v z; * resolution offered by the chosen discretization. As ξ 0 → ∞, the initial state becomes peaked around v z = 0, hence the v z; * discretization must include sufficient points around this value. We found that the influence of the initial value of ξ 0 on the observables is less significant at smaller values of 4πη/s. Thus, we employed progressively larger values of ξ 0 as we increased η/s, which were compatible with the discretization of v z; * , as shown in Table III. In this appendix, we discuss the numerical code needed for obtaining the linear order results discussed in Sec. III B. Sec. B 1 discusses the conceptual setup of the code and Sec. B 2 deals with the details of how the integration is performed.
Setup of the linear order code
The code is set up to compute the zeroth and first order contributions to the energy-momentum tensor, which is given in terms of the phase space density as
T µν = ν eff (2π) 3 d 2 p ⊥ dy p µ p ν f .(B1)
For simplicity, observables that are nonlinear in T µν with contributions from both zeroth and first order in the opacity expansion were computed only to zeroth order. The code is set up as follows. For an arbitrary initial energy density distribution 0 (τ 0 , x ⊥ ), the free-streaming energy momentum tensor is given as
T (0)µν = τ 0 τ dφ v 2π v µ ⊥ v ν ⊥ (τ 0 , x ⊥ − ∆τ v ⊥ ) ,(B2)
where ∆τ = τ − τ 0 . The integral over φ v is performed numerically, using the same stencils for all entries to prevent errors later on. Now, to go to first order in the opacity expansion, we first have to compute the zeroth order results for the restframe energy density (τ, x ⊥ ) and the flow velocity u µ (τ, x ⊥ ), as they are required for evaluating the RTA collision kernel. This is achieved by numerical diagonalization of T (0)µν . As computed before [71], the first order correction to the phase space distribution is given as an integral of the zeroth order collision kernel:
f (1) (τ, x ⊥ , p ⊥ , y − η) = τ τ0 dτ C[f (0) ] p τ (τ , x ⊥ , p ⊥ , y − η) ,(B3)
where f (0) is the free-streaming solution given in Eq. (24). The primes on the variables indicate the use of free-streaming coordinates,
x ⊥ = x ⊥ − v ⊥ t(τ, τ , y − η), y = η + arcsinh τ τ sinh(y − η) ,(B4)
with t(τ, τ , y − η) being given in Eq. (25). From this, the first order correction to the energymomentum tensor is obtained as
T (1)µν = ν eff (2π) 3 p ⊥ dy p µ p ν τ τ0 dτ × C[f (0) ](τ , x ⊥ , p ⊥ , y − η) p τ (p ⊥ , y − η) ,(B5)
where p ⊥ ≡ d 2 p ⊥ . As it turns out, the observables that are to be computed to first order in opacity depend only on transverse integrals of the components of T ij . Thus, we need to perform a 6D integral, which can be done in part analytically, reducing the complexity of the numerical integration. For further details of the analytical preparatory groundwork for the numerical implementation, see Appendix B 2.
The observables are now computed from these results in the following way. In the case of the transverse-plane energy, we have dE tr dη = τ
x ⊥ (T 11 + T 22 )
= τ
x ⊥ (T (0)11 + T (0)22 + T (1)11 + T (1)22 ) . (B6)
As elliptic flow is given as a quotient of two transverse integrals of components of T µν where the numerator vanishes at zeroth order, the first order result is given as
e 2iΨp ε p = x ⊥ (T 11 − T 22 + 2iT 12 ) x ⊥ (T 11 + T 22 ) = x ⊥ (T (1)11 − T (1)22 + 2iT (1)12 ) x ⊥ (T (0)11 + T (0)22 ) .(B7)
Both of these observables depend on the transverse integral of T (0)11 + T (0)22 , which using B2 can be straightforwardly evaluated to
x ⊥ T (0)11 + T (0)22 = 1 τ dE 0 ⊥ dη .(B8)
In particular, the quantity dE tr /dη, which we introduced as the analogue of dE ⊥ /dη = x ⊥ p τ p ⊥ , is in fact identical to dE ⊥ /dη to zeroth order. Furthermore it is constant in time, so only the first order correction has to be computed. We furthermore compute zeroth order results for the average transverse flow velocity and the average inverse Reynolds number as
u ⊥ = x ⊥ (0) u (0) 1 2 + u (0) 2 2 x ⊥ (0) ,(B9)Re −1 = x ⊥ 6π (0)µν π (0) µν x ⊥ (0) = x ⊥ 6T (0)µν T (0) µν − 24 3 (0) 2 x ⊥(0)
.
(B10)
Analytical and numerical integration in the computation of linear order results
As discussed in the previous appendix, obtaining numerical results for the linear order term in the energy momentum tensor requires the computation of a 6D integral. In this appendix, we explain what part of this integral is performed analytically and give the specific form of the remaining integral which the code computes numerically.
We start from the expression in Eq. (B5) for the purely spatial components of the energy momentum tensor, Table II) .
x ⊥ T (1)ij = ν eff (2π) 3 x ⊥ p ⊥ dy p i ⊥ p j ⊥ × τ τ0 dτ C[f (0) ] p τ (τ , x ⊥ , p ⊥ , y − η) ,(
is almost constant. At larger opacities, due to more work being performed against the longitudinal expansion, dE tr /dη decreases by a larger total amount. The opacity also sets the timescale for this cooling, as it sets in earlier for larger opacities. Elliptic flow ε p stays close to zero at small opacities 4πη/s ∼ 1000 and rises monotonically with opacity at each point in time. Qualitatively, the curves look the same at all opacities, with a buildup period at times 0.1R τ 2R and almost constant behaviour afterwards. The onset of this buildup is slightly earlier at larger opacities, but this difference is negligible.
As expected, the transverse flow velocity u ⊥ starts with the same early-time linear behaviour for all opacities. The proportionality constant with elapsed time ∆τ = τ − τ 0 can be computed according to Eq. (77) and evaluates to u ⊥ = 0.614∆τ /R. The larger the opacity, the earlier the system starts to deviate from this behaviour. For the largest opacities 4πη/s 0.1, the system is in its local Bjorken flow equilibrium state long enough for early time contributions to become negligible, such that it transitions to the late time pre-flow proportionality law. According to Eq. (79), in this regime, the flow velocity is given by u ⊥ = 0.658∆τ /R. All curves exhibit their strongest rise on the timescale of transverse expansion, τ ∼ R. The rise is stronger at smaller opacities and in all cases contributes the most to the buildup, such that the final (τ = 4R) values of transverse flow velocity are also larger at smaller opacities.
The inverse Reynolds number Re −1 stays almost constant at early times for small opacities 4πη/s ∼ 1000, but then slightly increases due to transverse expansion. At large enough opacities 4πη/s 10, interactions equilibrate the system and decrease its value. This process sets in earlier at larger opacities and brings the value of the inverse Reynolds number down to almost zero for the largest opacities 4πη/s 0.05. In these cases, the value stays close to zero even during transverse expansion. At slightly smaller opacities 0.05 4πη/s 1, there is a small rise in inverse Reynolds number due to transverse expansion. However, this sets in later than in the case of the smallest opacities. The curves for intermediate to small opacities 1 4πη/s 100 exhibit a bumpy behaviour during transverse expansion.
Appendix D: Time evolution in KøMPøST + viscous hydro simulations
In Sec. V C we considered hybrid simulation frameworks as a solution for alleviating problems with preequilibrium in hydrodynamic simulations and discussed the time evolution mainly in hybrid kinetic theory + viscous hydro simulations. The alternative hybrid framework using KøMPøST instead of full kinetic theory for the pre-equilibrium evolution has some limitations but, when applicable, yields results of similar accuracy. The time evolution of transverse energy dE tr /dη, elliptic flow ε p and transverse flow velocity u ⊥ in KøMPøST + viscous hydro simulations switching at fixed time τ s or fixed value of the inverse Reynolds number Re −1 is shown in Figure 15 for three different opacities 4πη/s = 0.5, 3 and 10.
The values of dE tr /dη at the time of switching are reproduced by KøMPøST almost perfectly. As one would expect, the time evolution afterwards follows a very similar behaviour to kinetic theory + viscous hydrodynamics, including the inaccuracies of hydrodynamic preequilibrium when switching too early.
As KøMPøST produces almost no elliptic flow, its value at switching time is close to zero. But the buildup during the hydro part of the simulation proceeds similarly to other simulation schemes, such that the discrepancy to kinetic theory in the final state (τ = 4R) is of similar size to the one at switching time. It is therefore larger at larger switching times.
The values of transverse flow velocity u ⊥ are in KøMPøST slightly underestimated for small switching times and slightly overestimated for large switching times. After switching, the curves seem to bend towards the hydrodynamic curve. This bending is mainly due to the division by ∆τ . u ⊥ in the hydro phase of hybrid simulations builds up similarly as in pure hydrodynamic simulations. The contributions from later times are much larger than those at early times, such that the discrepancy from early times becomes negligible. At late times, results from all switching times underestimate u ⊥ by almost the same amount, similarly to hybrid kinetic theory + viscous hydro simulations.
FIG. 2 .
2Comparison of KøMPøST (RTA) and full kinetic theory via results for the energy-momentum tensor on the line x = 0, represented at fixed times τ /R 0.1 in blue, 0.3 in yellow and 0.5 in green. The full kinetic theory results are plotted with points (+,×), while the KøMPøST ones obtained with and without energy perturbations are plotted with solid and dashed lines, respectively. Anisotropic observables are nonzero only with energy perturbations and are plotted with point-dashed lines.
FIG. 4 .
4Early time evolution of the transverse profile of the restframe energy density τ for an example event in the 30 − 40% centrality class of Pb-Pb collisions in naive viscous hydrodynamics (top), kinetic theory (middle) and scaled viscous hydrodynamics (bottom) at an opacity 4πη/s = 0.05.
FIG. 5 .
5Early time evolution of transverse energy dEtr/dη [top, cf. Eq.(13)] and ellipticity 2 [bottom, cf. Eq. (5)] in kinetic theory (blue), naive hydrodynamics (red) and scaled hydrodynamics (green). Hydrodynamics behaves differently in pre-equilibrium, such that differences to a kinetic theory description build up. This can be counteracted by scaling the initial condition.
which coincides with the initialization employed for RTA [shown in Eq. (1)] in the case when γ = 4/9. Since in hydrodynamics, γ > 4/9, the initial transverse-plane energy whenw 0 1 will be larger than in RTA:
FIG. 6 .
6Time evolution of transverse profiles of the restframe energy density τ in a heatmap plot together with transverse components of the flow velocity (u x , u y ) as a vector field plot for the averaged initial condition used in this work at different opacities 4πη/s = 0.5 (left), 3 (middle) and 10 (right). The snapshot times τ = 0.1R (top), τ = 1R (middle) and τ = 2R (bottom) were chosen as the beginning, peak and end of the buildup of elliptic flow εp.
FIG. 7 .
7Transverse profiles of the transverse anisotropy τ (T xx − T yy ) in kinetic theory at time τ = 1R for different opacities 4πη/s = 0.5 (left), 3 (middle), 10 (right).
FIG. 8 .
8Time evolution of (from top to bottom) transverse energy dEtr/dη [cf. Eq. (13)], elliptic flow εp [cf. Eq. (14)], transverse flow velocity u ⊥ [cf. Eq. (15)] and inverse Reynolds number Re −1 [cf. Eq. (16)] in kinetic theory (black) and scaled viscous hydrodynamics (purple). The time axis is scaled logarithmically in all plots. The plots showing elliptic flow εp feature an inset plot of the same quantity plotted in log-log scale. The plots of flow velocity also show the pre-flow result from
FIG. 9 .
9Opacity (γ = 11.3 4πη/s ) dependence of the characteristic times where the elliptic flow εp [cf. Eq. (14)] reaches 5% of its late time (τ = 4R) value (red), the transverse flow velocity [cf. Eq. (15)] builds up to a value of u ⊥ = 0.1 (purple), or the inverse Reynolds number [cf. Eq. (16)] drops to a value of Re −1 = 0.8 (pink, dashed), 0.6 (pink, solid) or 0.4 (pink, long-short dashed). The buildup in transverse flow velocity marks the transition from the Bjorken flow scaling regime to the regime of transverse expansion, while the drop in inverse Reynolds number marks the region where hydrodynamics is applicable. sion).
FIG. 10 .
10Time evolution of transverse energy dEtr/dη [top, cf. Eq. (13)], elliptic flow εp [middle, cf. Eq. (14)] and transverse flow velocity u ⊥ [bottom, cf. Eq. (15)] in hybrid kinetic theory + viscous hydro simulations at opacities 4πη/s = 0.5 (left), 3 (middle) and 10 (right) when switching at different values of the inverse Reynolds number [cf. Eq. (16)] Re −1 = 0.8 (light red), 0.6 (red) and 0.4 (dark red) or fixed time τ = 0.4 fm (light green) and τ = 1 fm (dark green). The switching points are marked with filled symbols. The time axis is scaled logarithmically. The plots showing elliptic flow εp feature an inset plot of the same quantity plotted in log-log scale. In the flow velocity plots, we also show the estimate u ⊥ ,RTA = 0.614∆τ /R for the early-time buildup of pre-flow in kinetic theory (see
FIG. 11 .
11Opacity-(η/s-) dependence of the final (τ = 4R) values of transverse energy dEtr/dη [top, cf. Eq. (13)], elliptic flow εp [middle, cf. Eq. (14)] and transverse flow velocity u ⊥ [bottom, cf. Eq. (
FIG. 12 .
12Opacity-(η/s-) dependence of the final (τ = 4R) values of transverse energy dEtr/dη [top, cf. Eq. (13)], elliptic flow εp [middle, cf. Eq. (14)] and transverse flow velocity u ⊥ [bottom, cf. Eq. (15)] in hybrid kinetic theory + viscous hydro (left) and KøMPøST + viscous hydro simulations (right) when switching at different values of the inverse Reynolds number [cf. Eq. (16)] Re −1 = 0.8, 0.6, 0.4 and 0.2 plotted in different shades of red from light to dark. Results from simulations with switching times after τ = 0.5R are plotted with smaller points (+) and dashed lines. The results are compared to kinetic theory (black), scaled hydro (purple) and the small opacity limits of an opacity-linearized result (blue) in the top two plots, the free-streaming result (blue, dashed) in the bottom plot as well as the large opacity limit of scaled ideal hydro (grey, dashed), which scales as (η/s) 4/9 in the top plot. The red shaded region shows the realistic values for QCD according to Bayesian estimates. The bottom part of each plot shows the ratios of all results to those from kinetic theory.opacities, we found that for suitable setups of hydrodynamics, results for final state transverse energy, elliptic flow and radial flow are in good agreement with kinetic theory up to shear viscosities 4πη/s 3 for the examined profile, which translates to opacity valuesγ 4.However, hydrodynamics is not suitable for describing out-of-equilibrium behaviour in the early pre-equilibrium stage and the late time period where the microscopic de-
FIG. 13 .
13Opacity-(η/s-) dependence of the final (τ = 4R) values of transverse energy dEtr/dη [top, cf. Eq. (13)], elliptic flow εp [middle, cf. Eq. (14)] and transverse flow velocity u ⊥ [bottom, cf. Eq. (
FIG. 14 .
14Time evolution of transverse energy dEtr/dη [top left, cf. Eq. (13)], elliptic flow εp [top right, cf. Eq. (14)], transverse flow velocity u ⊥ [bottom left, cf. Eq. (15)] and inverse Reynolds number Re −1 [bottom right, cf. Eq. (16)] in kinetic theory for a wide range of opacities (η/s) plotted in different colors. The plot of transverse flow velocity u ⊥ also shows the pre-flow result u ⊥ ,early = 0.614∆τ /R according to Eq. (77) and the late pre-flow result u ⊥ ,late = 0.658∆τ /R according to Eq. (79) (see also
FIG. 15 .
15Time evolution of transverse energy dEtr/dη [top, cf. Eq. (13)], elliptic flow εp [middle, cf. Eq. (14)] and transverse flow velocity u ⊥ [bottom, cf. Eq. (15)] in hybrid KøMPøST + viscous hydro simulations at opacities 4πη/s = 0.5 (left), 3 (middle) and 10 (right) when switching at different values of the inverse Reynolds number [cf. Eq. (16)] Re −1 = 0.8 (light red), 0.6 (red) and 0.4 (dark red) or fixed time τ = 0.4 fm (light green) and τ = 1 fm (dark green). The switching points are marked with filled symbols. The time axis is scaled logarithmically. The plots showing elliptic flow εp feature an inset plot of the same quantity plotted in log-log scale. Again, the flow velocity plots also show the pre-flow result u ⊥ = 0.614∆τ /R according to Eq. (77).
TABLE I. Characteristic properties of the initial condition for the energy density used in this work, corresponding to an average over profiles in the 30 − 40% centrality class of Pb-Pb collision at √ sNN = 5.02 TeV[112], as discussed in Sec. II A.dE 0
⊥ /dη [GeV] R [fm]γ × 4πη/s
2
4
6
1280
2.78
11.3
0.416 0.210 0.0895
elliptic flow εp [bottom left, cf. Eq. (14)] and inverse Reynolds number Re −1 [bottom right, cf. Eq.1
0.2
0.3
0.4
0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
.
Energy dE
tr /dη
[GeV]
Time τ /R
Time τ [fm/c]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.1
0.2
0.3
0.4
0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
.
Flow velocity ⟨u
⊥ ⟩
ϵ
Time τ /R
Time τ [fm/c]
Kinetic theory
4πη/s =10.0
2.0 (+0.05)
0.5 (+0.1)
KøMPøST
w/o perturbations
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.1
0.2
0.3
0.4
0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
.
Elliptic ow ϵ
p
Time τ /R
Time τ [fm/c]
Kinetic theory
4πη/s =10.0
2.0
0.5
KøMPøST
w/o perturbations
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
.
Inverse Reynolds number ⟨Re −1
⟩
ϵ
Time τ /R
Time τ [fm/c]
FIG. 1. Time evolution of transverse energy dEtr/dη [top left, cf. Eq. (13)], transverse flow velocity u ⊥ [top right, cf.
Eq. (15)],
Kinetic Naive hydro Scaled hydro theory Ideal Viscous Ideal Viscous TABLE II. Estimates for the pre-flow generated in kinetic theory, ideal hydrodynamics and viscous hydrodynamics (see Sec. IV D for details regarding the naive and scaled hydrodynamics setups).γ
4/9
0
0.526
0
0.526
α
1
2/3 1.071
2/3 1.071
R
τ
u ⊥ ,early
1 − (τ0/τ ) α 0.614 0.691 0.600
0.658 0.606
R
τ
u ⊥ ,late
1 − (τ0/τ ) 2/3 0.658 0.691 0.652
0.658 0.658
Table
II for the early-time limit for u ⊥ /(∆τ /R) (0.614 for kinetic theory and 0.606 for scaled hydrodynamics). Bjorken scaling results are shown with dashed lines for dEtr/dη (top) and Re −1 (bottom).
, H. Elfner, A. Mazeliauskas, H. Roch, A. Shark, and U. A. Wiedemann for valuable discussions. This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Stronginteraction matter under extreme conditions'-project number 315477589 -TRR 211. V.E.A. gratefully acknowledges the support through a grant of the Ministry of Research, Innovation and Digitization, CNCS -UE-FISCDI, project number PN-III-P1-1.1-TE-2021-1707, within PNCDI III. C.W. was supported by the program Excellence Initiative-Research University of the University of Wroc law of the Ministry of Education and Science. Numerical calculations presented in this work were performed at the Paderborn Center for Parallel Computing (PC2) and the Center for Scientific Computing (CSC) at the Goethe-University of Frankfurt and we gratefully acknowledge their support.Appendix A: Relativistic lattice Boltzmann
implementation details
TABLE III. Simulation parameters for the RLB solver, as employed for the ranges of 4πη/s displayed in the left column. The notation is explained in this appendix.4πη/s S ( δτ
τ )M δτM Qϕ Qz( * )τ0
ξ0
PL/PT
[0.01 : 0.5] 200 0.05 0.002 80 40(s)
10 −6 20 0.08
[1 : 5] 100 0.02 0.005 40 200(s) 10 −6 100 0.02
[10 : 1000] 100 0.1
0.005 40 1000(fs) 10 −3 1000 0.002
Commit number efa9e28d24d5115a8d8134852-32fb342b38380f0.
In general, the equilibration timescale scales with (η/s) 3(1−γ/4)/2 , with γ as defined in Eq.(50).Numerically, the exponent 1.30 for viscous hydrodynamics is close to the one for RTA.
For large evolution times, KøMPøST crashes in the setup stage when computing the Green's functions. This is because they are only implemented for a finite number of points in momentum space and have to be convolved with a Gaussian smearing kernel exp(−σ 2 |k|/2). But the Green's functions scale in |k|(τ − τ 0 ) such that for too large of an evolution time this smearing is no longer sufficient.
where the free-streaming coordinates x ⊥ and y were introduced in Eq. (B4).The integration variables can be changed from (x ⊥ , y) to (x ⊥ , y ), whereRight away and from this point on, we will drop the primes on all coordinates except τ for convenience. The specific form of the RTA kernel iswhere τ R = 5(η/s)T −1 is the relaxation time andwhere v ⊥ = p ⊥ /p ⊥ = (cos ϕ p , sin ϕ p ) is a unit vector in the transverse plane. Similarly, β ≡ u ⊥ /u τ = β(cos ϕ u , sin ϕ u ) is the transverse-plane fluid velocity and γ = 1/ 1 − β 2 is the local Lorentz factor. Plugging Eqs. (B13)-(B15) into Eq. (B11), we arrive atIn the above, all macroscopic quantities τ R , γ and β are computed from the zeroth order solution, f (0) . It is convenient to consider separately the contributions involving f (0) and f eq . In the case of the former, we plug inUsing the relationit is not difficult to obtain:where γ, τ R and β are evaluated at (τ , x ⊥ ).For the equilibrium buildup contribution, we can use the propertyleading towhere F ij eq = dy 2 dϕ p 2πIn the above, we introduced.(B23)It is understood that in Eq. (B21), the quantities (0) , τ R , γ and F ij eq are evaluated at (τ , x ⊥ ). Altogether, T (1)ij can be computed using the following formula:where γ, τ R , β, (0) and F ij eq are evaluated at (τ , x ⊥ ). Note that the zeroth order results for the flow velocity u (0) µ and the rest frame energy density (0) entering τ R via the temperature have been computed in the first step of diagonalizing T (0)µν . Thus, all quantities appearing in the above integrand are known and the remaining 4D integral can be performed numerically.Appendix C: Overview of time evolution at different opacitiesIn Sec. V B we compared the time evolution of the tracked observables in kinetic theory and scaled viscous hydro and pointed out some qualitative differences for results at three different opacities. To get a better overview of the opacity dependence in the time evolution, we can also compare results coming exclusively from kinetic theory on a wide range in opacity. This comparison for the time evolution of transverse energy dE tr /dη, elliptic flow ε p , transverse flow velocity u ⊥ and inverse Reynolds number Re −1 is presented inFigure 14for opacities ranging from 4πη/s = 0.01 to 1000.For very small opacities 4πη/s ∼ 1000, the system is close to free-streaming and transverse energy dE tr /dη
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| [] |
[
"QCD thermodynamics with two flavours of Wilson fermions on large lattices QCD thermodynamics with two flavours of Wilson fermions on large lattices",
"QCD thermodynamics with two flavours of Wilson fermions on large lattices QCD thermodynamics with two flavours of Wilson fermions on large lattices"
] | [
"Bastian B Brandt [email protected] ",
"Anthony Francis ",
"Harvey B Meyer ",
"Hartmut Wittig ",
"Owe Philipsen ",
"Cairns ",
"Australia ",
"Bastian B Brandt ",
"\nInstitut für Kernphysik\nPRISMA Cluster of Excellence\nInstitut für Kernphysik\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 4555099MainzGermany\n",
"\nGermany and Helmholtz Institut Mainz\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 4555099Mainz\n",
"\nInstitut für Theoretische Physik\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 36, Goethe-Universität, Max-von-Laue-Str. 155099, 60438Mainz, Frankfurt am MainGermany, Germany\n"
] | [
"Institut für Kernphysik\nPRISMA Cluster of Excellence\nInstitut für Kernphysik\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 4555099MainzGermany",
"Germany and Helmholtz Institut Mainz\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 4555099Mainz",
"Institut für Theoretische Physik\nJohannes Gutenberg-Universität Mainz\nJohann Joachim Becher-Weg 36, Goethe-Universität, Max-von-Laue-Str. 155099, 60438Mainz, Frankfurt am MainGermany, Germany"
] | [
"The 30th International Symposium on Lattice Field Theory"
] | We explore the phase diagram of two flavour QCD at vanishing chemical potential using dynamical O(a)-improved Wilson quarks. In the approach to the chiral limit we use lattices with a temporal extent of N t = 16 and spatial extent L = 32, 48 and 64 to enable the extrapolation to the thermodynamic limit with small discretisation effects. In addition to an update on the scans at constant κ, reported earlier, we present first results from scans along lines of constant physics at a pion mass of 290 MeV. We probe the transition using the Polyakov loop and the chiral condensate, as well as spectroscopic observables such as screening masses. | 10.22323/1.164.0073 | [
"https://arxiv.org/pdf/1210.6972v1.pdf"
] | 59,017,172 | 1210.6972 | 78ceef3d7f3d43af51221c823b3943f8327140d2 |
QCD thermodynamics with two flavours of Wilson fermions on large lattices QCD thermodynamics with two flavours of Wilson fermions on large lattices
June 24 -29, 2012 25 Oct 2012
Bastian B Brandt [email protected]
Anthony Francis
Harvey B Meyer
Hartmut Wittig
Owe Philipsen
Cairns
Australia
Bastian B Brandt
Institut für Kernphysik
PRISMA Cluster of Excellence
Institut für Kernphysik
Johannes Gutenberg-Universität Mainz
Johann Joachim Becher-Weg 4555099MainzGermany
Germany and Helmholtz Institut Mainz
Johannes Gutenberg-Universität Mainz
Johann Joachim Becher-Weg 4555099Mainz
Institut für Theoretische Physik
Johannes Gutenberg-Universität Mainz
Johann Joachim Becher-Weg 36, Goethe-Universität, Max-von-Laue-Str. 155099, 60438Mainz, Frankfurt am MainGermany, Germany
QCD thermodynamics with two flavours of Wilson fermions on large lattices QCD thermodynamics with two flavours of Wilson fermions on large lattices
The 30th International Symposium on Lattice Field Theory
June 24 -29, 2012 25 Oct 2012
We explore the phase diagram of two flavour QCD at vanishing chemical potential using dynamical O(a)-improved Wilson quarks. In the approach to the chiral limit we use lattices with a temporal extent of N t = 16 and spatial extent L = 32, 48 and 64 to enable the extrapolation to the thermodynamic limit with small discretisation effects. In addition to an update on the scans at constant κ, reported earlier, we present first results from scans along lines of constant physics at a pion mass of 290 MeV. We probe the transition using the Polyakov loop and the chiral condensate, as well as spectroscopic observables such as screening masses.
Introduction
The unresolved question about the order of the phase transition connected to chiral symmetry restoration in the chiral limit of two-flavour QCD is the remaining qualitative issue concerning the phase diagram in the {m ud , m s , T } parameter space at zero chemical potential (see [1] for a review). There are two possible scenarios [2,3]: In the first scenario the chiral critical line reaches the m ud = 0-axis at some tri-critical point m tric s and the transition at N f = 2 with m ud = 0 is of second order. Then the restoration of chiral symmetry belongs to the SU(2) × SU(2) O(4) universality class [4]. In the second scenario the chiral critical line never reaches the m ud = 0 axis and the transition remains first order for all values of the strange quark mass. In [2,3] it was shown that the realisation of one of the two scenarios can be linked to the strength of the anomalous breaking of the U A (1)-symmetry at the transition point in the chiral limit. The strength of the anomaly can be probed by looking at correlation functions in scalar and pseudo-scalar channels and the associated screening masses. Despite a number of recent studies aiming to extract information about the order of the transition in the chiral limit [5,6,7,8] no study has sufficient control over the systematic effects.
In these proceedings we present the current status of our study on the topic, first reported in [9,10], using non-perturbatively O(a)-improved Wilson fermions at N f = 2. We use lattices with a temporal extent of N t = 16 throughout to suppress discretisation effects, in particular the effect of the explicit breaking of chiral symmetry introduced by the Wilson term. We aim at simulations at several quark masses corresponding to zero-temperature pions with m π 300 MeV with three different volumes for each simulation points. This will eventually enable us to perform a scaling analysis with control over the main systematic effects. Here we present results on the transition temperatures, screening masses and on the strength of the anomalous breaking of the U A (1)-symmetry.
Setup
The simulations are done using non-perturbatively O(a)-improved Wilson fermions [11] with two degenerate dynamical quarks and the configurations are generated using DD-HMC [12,13] and MP-HMC [14] algorithms. For more details on the simulation setup see [9].
To extract information about the order of the transition in the chiral limit by means of a scaling analysis, it is important to control and disentangle the different systematic effects that might distort the scaling properties of the results at finite lattice spacing and volume. Of particular importance for Wilson fermions in this context is the suppression of the explicit breaking of chiral symmetry which shows up as a discretisation effect at finite lattice spacing. Therefore our simulations are done on large lattices of the size of 16 × 32 3 , 16 × 48 3 and 16 × 64 3 . The three different volumes at each simulation point allow for an extrapolation to infinite volume, and at this large temporal extent one can expect discretisation effects to be small (see also [15]).
We scan in the temperature by varying the bare coupling β while N t = 16 remains fixed either at fixed hopping parameter κ (for heavier quarks) or along lines of constant physics (for quarks with an associated m π 290 MeV). We set the scale by using an interpolation of the zero-temperature results for the Sommer parameter r 0 /a in the chiral limit obtained within the CLS effort in the region of 5.20 ≤ β ≤ 5.50 [16,17] and its continuum result r 0 = 0.503 (10) fm obtained in [17]. Similarly, the tuning of the bare parameters to lines of constant renormalised quark masses is done using the input from results obtained within CLS [17,18,19].
m M S ud (µ = 2 GeV) [MeV] T [MeV] m π = 200 MeV m π = 290 MeV m π = 540 MeV O(4) scaling B1 κ , V = 32 3 B3 κ , V = 64 3 C1, V = 32 3
We extract the critical temperature for the deconfinement transition and the chiral symmetry restoration using the real part of the APE-smeared Polyakov loop L SM ≡ Re [ L SM ] and the subtracted chiral condensate [20,21]
ψψ sub = 2 N f T V m PCAC d 4 x P(0) P(x) ,(2.1)
respectively. Here P(x) is the pseudoscalar density and m PCAC the PCAC mass. The transition temperatures are defined by the position of the peak of the associated susceptibilities
χ(O) ≡ N 3 s O 2 − O 2 , (2.2)
where O is any of the observables above. For the time being all those observables are unrenormalised. The error analysis has been done using the bootstrap method with 1000 bins.
Current status
The current set of simulation points in the {m ud , T }-parameter space 1 is shown in figure 1. The set of points consists of two different temperature scans, one at constant κ with two volumes (red and magenta circles) and another one at a line of constant quark mass of m ud = 14.5 MeV (orange diamonds). The associated parameters are listed in table 1. The results for scan B1 κ have in part already been reported in [9,10]. The main results remain unchanged even with increased statistics and a larger number of simulation points. The transition temperature in physical units, listed in table 2, changes slightly due to the updated scale determination reported in [17]. Note that the transition temperature is extracted from the peak position of a Gaussian fit to the Polyakov loop or condensate susceptibility peak. The uncertainties are estimated conservatively by the full spread of points included in the fit. The value given in table 2 for scan B1 κ is extracted from the Polyakov loop, but the condensate also shows a (weak) peak in its susceptibility at similar temperature. The additional simulations points from scan B3 κ probe the same transition only with twice the volume. The results are in good agreement with the results for the smaller volume but the limited number of three simulation points does not allow for a reliable determination of the critical temperature. The scan serves as a benchmark for the future simulations on larger volumes.
Scan C1 is our first scan along a line of constant quark mass of 14.5 MeV, which corresponds to a zero-temperature pion mass of about 290 MeV, and provides first results in the regime relevant for the future scaling analysis. The results for the smeared Polyakov loop, the subtracted condensate and their susceptibilities are shown in figure 2. As can be seen from the plot, both susceptibilities exhibit a peak at a similar temperature. The result given in table 1 is the one extracted from the condensate, which agrees with the one extracted from the Polyakov loop within errors. Table 2: Estimates for the transition temperatures, quark and pion masses. The first error bar reflects the uncertainty due to the extraction of the peak position. The second error denotes the uncertainty due to scale setting and renormalisation. For scan B1 κ and B3 κ the result for the transition temperature has been extracted from the Polyakov loop susceptibility, for scan C2 the value from the subtracted condensate has been used. The conversion from quark to pion masses is done using continuum χPT to NNLO [22] with the low energy constants from [19,23].
Screening masses
Further information about the chiral symmetry restoration pattern in a given scan can be extracted from the behaviour of screening masses [24]. In particular, the degeneracy of pseudoscalar and scalar screening masses signals the restoration of the anomalously broken U A (1)-symmetry. In this section we focus on scan C1 and study screening masses in the pseudoscalar (P), scalar (S), vector (V ) and axial vector (A) channels, measured on the stored configurations (separated by 40 MDUs) with a point source. Figure 3 illustrates the temperature dependence of the screening masses. The x-axis is normalised to the critical temperature listed in table 2. At 0.84 T C the screening masses are mostly in agreement with the expected splitting patterns of the zero-temperature meson masses. The screening mass in the pseudoscalar channel, M P , starts from a value of M P /(2π T ) = 0.33 (2), which means that at this point M P is a factor of 1.27 larger than the zero-temperature pion mass in the scan. Around T C it then starts to rise and at 1.16 T C it is roughly 30 % smaller than the asymptotic value 2π T . This is in the ballpark of what has been found for the Wilson action on pure gauge configurations [25], but larger than typical results for staggered fermions [26,27]. Note however, that regardless of the discretisation finite volume (here N s /N t = 2) and quark mass effects might still give a sizeable contribution around T C . The V and A-channels are accidentally already close to the asymptotic value of 2π T below T C and mainly fluctuate around this value in the whole interval. M V and M A fluctuate independently below T C and become degenerate above T C , consistent with chiral symmetry restoration.
The strength of the anomalous breaking of the chiral U A (1)-symmetry can be assessed via the splitting between the screening masses in P and S channel. Below and at T C the screening mass in the S-channel and shows large fluctuations. Above T C the signal becomes more stable and M S moves closer to M P , signalling a weakening of the breaking of U A (1). At 1.16 T C the symmetry is almost restored. This is in qualitative agreement with the findings from [28] where the symmetry has been found to be restored at about 1.25 T C .
Conclusions
This proceedings article contains a summary of the status of our ongoing study of the QCD deconfinement transition in the chiral limit at N f = 2. To date there are two scans available with N t = 16 and pion masses at the critical point of about 510 and 290 MeV. The critical temperatures are 245 and 211 MeV respectively. The scan at the pion mass of 290 MeV is the first scan in the region important for a future scaling analysis and is done along a line of constant renormalised quark mass of 14.5 MeV. To study the pattern of chiral symmetry restoration, we calculate mesonic screening masses. The pseudoscalar screening mass rises from a value close to the zero-temperature pion mass at 0.84 T C towards the asymptotic value of 2π T . At 1.16 T C it differs by roughly 30 % from this limit. This is in agreement with the results for screening masses extracted from pure gauge theory with Wilson fermions [25], but larger than typical results found in simulations with staggered fermions [26,27]. The results show the expected degeneracy for vector and axial vector channels around and above T C , signalling chiral symmetry restoration. At the same time, the U A (1)symmetry still appears to be broken. The large lattices used in our study also offer the possibility to study plasma properties in terms of temporal correlation functions in the vector channel and the associated spectral function (see [29]). A first study has already been reported at conferences and we refer to our future publication for the details.
Figure 1 :
1Simulation points in the {m ud , T }-parameter space. The black dashed line results from naive O(4)-scaling for the two transition temperatures indicated by the grey squares and is only shown to give a first impression on the possible location of the critical line.
1 :
1Scans at N t = 16 at constant κ, the ones with subscript κ, and at constant renormalised quark mass, C1. Listed are the size of the DD-HMC blocks, DD, the temperature range in MeV, the integrated autocorrelation time of the plaquette τ U P and the number of molecular dynamics units, MDUs, used in the analysis. The measurements have been done each 4 MDUs.
scan T C [MeV] m ud,C [MeV] m π,C [MeV] B1 κ and B3 κ 245 (7
Figure 2 :
2Results for the smeared Polyakov loop L SM (left) and the subtracted chiral condensate ψψ sub (right) and their susceptibilities for scan C1. The coloured areas are the transition regions extracted from the susceptibility. The curves are the Gaussian fits used to define the transition point.
Figure 3 :
3Temperature dependence of the screening masses in P, S, V and A channels. The dashed line corresponds to the asymptotic value of M = 2π T .
Table
Here m ud is the renormalised quark mass in the MS-scheme at a renormalisation scale of µ = 2 GeV.
AcknowledgmentsThe simulations where done on the WILSON cluster at the Institute for Nuclear Physics of the University of Mainz, on the FUCHS cluster at the Center for Scientific Computing of the University of Frankfurt and on JUROPA and JUGENE at FZ Juelich under project number HMZ21. We are grateful to the institutes for offering these facilities. B.B. is supported by DFG Grant ME 3622/2-1.
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| [] |
[
"Microscopic theory of Raman scattering for the rotational organic cation in metal halide perovskites",
"Microscopic theory of Raman scattering for the rotational organic cation in metal halide perovskites"
] | [
"Yu Cui \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Yi-Yan Liu \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Jia-Pei Deng \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Xiao-Zhe Zhang \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Ran-Bo Yang \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Zhi-Qing Li \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n",
"Zi-Wu Wang \nDepartment of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina\n"
] | [
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina",
"Department of Physics\nSchool of Science\nTianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology\nTianjin University\n300354TianjinChina"
] | [] | A gap exists in microscopic understanding the dynamic properties of the rotational organic cation (ROC) in the inorganic framework of the metal halide perovskites (MHP) to date. Herein, we develop a microscopic theory of Raman scattering for the ROC in MHP based on the angular momentum of a ROC exchanging with that of the photon and phonon. We systematically present the selection rules for the angular momentum transfer among three lowest rotational levels. We find that the phonon angular momentum that arising from the inorganic framework and its specific values could be directly manifested by Stokes (or anti-Stokes) shift. Moreover, the initial orientation of the ROC and its preferentially rotational directions could be judged in Raman spectra. This study lays the theoretical foundation for the high-precision resolution and manipulation of molecular rotation immersed in many-body environment by Raman technique. | 10.1103/physrevb.107.094306 | [
"https://export.arxiv.org/pdf/2209.13861v1.pdf"
] | 252,568,286 | 2209.13861 | e52706f79011e977b3539bccbd4df891f9fb605a |
Microscopic theory of Raman scattering for the rotational organic cation in metal halide perovskites
28 Sep 2022
Yu Cui
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Yi-Yan Liu
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Jia-Pei Deng
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Xiao-Zhe Zhang
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Ran-Bo Yang
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Zhi-Qing Li
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Zi-Wu Wang
Department of Physics
School of Science
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology
Tianjin University
300354TianjinChina
Microscopic theory of Raman scattering for the rotational organic cation in metal halide perovskites
28 Sep 2022
A gap exists in microscopic understanding the dynamic properties of the rotational organic cation (ROC) in the inorganic framework of the metal halide perovskites (MHP) to date. Herein, we develop a microscopic theory of Raman scattering for the ROC in MHP based on the angular momentum of a ROC exchanging with that of the photon and phonon. We systematically present the selection rules for the angular momentum transfer among three lowest rotational levels. We find that the phonon angular momentum that arising from the inorganic framework and its specific values could be directly manifested by Stokes (or anti-Stokes) shift. Moreover, the initial orientation of the ROC and its preferentially rotational directions could be judged in Raman spectra. This study lays the theoretical foundation for the high-precision resolution and manipulation of molecular rotation immersed in many-body environment by Raman technique.
Over the past few years, metal halide perovskites (MHP) as promising materials have aroused the intense interest from the worldwide research owing to their notable properties in the fields of photovoltaic cells, light emitting diodes, and photodetectors [1][2][3]. Traditionally, the consensus is that the species of organic cations are not directly involved in the formation of electronic transport levels [4,5]. However, the recent breakthroughs, studied by the spectral measurements [6][7][8] and the firstprinciples calculation [9][10][11][12], have shown that the dipole nature of the organic cation plays a critical role in the structure stabilizations and optoelectronic properties of MHP [13], e.g. the compositional engineering of organic cations to modulate the bandgap and to modify the crystal symmetry and phase. In particular, the rotational motion of organic cation [12][13][14][15][16] results in an effective Coulomb screening to affect the dynamic of charge carriers. Therefore, the manipulation of the rotational organic cation (ROC) not only gives an effective method to modify the properties of MHP, but provides a test bed to explore novel quantum phenomena in many-body physics [13,17].
Based on angular momentum exchange between photon and rotational particles, quantum control of the rotational atoms or molecules by laser have been studied both theoretically and experimentally in areas of atomic, molecular and optical physics as well as in physical chemistry [17][18][19][20]. While the corresponding studies on ROC in perovskite materials are still very few. On the other hand, ROC inevitably couples with the surrounding inorganic cage [see Fig. 1(a)]. Although the coupling effect of organic cation with inorganic sublattice by the hydrogen bonds have been analyzed widely [21,22], the role of phonons of inorganic cage on the rotational dynamic of organic cation received relatively little attention, partially because of the intricate angular-momentum algebra from the ROC coupled with many-body environment. Fortunately, Schmidt and Lemeshko undertook a critical step towards such a theory in 2015 by introducing the quasiparticle concept of the "angulon"-a quantum rotor dressed by a bath of harmonic oscillators [23], which provides a simple and effective model to study the angular momentum exchange between the rotational molecule and many-body environment, such as molecules rotating in superfluid helium, ultracold alkali dimers interacting with a Bose-Einstein condensate [24,25]. However, to the best of our knowledge, the theoretical model for the angular momentum of ROC exchanges with both photon and phonon in MHP has not yet been developed.
In this paper, we study the microscopic processes of Raman scattering mediated by a ROC in MHP based on ROC coupling with photon and phonon. We present the selection rules of quantum transitions among the rotational eigenstates |L, M of ROC (L and M denote the orbital angular quantum number and its projection on the laboratory-frame z-axis, respectively.), in which transitions from L = l to L = l ′ accompanying with and without the variation of the projection of angular quantum number are analyzed. We illustrate the Stokes and anti-Stokes shifting of Raman spectra for three lowest rotational levels according to different angular momenta provided by phonons. These results show that the transfer of phonon angular momentum, the initial orientation and the magnitude of the rotational angle of a ROC could be reflected by Raman scattering, both of which are the key problems for accurately quantum control of the ROC or molecules. More importantly, this model can be expanded to study molecules rotating in varieties of the cage-like structures, such as fullerene, carbon nanotube and so on.
In the frame of the classical model of Raman scattering mediated by elementary excitation, e.g. electron and exciton [26][27][28], Raman scattering for ROC in MHP could be divided into three steps as schemed in Fig. 1 Bi and B f represent the angular momenta (in the unit of rotational constant B) of the incident and scattering photon, respectively. (b) The direction of the organic cation in the laboratory-frame (x, y, z) and molecular-frame (x ′ , y ′ , z ′ ). (θ, φ) denotes the angular coordinates of organic cation in the laboratory frame. u is the molecular dipole moment and E is the electric field vector of the light. (ce) Clebsch-Gordan coefficients for different orbital quantum states, |L = l ′ , M = m ′ , coupling with phonon angular momentum states, |λ, µ . gular number L and its projection, M , on the laboratoryframe z-axis, with eigenenergies E L = BL(L + 1) [29,30], B is the rotational constant; (ii) the transition from |j = |L = l ′ , M = m ′ to |k = |L = l ′′ , M = m ′′ is accompanied by the exchange of angular momentum between ROC and phonons; (iii) ROC from |k comes back into the initial |i with the help of the angular momentum transfer B f by the scattering of a photon. So the cross section of Raman scattering for ROC can be expressed as [26][27][28]
|ℜ| 2 = jk i Ĥ opt j j Ĥ ph k k Ĥ opt i [B i − (E j − E i ) − iΓ] [B i − B f ± E 0 − iΓ] 2 ,(1)
where E i (E j ) is the eigenenergy of rotational state |i (|j ), E 0 is the rotational energy provided by phonons in unit of B, and Γ is the homogeneous line-width for the quantum transition. i|Ĥ opt |j ( k|Ĥ opt |i ) and j|Ĥ ph |k are the matrix elements for transitions between different rotational states, arising from ROC-photon and -phonon interaction, respectively. They are the key components to study the Raman scattering of ROC in the following.
For the sake of clarity, the ROC, such as CH 3 NH + 3 in MHP, is regarded as a linear molecule with frozen transitional motion. In the dipole approximation, the Hamiltonian for a ROC subjected to a linearly polarized light is given byĤ opt = −u · E, where u is the inherent dipole moment for ROC oriented along z ′ -axis in the molecularframe and E is the electric field vector of the light along the x-axis in the laboratory-frame [31][32][33], while the relative orientation of these two frames is given by the Euler angle (φ, θ) as shown in Fig. 1(b). The ROC-light interaction, in general, regarded as the perturbation to the system, gives rise to quantum transitions between different rotational states. After a series of mathematical processes (see the supplemental materials), the corresponding matrix element between |i and |j states is expressed as
i Ĥ opt j = F (l, l ′ , m, m ′ ) = −uEa l ′ −1,m ′ δ l,l ′ −1 δ m,m ′ +uEb l ′ −1,m ′ −1 δ l,l ′ −1 δ m,m ′ −1 −uEb l ′ −1,−(m ′ +1) δ l,l ′ −1 δ m,m ′ +1 ,(2)
where a l,m and b l,m are angular momentum dependent coefficients, shown in the supplemental materials. The constants u and E represent the magnitude of u and E. From Eq. (2), it can be inferred that the angular momentum of the incident photon results in the transition between orbital quantum states follows the selection rule of l ′ − l = 1 along with the exchange of angular momentum projection m ′ − m = 0, ±1 in the step (i) of Raman scattering. Similarly, the emission of photon in the step (iii) satisfies the selection rule of l − l ′′ = 1 along with m − m ′′ = 0, ±1.
In the frame of angulon model, the effective Hamiltonian describing the interaction between a ROC and phonon bath in the spherical basis is given by [23,[34][35][36]:
H ph = qλµ V λ (q) Ŷ * λµ (θ,φ)b † qλµ +Ŷ λµ (θ,φ)b qλµ . (3)
Here q = |q| is the scalar representation of the phonon wave vector, satisfying the relation q ≡ dq. λ and µ define, respectively, the phonon angular momentum and its projection onto the z-axis.Ŷ λµ (θ,φ) are the spherical harmonic operators, which is essential for the microscopic description of the transfer of phonon angular momentum. b † qλµ andb qλµ are the creation and annihilation operators of phonons in the angular momentum representation, respectively (see Refs. [23] and [34] for the detailed derivation). The angular momentum dependent interaction potential V λ (q) = v λ 8α c q 2 / (2λ + 1)
1/2 drr 2 f λ (r)j λ (qr)
is employed for an organic cation rotating in the cage-like phonon bath, where α c is the Fröhlich coupling constant, meaning that the exchange of angular momentum between longitudinal optical (LO) phonons and ROC are mainly taken into account [34][35][36], because LO phonon is the dominate mode caused by the lattice distortion of the cage-like structure both in the theoretical and experimental perspectives [37][38][39][40]; j λ (qr) is the spherical Bessel function; v λ and f λ (r) represent the strength and the shape of the potential in the respective angular momentum channel λ. Especially, the latter function describes the microscopic details of two-body interaction between ROC and phonon bath, whose expression is proposed as [41,42]
f λ (r) = ( r R ) λ , (r ≤ R) 0, (r > R) ,(4)
and the octahedral inorganic cage is approximated by the spherical cavity, based on the facts that (i) the rotating behavior of the organic cation in this cage demonstrated widely by recent experiments [15,16,43,44]; (ii) the strongest coupling strength (the potential distribution) is around the spherical boundary between ROC and inorganic cage [45,46]. R = a 0 /2 is the effective radius of this spherical space (a 0 is the length of the side of the octahedral cage). After proceeding some algebraic calculation, the matrix element for the transfer of phonon angular momentum between two different rotational states is given as
j Ĥ ph k = q V λ (q)g 1 C l ′′ 0 l ′ 0,λ0 C l ′′ m ′′ l ′ m ′ ,λµ ,(5)
where g 1 = {(2l ′ +1) (2λ + 1)/ [4π (2l ′′ + 1)]} 1/2 , and C l ′′ m ′′ l ′ m ′ ,λµ is the Clebsch-Gordan (C-G) coefficients [47]. Eventually, upon substitution of Eqs. (2) and (5), the cross section of Raman scattering is converted into
|ℜ| 2 = u 2 E 2 g 1 g 2 q V λ (q)C l ′′ 0 l ′ 0,λ0 C l ′′ m ′′ l ′ m ′ ,λµ (B i − B f ± E 0 ) 2 + Γ 2 2 .(6)
g 2 = F (l, l ′ , m, m ′ )F (l ′′ , l, m ′′ , m) summaries the roles of angular momentum transfer between ROC and photon in the absorption and emission processes, ensuring the classical process of Raman scattering; namely, the initial and final states are the same one. The redistribution of angular momentum between ROC and phonon bath in mediated process is determined by the C-G coefficients C l ′′ 0 l ′ 0,λ0 and C l ′′ m ′′ l ′ m ′ ,λµ . The values of |C l ′′ 0 l ′ 0,λ0 | 2 as functions of l ′ and λ are shown in Fig. 1(c), in which l ′′ = l ′ is assumed, implying the orbital angular quantum is unchanged during the angular momentum exchange between ROC and phonon. Thus, the phonon angular momentum only induces the change of the projection of orbital angular momentum, that is the variation of the orientation of ROC. This results in the distribution of the allowed transitions (red) as well as forbidden ones (white) depending on the phonon angular-momentum of quantum state λ, satisfying the following selection rule of l ′ + l ′ + λ = even. As a result, the lowest order of phonon angular-momentum that dominates the transfer of angular momentum between ROC and phonon is the quantum number λ = 2 shown in Fig. 1(c). Figs. 1(d) and (e) present the phonon angular-momentum state λ = 2 induces possible transition between the projections of angular momentum reflected by C-G coefficients |C 1m ′′ 1m ′ ,λµ | 2 and |C 2m ′′ 2m ′ ,λµ | 2 for angular quantum number l ′ = 1 and l ′ = 2, respectively. Two obvious features are shown that (1) the distribution of the coefficients reveals the symmetrical relations, rep-
resented as |C l ′′ m ′′ l ′ m ′ ,λµ | 2 = |C l ′′ −m ′′ l ′ −m ′ ,λ−µ | 2 ;
(2) the ROC is inclined to couple with phonons that can invert its angular momentum projection from m ′ to −m ′ . These results indicate that ROC has the preferential directions induced by phonon angular momentum.
In order to give the comprehensive comparison between different transfer of phonon angular momentum, the Raman spectra for three lowest rotational levels are illustrated in Fig. 2. Here, the typical example of MHP CH 3 NH 3 PbI 3 in cubic phase is selected, in which organic cation CH 3 NH + 3 rotating in the PbI 4− 6 octahedral cage as schemed in Fig. 1(a). The values for the related parameters in Eq. 6 are listed in Table S1 in the supplemental materials. These specific values of angular momentum dependent coefficients a l,m , b l,m and C-G coefficients involved in the angular momenta transfer among three lowest rotational levels are listed in Table S3 and Table S4 in the supplemental materials.
Firstly, Fig. 2 Fig. 1(d), and the optical coefficients given in Eq. (2). In 2015, Schmidt and Lemeshko proposed the phonon angular momentum couples with the rotating quantum molecule (or impurity) to form a new quasiparticle-angulon for the first time [23]. They pointed out this angulon induces a rich rotational fine structure in spectra of molecules, such as "rotational Lamb shift"; subsequently, they further revealed that the spectral function of the rotational molecule suddenly acquires the transfer of one quantum of phonon angular momentum from the many-body environment at a crit- ical rotational speed; however, the direct identification for the phonon angular momentum and its transfer is still a challenge task in experiments. For Raman scattering in Fig. 2, not only the phonon angular momentum arising from the octahedral-cage structure is proved, but also its specific values could be reflected directly by the Stokes and anti-Stokes shifts. In Fig. 2(a) Fig. 1(e), the asymmetrical intensity distribution should also be appeared for these scattering processes starting from the initial states (L = 1, M = −1) (see Fig. S2 in the supplemental materials). From these comparison, we can infer that the difference of the initial states between (L = 1, M = 0) and (L = 1, M = ±1), that is, the differently initial orientation of ROC, determines the features of Raman spectra. In turn, the initial orientation of ROC (or molecules) could be reflected by Raman spectra in experiments. In fact, a series of strategies to control the alignment and orientation of the rotational molecules have been proposed in the past decades, such as the linearly po-larized ultrafast laser pules [48][49][50], two-color and static fields [51], as well as two-color femtosecond lasers and terahertz field [52,53]. The molecular alignment and orientation are of crucial for a variety of applications ranging from chemical reaction dynamics to the design of molecular devices. For these applications, however, one of the most important prerequisites is to judge the initial orientation of the rotational molecules. Obviously, on the one hand, the Raman scattering of rotational molecule, provides an effective method to overcome this issue; on the other hand, the detailed dynamics and some novel physical phenomena related to the rotational particles in many-body bath should be analyzed deeply by Raman scattering even though the rotational structure and dynamics of molecules have been obtained widely from infrared spectroscopy [54,55]. Fig. 2(b) shows Raman scattering starting from the ground and first-excited states of ROC with the change of projection of angular momentum ∆M = ±2. One can see that the Stokes and anti-Stokes scattering starting from the initial states (L = 0, M = 0) and (L = 1, M = 0) have the same intensity. Moreover, the magnitude is much stronger than the corresponding scattering shown in Fig. 2(a). This indicates the rotation of organic cation has the preferentially orientation at certain external condition, which is very consistent with the prediction of C-G coefficient in Fig. 1(e). Therefore, this kind of Raman scattering would provide the anticipation for these chemical reactions depending on the high-precision control of the spatial orientation of the molecules, e.g. molecular imaging and selectivity [56], the enhancement of the in-teraction of a molecule with a surface in precise catalytic process [57,58]. For Raman scattering starting from the same states (L = 1, M = ±1) of ROC in Figs. 2(a) and (b), the spectral shapes show the significant difference since the change of projection of angular momentum follows ∆M = ±1 and ∆M = ±2, respectively. This means that the variational magnitude of the orientation of a ROC could be estimated by the spectral shape, which suggests the possibility to judge and modulate a molecule from a well-defined initial state to a target state.
The phonon angular momentum not only induces the variation of orientation of ROC at a given orbital quantum state, but induces transitions between different orbital states in Raman scattering when the second-order term of the photon interacting with ROC is considered. We illustrate the processes of (L Fig. S3 (see the supplemental materials). One can see that these scatterings have the similar features with the processes in Fig. 2, but the intensity become weaker by nearly one order of magnitude, demanding more accurate detection techniques. In fact, these scattering behaviors for three lowest rotational levels can be generalized to more quantum transitions between rotational levels assisted by photon and phonon angular momenta for ROC in MHP and could be explored by Raman scattering. Therefore, this type of Raman scattering could get deep into more complicated quantum transitions and reveal the fine spectroscopy of the rotational systems. Besides the rotational molecules or impurities as intermediaries for the Raman scattering, other elementary excitations in physics, such as the interlayer and intralayer excitons in van der Waals heterostructures [59,60], should have the similar Raman scattering when the rotational degree of freedom is considered. Lastly, we must emphasize that (1) only the angular momentum of LO phonon is considered in this study, other phonon modes have the similar effect and will play the important role at certain condition; (2) the perovskite materials undergo two structure phase transitions with temperature, the influence of which on the coupling strength between phonon bath and ROC, as well as the shape of the potential are also significant [61]. These effects are out of the scope of this study.
In summary, we develop a microscopic theory to describe the Raman scattering of an organic cation rotating in the octahedral cage of MHP. This theory predicts that the Raman spectra provide an effective and direct method to reflect the transfer of phonon angular momentum and its specific values in many-body environment.
Meanwhile, two key prerequisites for the alignment and orientation of the rotational molecules could be judged by Raman spectra, which may open a new door to explore quantum control of particle rotation in many-body physics.
This work was supported by National Natural Science Foundation of China (Grant Nos. 11674241 and 12174283).
FIG. 1 :
1(a): (i) the angular momentum transfer of an incident photon B i excites the ROC from the initial state |i = |L = l, M = m to |j = |L = l ′ , M = m ′ , where these rotational eigenstates are labeled by the orbital an-(a) The schematic diagram of Raman scattering for the rotational A + cation in the centre of BX 4− 6 octahedral cage, where A, B, and X correspond to the species of organic cation, metal ion, and halide anion, respectively, in MHP.
(a) illustrates the Raman scattering from the ground state of ROC (L = 0) to the first-excited state (L = 1) as well as from L = 1 to the second-excited state (L = 2) with the change of projection of angular momentum △M = ±1. One can see that the Stokes process of (L = 0, M = 0)→(L = 1, M = 1) (L = 1, M = 0)→(L = 0, M = 0) [→ and denote the transfer of angular momentum of photon and phonon, respectively.], and anti-Stokes process of (L = 0, M = 0)→(L = 1, M = −1) (L = 1, M = 0)→(L = 0, M = 0) follow a symmetrical distribution with the same intensity, which can be attributed to the symmetrical relations of C-G coefficient, shown in
FIG. 2 :
2The Stokes and anti-Stokes Raman scattering for three lowest rotational levels coupling with the phonon angular momentum states |λ = 2, µ = ±1 (a) and |λ = 2, µ = ±2 (b), respectively. The senarios of the angular momentum transfer for different Raman processes are shown in the insets, where L and M represent the quantum number of angular momentum and its projection onto the z-axis, respectively.
M = 0) and the anti-Stokes process of (L = 1, M = 0)→(L = 2, M = −1) (L = 2, M = 0)→(L = 1, M = 0) show the similar behaviors, however, with the smaller intensity, since the different values between C 1m ′′ 1m ′ ,2µ and C 2m ′′ 2m ′ ,2µ given in Figs. 1(d) and (e), respectively. But the intensity of the Stokes process (L = 1, M = 1)→(L = 2, M = 2) (L = 2, M = 1)→(L = 1, M = 1) is much stronger than that of the anti-Stokes process (L = 1, M = 1)→(L = 2, M = 0) (L = 2, M = 1)→(L = 1, M = 1). Following the rules of C-G coefficients in
M = 0) without the variation of orientation as well as the processes of (L = 0, M = 0)→(L = 2, M = 0) (L = 1, M = −1)→(L = 0, M = 0) and (L = 0, M = 0)→(L = 1, M = −1) (L = 2, M = 0)→(L = 0, M = 0) with the change of orientation ∆M = ±1 in
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| [] |
[
"Fuzzy α-cut and related structures",
"Fuzzy α-cut and related structures",
"Fuzzy α-cut and related structures",
"Fuzzy α-cut and related structures"
] | [
"Purbita Jana \nDepartment of Pure Mathematics\nUniversity of Calcutta\n\n",
"Mihir K Chakraborty \nSchool of Cognitive Science\nJadavpur University\n\n",
"Purbita Jana \nDepartment of Pure Mathematics\nUniversity of Calcutta\n\n",
"Mihir K Chakraborty \nSchool of Cognitive Science\nJadavpur University\n\n"
] | [
"Department of Pure Mathematics\nUniversity of Calcutta\n",
"School of Cognitive Science\nJadavpur University\n",
"Department of Pure Mathematics\nUniversity of Calcutta\n",
"School of Cognitive Science\nJadavpur University\n"
] | [] | This paper deals with a new notion called fuzzy α-cut and its properties. A notion called localic frame is also introduced. Algebraic structures arising out of the family of fuzzy α-cuts have been investigated. It will be seen that this family forms a localic frame. Some significance and usefulness of fuzzy α-cuts are discussed. | null | [
"https://arxiv.org/pdf/1806.02187v1.pdf"
] | 119,180,169 | 1806.02187 | 5e8bbb2a85d70cdd7ec75d6b3acba48c5f95a92f |
Fuzzy α-cut and related structures
4 Jun 2018
Purbita Jana
Department of Pure Mathematics
University of Calcutta
Mihir K Chakraborty
School of Cognitive Science
Jadavpur University
Fuzzy α-cut and related structures
4 Jun 2018L-fuzzy setα-cutFuzzy α-cutFrameGraded frameGödel arrow
This paper deals with a new notion called fuzzy α-cut and its properties. A notion called localic frame is also introduced. Algebraic structures arising out of the family of fuzzy α-cuts have been investigated. It will be seen that this family forms a localic frame. Some significance and usefulness of fuzzy α-cuts are discussed.
Introduction
Though well known now-a-days, we would like to start with a little bit of history. Fuzzy set was first introduced and studied by Lotfi Zadeh [23] in 1965, which can be considered in a sense a generalisation of ordinary set. It is well known that in informal set theory a (crisp) set A is considered as a subset of a universal set U and is fully determined by a function from U to {0, 1} called the characteristic function of A (denoted by χ A ). Whereas a fuzzy set is a function from U to [0,1], and in this case the function is known as membership function. In 1967, J. Goguen [12] generalised this notion one step further by considering the function from U to L (a complete lattice) and called it L-fuzzy set. Subsequently, there had been many other generalisations of the original proposal of Zadeh [1,3,9]. In this paper we will consider L as a frame (c.f. Definition 1.1), 1 L and 0 L being the top and the bottom elements respectively.
In 1971, Zadeh proposed a representation theorem of fuzzy sets using the notion of α-cuts (c.f. Definition 1.2 considering L as [0,1]), known as first decomposition theorem [16] in the literature. α-cuts of a fuzzy set are crisp sets. In this paper we delve into the notion of fuzzy α-cut (c.f. Definition 2.1), which was introduced by the present authors in [15]. Recently we have noticed that the notion of fuzzy α-cut exists in the literature as 'level fuzzy sets' introduced in [19]. In this regard the authors are grateful to the editor of this journal for his valuable advise on an earlier version of this paper. A fuzzy α-cut of a fuzzy set gives a fuzzy subset of the given fuzzy set. We will see that fuzzy α-cut of a fuzzy mathematical structure is a fuzzy mathematical substructure.
In this paper we have proved that a family of fuzzy α-cuts over a frame forms a localic frame. Consequently they generate a model of so called fuzzy geometric logic with graded consequence [7]. The notion of graded frame and fuzzy geometric logic with graded consequence was introduced in [7]. Such an algebraic structure and a logic were proposed to serve the purpose of giving an answer to the question-"From which logic fuzzy topology can be generated?"
This question came up parallel to a similar idea provided in Vickers's book [21] "Topology via logic", viz. from which logic classical topology can be generated?. It is to be noted that as an answer, fuzzy geometric logic was invented and as a further generalisation fuzzy geometric logic with graded consequence was introduced. Notion of graded frame came into the picture as Lindenbaum type algebra of fuzzy geometric logic with graded consequence [7]. The notion of localic frame (c.f. Definition 3.1) is introduced here which is a further generalisation of graded frame by taking a frame-valued binary relation instead of [0, 1] -valued binary relation as [0, 1] is a particular frame.
As for usefulness and significance of the notion of fuzzy α-cuts, we claim that they give natural substructures of fuzzy topological spaces and fuzzy al-gebraic structures. Classical fuzzy topological spaces are defined by taking a crisp set and fuzzy open sets. There are two major streams of research, one following Chang's definition [8] and the other following Lowen's definition [17].
In both cases, subspaces are defined on crisp subsets of the base set. On the other hand in case of fuzzy algebraic structures (e.g. Rosenfeld [20]), one starts with a classical algebraic structure and defines fuzzy substructures. In neither construction topological or algebraic, the starting base set is taken to be fuzzy.
While in the topological case, the base set as well as all the subspaces are to be taken crisp, in the algebraic case though the sub algebras are fuzzy, one has to begin with a classical crisp algebraic structure.
In [4] and [6] there had been proposals to develop both the kinds of structures on fuzzy sets. Besides, in [6] the algebraic compositions also have been fuzzy right from the start. Fuzzy α-cuts being fuzzy subsets of a fuzzy set, using the above proposals it would be possible to define topological and algebraic substructures on them. These will be quite natural substructures. We shall present these constructions in section 4.
Secondly, fuzzy α-cuts will provide fuzzy sets as lower and upper approximations in the probabilistic rough set framework [22]. We expects these kinds of approximation will be useful in the domain of application of rough set theory [18]. This paper is organised as follows. Section 2 emphasises upon the properties of fuzzy α-cut. In section 3, various algebraic structures and a notion of Gödellike arrow along with its properties are discussed. Algebraic structures formed by the family of fuzzy α-cuts are also studied in this section. The significance of fuzzy α-cuts is provided in section 4. Section 5 presents some concluding remarks.
We give below some preliminary definitions that would be required in the sequel. Definition 1.1 (Frame). A frame is a complete lattice such that,
x ∧ Y = {x ∧ y : y ∈ Y }.
i.e., the binary meet distributes over arbitrary join.
Definition 1.2 (α-cut of a fuzzy set). Let (X,Ã) be an L-fuzzy set, where X is the base set and L is a frame. Then for α ∈ L, the α-cut of (X,Ã) is the ordinary set {x ∈ X |Ã(x) ≥ α} and is denoted by αÃ. Definition 1.3 (Gödel arrow). [16] Gödel arrow is defined as follows:
a → b = 1 if a ≤ b b if a > b.
for all a, b ∈ [0, 1].
Gödel arrow can be generalised to the following.
Definition 1.4 (Gödel-like arrow). Let L be any frame. Then the Gödel-like arrow is defined as follows:
a → b = 1 L if a ≤ b b otherwise.
for all a, b ∈ L.
There is another kind of implication in L called residuated implication defined as below. The relationship between these two types of implications is discussed in subsection 3.2.
Fuzzy α-cut and its properties
In this section we will define the notion of fuzzy α-cut and provide some of the algebraic properties of fuzzy α-cut. For the corresponding classical notion we refer to [16].
Definition 2.1 (Fuzzy α-cut of a fuzzy set). [15] Let (X,Ã) be an L-fuzzy set.
Then for α ∈ L, the fuzzy α-cut of (X,Ã) is the fuzzy subset (X,Ãα) wherẽ A α is defined as follows:Ã
α(x) = Ã (x) ifÃ(x) ≥ α 0 L otherwise.
We will denote fuzzy α-cut of an L-fuzzy set (X,Ã) simply byÃα if the base set X is understood.
It is to be noted that fuzzy α-cut of a fuzzy set is also known as level fuzzy set [19]. As mentioned in the introduction, present authors were not aware of this paper published long back in 1977 and not used frequently in subsequent literature. In the paper [19] the author defined the algebraic operations viz.
intersection, union, complementation of level sets as is done in fuzzy set theory by min, max and 1−(·), in the value set [0, 1] and established certain elementary properties. In our paper, however, these operations are presumed since these are none else than the corresponding operations of fuzzy subsets. We have rather proved some non-trivial results in this section where the value set L is taken to be a frame. As we are dealing with L-fuzzy sets, we are considering the generalised version of the definition of α-cut and fuzzy α-cut by generalising the value set [0, 1] to a frame L.
Theorem 2.2. Let (X,Ã), (X,B) be two fuzzy sets. Then for any α, α 1 , α 2 ∈ L the following properties hold: 1.Ãα ⊆ χ αà ; 2. α 1 ≤ α 2 impliesÃα 1 ⊇Ãα 2 ;
3. (Ã∩B) α =Ãα ∩Bα and (Ã∪B) α =Ãα ∪Bα.
Proof. We demonstrate the proof of 3 only. For the first part of 3 we proceed as follows:
(Ã∩B) α(x) = (à ∩B)(x) if (à ∩B)(x) ≥ α 0 L otherwise. = à (x) ∧B(x) ifÃ(x) ∧B(x) ≥ α 0 L otherwise. = à (x) ∧B(x) ifÃ(x) ≥ α andB(x) ≥ α 0 L otherwise. = à (x) ∧B(x) ifÃ(x) ≥ α andB(x) ≥ α 0 L ifÃ(x) ≥ α andB(x) < α 0 L ifÃ(x) < α andB(x) ≥ α 0 L ifÃ(x) < α andB(x) < α. = à (x) ∧B(x) ifÃ(x) ≥ α andB(x) ≥ α A(x) ∧ 0 L ifÃ(x) ≥ α andB(x) < α 0 L ∧B(x) ifÃ(x) < α andB(x) ≥ α 0 L ifÃ(x) < α andB(x) < α. =Ãα(x) ∧Bα(x) = (Ãα ∩Bα)(x).
Similarly the second equality holds.
For any mapping f : X −→ Y , the image of the fuzzy subset (X,Ã) of X is the fuzzy subset (Y, f (Ã)) and defined by [16]
f (Ã)(y) = x∈X {Ã(x) | y = f (x)}.
Thus f (Ãα) gives the fuzzy subset (Y,Ãα) of Y . We now have the following theorem.
Theorem 2.3. Let f : X −→ Y be a mapping.
Then for any L-fuzzy set (X,Ã)
and α ∈ L, f (Ãα) = (f (Ã)) α.
Proof. For any y ∈ Y , we have the following:
f (Ãα)(y) = x {Ãα(x) | y = f (x)} = 0 L ifÃ(x) < α for all x ∈ X x {Ã(x) | y = f (x)} otherwise. = x {Ã(x) | y = f (x)} if x {Ã(x) | y = f (x)} ≥ α 0 L otherwise. = (f (Ã))(y) if (f (Ã))(y) ≥ α 0 L otherwise. = (f (Ã)) α(y).
Hence f (Ãα) = (f (Ã)) α.
L, (g • f )(Ãα) = (g•f (Ã)) α. Proof. (g • f )(Ãα) = g(f (Ãα)) = g( (f (Ã)) α) = (g(f (Ã))) α = (g•f (Ã)) α. (Z, (g • f )Ãα) is a fuzzy subset of Z. Proposition 2.3. Let f : X −→ Y , g : Y −→ Z and h : Z −→ W . Then for any L-fuzzy set (X,Ã) and α ∈ L, (h • (g • f ))(Ãα) = ((h • g) • f )(Ãα).
Proposition 2.4. Let f : X −→ Y be a mapping and id X : X −→ X be the identity mapping. Then for any L-fuzzy set (X,Ã) and α ∈ L,
(f • id X )(Ãα) = f (Ãα).
Proposition 2.5. Let f : X −→ Y be a mapping and id Y : Y −→ Y be the identity mapping. Then for any L-fuzzy set (X,Ã) and
α ∈ L, (id Y • f )(Ãα) = f (Ãα). Proof. α ≤ β ⇒Ãα ⊆Ãβ ⇒ f (Ãα) ⊆ f (Ãβ).
Algebraic structure of the family of fuzzy α-cuts
We shall establish that the family of fuzzy α-cuts forms a localic frame [c.f.
Definition 3.1 (Localic Frame). A localic frame is a 5-tuple (A, ⊤, ∧, , R L ),
where A is a non-empty set, ⊤ ∈ A, ∧ is a binary operation, is an operation on arbitrary subset of A, R L is an L-valued fuzzy binary relation on A satisfying the following conditions:
1. R L (a, a) = 1 L (fuzzy reflexivity); 2. R L (a, b) = 1 L = R L (b, a) ⇒ a = b (fuzzy antisymmetry); 3. R L (a, b) ∧ R L (b, c) ≤ R L (a, c) (fuzzy transitivity); 4. R L (a ∧ b, a) = 1 L = R L (a ∧ b, b); 5. R L (a, ⊤) = 1 L ; 6. R L (a, b) ∧ R L (a, c) = R L (a, b ∧ c); 7. R L (a, S) = 1 L if a ∈ S; 8. inf {R L (a, b) | a ∈ S} = R L ( S, b); 9. R L (a ∧ S, {a ∧ b | b ∈ S}) = 1 L ;
for any a, b, c ∈ A and S ⊆ A. We will denote a localic frame by (A, R L ).
In particular, (A, ⊤, ∧, , R [0,1] ) is a graded frame [7]. Thus localic frame is a generalisation of graded frame. It is to be noted that an algebraic structure satisfying the first three conditions of Definition 3.1 is known as localic poset [10]. That is, a localic poset is a set endowed with fuzzy partial order relation.
However, this is not the only definition of fuzzy partial order. For more general definitions see [11].
Theorem 3.2. ({Ãα | α ∈ L}, ⊆, ∩, ) is a frame.
Proof. Here we only show the distributive property i.e.,
A α ∩ ià α i = i (Ãα ∩Ãα i ). (Ãα ∩ ià α i )(x) =Ãα(x) ∧ (( ià α i )(x)) =Ãα(x) ∧ ( ià α i (x)) = i (Ãα(x) ∧Ãα i (x)) [as L is a frame] = i (Ãα ∩Ãα i )(x)) = ( i (Ãα ∩Ãα i ))(x).
This completes the proof.
Prelinear and Semilinear Frame
In this subsection we will consider prelinear frame and semilinear frame.
This subsection includes detailed study of the above mentioned notions with examples.
Definition 3.3 (Prelinear Frame). [2]
A frame L together with a binary oper-
ation → is said to be a prelinear if for each l 1 , l 2 ∈ L, (l 1 → l 2 ) ∨ (l 2 → l 1 ) = ⊤,
where ⊤ is the top element of L.
For our purpose we will take → as Gödel-like arrow. In this section henceforth all the arrows are Gödel-like arrow.
Note: For the notion of prelinearity in more general set up we refer to [13].
Whenever there is an → satisfying the property l 1 ≤ l 2 implies l 1 → l 2 = ⊤, linearity of the order implies prelinearity. Our purpose here will be served under the assumption of a notion more general than prelinearity viz. semilinearity (c.f.
Definition 3.4).
The purpose is to generalize the notion of graded frame [7] where the value set is taken as [0, 1].
Definition 3.4 (Semilinear Frame). A semilinear frame L = (L, ∧, , →) is a frame (L, ∧, )
together with a binary operation → such that for all l 1 , l 2 , l 3 ∈ L,
(l 1 → l 2 ) ∧ (l 1 → l 3 ) = (l 1 → l 2 ∧ l 3 ).
It can be verified by considering all possible cases that any frame with up to 4-elements is always preilinear. We give below an example of a 5-element lattice which is the smallest semilinear but not prelinear frame.
Example 3.1. The following frame is not prelinear but semilinear.
⊤ a b c ⊥ For this frame (b → c) ∨ (c → b) = c ∨ b = a = ⊤.
Hence it is not prelinear.
The following is an example of a frame with six elements which is not semilinear. It is to be noted that the following frame is the smallest non semilinear frame. In other words a non-semilinear distributive lattice contains at least six elements.
Example 3.2. The following frame is not semilinear.
⊤ a d b c ⊥ For this frame (b → a) ∧ (b → c) = ⊤ ∧ c = c, whereas b → (a ∧ c) = b → ⊥ = ⊥.
So, Property 3.5 fails. Before proceeding to the next theorem let us enlist below some properties of Gödel-like arrow [16] that would be used in the sequel.
Properties of Gödel-like arrow
In this subsection some required properties of Gödel-like arrow are listed along with verification of some of them. Now the following cases may arise:
Case 1: a → b = 1 L and b → c = 1 L .
Here a ≤ b and b ≤ c and hence as L is transitive, a ≤ c. Consequently When b∧c > a then (b∧c)∨a = b∧c = b as if b∧c = b then b ≤ c, a contradiction.
a → c = 1 L . Therefore (a → b) ∧ (b → c) = 1 L ∧ 1 L = 1 L = (a → c). Case 2: a → b = 1 L and b → c = c. We have (a → b) ∧ (b → c) = 1 L ∧ c = c ≤ c (or 1 L ) = (a → c). Case 3: a → b = b and b → c = 1 L . As b ≤ c, (a → b) ∧ (b → c) = b ∧ 1 L = b ≤ c (or 1 L ) = (a → c). Case 4: a → b = b and b → c = c. In this case (a → b) ∧ (b → c) = b ∧ c = c ≤ c (or 1 L ) = (a → c).As L is prelinear (a → (b ∧ c)) ∨ ((b ∧ c) → a) = ⊤. When the pair (b ∧ c, a) is incomparable then (a → (b ∧ c)) ∨ ((b ∧ c) → a) = (b ∧ c) ∨ a = ⊤ = b (as b and
c are incomparable).
Hence for either cases
b ∧ (c ∨ a) = b = (b ∧ c) ∨ (b ∧ a), but L is distributive.
Similarly for Case 2 also we get a contradiction.
Corollary 3.5. If L is totally ordered frame then (a → b) ∧ (a → c) = a → (b ∧ c), for any a, b, c ∈ L. Property 3.6. inf i {(a i → b)} = sup i {a i } → b, for any a i , b ∈ L. Proof. sup i {a i } → b = 1 if sup i {a i } ≤ b b otherwise. Now for sup i {a i } ≤ b we have a i ≤ sup i {a i } ≤ b.i → b = 1. As b ≤ 1, inf i {a i → b} = b. If sup{a i } and b
are incomparable then atleast one of the a i 's, say a j is incomparable to b and
consequently a j → b = b. Hence inf i {a i → b} = b. Property 3.7. a ≤ b iff a → b = 1 L . Property 3.8. a ∧ (a → b) ≤ b.
It is to be noted that all the above properties are true for generalised Gödel arrow as these properties are satisfied by any arrow with residuation property.
That means residuated arrow satisfies semilinear property but there are semilinear arrows which are not residuated arrows. We now proceed to the main theorem which constitutes subsection 3.3
Main Theorem
Let
R L (Ãα 1 ,Ãα 2 ) = 1 L = R L (Ãα 2 ,Ãα 1 ). So, inf x {Ãα 1 (x) →Ãα 2 (x)} = 1 L = inf x {Ãα 2 (x) →Ãα 1 (x)}. ThereforeÃα 1 (x) ≤Ãα 2 (x) andÃα 2 (x) ≤ A α 1 (x)
, for all x. So,Ãα 1 (x) =Ãα 2 (x), for any x. HenceÃα 1 =Ãα 2 .
From Property 3.2, we have (Ãα
1 (x) →Ãα 2 (x)) ∧ (Ãα 2 (x) →Ãα 3 (x)) ≤ (Ãα 1 (x) →Ãα 3 (x)), for all x. Hence inf x {(Ãα 1 (x) →Ãα 2 (x))∧(Ãα 2 (x) → A α 3 (x))} ≤ (Ãα 1 (x) →Ãα 3 (x)), for any x and consequently inf x {(Ãα 1 (x) → A α 2 (x)) ∧ (Ãα 2 (x) →Ãα 3 (x))} ≤ inf x {Ãα 1 (x) →Ãα 3 (x)}. Therefore, R L (Ãα 1 ,Ãα 2 ) ∧ R L (Ãα 2 ,Ãα 3 ) = inf x {Ãα 1 (x) →Ãα 2 (x)} ∧ inf x {Ãα 2 (x) →Ãα 3 (x)} ≤ inf x {(Ãα 1 (x) →Ãα 2 (x)) ∧ (Ãα 2 (x) →Ãα 3 (x))} ≤ inf x {Ãα 1 (x) →Ãα 3 (x)} [using Property 3.2] = R L (Ãα 1 ,Ãα 3 ). 4. R L (Ãα 1 ∧Ãα 2 ,Ãα 1 ) = inf x {(Ãα 1 ∧Ãα 2 )(x) →Ãα 1 (x)} = 1 L , asÃα 1 ∩ A α 2 ⊆Ãα 1 . Similarly R L (Ãα 1 ∧Ãα 2 ,Ãα 2 ) = 1 L . 5. R L (Ãα 1 ,Ã0 L ) = inf x {Ãα 1 (x) →Ã0 L (x)} = 1 L , asÃ0 L (x)
=Ã(x) and
A α 1 (x) ≤Ã(x), for any x.
6. We have,
R L (Ãα 1 ,Ãα 2 ) ∧ R L (Ãα 1 ,Ãα 3 ) = inf x {Ãα 1 (x) →Ãα 2 (x)} ∧ inf x {Ãα 1 (x) →Ãα 3 (x)} = inf x {(Ãα 1 (x) →Ãα 2 (x)) ∧ (Ãα 1 (x) →Ãα 3 (x))} = inf x {Ãα 1 (x) → (Ãα 2 (x) ∧Ãα 3 (x))} [using Property 3.5] = inf x {Ãα 1 (x) → (Ãα 2 ∩Ãα 3 )(x)} = R L (Ãα 1 ,Ãα 2 ∩Ãα 3 ). 7. LetÃα ∈ {Ãα i } i , then R L (Ãα, ià α i ) = inf x {Ãα(x) → ( ià α i )(x)} = inf x {Ãα(x) → ià α i (x)} = 1 L [using Property 3.7].
8. Here we have,
inf i {R L (Ãα i ,Ãα)} = inf i {inf x {Ãα i (x) →Ãα(x)}} = inf x {inf i {Ãα i (x) →Ãα(x)}} = inf x { i (Ãα i (x)) →Ãα(x)} [usingP roperty3.6] = inf x {( ià α i )(x) →Ãα(x)} = R L ( ià α i ,Ãα).
From Theorem 3.2 we haveÃα
∩ ià α i = i (Ãα ∩Ãα i ). So, R L (Ãα ∩ ià α i , i (Ãα ∩Ãα i )) = inf x {(Ãα ∩ ià α i )(x) → ( i (Ãα ∩Ãα i ))(x) = 1 L .
Hence ({Ãα | α ∈ L},Ã0 L , ∩, , R L ) is a localic frame.
α 2 ∈ [0, 1].
It may be noted that the 5-tuple ({Ãα | α ∈ L},Ã0 L , ∩, , R L ) is a localic preordered set, where L is a frame, R L (Ãα 1 ,Ãα 2 ) = inf x {Ãα 1 (x) →Ãα 2 (x)} for α 1 , α 2 ∈ L and '→' is the Gödel-like arrow, as it satisfies all the properties to be a localic frame except the property namely
Some applications of fuzzy α-cut
In this section we shall show some usages of the notion of fuzzy α-cuts.
Topological Structure
Usually a fuzzy topological space is defined as a crisp set having fuzzy open sets [8,17]. In 1992, Chakraborty and Ahsanullah proposed a notion of fuzzy topology on fuzzy sets [4]. This generalisation allows for defining topological subspaces on fuzzy subsets of the original fuzzy topological space. With respect to the classical definition subspaces have to be defined on crisp subsets of the original set. In our recent work on fuzzy topological systems [15] we needed to use L -topological spaces where the value set L is a frame. This is one further step towards generalisation of [4]. To make this paper self contained we give the definition below. 2. (X,Ã 1 ), (X,Ã 2 ) are in τ implies (X,
à 1 ∩à 2 ) is in τ , where (à 1 ∩à 2 )(x) =à 1 (x) ∧à 2 (x), for all x ∈ X;
3. (X,Ã i ) ∈ τ implies (X, i∈IÃ i ) ∈ τ , where i∈IÃ i : X −→ L is such that ( i∈IÃ i )(x) = i∈IÃ i (x), for all x ∈ X.
Then (X,Ã, τ ) is an L -topological space.
One can easily see that the fuzzy α-cuts of a fuzzy topological space are fuzzy topological subspaces. More specifically, we have. Hence fuzzy α-cuts provide us with a natural class of fuzzy substructure of L -topological space.
It should be noted that in subsequent years there has been a lot of serious work on fuzzy topological spaces from the angle of category theory [4,5,14,17].
But to our knowledge, the notion of fuzzy α-cuts as substructures has not been discussed. It would be interesting to investigate what kind of sub objects these fuzzy α-cuts give rise to.
Algebraic Structure
A similar approach was initially adopted in developing fuzzy algebraic structures. Rosenfeld's pioneering work in fuzzy groups starts with an ordinary group and proceeds to define fuzzy subgroups of that group. On the other hand in [6], Chakraborty and Banerjee defined fuzzy operations on fuzzy sets thus obtaining a generalisation that was intended. They, however, placed their work in categorical framework. We shall adopt their idea basically but avoiding categorical language and then show the role of fuzzy α-cuts in this context. It is to be noted that a binary operation on a crisp set A (e.g. the group operation) is a mapping from A × A to A. We shall define a fuzzy binary operation on a fuzzy set (X,Ã) using fuzzy equality. It is also to be noted that the Cartesian product of two L-fuzzy sets (X,Ã) and (Y,B) is the L-fuzzy set (X × Y,à ×B) where (à ×B)(x, y) =Ã(x) ∧B(y). So a fuzzy binary composition on (X,Ã) has to be a kind of mapping from (X × X,à ×Ã) to (X,Ã) where the pre-image is mapped to the image to some grade belonging to the frame L. For any L-fuzzy set (X,Ã) by |à | is meant the support viz. {x ∈ X |Ã(x) > 0 L }. Instead of using pre-fix notation for the operator ⊕ we shall use infix notation, i.e. we write
x 1 ⊕ x 2 = x 3 , for ⊕ (x 1 , x 2 ) = x 3
and additionally equality relation (=) is graded. That is, for any x ∈ X, y ∈ Y , the expression x = y gets a degree from L. We shall write gr(x ≃ y) to make a distinction between fuzzy equality and ordinary equality. Formally, we have It is to be noted that property 1 of being an L-fuzzy group represents the fuzzy version of associativity.
Equipped with this definition of an L-fuzzy group, it will be observed how does fuzzy α-cuts play a role.
gr(x 1 ⊕ ′ x 2 ≃ x 3 ) = gr(x 1 ⊕ x 2 ≃ x 3 ) ∧B(x 1 ) ∧B(x 2 ) ∧B(x 3 ),
for any x 1 , x 2 , x 3 ∈ X is called a fuzzy subgroup of (X,Ã, ⊕) if (X,B, ⊕ ′ ) is itself an L-fuzzy group, [⊕ ′ is the restriction of ⊕ on the fuzzy subset (X,B) of (X,Ã)]. Proof. Given that (X,Ã, ⊕) is an L-fuzzy group. Let us restrict the function ⊕ onÃα. Then for x, y, z ∈ X, if gr(x ⊕ y ≃ z) =Ã(x) ∧Ã(y) then gr(x ⊕ ′ y ≃ z) =Ã(x) ∧Ã(y) ∧Ãα(x) ∧Ãα(y) ∧Ãα(z). Now asÃ(x) ∧Ã(y) ≤Ã(z), we haveÃα(x) ∧Ãα(y) ≤Ãα(z). ThereforeÃα(x) ∧ A α(y) ∧Ãα(z) =Ãα(x) ∧Ãα(y). Hence for any x, y, z ∈ X if gr(x ⊕ y ≃ z) = A(x) ∧Ã(y) then gr(x ⊕ ′ y ≃ z) =Ãα(x) ∧Ãα(y).
To show that (X,Ãα, ⊕ ′ ) is an L-fuzzy subgroup for α such that |Ãα | = ∅ first of all notice that ⊕ ′ is indeed an L-fuzzy binary operation as the following holds.
(i) gr(x ⊕ ′ y ≃ z) = gr(x ⊕ y ≃ z) ∧Ãα(x) ∧Ãα(y) ∧Ãα(z) ≤ (Ãα(x) ∧ A α(y)) ∧Ãα(z).c ∈|Ãα | with gr(a ⊕ ′ b ≃ c) =Ãα(a) ∧Ãα(b) and gr(a ⊕ ′ b ≃ c ′ ) = 0 L if c ′ ( = c) ∈|Ãα |.
As for any x, y, z ∈ X if gr(x ⊕ y ≃ z) =Ã(x) ∧Ã(y) then gr(x ⊕ ′ y ≃ z) = A α(x) ∧Ãα(y), associativity holds good. Similarly it can be shown that for any a ∈|Ãα |, there exist a −1 ∈|Ãα | such that gr(a ⊕ ′ a −1 ≃ e) =Ãα(a) ∧Ãα(a −1 ), asÃ(a −1 ) =Ã(a) ≥ α.
Thus fuzzy α-cuts form natural fuzzy subgroups of the fuzzy group.
This method of defining fuzzy algebraic structures and their sub structures may be adopted for any kind of algebraic structure not necessarily fuzzy groups only.
gr(x 1 ⊕ x 1 ≃ x 4 ) = l 1 gr(x 1 ⊕ x 2 ≃ x 3 ) = l 3 gr(x 1 ⊕ x 3 ≃ x 2 ) = l 3 gr(x 1 ⊕ x 4 ≃ x 1 ) = l 1 gr(x 2 ⊕ x 1 ≃ x 3 ) = l 3 gr(x 2 ⊕ x 2 ≃ x 4 ) = l 2 gr(x 2 ⊕ x 3 ≃ x 1 ) = l 3 gr(x 2 ⊕ x 4 ≃ x 2 ) = l 2 gr(x 3 ⊕ x 1 ≃ x 2 ) = l 3 gr(x 3 ⊕ x 2 ≃ x 1 ) = l 3 gr(x 3 ⊕ x 3 ≃ x 4 ) = l 3 gr(x 3 ⊕ x 4 ≃ x 3 ) = l 3 gr(x 4 ⊕ x 1 ≃ x 1 ) = l 1 gr(x 4 ⊕ x 2 ≃ x 2 ) = l 2 gr(x 4 ⊕ x 3 ≃ x 3 ) = l 3 gr(x 4 ⊕ x 4 ≃ x 4 ) = l 4
Then clearly x 4 = e, x −1
1 = x 1 , x −1 2 = x 2 , x −1 3 = x 3 and x −1 4 = x 4 .
Hence (X,Ã, ⊕) is an L-fuzzy group.
Let α = l 1 , then |Ãl 1 |= {x 1 , x 4 } = ∅ and (X,Ãl 1 , ⊕ ′ ) forms an L-fuzzy group and consequently becomes an L-fuzzy subgroup of (X,Ã, ⊕). Similarly for other α ∈ L, where |Ãα | = ∅, it can be shown that (X,Ãα, ⊕ ′ ) forms L-fuzzy subgroups of (X,Ã, ⊕).
Probabilistic Rough Set Theory
We will now observe another kind of usefulness of the notion of fuzzy αcuts in the context of rough set theory [18,22]. An approximation space is a tuple (X, R), consisting of a set of objects X and an equivalence relation R, known as indiscernibility relation on X. For any A ⊆ X, the lower and upper approximations of A in the approximation space (X, R) are denoted by A and A respectively and defined as follows.
A = {[x] | [x] ⊆ A}; A = {[x] | A ∩ [x] = ∅}.
A rough membership function of A, denoted by µ A , is a function from X to
[0, 1] such that µ A (x) = |[x]∩A| |[x]| ≤ 1,
where | S | stands for the cardinality of the set S and [x] stands for the equivalence class of x ∈ X. In this definition X is taken to be a finite set.
In [22], we notice that for generalised probabilistic approximations, they considered a pair of parameters α, β ∈ [0, 1] with α ≥ β to ensure that the lower approximation is smaller than the upper approximation in order to be consistent with existing approximation operators.
In the theory of probabilistic rough sets a weight or grade from the set [0, 1] is attached with each granule. The grades of granules are obtained with the help of some rough membership function. In particular the grade of granule may be determined with the help of above described rough membership function. Notice that in [22], while defining lower and upper approximations of a set A, α-cuts and strict β-cuts are used with 0 ≤ β < α ≤ 1 in the following way.
A α = {x ∈ X | µ A (x) ≥ α}; A β = {x ∈ X | µ A (x) > β}.
These are crisp sets. Hence the grade disappears in the final approximations.
But while defining lower and upper approximations of a set A, if we use the concept of fuzzy α-cuts and fuzzy β-cuts instead of α-cuts and strict β-cuts then we will able to end up with the final approximations having grades. That is, the lower and upper approximations becomes fuzzy sets and defined as follows.
A α : X −→ [0, 1] s.t. A α(x) = µ A (x) if µ A (x) ≥ α 0 otherwise. A β : X −→ [0, 1] s.t. A β(x) = µ A (x) if µ A (x) ≥ β 0 otherwise.
It is quite expected that the above described notion of lower and upper approximations will play a significant role in probabilistic rough set theory. Here instead of two different types of cuts viz. α-cuts and strict β-cuts one type of cut has been used uniformly in determining lower and upper approximations.
In this paper we will not delve into this topic, but it will be considered in our future research.
Concluding Remarks
In this paper we have dealt with the notion of fuzzy α-cut and its significance.
Study of the family of fuzzy α-cuts provides an example of graded frame which was introduced in [7]. Moreover in this paper we generalise the notion of graded frame one step further and call it 'semilinear frame'. It is to emphasise that the notion semilinearity introduced in this paper is more general than 'prelinearity';
while the latter notion has been widely discussed in literature, the former notion is not. We also proposed the notion of localic frame in this work. A detailed study of Gödel-like arrow provides a nice result about the relation between prelinearity and semilinearity property. The algebraic notion of semilinear frame needs to be studied in more detail. Taking a general fuzzy arrow instead of Gödel arrow may also be considered as an interesting future project.
Definition 1. 5 (
5Residuated implication). Let L be any any frame. Then the residuated implication is defined by a → b = sup{c ∈ L | c ∧ a ≤ b} for all a, b ∈ L.
Example 2. 1 .
1Consider the fuzzy setà defined on the interval X = [0, 10] of real numbers by the membership functionÃ(x) Let (X,Ã), (X,B) be two L-fuzzy sets. Thenà ⊆B if and only ifÃ(x) ≤ B(x), for any x ∈ X.
Proposition 2. 1 .
1Let f : X −→ Y be a mapping. Then for any L-fuzzy set (X,Ã) and α, β ∈ L, α ≤ β ⇒ f (Ãα) ⊆ f (Ãβ). Proposition 2.2. Let f : X −→ Y and g : Y −→ Z. Then for any L-fuzzy set (X,Ã) and α ∈
Definition 3.1] with respect to an L-fuzzy relation R defined in terms of the Gödel-like arrow [c.f. Definition 1.4] in L.
Example 3. 3 .
3The following frame is the smallest Boolean algebra which isnot semilinear. frame (a → c)∧(a → d) = c∧⊤ = c, whereas a → (c∧d) = a → ⊥ = ⊥.So, Property 3.5 fails.One can see that the concepts of prelinearity and semilinearity are based on the underlying lattice of the frame which is distributive. While prelinearity is an well known concept, semilinearity is not so and which is a more general concept [c.f.Property 3.5]
Property 3. 1 .
1a → a = 1 L , for any a ∈ L.Property 3.2. (a → b) ∧ (b → c) ≤ (a → c), for any a, b, c ∈ L.Proof. It may be observed that the values of a → b is either 1 L or b. Similarly for b → c the values are either 1 L or c and for a → c values are either 1 L or c.
Property 3. 3 .
3a ≤ b implies (a → x) ≥ (b → x), for any a, b, x ∈ L.Property 3.4. a ≤ b implies (x → a) ≤ (x → b), for any a, b, x ∈ L.Property 3.5. If L is prelinear then it is semilinear. Proof. If possible let L is prelinear i.e., (a → b) ∨ (b → a) = ⊤, for any a, b ∈ L and for some a, b, c ∈ L, (a → b) ∧ (a → c) = a → (b ∧ c). Then two cases may arise. Case 1: a < b, (a, c) and (b, c) are incomparable [where (a, b) represents the pair of points from L]. Case 2: a < c, (a, b) and (b, c) are incomparable. Case 1: In this case notice that a ∨ c = ⊤ as (c → a) ∨ (a → c) = ⊤ and a, care incomparable. Hence b ∧(c∨a) = b ∧⊤ = b. Now (b ∧c)∨(b ∧a) = (b ∧c)∨a.The following three cases may arise under this situation. Either b ∧ c ≤ a or b∧c > a or the pair (b∧c, a) is incomparable. If b∧c ≤ a, then (b∧c)∨a = a = b.
Hence for this case (a i → b) = 1, for each i and consequently inf i {a i → b} = 1.If sup i {a i } > b then there exist atleast one a i such that a i > b and rest will be either bellow b or equal to b. Now for the case a i > b, a i → b = b and for all other cases a
Theorem 3. 6 .
6Let L be semilinear frame. Then ({Ãα | α ∈ L},Ã0 L , ∩, , R L ) is a localic frame, where R L (Ãα 1 ,Ãα 2 ) = inf x {Ãα 1 (x) →Ãα 2 (x)} for α 1 ,α 2 ∈ L and '→' is the Gödel-like arrow.Proof. Let us check the properties for ({Ãα | α ∈ L},Ã0 L , ∩, , R L ) to be a localic frame.
1 .
1R L (Ãα,Ãα) = inf x {Ãα(x) →Ãα(x)} = 1 L [from Property 3.1Ãα(x) → A α(x) = 1 L , for all x].
Corollary 3.7. ({Ãα | α ∈ [0, 1]},Ã0, ∩, , R [0,1] ) is a graded frame, where '→' is the Gödel arrow and R [0,1] (Ãα 1 ,Ãα 2 ) = inf x {Ãα 1 (x) →Ãα 2 (x)} for α 1 ,
Definition 4.1 (L -Topological Space).[4] Let (X,Ã) be an L-fuzzy set and τ a collection of fuzzy subsets of (X,Ã) such that 1. (X,∅) and (X,Ã) are in τ , where∅ : X −→ L is such that∅(x) = 0 L , for all x ∈ X, where 0 L is the least element of the frame L;
Theorem 4.2.[15] Let (X,Ã, τ ) be an L -topological space and a fuzzy α-cut of (X,Ã) i.e., (X,Ãα) be taken. Let τ ′ be defined by τ ′ = {(X,T ′ ) |T ′ = A α ∩T ,T ∈ τ }. Then (X,Ãα, τ ′ ) also forms an L -topological space and is an L -topological subspace.
e
∈|Ã | described in 2 is known as the identity whereas a −1 ∈|Ã | for each a ∈|Ã | illustrated in condition 3 are known as inverse of a in the L-fuzzy group (X,Ã, ⊕). It is possible to show that identity and inverse of an element in the L-fuzzy group are unique.
Definition 4.5 (L-fuzzy subgroup). Let (X,Ã, ⊕) be an L-fuzzy group and (X,B) be a L-fuzzy subset of the L-fuzzy set (X,Ã). Then the fuzzy substructure (X,B, ⊕ ′ ) where ⊕ ′ is an L-fuzzy binary operation on (X,B) defined by,
Proposition 4 . 1 .
41Let (X,Ã, ⊕) be an L-fuzzy group. Then for any a ∈|à |, A(a) ≤Ã(e).Proof. For any a ∈|à |, there exist a −1 ∈|à | such that gr(a ⊕ a −1 ≃ e) = A(a)∧Ã(a −1 ) =Ã(a), asÃ(a) =Ã(a −1 ). Also we know gr(a⊕ a −1 ≃ e) ≤Ã(e) and consequentlyÃ(a) ≤Ã(e), for any a ∈|à |.
Theorem 4. 6 .
6Let (X,Ã, ⊕) be an L-fuzzy group. Then (X,Ãα, ⊕ ′ ) is an Lfuzzy sub group for any α ∈ L such that |Ãα | = ∅.
(
ii) If a, b ∈|Ãα | thenÃα(a) > 0 L ,Ãα(b) > 0 L . HenceÃ(a) ≥ α and A(b) ≥ α. As ⊕ is an L-fuzzy binary operation so for any a, b ∈|Ãα |⊆|à | there exist unique c ∈|à | with gr(a ⊕ b ≃ c) =Ã(a) ∧Ã(b) ≤Ã(c) and gr(a ⊕ b ≃ c ′ ) = 0 L , if c ′ ( = c) ∈|à |. ThereforeÃ(c) ≥ α and henceÃα(c) = A(c). c ∈|Ãα |. Consequently for (a, b) ∈|Ãα ×Ãα |, there exist unique
Now (X,Ã, ⊕) is an L-group and so there exist e ∈|à | such that for any a ∈|à |, gr(a ⊕ e ≃ a) =Ã(a) ∧Ã(e). Hence for any a ∈|Ãα |⊆|à |, gr(a ⊕ ′ e ≃ a) =Ãα(a) ∧Ãα(e). Now a ∈|Ãα | impliesÃα(a) > 0 L and henceÃ(a) ≥ α. Using Proposition 4.1, we have α ≤Ã(a) ≤Ã(e). Hencẽ A α(e) =Ã(e). So, e ∈|Ãα |.
Example 4 . 1 .
41Let X = {x 1 , x 2 , x 3 , x 4 , x 5 },Ã : X −→ (x i ) = l i , for i = {1, 2, 3, 4} andÃ(x 5 ) = 0 L . Here |Ã |= {x 1 , x 2 , x 3 , x 4 }.Let us define ⊕ as follows:
R L (Ãα 1 ,Ãα 2 ) ∧ R L (Ãα 1 ,Ãα 3 ) = R L (Ãα 1 ,Ãα 2 ∩Ãα 3 ).
⊕ x 2 ≃ x 3 ) ≤à ×Ã(x 1 , x 2 ) ∧Ã(x 3 ) for any x 1. ×ã X × X) × X −→ L Andã, x 2 , x 3 ∈ X, where grgr(x 1 ⊕ x 2 ≃ x 3 ) ≤à ×Ã(x 1 , x 2 ) ∧Ã(x 3 ) for any x 1 , x 2 , x 3 ∈ X, where gr : (X × X) × X −→ L andà ×Ã(x 1 , x 2 ) =Ã(x 1 ) ∧Ã(x 2 );
∈|à ×à | there exist a unique a ∈|à | with gr(a 1 ⊕ a 2 ≃ a) =Ã(a 1 ) ∧Ã(a 2 ) and gr(a 1 ⊕ a 2 ≃ a ′ ) = 0 L , if a ′ ( = a) ∈|à |. the operation ⊕. Also gr(x 1 ⊕ x 2 ≃ x 3 ) represents the fuzzy equality that is, the grade in which the pair (x 1 , x 2 ) equals to x 3 by the fuzzy composition ⊕. That is, here we will talk about the degree of equality between x 1 ⊕ x 2 and x 3 of X2. for any (a 1 , a 2 ) ∈|à ×à | there exist a unique a ∈|à | with gr(a 1 ⊕ a 2 ≃ a) =Ã(a 1 ) ∧Ã(a 2 ) and gr(a 1 ⊕ a 2 ≃ a ′ ) = 0 L , if a ′ ( = a) ∈|à |. the operation ⊕. Also gr(x 1 ⊕ x 2 ≃ x 3 ) represents the fuzzy equality that is, the grade in which the pair (x 1 , x 2 ) equals to x 3 by the fuzzy composition ⊕. That is, here we will talk about the degree of equality between x 1 ⊕ x 2 and x 3 of X.
sisting of an L-fuzzy set (X,Ã) with |Ã | = ∅ and an L-fuzzy binary operation ⊕ such that 1. if for any a, b, a 1 , a 2 , a 3 , b 1 , b 2 ∈|Ã |, gr(a 1 ⊕ a 2 ≃ b 1 ) =Ã(a 1 ) ∧Ã(a 2 ), gr(b 1 ⊕ a 3 ≃ a) =Ã(b 1 ) ∧Ã(a 3 ). X , Ã , ⊕ ) , An L-fuzzy group is a triple. Definition 4.4 (L-fuzzy group. gr(a 2 ⊕ a 3 ≃ b 2 ) =Ã(a 2 ) ∧Ã(a 3 )Definition 4.4 (L-fuzzy group). An L-fuzzy group is a triple (X,Ã, ⊕) con- sisting of an L-fuzzy set (X,Ã) with |Ã | = ∅ and an L-fuzzy binary operation ⊕ such that 1. if for any a, b, a 1 , a 2 , a 3 , b 1 , b 2 ∈|Ã |, gr(a 1 ⊕ a 2 ≃ b 1 ) =Ã(a 1 ) ∧Ã(a 2 ), gr(b 1 ⊕ a 3 ≃ a) =Ã(b 1 ) ∧Ã(a 3 ), gr(a 2 ⊕ a 3 ≃ b 2 ) =Ã(a 2 ) ∧Ã(a 3 ) and
A(a) ∧Ã(a −1 ) = gr(a −1 ⊕ a ≃ e) andÃ(a) =Ã(a −1 ). A(a) ∧Ã(a −1 ) = gr(a −1 ⊕ a ≃ e) andÃ(a) =Ã(a −1 ).
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| [] |
[
"A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters",
"A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters",
"A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters",
"A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters"
] | [
"Krishnakanta Bhattacharya \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia\n",
"Krishnakanta Bhattacharya \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia\n"
] | [
"IUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia",
"IUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia"
] | [] | The Einstein-Hilbert (EH) action is peculiar in many ways. Some of the Peculiar features have already been highlighted in literature. In the present article, we have discussed some peculiar features of EH action which has not been discussed earlier. It is well-known that there are several ways of decomposing the EH action into the bulk and the surface part with different underlying motivations. We provide a review on all of these decompositions. Then, we attempt to study the static coordinate as a limiting case of a time-dependent coordinate via dynamic upgradation of the constant metric parameters. Firstly, we study the consequences when the constant parameters, present in a static and spherically symmetric (SSS) metric, are promoted to the time dependent variables, which allows us to incorporate the time-dependence in the static coordinate. We find that, in every sets of decomposition, the expression of the bulk term remains invariant, whereas the surface term changes by a total derivative term. Finally, when we obliterate the time dependence of the metric parameters, we find that the expression of the Ricciscalar (or the EH action) does not go back to its original value. Instead, we find that the curvature becomes singular on the horizon, which implies a topological change from the original spacetime. * [email protected] 1 arXiv:2207.08199v1 [gr-qc] 17 Jul 2022 | 10.1007/s10714-022-02958-9 | [
"https://export.arxiv.org/pdf/2207.08199v1.pdf"
] | 250,627,477 | 2207.08199 | ba6a1eb6ec76cd19d4a47afe5f4c465eeabc0257 |
A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters
July 19, 2022
Krishnakanta Bhattacharya
IUCAA
Post Bag 4, Pune -411 007GaneshkhindIndia
A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters
July 19, 2022
The Einstein-Hilbert (EH) action is peculiar in many ways. Some of the Peculiar features have already been highlighted in literature. In the present article, we have discussed some peculiar features of EH action which has not been discussed earlier. It is well-known that there are several ways of decomposing the EH action into the bulk and the surface part with different underlying motivations. We provide a review on all of these decompositions. Then, we attempt to study the static coordinate as a limiting case of a time-dependent coordinate via dynamic upgradation of the constant metric parameters. Firstly, we study the consequences when the constant parameters, present in a static and spherically symmetric (SSS) metric, are promoted to the time dependent variables, which allows us to incorporate the time-dependence in the static coordinate. We find that, in every sets of decomposition, the expression of the bulk term remains invariant, whereas the surface term changes by a total derivative term. Finally, when we obliterate the time dependence of the metric parameters, we find that the expression of the Ricciscalar (or the EH action) does not go back to its original value. Instead, we find that the curvature becomes singular on the horizon, which implies a topological change from the original spacetime. * [email protected] 1 arXiv:2207.08199v1 [gr-qc] 17 Jul 2022
Introduction
All the fundamental theories are expected to be obtained via a well-defined action principle. In general relativity, the Einstein-Hilbert (EH) action is widely accepted as one which provides the dynamics of the gravitational system. However, there have been debates for long regarding the well-posedness of the action principle while obtaining the Einstein's field equation from the EH action. In addition, there are several peculiar features of EH action which makes it distinctively different from the actions of other fundamental theories. In this case, the action contains the second order derivative of the metric. Therefore, one has to fix both the metric as well as its first order derivative on the boundary (i.e. one has to impose both Dirichlet as well as Neumann boundary condition simultaneously), which creates the action principle to be devoid of being a well-posed formulation. This issue can be resolved by adding a suitable boundary term [1] along with the EH action. The most popular boundary term in literature is the Gibbons-Hawking-York (GHY) boundary term [2][3][4] (there are several other popular boundary term, for ref see [1]). In addition, the peculiar structure of the EH action helps to get rid of such problem. The second order derivative terms of the EH action can be expressed as a total derivative as a whole and, therefore, does not contribute in the dynamics. Thus, the EH action can be decomposed into bulk part as well as the surface part, where the bulk part contains the first order derivative of the metric and the surface term contains the second order derivative of the metric. Most importantly, it has been found by Padmanabhan and collaboration [5][6][7][8][9][10] that the bulk part and the surface part of the EH action are not independent of each other. They are related by the "holographic relation", which makes gravity intrinsically holographic as this relation suggests that the surface degrees of freedom participates in the dynamics of the system. Apart from this decomposition, there are several other decomposition in literature such as the ADM decomposition and the decomposition in terms of G 0 0 and R 0 0 . Each of these decompositions of EH action has its own significance, which has been discussed briefly in this article.
The scope of the present paper is to discuss about the EH action, its decompositions and its peculiar features which has not been discussed earlier. The effect of coordinate transformation in the expression of EH action and in bulk/ surface part of different decompositions are quite known (for example see [11]). In this paper, we formulate a novel approach to probe the EH action and its decomposed parts. The idea is that a static metric can be considered as a limiting case of a time-dependent one. For example, there are several constant parameters which appears in the spacetime metric. We can promote (some/ all of) those constant parameters as time-dependent variables and think the static metric as a limiting case of this time-dependent metric, where the static metric can be obtained in the limit where the promoted time-dependent parameters corresponds to its earlier constant value. Of course, these two metrics (one with constant metric parameters and other with the time-dependent parameter(s)) will correspond to different spacetime geometry and different expressions of the dynamical equations. However, we can check how the expression of the EH action (and the bulk or the surface term of different decompositions) changes (change) due to such change in the parameters. More importantly, it will be interesting to see whether the spacetime geometry/ EH action boils down to its static expression in the limit where the promoted parameters correspond to their constant value (i.e. the question we ask is that if we make a reversible change in the metric (via dynamic upgradation), whether the spacetime geometry changes reversibly).
Here, it will be shown that in static and spherically symmetric (SSS) metric, if one incorporates the time-dependence by promoting the underlying constant parameters as time-dependent variables, the expression of the EH action changes by a total timederivative term. When we make a deeper investigation, we find that the bulk part of each decomposition remains unchanged and only the surface part of each decomposition changes by that total derivative term. This characteristics of the EH action (i.e the bulk part of the Lagrangian in different decomposition being invariant under the dynamic upgradation of constant metric parameters) can be attributed to spherically symmetric metric. On the other hand, as we show later, in Painleve coordinate, both the bulk as well as the surface part changes under such dynamic upgradation of parameters and the invariance of the bulk Lagrangian is no longer preserved. When the time dependence of the constant parameters are removed (by introducing a smallness parameter and taking the limit → 0, in which the time dependent parameter will again boil down to its constant value), one expects that the expression of the EH action will reduce to its original expression. However, we have found that the curvature (or the EH Lagrangian) becomes singular with infinite discontinuity near the horizon. This implies a permanent and non-reversible shift of the spacetime geometry due to dynamic upgradation.
The paper is organized as follows. In the following section (i.e. section 2) we provide all the decompositions of the EH action and mention its significance briefly. In section 3 we describe the consequence (in EH action) when the constant parameters in the metric is promoted to a time-dependent variables in SSS metric. In section 4, we obliterate the time dependence and check whether the EH action reduces to its original value. Finally, in section 5 we provide a comparative discussion on coordinate transformation and dynamic upgradation. The conclusion of the analysis is provided in section 6.
Mathematical background: Avatars of Einstein-Hilbert action
The starting point of the present analysis is the Einstein-Hilbert (EH) action, which is given as
A EH = ν √ −g Ld 4 x = 1 16π ν √ −g Rd 4 x .(1)
Not only the above action describes the dynamics in the curved spacetime, several other features of gravity are also encoded in the above action. To explore those features, the above form of the action is expressed in several ways. In this section, we briefly discuss about those several forms of the action, their implications, and, make a comparative analysis of those different forms of the action. The Einstein-Hilbert action (1) contains the first-derivative as well as the secondderivative of the metric tensor g ab . As a result, the principle of extremal action becomes ill-defined when it is followed from the action (1). This is because, in that case, one has to fix both the metric and its first-derivative on the boundary. To resolve this issue, there are two different routes. The first one is the addition of a suitable surface term with the action (1) so that its variation cancels the aforementioned first-derivative, which were required to be fixed on the boundary. The most popular surface term in the literature is the Gibbons-Hawking-York (GHY) boundary term [2][3][4]. On a surface, which has normal u a , the GHY term is defined as
A GHY =¯ 8π ∂ν h ⊥ Kd 3 y ,(2)
where,¯ = u i u i = ±1 (+1 for time-like and −1 for space-like surface respectively), h ⊥ is the determinant of the induced metric on the surface and K = −∇ i u i is the trace of the extrinsic curvature on the surface. The above surface term, can be equivalently written as a bulk term as
A GHY = (1/8π) ν √ −g ∇ i (Ku i )d 4
x which, by using Gauss' theorem, boils down to Eq. (2). For a space-like hypersurface, we denote the normal as u i ≡ n i , h ⊥ ≡ h and K = −∇ i n i (this notational convention we shall follow during the discussion of ADM decomposition). Usually, people add the GHY term (2) with the action (1) and show that the problematic part (i.e. the first order derivative of the metric, which were required to be fixed on the boundary) gets cancel with the variation of the GHY term. However, it has be systematically proved by Padmanabhan [12] that if one starts from the Einstein-Hilbert action (1), one needs to add the GHY term (2) in order to make the principle of extremal action well-defined. More importantly, it can also be shown that all components of the metric are not required to be fixed on the boundary. For instance, for t =constant boundary, only the spatial part of the metric (i.e. g αβ , where the Greek indices α, β etc. indicates the spatial components) are required to be fixed, whereas the components g 00 and g 0α can be left free.
Although, the GHY boundary term is a popular boundary term, it cannot be applied for the null surfaces as the construction of the GHY term requires non-degenerate induced metric on the boundary and unit normals. Thus, the definition of a suitable boundary term for a null surface is more non-trivial, which has been resolved from the works of Padmanabhan and collaboration, who defined a proper counter term on the null surface as 2 √ q(Θ+κ) [13], where Θ is the expansion parameter of the null-geodesics on the null surface, q is the determinant of the two-metric on the null surface and κ is the surface gravity. This has been further generalized, in order to make it applicable for both null and non-null surfaces [14][15][16][17]. Above, we have discussed the procedure of adding a proper counter term which cancels the problematic term on the boundary while taking variation of the EH action. In literature, there is another procedure, which we discuss in more detail in the following. There are several prescriptions in literature to decompose the EH action into the bulk and the surface part. In the following, we shall discuss those decompositions one-by-one along with their significances.
Decomposition I
The above action (1) can be decomposed into the two parts: one is the quadratic part, which contains the first derivative of the metric tensor g ab and another one is the surface part, which is a total-derivative term and contains the second derivative of g ab (extensively discussed in the chapter 6 of [18]). It can be shown that this decomposition helps us to obtain the Einstein's equation in a well-defined manner in two different ways. 1. One can drop the surface part of the action (as it is a total derivative term and, hence, presence of this part does not alter the dynamics of the system) and the equation of motion is obtained only from the bulk part. 2. One can define proper conjugate variables and can obtain Einstein's equation from the whole EH action by fixing the conjugate momenta on the boundary. The decomposition of the Einstein-Hilbert action, in terms of the quadratic part and the surface part, is described as follows.
16π
√ −g L = √ −g R = √ −g L quad + L sur .(3)
The quadratic term, containing the first-order derivative of the metric tensor, is given by the following expression.
L quad = g ab Γ i ja Γ j ib − Γ i ab Γ j ij = 2Q bcd a Γ a dk Γ k bc ,(4)
where
Q bcd a = ∂R ∂R a bcd = 1 2 (δ c a g bd − δ d a g bc ) ,(5)
which has the same symmetric/anti-symmetric property as of the Riemann tensor due to the exchange of indices. The surface part of the Einstein-Hilbert action, as mentioned in Eq. (3) is given as
L sur ≡ ∂ c ( √ −g V c ), where V c ≡ (g ik Γ c ik − g ck Γ m km ) = − 1 g ∂ b (gg bc ) ,(6)
where g is the determinant of the metric tensor g ab . Since all the second order derivatives of the metric is within L sur , which is a total derivative term, one can obtain the Einstein's equation from the bulk part only (see chapter 6 of [18]). Thus, this peculiar structure of EH Lagrangian helps us to formulate a well-posed action principle. Remember, that the decomposition of the Einstein-Hilbert action in terms of the quadratic part (also known as the bulk Lagrangian) and the surface part is not a covariant one and the quadratic and the surface Lagrangian are not covariant scalars. Interestingly, the quadratic and the surface Lagrangian are not independent of each other. Instead, those are connected to each other by the following relation, which is known as the holographic relation in the work of Padmanabhan et. al. [5][6][7][8][9][10]19], which is given as
L sur = −∂ c g ab ∂( √ −g L quad ) ∂(∂ c g ab ) .(7)
The above holographic relation also implies that the gravity is intrinsically Holographic as the surface degrees of freedom are related to the dynamical degrees of freedom . The above connection of the quadratic and the surface Lagrangian helps us to interpret the Einstein-Hilbert Lagrangian as the Lagrangian of the momentum space by the following analogy with the classical mechanics. Consider the Lagrangians L 1 ≡ L 1 (q A , ∂q A ) and L 2 ≡ L 2 (q A , ∂q A , ∂ 2 q A ). If L 1 and L 2 are connected by the relation
L 2 (q A , ∂q A , ∂ 2 q A ) = L 1 (q A , ∂q A ) − ∂ i (q A P i A ) (where, the canonical momenta P i A is de- fined as P i A = ∂L 1 /∂(∂ i q A ))
, it can be shown that L 1 and L 2 yields the same equation of motion. However, when the Lagrangian is L 1 , one has to fix q A on the boundary whereas, when the Lagrangian is L 2 , one has to fix P i A on the boundary. It can be shown that the Einstein-Hilbert Lagrangian L corresponds to L 2 (q A , ∂q A , ∂ 2 q A ) and the quadratic Lagrangian √ −g L quad corresponds to L 1 (q A , ∂q A ) of the above discussion. Also, ∂( √ −g L quad )/∂(∂ c g ab ) = √ −g M cab corresponds to the canonical momentum of g ab . Similarly − √ −g M a bc corresponds the momentum of g bc (for detail discussion, see [20]). Importantly, one can either obtain equation of motion from the L quad only by fixing g ab at the boundary, or one can take the whole Einstein-Hilbert Lagrangian and obtain equation of motion by fixing √ −g M c ab at the boundary. Thus, g ab and √ −g M c ab act as a pair of holographically conjugate variables (HCV's) [20]. Apart from g ab and √ −g M c ab , another pair of HCV's can be introduced as f ab and N c ab , where f ab = √ −gg ab and N c ab is the conjugate momentum (see [20], also see [21] for the generalization to Lanczos-Lovelock gravity). These definitions also interprets the EH action as the action of momentum space. In addition, these definition of f ab and N c ab also help us to obtain the field equations in a similar way as of the Hamilton's equation of classical mechanics i.e. ∂ c f ab = ∂H/∂N c ab and p c N c ab = −∂H/∂f ab , where H represents the suitably defined Hamiltonian [20,21].
Although this method of decomposition helps us to define a well-posed action principle and to illuminate the holographic nature of gravity, this method is not a covariant one. In the following, we shall discuss about other decompositions which are covariant but also are foliation dependent.
Decomposition II
A proper canonical formalism of general relativity is provided by the ADM (named after Arnowitt, Deser and Misner) formalism [22], which depends on the foliation of the spacetime (also see [14,18]). In this case, it is considered that the spacetime is foliated by the family of space-like surfaces with time-like normal n a (which is normalized i.e. n a n a = −1) and induced metric on the spacelike surface is defined as h ab = g ab + n a n b . Then, the Einstein-Hilbert action can be written in terms of the ADM Lagrangian as
16π √ −g L = √ −g R = √ −g L ADM + L (ADM ) sur ,(8)
where, the ADM Lagrangian L ADM is given by
L ADM = (3) R + K ab K ab − K 2 ,(9)
and the surface term L (ADM ) sur is given as
L (ADM ) sur = −2 √ −g ∇ i (Kn i + a i ) .(10)
Here (3) R is the three-Ricci scalar corresponding to the metric h αβ (where α, β denote the spatial indices); K ab is the extrinsic curvature tensor of the spacelike surface, which is given by K ab = −h m a h n b ∇ m n n ; K is the trace of K ab and a i = n a ∇ a n i . Again, in this case, one can obtain Einstein's equation either from the bulk ADM action (where h αβ are required to be fixed on the boundary) or from the total action (by fixing the corresponding momenta of h αβ on the boundary). This method is also important because this provides the Hamiltonian formalism of general relativity, which plays an important role in canonical quantum gravity and numerical relativity.
As we have discussed, one can obtain the Einstein's equation from the bulk part of these decompositions (L quad and L ADM ). Whereas both L sur and L (ADM ) sur are the surface terms,Presence of which does not alter the dynamics of the system. However, presence of these surface terms in the action (in both the decomposition) interprets the EH action as the action of the momentum space as one has to fix the corresponding momentum in the boundary. However, note that L sur and L (ADM ) sur are not equal and, therefore, L quad = L ADM . The spacetime foliated by t =constant hypersurfaces are properly described by the following metric:
ds 2 = −N 2 dt 2 + h αβ (dx α + N α dt)(dx β + N β dt) .(11)
For this metric, one can obtain
L sur − L (ADM ) sur = √ −g (L ADM − L quad ) = ∂ 0 √ h N (∂ α N α ) − ∂ α √ h N (∂ 0 N α ) −∂ α N √ h ∂ β (hh αβ ) + ∂ α √ h N N β ∂ β N α − N α ∂ β N β + √ h N (∂ α N β )(∂ β N α ) − (∂ α N α )(∂ β N β ) . (12)
Also note that the GHY term is not the same as either of L sur or L (ADM ) sur
. The GHY Lagrangian, on a space-like surface can be written as 16πL GHY
= 2∇ i (Kn i ) (where A GHY = ν √ −g L GHY d 4 x)
and for the general metric (11), it can be shown that it is not the same as the L sur or L (ADM ) sur .
Decomposition III
There is another way of decomposing the Einstein-Hilbert Lagrangian with the help of the normal n a , which is given by [8,18] 16π
√ −g L = √ −g R = 2 √ −g (G ab − R ab )n a n b(13)
From the Gauss-Codazzi relations we obtain
2G ab n a n b = (3) R − K ab K ab + K 2 .(14)
When Einstein's equation holds, 2G ab n a n b = 16πρ can be identified as the numerical value of ADM Hamiltonian density. Therefore, it can be considered as the energy density. Now, R ab n a n b is given by
R ab n a n b = ∇ i (Kn i + a i ) − K ab K ab + K 2 .(15)
In a static spacetime, above decomposition of the Einstein-Hilbert action (as mentioned in (13)), has a particular significance from thermodynamic viewpoint. It interprets the Einstein-Hilbert action as the free energy of the spacetime [8]. The near-horizon geometry of a static spacetime can be described by the metric [23]
ds 2 = −N 2 dt 2 + dl 2 + σ AB dx A dx B ; N = κl + O(l 3 ); σ AB = µ AB (x A ) + O(l 2 ) ,(16)
where, the location of the horizon is given by l = 0 ; A, B etc. are the indices of the two surface and κ is the surface gravity. For the above metric (16) (in general, it is true for any arbitrary static metric, expressed in coordinate system with no shift, i.e. for N α = 0) one can obtain that all the components of K ab vanishes. Therefore, one finds
R ab n a n b = −R 0 0 = ∇ i a i = 1 √ −g ∂ α ( √ −g a α ) ,(17)
which implies that R 0 0 is a total derivative term. In addition, the fact, that R 0 0 is a total derivative term in a static spacetime, can also be proved generally in a metric independent way. For static spacetime, one can consider the presence of time-like Killing vector ξ a = {1, 0, 0, 0}. Therefore, one can obtain [8]
R a i ξ i = R a 0 = ∇ b (∇ a ξ b ) = 1 √ −g ∂ b (∇ a ξ b ) .(18)
It yields that all the components of R a 0 are total-derivative term in static spacetime and, the expression of R 0 0 is given as
R 0 0 = ∂ α ( √ −g g i0 Γ α i0 )/ √ −g .
Therefore, for any static metric with no shift (such as (16)), a α = −g 0i Γ α 0i ). It is also noteworthy that for any static spacetime, expressed in coordinate system with no shift, one obtains
L (ADM ) sur = −2 √ −g R ab n a n b = 2 √ −g R 0 0 = −2 √ −g ∇ i a i = 2∂ α ( √ −g g i0 Γ α i0 ) . (19)
In that case (i.e. in a static spacetime with no shift), one also finds L ADM = 2G ab n a n b . However, even that case, L sur does not coincide with either L (ADM ) sur or with −2 √ −g R ab n a n b . Interestingly, in spacetime (16), L sur , L (ADM ) sur and −2 √ −g R ab n a n b -all of the surface terms have the thermodynamic interpretation as the entropy on the horizon surface (l = 0). Also, in any spacetime, 2G ab n a n b can be identified as the energy density, which we have discussed earlier. Therefore, the decomposition of Einstein-Hilbert action, as stated in equation (13), interprets the Einstein-Hilbert action as the free energy of the spacetime [8,19] (i.e. A EH ≡ βE − S), where the integration containing G ab n a n b can be interpreted as βE (with β being the periodicity of Euclidean time, which is identified as inverse temperature) and the integration containing R ab n a n b can be interpreted as the entropy. Furthermore, when A sur = L sur d 4 x (where L sur has been defined while discussing "Decomposition I") when integrated over the horizon of a static metric, provides τ T S, where τ is the range of the time-integration, T is the Hawking temperature and S is the Bekenstein-Hawking entropy [10,24]. This leads to the further definition of "surface Hamiltonian" as H sur = −∂A sur /∂τ = T S [24] . In addition, the term pdq of "Decomposition I" (i.e. N c ab δf ab or √ −gM c ab δg ab ) integrated over the transverse surface (for the general static metric (16)) has the thermodynamic interpretation of T δS, while qdp (i.e. f ab δN c ab or δ( √ −gM c ab )g ab ) has the thermodynamic interpretation of SδT [20] (also see [21] for the generalization to Lanczos-Lovelock gravity). Also, since the total EH action can be expressed as the free energy of the spacetime, the action principal can be interpreted as the thermodynamical extremum principal (for example, see ref. [8,20]).
Note that both "Decomposition II" and "Decomposition III", which we have discussed above, crucially depends upon the timelike (or spacelike) nature of the boundary surface and, these are not adapted for the null-surfaces. For Hamiltonian formulation on the null-surfaces, see [25,26]. Earlier, we have commented that, in static spacetime, the surface parts of each decomposition (including "Decomposition II" and "Decomposition III") have the thermodynamic interpretation as the entropy on the horizon surface (and the bulk as thermodynamic energy). Now, the horizon is a null surface in general. Nevertheless, the thermodynamic interpretation of the bulk and the surface terms remain valid (including in the case of "Decomposition II" and "Decomposition III"). This is because, the black hole horizon in static spacetime is also the Killing horizon (as per Hawking's rigidity theorem). The normal of a Killing horizon, i.e. the Killing vector, is usually non-null everywhere except on the horizon. Thus, "Decomposition II" and "Decomposition III" is valid for the Killing normals and can have thermodynamic interpretations on the horizon.
Before moving on to the next part, we mention that the similar decompositions and their connection with each other of the Einstein-Hilbert action has been presented earlier in literature (for example see [14,20] etc.). So far, we have mentioned how the Einstein-Hilbert action is decomposed in several ways and also have mentioned about the implications of each form of the action. In the following section, we shall investigate what happens when the constant parameters in a static spacetime is promoted to timedependent variables.
Dynamic upgradation of metric parameters
For static metric, all the time-derivatives vanish and all the expressions (Einstein-Hilbert action and its several decomposed forms) appear in terms of the spatial derivatives. Therefore, one can think that these static metric can be viewed as a limiting case of the time-dependent metric. From now onward, we want to probe how the expression of the EH action of a time-dependent metric differs from its static counterpart. Also, it will help us to find whether it is a right way to think the static metric as a limiting case of the time-dependent one. To analyse these, the method we adopt is the following, which we call as the dynamic upgradation of metric parameters. In general, the spacetime metric contains several parameters. It is therefore, interesting to ask that what will happen if we promote those constant parameters to the time-dependent variables. This will allow us to introduce the time-dependence in the static spacetime. In addition, it will allow us to check the consequence in the limit where the promoted parameters are reduced to the constants again (i.e. the metric again becomes static). Note, here we are interested in the level of EH action (or the reduced EH action) and we do not bother about the underlying dynamics (or EH equation). In other words, the analysis which we follow here is off-shell.
Let us start from a particular example. A Schwartzschild metric is described by the metricḡ ab ≡ḡ ab (r, M ) (from now onward, a bar overhead will imply a static metric). If we promote M −→ M (t), the metric will be given as
ds 2 = g ab dx a dx b = − 1 − 2M (t) r dt 2 + dr 2 1 − 2M (t) r + r 2 dΩ 2 .(20)
The Ricci-scalar corresponding toḡ ab (r, M ) isR = 0. However, the same for the metric (20) is given by
√ −g R = d dt 2Ṁ r sin θ (1 − 2M r ) 2 .(21)
Clearly, the Schwartzschild metric and the metric mentioned in (20) describe different spacetimes with different geometries. Also, it must be mentioned that the source of the metric (20) is rather unphysical as the energy-momentum tensor (T ab ) corresponding to the metric has only non-vanishing components T θθ , T φφ and T tr . Since T tt and T rr vanishes, it corresponds to a system with zero energy and non-zero transverse pressure, which implies that the source is unphysical. However, since we are not concerned about solving Einstein's equation, we do not bother about it. On the other hand, another example of dynamic upgradation of mass, which lead to the physical solution, is the Vaidya metric [27]. Vaidya metric can be obtained in the following way. From Schwarzschild metric, one can make a coordinate transformation by replacing the (Schwarzschild) time coordinate with the advanced/ retarded null coordinate. Thereafter, the constant mass parameter is promoted to the function of the advanced/ retarded null coordinate. Physically, Vaidya metric represents spherically symmetric body radiating or absorbing null dusts.
Above conclusion (i.e. the reduced EH action becoming a total-derivative) can also be drawn for Reissner-Nördstrom (RN) spacetime. If we take the RN metricḡ ab (r, M, Q) and promote the M and Q as M −→ M (t) and Q −→ Q(t), the metric will be given as
ds 2 = g ab dx a dx b = − 1 − 2M (t) r + Q(t) 2 r 2 dt 2 + dr 2 1 − 2M (t) r + Q(t) 2 r 2 + r 2 dΩ 2 . (22)
The Ricci-scalar of RN metricḡ ab (r, M, Q) isR = 0, whereas for metric (22) it is given by
√ −g R = d dt 2(Ṁ r − QQ) sin θ (1 − 2M r + Q(t) 2 r 2 ) 2 .(23)
So far, all the results are obtained for particular spacetimes such as Schwartzschild spacetime, RN spacetime. It can generally be proved that for a general static and spherically symmetric spacetime, one can obtain a similar conclusion. A general static and spherically symmetric (SSS) metric with constant parameters (κ's and λ's) is given as
ds 2 = −f (r, κ)dt 2 + dr 2 g(r, λ) + r 2 dΩ 2 .(24)
The Ricci-scalar corresponding to the metric (24) is given as
R = 1 2r 2 f 2 r 2 gf 2 − 4f 2 (−1 + g + rg ) − rf (rf g + 2g(2f + rf )) .(25)
Now, if we promote the constant parameters to time-dependent variables, one can obtain the metric as
ds 2 = −f (r, κ(t))dt 2 + dr 2 g(r, λ(t)) + r 2 dΩ 2 .(26)
The expression of Ricci-scalar corresponding to the metric (26) is provided as
√ −g R = √ −gR κ,λ−→κ(t),λ(t) − d dt r 2 sin θλ i (∂g/∂λ i ) √ f g 3/2 ,(27)
whereR| κ,λ−→κ(t),λ(t) implies the expression ofR (as provided in Eq. (25)) with κ's and λ's been promoted to κ(t)'s and λ(t)'s respectively. Therefore, one can conclude that the expression of Einstein-Hilbert Lagrangian differ by a total time-derivative term when the constant parameters of a static & spherically symmetric metric is promoted to time-dependent variables. Interestingly, as we discuss it in the later part of the section, the total derivative term (i.e. the last term of (27)) can be identified as −2 √ −g ∇ i (Kn i ) for the metric (26). Therefore, one obtains 1 16π
√ −g R = 1 16π √ −gR κ,λ−→κ(t),λ(t) − 1 8π √ −g ∇ i (Kn i ) .(28)
The last term of (28) can be identified as the negative of GHY Lagrangian on a spacelike hypersurface. Also, note that the difference in the Lagrangian, as it has been mentioned in (27), appears due to the fact that the constant parameters (λ's) in g(r, λ) are promoted to time-dependent variables and it does not depend on whether the constant parameters in f (r, κ) are promoted to time dependent variables. Therefore, in metric (24), if we promote κ's as κ −→ κ(t) and keep λ's unchanged, the extra total-time-derivative term in (27) will not appear and the Einstein Hilbert Lagrangian will be exactly same even after promoting κ's to time-dependent variables. Earlier, we have discussed how Einstein-Hilbert action can be decomposed in several forms. We shall now investigate how the decomposed components of Einstein-Hilbert action changes due to the dynamic upgradation of constant parameters. As we have discussed earlier, the quadratic part of Einstein-Hilbert Lagrangian plays the pivotal role while obtaining the Einstein's equation from the action principle. The quadratic part of the Lagrangian corresponding to the metric (24) is obtained as
L quad = 2g(f + rf ) r 2 f .(29)
After promoting constant parameters to time-dependent variables, one can obtain the same quantity corresponding to the metric as
L quad =L quad κ,λ−→κ(t),λ(t) ,(30)
which means that the quadratic part of the Einstein-Hilbert Lagrangian remains invariant even after upgradation of parameters to time-dependent variables in a SSS coordinate. Since the quadratic part remains unchanged, and the Einstein-Hilbert Lagrangian differ by a total time-derivative term (as mentioned in (27)), the surface part of the Lagrangian will also differ by the same total derivative term i.e.,
L sur =L sur κ,λ−→κ(t),λ(t) − d dt r 2 sin θλ i (∂g/∂λ i ) √ f g 3/2 .(31)
Again, if we upgrade only κ's to time-dependent variables and leave λ's unchanged, the surface term will also remain unchanged due to κ −→ κ(t).
Let us investigate now about the ADM decomposition of the Einstein-Hilbert action. If we compute the ADM Lagrangian, which is defined in (9), for the SSS metric (24), one can find, the expression of the ADM Lagrangian is given as L ADM = (3) R (due to the fact that all components of K ab = 0.) Since the expression of (3) R does not contain any time-derivative, its expression does not change when the parameters are promoted to time-dependent variables. On the contrary, when the parameters are promoted, the only non vanishing component of K ab (corresponding to the metric (26)) is K rr . As a result, K ab K ab − K 2 = (g rr ) 2 (K rr K rr − K rr K rr ) = 0 . Therefore, we obtain that the expression of the ADM Lagrangian is also unchanged when the parameters are promoted to time-dependent variables in a general SSS spacetime, i.e.,
L ADM =L ADM κ,λ−→κ(t),λ(t)
.
(32)
Following the same argument, as been provided for L sur , we obtain that the surface part of ADM decomposition (L (ADM ) sur ) changes in a similar way of L sur i.e.,
L (ADM ) sur =L (ADM ) sur κ,λ−→κ(t),λ(t) − d dt r 2 sin θλ i (∂g/∂λ i ) √ f g 3/2 .(33)
The expression of L (ADM ) sur is provided in Eq. (10). Now, the metric (24) is static with no shift. Therefore, as we have mentioned earlier, all the components of K ab and its trace vanishes (i.e. K ab = 0 and K = 0). As a result,L (ADM ) sur is given as L
(ADM ) sur = −2∂ α ( √ −g a α ) (also,L (ADM ) sur | κ,λ−→κ(t),λ(t) = −2∂ α ( √ −g a α )| κ,λ−→κ(t),λ(t)
). Now, for metric (26), the only non-vanishing component of K ab is K rr . Therefore, when the parameters are promoted to variables, K does not vanish any more. Therefore, from (10) and (33), we obtain d dt
r 2 sin θλ i (∂g/∂λ i ) √ f g 3/2 = 2 √ −g ∇ i (Kn i ) ,(34)
which we have used in (28). We also have shown how the Einstein-Hilbert Lagrangian is decomposed in terms of G ab n a n b and R ab n a n b in (13). We have mentioned that this decomposition, in static spacetime, enables us to interpret the Einstein-Hilbert action as the free energy of the spacetime, where the integration over G ab n a n b contributes as βE and the integration over R ab n a n b contributes as the entropy. Therefore, it motivates us to ask what happens in this decomposition of Einstein-Hilbert action if we promote the parameters to the time-dependent variables for a general SSS spacetime.
We find that G ab n a n b does not change if we promote the constant parameters to variables in a SSS spacetime i.e., G ab n a n b =Ḡ abn anb κ,λ−→κ(t),λ(t)
,
and following the same argument as earlier we obtain R ab n a n b =R abn anb κ,λ−→κ(t),λ(t)
+ d dt r 2 sin θλ i (∂g/∂λ i ) 2 √ f g 3/2 .(36)
Thus, while interpreting Einstein-Hilbert action as the free energy of SSS spacetime, the term which is interpreted as βE, will remain invariant but, the term which is identified as the entropy, will change when the constant parameters are promoted to time-dependent variables.
Let us now summarize all the results which we have obtained so far. The Einstein-Hilbert action can be decomposed in several forms. Each of the forms have different significances. Interestingly, for a general SSS spacetime, if we promote the constant parameters to time-dependent variables, one component in each of the decompositions remain invariant (such as L quad , L ADM and 2G ab n a n b ). On the contrary, the other component in each decompositions (and the Einstein-Hilbert action itself) differ by the same total time-derivative term. The total-time derivative can be identified as −2 √ −g ∇ i (Kn i ), which is equivalent to the negative of GHY surface term (considering the 16π factor properly) on a spacelike hyper-surface.
Once again, let us note that the difference in the expression of √ −g R, L sur , L (ADM ) sur and R ab n a n b appears only when the parameters of g(r, λ) (of metric (24)) are promoted to time-dependent variables. If the parameters of f (r, κ) are changed to time-dependent variables and the parameters of g(r, λ) are left unchanged, there will be no difference in the expression of √ −g R, L sur , L (ADM ) sur and R ab n a n b as well. In fact, it can generally be proved that for any diagonal metric (say described by the coordinates {t, x, y, z}) if the parameters of any one of the spatial metric components (i.e. one among g xx , g yy and g zz ) are promoted to time-dependent variables, it can be shown that L bulk and L
ds 2 = −f (x, λ)dt 2 + dx 2 + dy 2 + dz 2 ,(37)
and promote λ's to λ(t)'s it can be shown that neither √ −g R changes nor any component of the decomposition (such as L bulk , L etc.) remains unchanged (in this particular case, G ab n a n b = 0 and R = −2R ab n a n b , which remain unchanged when λ −→ λ(t)). Note that when f (x, λ) = (1 + αx) 2 or f (x, λ) = α 2 x 2 , it corresponds to the Rindler metric with vanishing Ricci-scalar. As the above analysis suggests, one can check that if one replaces the constant α by a time dependent function α(t), all the aforementioned comments will be valid and the spacetime still remains flat.
We particularly mention about the special case of SSS metric (24) when f (r, κ) = g(r, λ). In that case, the metric is given as
ds 2 = −f (r, λ)dt 2 + dr 2 f (r, λ) + r 2 dΩ 2 .(38)
Interestingly, for static metrics (37) and (38), the Einstein-Hilbert action is itself a totalderivative term. For metric (37), the expression of Einstein-Hilbert Lagrangian is given as 16π
√ −ḡL = √ −ḡR = −d/dx(f / √ f )
and for metric (38) √ −ḡR = d 2 /dr 2 (r 2 sin θ(1− f )). Therefore, for these metrics, if we dynamically upgrade λ's to time dependent as well as space-dependent function (i.e., λ(t, x) for metric (37) and λ(t, r) for metric (38) respectively), the Einstein-Hilbert action will be modified by extra space-derivative for metric (37) and by extra time-derivative plus a space-derivative for metric (38). 4 Obliterating the time-dependence from the metric parameters: Back to square one?
In several ways the geometry of spacetime can be changed significantly by a little modification in the metric. Dynamic upgradation of constant parameters in the metric is one such way, which we have discussed thoroughly in the previous section for the SSS spacetime. Another possibility can be the addition of extra parameters in the metric. In both the cases, the spacetime geometry will be different from the original spacetime.
Then the question arises, if we go from the transformed spacetime to the original one (by obliterating the time-dependence from the metric parameters or by setting the extra parameters zero), does the geometry of the transformed spacetime reduces to the geometry of the original one? We investigate it by studying the properties of the Ricci-scalar. We start this analysis with a simple example, which has been studied earlier in the literature [8,28]. We take the metric (37) with f (x, λ) = 2 + a 2 x 2 . When a = 0, it corresponds to the flat spacetime. However, when = 0, it corresponds to the flat spacetime in Rindler coordinates, which has horizon at x = 0 and R = 0 everywhere. On the contrary, if we account 2 term in f (x, λ) the corresponding expression of the Ricci-scalar is given as
R = − 2 2 a 2 ( 2 + a 2 x 2 ) 2 .(39)
Now, at −→ 0 limit, one expects to return back to the flat spacetime with R = 0. However, the expression of Ricci-scalar in (39) suggests that for −→ 0 one obtains
lim →0 R = −2δ(x 2 ) ,(40)
which means the Ricci-scalar vanishes everywhere except on the horizon. This phenomenon has been mentioned as the concentration of curvature (for details about this phenomenon and its resemblance with an electromagnetic phenomenon, see [8,28]).
There are other such examples, which are not yet highlighted in literature. Which we have discussed in the following. In Schwarzschild metric, one can introduce smallness parameter and write the spacetime metric as
ds 2 = − 1 − 2M r + 2 dt 2 + dr 2 1 − 2M r − 2 + r 2 dΩ 2 .(41)
Note that after introducing the parameter 2 , g tt = −g rr in the metric (41) 1 . However, when = 0, the above metric (41) boils down to the usual Schwarzschild one, in which case R = 0. However, the expression of scalar curvature corresponding to the metric (41) is given as
R = 2 2 r(1 + 2 ) − 2M 2 − 2M 2 r 2 r(1 + 2 ) − 2M 2 .(42)
It can be proved that
lim →0 R = 2 2 r 2 − 2M r 2 δ(r − 2M ) .(43)
Therefore, only last term becomes significant near r = 2M . When −→ 0, one finds that R −→ 0 everywhere except very near to the horizon. In figure 1, we describe the behaviour of the Ricci scalar R(as given in (42)) near the horizon r H = 2M . In [28], it has been mentioned that any metric, which can be approximated to the Rindler metric near the horizon, can have a diverging Ricci-scalar near the horizon when the smallness parameter goes to zero. This statement can be verified for Schwarzschild metric. It can be shown that, if we define l = r − r H (where r H = 2M is the horizon), in the limit l << r H (i.e. near the horizon) f (r) → h(l) where h(l) = 2al, with a = f (r H )/2 = 1/2r H . Thus, in this limit, the Schwarzschild metric takes the form of Rindler metric in the t − l coordinates. Therefore, adopting the similar prescription to that of [28] (where, h(l) is deformed as h(l) −→ 2 + h(l) 2 , see Eq. (19) of [28]), here we deform f (r) as f (r) −→ 2 + f (r) 2 . Thus, the asymptotic Schwarzschild metric looks
ds 2 = − 2 + (1 − 2M r ) 2 dt 2 + dr 2 2 + (1 − 2M r ) 2 + r 2 dΩ 2 ,(44)
which reduces to the Schwarzschild metric in the limit → 0. The Ricci scalar of the above metric is given as It can be checked that, away from the horizon (i.e. for r = 2M ), the Ricci scalar vanishes in the limit → 0. At the horizon (i.e. r = 2M ), it can be checked that the behaviour of R is given as
R = −4M 2 2 r 4 (1 − 2M r ) 2 + 2 3 2 − 2 1 − 2M r + 2 r 2 1 − 2M r 2 + 2 + 2 r 2 .(45)lim →0 R = − 1 4M 2 δ(1 − 2M r ) − 2M 2 + 1 2M 2 .
(46)
Thus, in the limit r = 2M , the first term of Eq. (45) is the leading order term, which diverges near the horizon. This justifies the statement of [28]. In addition, this also suggests that there can be many possible ways to obtain divergence in the Ricci scalar for the same metric, deformed by smallness parameters in different ways.
Now we come back to the time-dependent cases. We found that the divergence in the Ricci scalar also happens when we remove the time-dependence of the parameters. For instance, the Schwatzschild spacetime is Ricci flat. If we promote the parameter M −→ M (t), the expression of Ricci-scalar is given a total derivative term as given by Eq. (21), which can be expressed in a more convenient form as
R = 8rṀ (t) 2 (r − 2M (t)) 3 + 2rM (t) (r − 2M (t)) 2 .(47)
For simplicity, first we consider
M (t) = M exp( t) ,(48)
where M is independent of t and is the smallness parameter. If = 0, the metric will boil down to the Schwarzschild one. If one considers M (t) of the form given in (48), the expression of the Ricci-scalar will be given as
R = 8r 2 M 2 exp( t) (r − 2M exp( t)) 3 + 2r 2 M exp( t) (r − 2M exp( t)) 2 .(49)
When r = 2M , one obtains R = 0 for = 0 . However, near the horizon of static Schwarzschild black hole (i.e. r H = 2M ), one obtains
lim →0 R ∼ O( 2 ) r − 2M + O( ) 3(50)
Therefore, near the horizon, the Ricci-scalar appears to be a diverging quantity. However, this divergence on the horizon is not exactly similar as it has been the case in Eq.
(39). This is because, the denominator in Eq. (50) changes its sign on the point r = 2M for −→ 0. In this case, the Ricci-scalar shows infinite discontinuity on the horizon. In figure 2 we have shown the behaviour of the Ricci scalar (as given by Eq. (49)) near the horizon. Although the infinite discontinuity of curvature has been shown for M (t) having a particular (exponential) time dependence (as shown in Eq. (48)), this phenomenon is not limited to that time-dependence only. In general, the smallness parameter should be associated with time in such a way that when = 0, M ( = 0, t) = M , where M is the mass of the static Schwarzschild black hole. Also, we must have M (n) ≤ O( ), ∀n ∈ N; where M (n) corresponds to the n-th order derivative of M (t) with respect to t. This is because, when = 0, M (n) should vanish. Now, for a general M (t) ≡ M ( , t) can be expanded in terms of Maclaurin's series as
M ( , t) = M + ∞ i=1 i m i (t) ,(51)
where
m k (t) = 1 k! ∂ k M ( , t) ∂ k =0(52)
Therefore, from (51), we obtaiṅ
M ( , t) = ṁ 1 (t) + O( 2 ) + ... ,(53)
andM
( , t) = m 1 (t) + O( 2 ) + ....(54)
Note that (53) and (54) suggest that M (n) vanishes at the limit → 0. Therefore, generally, we haveṀ ( , t) ∼ O( ) and the expression of R (given in (49)) boils down to the expression of (50). Hence, for a general M ( , t), it can be concluded that the nearhorizon expression of the Ricci-scalar is given by the form (50) and the near horizon behaviour of the Ricci-scalar is given by the figure 2. This phenomena can be generalized to the spherically symmetric metrics. Consider the metric (38). When λ −→ λ(t), the scalar curvature is obtained as
R =R λ−→λ(t) + R ,(55)
whereR corresponds to the Ricci-scalar of static and spherically symmetric metric (38) and
R = − d dt 1 f 2 λ ∂f ∂λ = −λ ∂f ∂λ +λ 2 ∂ 2 f ∂λ 2 f (r, λ(t)) 2 + 2λ 2 ( ∂f ∂λ ) 2 f (r, λ(t)) 3 .(56)
Now, we consider λ(t) contains the smallness parameter such a way that for = 0, λ(t) ≡ λ(t, ) = λ (where λ is the constant parameter before the dynamic upgradation) and λ (n) (t) ≤ O( ), ∀n ∈ N; where λ (n) (t) corresponds to the n-th order derivative of λ(t) with respect to t. Now, one can make Maclaurin series expansion of f (r, λ( , t)) as
f (r, λ( , t)) = f (r, λ) + ∂f ∂λ ∂λ( , t) ∂ =0 + O( 2 ) + ..(57)
Again, as we did in (51), we can make Maclaurin series expansion of λ(t) and generally show thatλ ∼ O( ). As a result, we obtain that for −→ 0, R vanishes everywhere except near the horizon (defined by f (r H , λ) = 0), where the expression is given as (the last term of (56) becomes the leading order term)
lim →0 R ∼ O( 2 ) (f (r, λ) + O( )) 3 ,(58)
which is a diverging quantity near the horizon. Also, since f (r, λ) changes its sign at r = r H , the near horizon behaviour of R should be the same as described in figure 2. For a general SSS metric of the form given by (24), the situation is a little different. When λ are κ are promoted to λ(t) are κ(t), the expression of √ −g R is given by
√ −g R = √ −gR| κ,λ−→κ(t),λ(t) + √ −g R , whereR
is given by (25) and √ −g R is defined in (27), which can be further expressed in a more convenient form as
√ −g R = − d dt r 2 sin θλ i (∂g/∂λ i ) √ f g 3/2 = r 2 sin θ λκ 2(f g) 3 2 ∂f ∂κ ∂g ∂λ + 3 2λ 2 √ f g 5 2 ∂g ∂λ 2 − 1 √ f g 3 2 λ ∂g ∂λ +λ 2 ∂ 2 g ∂λ 2 .(59)
For simplicity, we consider λ(t) = λ exp( t) and κ(t) = κ exp( t). With this, we can show, for −→ 0, R −→ 0 everywhere except those places where g(r, λ) = 0 and/or f (r, κ) = 0. For the static & spherically symmetric metric (24), g(r, λ) = 0 corresponds to the apparent horizon (defined by g rr | r=r AH = g(r AH , λ) = 0, where r AH stands for the radius of the apparent horizon). Near r = r AH , we can again obtain the Maclaurin series of g(r, λ(t)) and show that √ −g R diverges as
√ −g R ∼ O( 2 ) (g(r, λ) + O( )) 5 2 ,(60)
where we have considered the fact that f (r AH , κ) is non-zero finite. Note that in this case, R alone becomes a non-zero finite quantity i.e. lim r→r H R ∼ O(1) (again, considering f (r AH , κ) is non-zero finite). Therefore, we observe that if we put the smallness parameter →, the Ricci-scalar do not agree with its static valueR on the apparent horizon and there we found the divergence of Einstein-Hilbert Lagrangian. For metric (24), f (r, κ) = 0 corresponds to the Killing horizon (defined by g tt | r=r KH = f (r KH , κ) = 0, where r KH stands for the radius of the Killing horizon). In that case, for λ(t) = λ exp( t) and κ(t) = κ exp( t) one obtains R ∼ O(1) for r = r KH (considering g(r KH , λ) = 0). As a result √ −g R vanishes. Therefore, again we found that, if we put the smallness parameter = 0, the metric reduces to the static one. But the Ricci-scalar do not agree with its static valueR. However, the Einstein-Hilbert Lagrangian, unlike on the apparent horizon, agree to its static value on the Killing horizon.
Thus, in this section, we have found that even after obliterating the time dependence (by setting the smallness parameter associated with time to zero), the curvature and the EH action does not (usually) boil down to its static value. Instead, it becomes singular near the horizon. Before that, we have shown that even in static cases (Rindler and Schwarzschild metric), if one deform the metric with a smallness parameter, the curvature and the EH action becomes a Dirac-delta function on the horizon. This implies a permanent topological change which is irreversible. The cause of such peculiar behaviour is not known yet and should in investigated in future.
Coordinate transformation vs dynamic upgradation /reduction of parameters
The coordinate transformation does not change the geometry of the spacetime. On the contrary, as we have discussed earlier, the spacetime geometry indeed gets changed if one promotes the constant parameters to variables. Therefore, for a same metric expressed in different coordinates, if we promote the constant parameters to time-dependent variables, the change of Einstein-Hilbert action and its various components are not the same.
In the following, we provide some simple examples to demonstrate this in a more detail. A flat metric in Rindler coordinate is given as
ds 2 = −(1 + ax) 2 dt 2 + dx 2 + dy 2 + dz 2 .(61)
By a coordinate transformation (1 + ax) = ax, the metric (61) can be written alternatively as
ds 2 = −a 2x2 dt 2 + dx 2 + dy 2 + dz 2 .(62)
One can make another coordinate transformation (1 + ax) = √ 2al in metric (61) and obtain
ds 2 = −2aldt 2 + dl 2 2al + dy 2 + dz 2 .(63)
Among these three expressions of the Rindler metric, the first two, provided in Eqs.
(61) and (62), corresponds to the metric (37). Therefore, from our earlier analysis, we obtain that the spacetime is still flat if we promote a −→ a(t) in (61) and (62). In addition, various components of the decompositions of Einstein-Hilbert action will remain unaffected for a −→ a(t). Since, the metrics (61) and (62) are flat, even after promoting a −→ a(t), we provide the transformation relation of those coordinates with the Minkowski one (say, described by the coordinates T , X, Y , Z). The metric (61), with a −→ a(t), is related to the Minkowski coordinates as follows (see project 3.4 of [18])
X = sinh χ(t)dt + x cosh χ(t) , T = cosh χ(t)dt + x sinh χ(t) .(64)
Also, a(t) is given by a(t) = dχ/dt . When a is constant (i.e. in metric (61)), the transformation relations can simply be obtained from (64) as X = (1 + x) cosh(at) and T = (1 + x) sinh(at) .
The metric (62), with a −→ a(t), is related to the Minkowski coordinates as
X =x cosh χ(t) , T =x sinh χ(t) ,(65)
and a(t) = dχ/dt . When a is constant, the transformation relation can simply be obtained from (65) as X =x cosh(at) and T =x sinh(at) . On the other hand, the metric (63) is Ricci-flat. Now, if we promote a −→ a(t), the spacetime will no longer remain flat. One can obtain the expression of the Ricci scalar as
R = 2ȧ(t) 2 − a(t)ä(t) 2la(t) 3 = − ∂ ∂t ȧ(t) 2la(t) 2 .(66)
In addition, it can be shown that there will be a divergence of the Ricci-scalar (or EH action) near the horizon if the time dependence is removed by incorporating the smallness parameter. For example, if we consider a(t) is given as a(t) = a exp[ t], the expression of the Ricci scalar (given by the Eq. (66)), at the limit −→ 0, can be obtained as
lim →0 R = 2 2la ,(67)
which vanishes everywhere in the limit −→ 0 except very near the horizon 0 ≤ l < 2 (note that for the metric (63), the horizon is at l = 0). Thus, in the above, we have argued how the geometry of the same metric, expressed in different coordinates, changes after promoting the constant parameters to time-dependent variables. Note that in the above three metrics (61), (62) and (63) the time coordinate does not transform. Nevertheless, if the constant parameter a is promoted to a(t), the geometry of the spacetime changes in different way in different coordinates.
We provide another example in the following. We have mentioned earlier how the Ricci-scalar changes in Schwarzschild metric, when we promote M −→ M (t). Also we have mentioned how the curvature gets concentrated on the horizon if we remove the time-dependence using a smallness parameter. Now, the Schwarzschild metric can be expressed in several other coordinates. For example, we can express the Schwarzschild metric in the isotropic coordinate as
ds 2 = − 1 − M 2ρ 1 + M 2ρ 2 dt 2 + 1 + M 2ρ 4 (dρ 2 + ρ 2 dΩ 2 ) ,(68)
where the radial coordinate of Schwarzschild coordinate is connected to the same in isotropic coordinate as r = ρ(1 + (M/2ρ)) 2 . The Ricci-scalar of the metric (68), of course, vanishes. However, if we now promote M −→ M (t) in the metric (68), the expression of the Ricci-scalar turns out as
R = 12[Ṁ 2 (10ρ − 3M (t)) +M (4ρ 2 − M (t) 2 )] (2ρ − M (t)) 3 .(69)
Note that the expression of the Ricci-scalar, as provided in (69) Ricci-scalar corresponding to the metric (68) with M −→ M (t) is similar to that of the metric (20). The Schwarzschild metric can also be expressed in Painleve-Gullstrand coordinates as
ds 2 = − 1 − 2M r dT 2 − 2 2M R dT dr + dr 2 + r 2 dΩ 2 ,(70)
where T is connected to the Schwarzschild time coordinate t as
T = t + 4M r 2M + 1 2 ln r/2M − 1 r/2M + 1 .(71)
Again, the Ricci-scalar corresponding to the metric (70) vanishes. However, if we pro-mote M −→ M (T ), the corresponding expression of the Ricci-scalar is given as
R = 3Ṁ r 2 2M (T ) r .(72)
Here,Ṁ ≡ ∂M/∂T . Now, if we consider M (T ) = M exp( T ), we find that R vanishes everywhere (even on the horizon r = 2M ) when −→ 0. We end up this section by providing a more general analysis. A general SSS metric (of the form(38)) can be written in painleve-Gullstrand coordinates as
ds 2 = −f (r, λ)dT 2 − 2 1 − f (r, λ)dT dr + dr 2 + r 2 dΩ 2 ,(73)
where, dt = dT + ( 1 − f (r, λ)/f (r, λ))dr. Now, it is obvious that the expression of the Ricci scalars corresponding to the metrics (38) and (73) are the same. However, the interesting fact is that the quadratic part of the Einstein-Hilbert Lagrangian is also the same of the metrics (38) and (73). Therefore, the surface part of the Lagrangian is also the same for the metrics (38) and (73). We have earlier mentioned that the decomposition of the Einstein-Hilbert Lagrangian in terms of quadratic and the surface parts are not covariant one and the quadratic and the surface part are not covariant scalars. But, for any arbitrary static spacetime (described by the coordinates say {t, x α }) if we change the time coordinate by t = t +f (x α ) then, in new coordinate system {t , x α }, one can prove that the quadratic and the surface part of the Einstein-Hilbert Lagrangian remains invariant due to that specific coordinate transformation. The transformation from SSS spacetime (38) to the Painleve metric (73) is a special case of this result. The expression of the Ricci scalar corresponding to the metric (73) is the same as of the metric (38), which is given as
−ḡ (P G)R (P G) = d 2 dr 2 r 2 sin θ(1 − f )(74)
In the above equation, P G stands for painleve-Gullstrand coordinates. Now, if we promote the constant parameters λ's to λ(T )'s in (73), the corresponding Ricci-scalar can be obtained as
−g (P G) R (P G) = −ḡ (P G)R (P G) λ−→λ(T ) + 2∂ r ∂ t −g (P G) 1 − f ,(75)
where ∂ r ≡ ∂/∂r and ∂ t ≡ ∂/∂t. Again, in Painleve-Gullstrand coordinates, the change in Einstein-Hilbert Lagrangian comes as a total derivative term (can be expressed as either total space or total space (r) derivative) if we promote the constant parameters to the time-dependent variables. Therefore, the expression on the RHS of Eq. (72) can be expressed in terms of either total time or total space (r) derivative. Note that although the change in the expression of the EH action appears to be a total derivative term after the dynamic upgradation of the constant parameters (like the SSS metric case), once the time-dependence is obliterated, the expression of EH action (or the Ricci-scalar) boils down to its earlier expression (given by Eq. (74)) and there is no divergence in this case near the horizon. This can be shown from the following mathematical analysis. The extra term of (75) can be written explicitly as
2∂ r ∂ t −g (P G) 1 − f = −λ i sin θ √ 1 − f 2r ∂f ∂λ i + r 2 ∂ 2 f ∂r∂λ i + r 2 2(1 − f ) ∂f ∂λ i ∂f ∂r .(76)
In the above expression, the extra term is proportional toλ(t), which vanishes when we remove the time dependence (for example, if we consider λ(t) = λ exp t,λ(t) ∼ O( )).
On the other hand, denominator is non-zero finite near the horizon. Therefore, at the limit → 0, we obtain −g (P G) R (P G) −→ −ḡ (P G)R(P G) and the extra term (of Eq. (76)) vanishes. This is true for Eq. (72) as well i.e. R of Eq (72) vanishes for → 0. There is another difference of SSS and the metric in Painleve-Gullstrand coordinates is the following. For SSS metric (38), if we promote λ −→ λ(t), earlier we have found that the quadratic part of the Einstein-Hilbert Lagrangian remains invariant. However, for the metric in Painleve-Gullstrand coordinates (73), we obtain that the quadratic part of the Lagrangian is no more invariant when λ −→ λ(T ). It changes as follows
L (P G) quad =L (P G) quad λ−→λ(T ) − 2 ∂ ∂t √ 1 − f r .(77)
Finally, we comment on the Kruskal coordinates. The Schwarzschild metric in Kruskal coordinates (T , X, θ, φ) are given as [18]
ds 2 = 32M 3 r exp (−r/2M )(−dT 2 + dX 2 ) + r 2 dΩ 2 .(78)
where r is an implicit function of X and T determined by the following relation
r 2M − 1 exp (r/2M ) = X 2 − T 2 .(79)
Now, if we promote M → M (t), we obtain the expression of the Ricci-scalar as
R = 1 64M (T ) 6 r 2 M (T ) 2 r 3 e r 2M (T ) −∂ 2 T r + ∂ 2 X r + 6M − 2Ṁ 2 r 4 e r 2M (T ) +M (T )r 3 e r 2M (T ) M r + 2Ṁ ∂ T r − 6Ṁ 2 + 6M (T ) 3 r(∂ T r 2 − ∂ X r 2 +r ∂ 2 T r − ∂ 2 X r )e r 2M (T ) + 128M (T ) 6(80)
Since the denominator does not vanish when the time-dependence is removed by introducing smallness parameter, it can be said that the divergence of the Ricci-scalar does not happen in this case. Thus, in this section, we see that for the same metric when it is expressed in different coordinates, they imply same geometry. But, after the promotion of constant parameters to time dependent variables, they imply different geometry. This is because, the dynamically upgraded metrics are no longer connected by a coordinate transformation.
Conclusions
As it has been mentioned, the Einstein-Hilbert action is peculiar and different in many ways as compared to the actions of other fundamental theories. In the present paper, we have discussed the EH action and its different decompositions in great details and also have discussed some of the peculiar features of the EH action, which has not been reported earlier. Thus, the present paper can be viewed as a review of EH action along with some new findings.
In section 2, we have provided a concise review on various decomposition of the Einstein-Hilbert action and their underlying motivations. Here, we have discussed three ways of decomposing the EH action into the bulk and the surface parts and their connection with the gravitational thermodynamics. In addition, we also have discussed how the expressions of the bulk (or the surface) parts of these decompositions differ from one another for a given metric.
In section 3, we have discussed how the expression of the Ricci-scalar (or the EH action) changes when the constant parameters of a static spacetime metric are promoted to time-dependent variables, which we call as dynamical upgradation of the constant parameters. We have found that for static and spherically symmetric metric, after the dynamical upgradation, the bulk parts of each decomposition remain invariant, whereas the surface part (of each decomposition) change by a total time-derivative. Therefore, the expression of EH action (or the Ricci-scalar), as a whole, changes by the same total time-derivative. As it has been noticed, the invariance of the bulk part in each decomposition, is a feature of spherically symmetric metric (i.e. not general) as it no longer remains invariant for other coordinates (such as Painleve coordinate or in the isotropic coordinate). In this context, we also have obtained that for the metric of the form (37), if we dynamically upgrade the constant parameters (making them timedependent variable), the expression of scalar curvature (or the EH action) does not change at all. This has been explicitly shown for the Rindler metrics (61) and (62), where if we promote a −→ a(t), the Ricci scalar still vanishes. On the other hand, for the Rindler metric (63), the dynamic upgradation a −→ a(t) leads to the singular behaviour of the Ricci scalar when we remove the time-dependence introducing the smallness parameter. Furthermore, we have noticed that for the static and spherically symmetric spacetime metric, the dynamic upgradation of the parameters which are present in g rr (we have distinguished them as λ i 's) are responsible for the change of the Ricci-scalar in terms of a total time-derivative. Whereas, the dynamic upgradation of the parameters present in g tt , which we identify as κ i 's does not alter the expression of the Ricci scalar or its decompositions.
In section 4 and 5, we explore on the consequences after the removal of the timedependence on the dynamically upgraded parameters. Also, we discuss how the dynamic upgradation results differently for the same metric expressed in different coordinates. After the dynamic upgradation, we introduce a smallness parameter in such a way that when −→ 0, the upgraded parameter reduces to its constant value. However, we find that in the limit −→ 0, the curvature (or the EH Lagrangian) becomes singular near the horizon for a general spherically symmetric metric, Rindler metric (of form (63)) and for Schwarzschild metric in isotropic coordinates. This means that the change in geometry in the spacetime (due to the dynamic upgradation of the constant metric parameters) is permanent and not reversible even after setting the constant parameters to its constant value. We also have discussed a few examples in the static spacetime metrics, which are Ricci-flat (such as the Rindler metric, Schwarzschild metric). We deform those metrics by adding smallness parameters. Even in those cases, it was found that the expression of the Ricci-scalar does not vanish after setting the smallness parameter to zero and we obtain the singular behaviour of the scalar curvature. In addition, we have shown that for the Schwarzschild metric (which is also true for a general SSS metric), when it expressed in different coordinates, it respond differently due to the dynamic upgradation of the constant parameters. When the metric is expressed in Schwarzschild coordinate, the dynamic upgradation of parameters lead to the change in the Ricci-scalar by a total time-derivative. However, the bulk term of each decomposition remains unchanged. If the time-dependence is obliterated introducing a smallness parameter, we obtain singular Ricci-scalar near the horizon. When the Schwarzschild metric is expressed in terms of isotropic coordinate, the change in the expression of the Ricci-scalar is not a total derivative term. However, we obtain that the Ricci-scalar still diverges upon obliteration of the time-dependence by the smallness parameter. When the Schwarzschild metric is expressed in terms of the Painleve coordinate, the bulk part of the EH Lagrangian changes (due to the dynamic upgradation) and the change of the whole EH Lagrangian can be expressed in terms of either total space-or total time-derivative. However, in this case, there is no divergence in the Ricci scalar if we remove the time dependence introducing the smallness parameter. Also, it can be shown that the divergence in the Ricci-scalar is also not present for the Kruskal coordinates.
Finally, we make some comments on the divergence of the scalar curvature. Although the exact reason for the divergence of the Ricci-scalar might be revealed in the future work; nevertheless, some comments can be provided as follows. While, computing the Ricci-scalar we take derivative with respect to the proper time of a static observer at infinity. For a distant observer, the horizon is an infinite redshift surface, which implies that it takes infinite time (from the perspective of a distant observer) to see an infalling observer to cross the horizon. This may be the reason of divergence as the derivatives (such asṀ ) are computed with respect to the proper time of the distant observer. Moreover, the divergence in the Ricci-scalar may be a reflection of the coordinate singularity. All the metrics, for which we have obtained the divergence of the Ricci-scalar, have coordinate singularity on the horizon. In every case, when the divergence happens, the metric component which vanishes on the horizon, appears on the denominator. Therefore, one is required to explore further along this direction to determine precisely the reason for this divergence. May be, this singular behaviour might have some other significance. For example in the work of Padmanabhan [28], he has correlated this phenomenon with the Bohm-Aharanov effect. We hope to report soon in this aspect to obtain more insight in this topic.
R, L sur and L (ADM ) sur change by a total timederivative term. If the parameters in g tt are promoted to time-dependent variables and the parameters of the spatial components of the metric are left unchanged, the Einstein-Hilbert Lagrangian as well as the various components of decomposition (L bulk , L (ADM ) bulk , L sur , L (ADM ) sur etc.) remains unchanged. Therefore, if we consider the metric of the following form,
Figure 1 :
1Behaviour of Ricci scalar R (given by Eq. (42)) near the horizon. For simplicity, we have assumed M = 1 and = 10 −10 .
Figure 2 :
2Behaviour of Ricci scalar R (given by Eq. (49)) near the horizon. For simplicity, we have assumed M = 1, t = 1 and = 10 −10 .
Figure 3 :
3, does not boil down to the expression provided in (47) using the transformation relation between ρ and r. The horizon of the static metric (68) is located as ρ H = M/2. If one takes the form of M (t) of the form (48), one can again show that he Ricci-scalar everywhere when → 0 except on the horizon, where it diverges as R ∼ O(1/ ) . The near-horizon behaviour of the Ricci-scalar is shown infigure 3. As figure 3shows, the near-horizon behaviour of the Behaviour of Ricci scalar R (given by Eq. (69) with M (t) given by Eq. (48)) near the horizon. For simplicity, we have assumed M = 1, t = 1 and = 10 −10 .
If we consider −gtt = g rr = 1 − 2M/r + 2 , we obtain R = −2 2 /r 2 , which immediately vanishes when → 0 and, therefore, do not have any diverging behaviour near the horizon.
AcknowledgementI had started this work under the mentorship of Prof. T. Padmanabhan, who passed away after a few months after I had started to work with him. I sincerely acknowledge his guidance in this work. This work is dedicated to his memory.Data availability: This manuscript has no associated data or the data will not be deposited.
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| [] |
[
"Overlay Accuracy Limitations of Soft Stamp UV Nanoimprint Lithography and Circumvention Strategies for Device Applications",
"Overlay Accuracy Limitations of Soft Stamp UV Nanoimprint Lithography and Circumvention Strategies for Device Applications"
] | [
"P J Cegielski \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n\nRWTH Aachen University\n52074AachenGermany\n",
"J Bolten \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"J W Kim \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"F Schlachter \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"C Nowak \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"T Wahlbrink \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"A L Giesecke \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n",
"M C Lemme \nAMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany\n\nRWTH Aachen University\n52074AachenGermany\n"
] | [
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"RWTH Aachen University\n52074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"AMO GmbH\nOtto-Blumenthal-Str. 2552074AachenGermany",
"RWTH Aachen University\n52074AachenGermany"
] | [] | In this work multilevel pattering capabilities of Substrate Conformal Imprint Lithography (SCIL) have been explored. A mix & match approach combining the high throughput of nanoimprint lithography with the excellent overlay accuracy of electron beam lithography (EBL) has been exploited to fabricate nanoscale devices.An EBL system has also been utilized as a benchmarking tool to measure both stamp distortions and alignment precision of this mix & match approach. By aligning the EBL system to 20 mm × 20 mm and 8 mm × 8 mm cells to compensate pattern distortions of order of 3 µm over 6 inch wafer area, overlay accuracy better than 1.2 µm has been demonstrated. This result can partially be attributed to the flexible SCIL stamp which compensates deformations caused by the presence of particles which would otherwise significantly reduce the alignment precision. | 10.1016/j.mee.2018.06.004 | [
"https://arxiv.org/pdf/1808.01320v1.pdf"
] | 116,025,662 | 1808.01320 | f2548bb681d8466edd1c30f9a59b76c97e879078 |
Overlay Accuracy Limitations of Soft Stamp UV Nanoimprint Lithography and Circumvention Strategies for Device Applications
P J Cegielski
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
RWTH Aachen University
52074AachenGermany
J Bolten
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
J W Kim
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
F Schlachter
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
C Nowak
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
T Wahlbrink
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
A L Giesecke
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
M C Lemme
AMO GmbH
Otto-Blumenthal-Str. 2552074AachenGermany
RWTH Aachen University
52074AachenGermany
Overlay Accuracy Limitations of Soft Stamp UV Nanoimprint Lithography and Circumvention Strategies for Device Applications
1nanoimprint lithographye-beam lithographyoverlay accuracynanophotonics
In this work multilevel pattering capabilities of Substrate Conformal Imprint Lithography (SCIL) have been explored. A mix & match approach combining the high throughput of nanoimprint lithography with the excellent overlay accuracy of electron beam lithography (EBL) has been exploited to fabricate nanoscale devices.An EBL system has also been utilized as a benchmarking tool to measure both stamp distortions and alignment precision of this mix & match approach. By aligning the EBL system to 20 mm × 20 mm and 8 mm × 8 mm cells to compensate pattern distortions of order of 3 µm over 6 inch wafer area, overlay accuracy better than 1.2 µm has been demonstrated. This result can partially be attributed to the flexible SCIL stamp which compensates deformations caused by the presence of particles which would otherwise significantly reduce the alignment precision.
due to their flexibility, which can follow a target substrate's topography, e.g. bow and warp of a standard silicon wafer [5]. At the same time this flexibility can lead to significant distortions of the stamp and replicated patterns, potentially limiting alignment precision of imprint to imprint or mix & match lithography using soft UV-NIL.
These limitations of the alignment accuracy achievable with SCIL have not been fully assessed until
now. An alignment with overlay errors of 1 µm between two imprinted patterns by two different stamps was achieved by correction of a systematic error introduced by lateral displacement of the working stamp occurring during imprinting [6]. Such limited overlay precision reduces the usefulness of SCIL for micro-and nanoscale devices, despite its general potential for high resolution large area patterning. As a consequence, SCIL is currently mostly limited to large area patterning of periodic structures. The overlay precision of 1 µm demonstrated in the referenced work is thought to be limited by the mechanical precision of the mask aligner used for these experiments, which is approximately 1 µm [6]. In order to obtain a more complete picture of SCIL capabilities and limitations for multilevel patterning, we utilized mix & match lithography using SCIL and EBL. This approach ensures that alignment errors originate nearly exclusively from the SCIL process because EBL can achieve an alignment precision better than 5 nm, if the set of EBL alignment marks is well defined and the substrate is conductive [7]. Hence, errors introduced by the EBL system can be neglected compared to errors introduced by other process steps. Such evaluation is of great importance for broadening the application range of SCIL. Hence, in this work we explore those limitations and present one potential application for SCIL/EBL mix & match lithography in nanophotonics.
Experiment
Many EBL systems allow high precision automated marker detection at predefined positions.
This makes them ideally suited to acquire marker position data, which can be translated into a distortion map of patterns defined by a SCIL process. Furthermore, the marker position data can be compared with the expected marker positions and used to calculate both global and local rotation, scale and keystone corrections. These corrections enable the definition of patterns which partially compensate the positioning errors introduced by SCIL.
In order to evaluate the distortion of imprinted patterns, as well as EBL to SCIL alignment precision, a special layout was designed containing twelve 20 mm × 20 mm cells covering a full 6'' wafer area (Fig. S1 of Supporting Information). Each cell contained four 10 µm × 10 µm rectangular EBL alignment markers and nine evenly spaced Vernier scales for both X and Y axes, which allow reading the overlay error with 100 nm precision (Fig. 1). In addition, each 20 mm × 20 mm field contained four 8 mm × 8 mm sub-cells with individual sets of 4 EBL alignment markers and 9 Vernier scales. These sub-cells allowed to investigate whether a reduction of the cell size and hence a more localized alignment and compensation of local distortions of the pattern influences the alignment precision. Fig. 1 Vernier scale used to evaluate the alignment precision in Y direction. The left side was patterned by EBL and the right side by SCIL.
The fabrication of an imprint working stamp started with preparation of a Si master template. For this purpose a Si wafer was coated with PMMA resist, which was exposed with an e-beam and subsequently developed using a 7:3 mixture of IPA and water. The pattern was then transferred by inductively coupled plasma reactive ion etching (ICP RIE) with SF 6 and C 4 F 8 gases 100 nm deep into the master wafer. After resist removal, the obtained Si master was coated with antiadhesive C 4 F 8 polymer and ~1.5 µm of xPDMS. Afterwards xPDMS was joined with a support glass plate (Schott AF32) by a PDMS buffer layer. The resulting three layer SCIL stamp was cured for 48 h in 50 °C, followed by a release from the master.
EBL alignment marks and the right side of the Vernier scales ( Fig. 1) were defined in a single SCIL process. As the alignment precision of the EBL is highly depending on the quality of the markers, special care has been taken to ensure the definition of high quality and high contrast markers in the alignment test wafer. The marker features had to be transferred ~500 nm deep into silicon.
Therefore, SCIL was utilized to pattern a 40 nm thick highly selective Cr hard mask by a bi-layer resist lift-off process: AMONIL imprint resist was spun on a 50 nm thick PMMA layer. Imprinting was performed using a SCIL imprint tool based on the MA8 mask aligner series (SÜSS Microtec GmbH) with pressure of 30 mbar at a speed of the contact front of 1 s per groove. After imprinting and etching of the AMONIL residual layer, this sacrificial PMMA was etched in O 2 plasma exposing the surface of the Si wafer. Next, Cr was evaporated and the wafer was placed in acetone, which dissolved PMMA allowing a lift-off of the otherwise insoluble AMONIL and Cr layers. Afterwards the pattern was transferred by SF 6 and C 4 F 8 ICP RIE and the residual Cr mask was removed by wet etching using a commercially available mixture of perchloric acid and ceric ammonium nitrate. In a next step the second half of Vernier scales was defined in HSQ resist by EBL. The marker structures defined in the SCIL master and replicated by the SCIL process discussed above were used to align both sets of Vernier scales. The alignment precision was then measured by analyzing scanning electron micrographs of the Vernier scales.
Results and discussion
The EBL system successfully aligned to every design cell by the automated marker search routine. At the same time it provided exact data regarding the absolute and relative positions of the alignment markers defined by NIL. A few markers were not defined by SCIL due to imprint defects leading to uncharacteristically large alignment errors in some of the cells, which have therefore been excluded from further considerations. The data set was used to calculate the difference between the expected positions and observed positions of the markers with respect to a reference marker located close to the wafer center including corrections for the wafer rotation (Fig. 2). The markers displacements in both X and Y directions evidence a global compression of the imprinted pattern with a slope of 0.01µm/mm (Fig. S2 of Supporting Information), and reached 3 µm in the outermost design cells. Taking into the account that the stamp was cured at 50 °C and then cooled to room temperature it is clear that the imprinted pattern was scaled down with an effective thermal expansion coefficient (α) of 0.33 ppm/K (in agreement with previously reported value [8]), which is two times smaller than the actual difference between α of Si master wafer (2.6 ppm/K) and AF32 stamp support glass (3.2 ppm/K). The marker displacement reached 3 µm and was 1.65 µm on average therefore it could not be explained only by the thermal mismatch of Si and AF32 glass, which could distort the pattern only by 600 nm over 60 mm distance. Hence, contributions of the imprint process must be accounted for. 20 mm × 20 mm cells dimensions differed from the nominal values (ΔX and ΔY) by max ± 3µm.
The average ΔX was close to 0 with σ of 1.5 µm while in Y the cells were mostly stretched (average ΔY of 0.54 µm) and the error was more systematic indicated by lower σ = 1 µm. In The pattern distortions can be divided into 3 categories depending on their source: i) displacement due to the thermal contraction of the stamp after curing, ii) stretching of cells in the Y direction due to imprinting and iii) random, local cell deformations in X and Y. Marker displacement was also measured on the Si master wafer, resulting in an average value of 115 nm (Fig. S3). This confirms that discussed displacements are not present in the initial master wafer and mainly originate from the imprinting process. x,x -marker not found Fig. 3 Map of alignment error extracted from the Vernier scales of the right most upper 20 mm × 20 mm cell (red) including 4 8 mm ×8 mm sub cells (black). Arrows (scaled 100 000 times indicated by 300 nm scale arrow) illustrate the direction and magnitude of misplacement of pattern defined by EBL relative to pattern defined by NIL. A map of the full wafer is available in the supporting information in the figure S4.
x x Y[mm] X [mm] 3 µm imprinting direction x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x .
The statistical analysis of the measured overlay error obtained from the cells, in which alignment markers were found by EBL system (Fig. 4(a)) shows a standard deviation of σ = 361 nm and σ = 381 nm for X and Y directions for 20 mm × 20 mm cells, with a small offset of 120 nm in X direction and 43 nm in Y direction. Reducing the cell size to 8 mm × 8 mm resulted in an overlay error σ = 376 nm in X and σ = 322 nm in Y directions, with only slight offsets of 6 nm in X and -64 nm in Y direction ( Fig. 4 (b)). The 3σ value, which corresponds to probability of 99.7 %, is in the range of 1.2 µm.
The alignment precision was similar in both X and Y directions and also across the wafer indicating that the systematic pattern distortions, caused by thermal effects and those related to the imprint direction, were well compensated by the EBL system. Therefore the errors that could not be corrected must have their source in local and nonlinear pattern distortions. Particle contamination was initially expected to be the cause of those deformations. However, it was observed that errors red out from Vernier scales which were located directly next to particles (Fig. S6 of Supporting Information), which locally deformed the stamp preventing contact with the substrate, were often very low. Hence, the flexible SCIL stamp is capable of compensating the particle contamination to maintain the alignment precision.
Device application
Nanophotonic devices based on planar waveguides often require patterning with ~100 nm resolution to ensure critical dimension control, which is important for correct realization of optical functions. In many applications the waveguide layer, which is patterned first, is also the only layer requiring high patterning precision and additional layers such as metallization (e.g. electrodes [9]) have relaxed requirements. Both hold true for the demonstrator presented here. It consists of a silicon nitride double slot waveguide with minimum CD features of ~100 nm (width of the slots) and two large Al electrodes with a 4 µm wide gap, which can be used to control optical properties of an organic functional material deposited on the waveguide. The alignment tolerance for waveguides and electrode layers is ±1 µm, which ensures that light propagating in the waveguide cannot reach the electrode. The waveguides were patterned in a 250 nm thick Si 3 N 4 layer on 2.2 µm thermal SiO 2 on a 6'' Si handle wafer by using a 20 nm Cr hard mask patterned by lift-off with double AMONIL/PMMA resist, as described above. Aluminum electrodes were then patterned by liftoff utilizing a PMMA/MMA bi-layer resist stack patterned by EBL (Fig. 5). The alignment precision better than 0.8 µm was achieved by using 20 mm × 20 mm cells, which was within the devices tolerance. However for some cells and devices much better results were achieved; alignment error of order of 50 nm could be measured as can be seen in Fig. 5.
Conclusions
Large global distortions of the imprinted pattern were quantified and reached values as high as 3 µm caused by thermal expansion coefficient mismatch between the Si master and the stamp support glass and by stamp deformations during imprinting. Such distortions are strongly limiting the possibility of full wafer alignment without making corrections, such as those performed by EBL systems, which can efficiently compensate such systematic errors. Local distortions of the pattern evaluated by EBL exposure yielded 3σ of 1.2 µm. Since the systematic errors were corrected by the EBL system and EBL is capable of achieving 5 nm alignment precision the obtained 3σ value arises exclusively from nonlinear distortions of the imprinted pattern. Hence, 3σ of 1.2 µm is the practical limit of alignment precision of SCIL, which can be improved only by a careful optimization of the imprinting process.
If the alignment tolerance of a specific application allows SCIL can be used for high resolution patterning of the first layer to reduce the total cost per device, once the design is finalized so that fabrication of the imprint master is justified. Defining a second layer by EBL as a flexible and fast prototyping technique can then significantly speed up development cycles. This approach is particularly useful in nanophotonics prototyping, where a single waveguide design can be used as a basic building block which can be extended to different functional devices by adding further device features using other lithographic techniques such as EBL. S 6 A Vernier structure imprinted next to a particle. Around the particle the stamp could not contact the wafer surface leaving a bubble of ~ 100 µm radius. Despite that the alignment error measured by this Vernier scale was less than 100 nm in X and less than 200 nm in Y directions.
8 mm × 8 mm cells the max ΔX and ΔY were similar to maximal values observed in the larger cells; however the standard deviation was smaller by one third meaning those deformations have mostly local nature.
Fig. 2
2Displacement of EBL alignment markers of 20 mm × 20 mm (red) and 8 mm × 8 mm (black) cells defined by SCIL with respect to the reference marker (red box). The displacement vector tail marks the expected position and the arrow head marks the observed position. The vectors are scaled up by a factor of 4000 compared to the wafer position (indicated by the 3 µm scale arrow). The left most and right most cells were not fully imprinted because the SCIL tool could not provide full contact of the stamp with the wafer. 6''wafer is outlined with a black line.On the overlay error map of a cell located in the rightmost upper side of the wafer obtained by extracting the values from the Vernier scales(Fig. 3, full mapping for all 12 cells is available inFig S4of the supporting information) one can see that alignment precision was different for each 8 mm × 8 mm sub cell. Sub cells located in the left column showed a higher alignment error which was directly related to higher cell deformations than in the right column sub cells. On a scale of the entire wafer it can be seen that the alignment error is directly correlated to the average of |ΔX| and |ΔY| in each cell. The alignment error σ was growing by ~120 nm for every µm of average cell deformation(Fig. S5 of supporting information). No correlation was found when deformations in only X or Y directions were taken into account. Whether the cell was mostly stretched or compressed was not correlated with the alignment error as well.
Fig. 4
4Statistical distribution of alignment error measured with Vernier scales in (a) 20 mm × 20 mm cells (b) 8 mm × 8 mm.
Fig. 5
5Nanophotonic device with double slot silicon nitride (Si 3 N 4 ) waveguides fabricated by SCIL and EBL mix & match lithography.
of a 20 mm × 20 mm cell including 4 8 mm ×8 mm sub cells. Each cell contains 9 Vernier scales (Inset) for measuring alignment error in X and Y directions with 100 nm precision deviation (σ) of the alignment error in 8 mm × 8 mm cells vs. the average of absolute values of cell deformation in both X and Y directions. Solid lines are linear fits with slope of 166 nm and 132 nm per 1 µm.
S 4 Map of the overlay error of EBL structures aligned to pattern defined by SCIL using 20 mm × 20 mm alignment cells (red) and 8 mm by 8 mm sub cells (black). In the cell marked C33 alignment markers were not found by EBL system in each 3 out of 4 sub cells, therefore distortions were not properly compensated.Scaling factor: 8000 C33
. S Y Chou, P R Krauss, P J Renstrom, Science. S.Y. Chou, P.R. Krauss, P.J. Renstrom, Science (80-. ). 272 (1996) 85-87.
. J Haisma, J. Vac. Sci. Technol. B Microelectron. Nanom. Struct. 144124J. Haisma, J. Vac. Sci. Technol. B Microelectron. Nanom. Struct. 14 (1996) 4124.
. M Colburn, S C Johnson, M D Stewart, S Damle, T C Bailey, B Choi, M Wedlake, T B , M. Colburn, S.C. Johnson, M.D. Stewart, S. Damle, T.C. Bailey, B. Choi, M. Wedlake, T.B.
S V Michaelson, J G Sreenivasan, C G Ekerdt, Willson, SPIE Proc. Y. Vladimirsky379Michaelson, S. V. Sreenivasan, J.G. Ekerdt, C.G. Willson, in:, Y. Vladimirsky (Ed.), SPIE Proc., 1999, p. 379.
Substrate Conformal Imprint Lithography for Nanophotonics. M Verschuuren, M. Verschuuren, Substrate Conformal Imprint Lithography for Nanophotonics, 2009.
. R Ji, M Hornung, M A Verschuuren, R Van De Laar, J Van Eekelen, U Plachetka, M Moeller, C Moormann, Microelectron. Eng. 87R. Ji, M. Hornung, M.A. Verschuuren, R. van de Laar, J. van Eekelen, U. Plachetka, M. Moeller, C. Moormann, Microelectron. Eng. 87 (2010) 963-967.
. R Fader, M Rommel, A Bauer, M Rumler, L Frey, M Verschuuren, R Van De Laar, R Ji, U Schömbs, J. Vac. Sci. Technol. B, Nanotechnol. Microelectron. Mater. Process. Meas. Phenom. 31R. Fader, M. Rommel, A. Bauer, M. Rumler, L. Frey, M. Antonius Verschuuren, R. van de Laar, R. Ji, U. Schömbs, J. Vac. Sci. Technol. B, Nanotechnol. Microelectron. Mater. Process. Meas. Phenom. 31 (2013) 06FB02.
. J Bolten, N Koo, T Wahlbrink, H Kurz, Microelectron. Eng. 110J. Bolten, N. Koo, T. Wahlbrink, H. Kurz, Microelectron. Eng. 110 (2013) 224-228.
. R Fader, M Förthner, M Rumler, M Rommel, A J Bauer, L Frey, M A Verschuuren, J , R. Fader, M. Förthner, M. Rumler, M. Rommel, A.J. Bauer, L. Frey, M.A. Verschuuren, J.
M Butschke, E Irmscher, R Storace, U Ji, Schömbs, NNT 2014, 13th Int. Conf. Nanoimprint Nanoimprint Technol. Kyoto, JapanButschke, M. Irmscher, E. Storace, R. Ji, U. Schömbs, in:, NNT 2014, 13th Int. Conf. Nanoimprint Nanoimprint Technol., Kyoto, Japan, 2014.
. F Qiu, A M Spring, D Maeda, M Ozawa, K Odoi, A Otomo, I Aoki, S Yokoyama, Sci. Rep. 58561F. Qiu, A.M. Spring, D. Maeda, M. Ozawa, K. Odoi, A. Otomo, I. Aoki, S. Yokoyama, Sci. Rep. 5 (2015) 8561.
Measurement of positions of alignment markers on Si master with respect to a marker located in the bottom left corner. Initially only the rotation originating from placing the wafer in the wafer holder is visible. Red: Measured positions after subtracting rotation. The mean marker misplacement in X and Y is -167 and -344 nm while standard deviation is 115 nm for both axes. S 3 Black, Since mean values are greater than the standard deviation it is clear that the rotational error was not fully corrected and the real marker displacement is smallerS 3 Black: Measurement of positions of alignment markers on Si master with respect to a marker located in the bottom left corner. Initially only the rotation originating from placing the wafer in the wafer holder is visible. Red: Measured positions after subtracting rotation. The mean marker misplacement in X and Y is -167 and -344 nm while standard deviation is 115 nm for both axes. Since mean values are greater than the standard deviation it is clear that the rotational error was not fully corrected and the real marker displacement is smaller.
| [] |
[
"Courcelle's Theorem Made Dynamic *",
"Courcelle's Theorem Made Dynamic *"
] | [
"Patricia Bouyer \nLSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance\n",
"Vincent Jugé \nLSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance\n",
"Nicolas Markey \nLSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance\n\nLeibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum für Informatik, Dagstuhl Publishing\nIRISA\nCNRS & Inria & Univ\nRennes 1France, Germany\n"
] | [
"LSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance",
"LSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance",
"LSV\nCNRS & ENS Cachan\nUniv. Paris-Saclay\nFrance",
"Leibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum für Informatik, Dagstuhl Publishing\nIRISA\nCNRS & Inria & Univ\nRennes 1France, Germany"
] | [] | Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of model checking a fixed monadic secondorder formula over evolving subgraphs of a fixed maximal graph having bounded tree-width; here the subgraph evolves by losing or gaining edges (from the maximal graph). We show that this problem is in DynFO (with LOGSPACE precomputation), via a reduction to a Dyck reachability problem on an acyclic automaton.IntroductionMonadic second-order logic, tree-width of graphs, and Courcelle's theorem. Monadic second-order logic (MSO) is a powerful formalism for expressing and checking properties of graphs. It allows first-order quantification (over states of the graph) as well as monadic second-order quantification (over sets of states), and can thus express properties such as connectivity or 3-colorability of finite graphs. While the satisfiability problem of this logic is in general undecidable, model checking (i.e., deciding if a formula holds true in a given graph) is PSPACE-complete [25]; when the MSO formula is fixed, the problem is hard for every level of the polynomial hierarchy over finite graphs[24], and can be performed in PTIME over finite trees. The tree-width of graphs has been defined by Robertson and Seymour [21] as a measure of the complexity of graphs-intuitively, of how close a graph is to a tree. Many classes of graphs have been shown to have bounded tree-width[3]. Over such graphs, several (NP-)hard problems can be solved in polynomial time[3]. Model checking a fixed MSO formula is such an example, as proven by Courcelle in 1990 [5].Dynamic problems and dynamic complexity. In this paper, we focus on the dynamic complexity of MSO model checking over finite graphs. Dynamic-complexity theory aims at developing algorithms that are capable of efficiently updating the output of a problem after a slight change in its input[10,19]. Such algorithms would keep track of auxiliary information about the current instance, and update it efficiently when the instance is modified. Consider the problem of reachability in directed graphs, and equip such graphs with two operations, for respectively inserting and deleting edges (one at a time). It has recently been proven that this problem is in the class DynFO [8], which was a long-standing open problem. Roughly speaking, a problem is in DynFO when it admits an algorithm updating the solution and some auxiliary information through FO formulas (or, equivalently by AC 0 circuits satisfying some uniformity requirements) after a small change in the input.Our contributions.We study the MSO model-checking problem from a dynamic perspective, considering the following basic operations on graphs: insertion and deletion of an edge. * This work is supported by EU under ERC EQualIS (FP7-308087) and STREP Cassting (FP7-601148).We assume that we are given a maximal graph, which embeds all constructed game graphs along the dynamic process: this maximal graph represents the set of all possible connections in the subgraphs we will consider. We first realize that, since the MSO model-checking problem over arbitrary finite graphs is NP-hard (see[16,24]), the MSO model-checking problem over arbitrary maximal graphs is unlikely to be solvable in DynFO, even allowing PTIME precomputation (unless Dyn(PTIME,FO)=PTIME =NP).We therefore make a standard restriction and assume that the maximal graph has bounded tree-width. Under this hypothesis, we show that the MSO model checking over such graphs can be solved in DynFO with LOGSPACE precomputation. To obtain this result, we rely on (and extend) a DynFO algorithm for finding a Dyck path in an acyclic automaton[26], and build a transformation of our model checking problem into such a Dyck reachability problem. The latter transformation is performed by using Courcelle's theorem, and by realizing that the runs of bottom-up, deterministic tree automata can be computed step-by-step. These simple steps can be stored in an auxiliary graph, in which only few edges depend on the real edges that exist in the original game; the correctness of the construction then goes through the search for paths labeled with Dyck words.Related works.MSO has been extensively studied over various classes of structures in the last 40 years, both regarding its expressiveness and regarding the algorithmic properties of its decision problems (see[6]and references therein). Similarly, numerous measures of the complexity of graphs, such as tree-width [21], clique-width[7]or entanglement [2], have been defined and studied; they provide large classes of graphs in which different kinds of hard problems become tractable (see[20]and references therein).On the other hand, dynamic complexity is much less developed: while the main dynamic complexity classes were defined and studied 20 years ago[10,19], only few problems have been considered from that point of view[26,8,27]. As cited above, directed-graph reachability has recently been proven in DynFO, which was an important open problem in the area.Finally, let us mention that the results reported in this paper were originally presented in the setting of parity games played on a graph having bounded tree-width [4].DefinitionsMonadic second-order logic over graphsA graph is a pair G = V, E where V is a finite set of vertices, and E ⊆ V × V is a finite set of edges. The size of G is the cardinality of V . Formulas of the monadic second-order logic (denoted MSO) are built using first-order and (monadic) second-order variables, used respectively to quantify over vertices and sets of vertices; formulas may also use equality, and the edge relation E of the graph. As an example, connectivity of a graph can be expressed asThe standard static question regarding MSO over graphs is to decide whether a given formula is satisfiable, or to check whether it is satisfied in a given graph model. These problems have been extensively considered in the literature; in particular, satisfiability is undecidable [23], while model checking is PSPACE-complete [25]. We refer to [17] for a survey. P. Bouyer, V. Jugé and N. Markey 3Tree decompositionThe notion of tree decomposition[21,22]was introduced by Robertson and Seymour. It gives rise to classes of graphs on which many problems that are NP-hard in general become tractable.Definition 1. An ordered tree decomposition of G is a pair D = T , T , where T = N , E is an ordered tree, and T : N → 2 V is a function such that:for each edge e ∈ E, there exists a node n ∈ N such that e ∈ T(n) 2 ; for each vertex v ∈ V , the set N v = {n ∈ N | v ∈ T(n)} is non-empty, and the restriction of T to N v is connected. The width of D is defined as the integer max{|T(n)| | n ∈ N } − 1, and the tree-width of G is the least width of all tree decompositions of G.Tree automatonThe notion of (deterministic, bottom-up) tree automaton is a powerful tool for expressing and checking properties of finite trees.Definition 2.A tree automaton is a tuple A = Q, Σ, ι, Q end , δ where Q is a finite set of states, Σ is a finite input alphabet, ι ∈ Q is the initial state, Q end ⊆ Q is the set of accepting states and δ : Q 2 × Σ → Q is the transition function.Let T = N , E be a binary ordered labeled tree, with label set Σ. The run of A over T is the function ρ : N → Q such that:for every leaf n of T with label λ, we have ρ(n) = δ(ι, ι, λ); for every internal node n of T with label λ and with children m 1 and m 2 , we have ρ(n) = δ 2 (ρ(m 1 ), ρ(m 2 ), λ). If, furthermore, the run ρ maps the root of T to an accepting state q ∈ Q end , then we say that ρ is accepting, and that the automaton A accepts the tree T . | null | [
"https://arxiv.org/pdf/1702.05183v1.pdf"
] | 14,126,834 | 1702.05183 | ef923af4a425b4943ede7fdd046e57d5884ad9a1 |
Courcelle's Theorem Made Dynamic *
16 Feb 2017
Patricia Bouyer
LSV
CNRS & ENS Cachan
Univ. Paris-Saclay
France
Vincent Jugé
LSV
CNRS & ENS Cachan
Univ. Paris-Saclay
France
Nicolas Markey
LSV
CNRS & ENS Cachan
Univ. Paris-Saclay
France
Leibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum für Informatik, Dagstuhl Publishing
IRISA
CNRS & Inria & Univ
Rennes 1France, Germany
Courcelle's Theorem Made Dynamic *
16 Feb 2017
Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of model checking a fixed monadic secondorder formula over evolving subgraphs of a fixed maximal graph having bounded tree-width; here the subgraph evolves by losing or gaining edges (from the maximal graph). We show that this problem is in DynFO (with LOGSPACE precomputation), via a reduction to a Dyck reachability problem on an acyclic automaton.IntroductionMonadic second-order logic, tree-width of graphs, and Courcelle's theorem. Monadic second-order logic (MSO) is a powerful formalism for expressing and checking properties of graphs. It allows first-order quantification (over states of the graph) as well as monadic second-order quantification (over sets of states), and can thus express properties such as connectivity or 3-colorability of finite graphs. While the satisfiability problem of this logic is in general undecidable, model checking (i.e., deciding if a formula holds true in a given graph) is PSPACE-complete [25]; when the MSO formula is fixed, the problem is hard for every level of the polynomial hierarchy over finite graphs[24], and can be performed in PTIME over finite trees. The tree-width of graphs has been defined by Robertson and Seymour [21] as a measure of the complexity of graphs-intuitively, of how close a graph is to a tree. Many classes of graphs have been shown to have bounded tree-width[3]. Over such graphs, several (NP-)hard problems can be solved in polynomial time[3]. Model checking a fixed MSO formula is such an example, as proven by Courcelle in 1990 [5].Dynamic problems and dynamic complexity. In this paper, we focus on the dynamic complexity of MSO model checking over finite graphs. Dynamic-complexity theory aims at developing algorithms that are capable of efficiently updating the output of a problem after a slight change in its input[10,19]. Such algorithms would keep track of auxiliary information about the current instance, and update it efficiently when the instance is modified. Consider the problem of reachability in directed graphs, and equip such graphs with two operations, for respectively inserting and deleting edges (one at a time). It has recently been proven that this problem is in the class DynFO [8], which was a long-standing open problem. Roughly speaking, a problem is in DynFO when it admits an algorithm updating the solution and some auxiliary information through FO formulas (or, equivalently by AC 0 circuits satisfying some uniformity requirements) after a small change in the input.Our contributions.We study the MSO model-checking problem from a dynamic perspective, considering the following basic operations on graphs: insertion and deletion of an edge. * This work is supported by EU under ERC EQualIS (FP7-308087) and STREP Cassting (FP7-601148).We assume that we are given a maximal graph, which embeds all constructed game graphs along the dynamic process: this maximal graph represents the set of all possible connections in the subgraphs we will consider. We first realize that, since the MSO model-checking problem over arbitrary finite graphs is NP-hard (see[16,24]), the MSO model-checking problem over arbitrary maximal graphs is unlikely to be solvable in DynFO, even allowing PTIME precomputation (unless Dyn(PTIME,FO)=PTIME =NP).We therefore make a standard restriction and assume that the maximal graph has bounded tree-width. Under this hypothesis, we show that the MSO model checking over such graphs can be solved in DynFO with LOGSPACE precomputation. To obtain this result, we rely on (and extend) a DynFO algorithm for finding a Dyck path in an acyclic automaton[26], and build a transformation of our model checking problem into such a Dyck reachability problem. The latter transformation is performed by using Courcelle's theorem, and by realizing that the runs of bottom-up, deterministic tree automata can be computed step-by-step. These simple steps can be stored in an auxiliary graph, in which only few edges depend on the real edges that exist in the original game; the correctness of the construction then goes through the search for paths labeled with Dyck words.Related works.MSO has been extensively studied over various classes of structures in the last 40 years, both regarding its expressiveness and regarding the algorithmic properties of its decision problems (see[6]and references therein). Similarly, numerous measures of the complexity of graphs, such as tree-width [21], clique-width[7]or entanglement [2], have been defined and studied; they provide large classes of graphs in which different kinds of hard problems become tractable (see[20]and references therein).On the other hand, dynamic complexity is much less developed: while the main dynamic complexity classes were defined and studied 20 years ago[10,19], only few problems have been considered from that point of view[26,8,27]. As cited above, directed-graph reachability has recently been proven in DynFO, which was an important open problem in the area.Finally, let us mention that the results reported in this paper were originally presented in the setting of parity games played on a graph having bounded tree-width [4].DefinitionsMonadic second-order logic over graphsA graph is a pair G = V, E where V is a finite set of vertices, and E ⊆ V × V is a finite set of edges. The size of G is the cardinality of V . Formulas of the monadic second-order logic (denoted MSO) are built using first-order and (monadic) second-order variables, used respectively to quantify over vertices and sets of vertices; formulas may also use equality, and the edge relation E of the graph. As an example, connectivity of a graph can be expressed asThe standard static question regarding MSO over graphs is to decide whether a given formula is satisfiable, or to check whether it is satisfied in a given graph model. These problems have been extensively considered in the literature; in particular, satisfiability is undecidable [23], while model checking is PSPACE-complete [25]. We refer to [17] for a survey. P. Bouyer, V. Jugé and N. Markey 3Tree decompositionThe notion of tree decomposition[21,22]was introduced by Robertson and Seymour. It gives rise to classes of graphs on which many problems that are NP-hard in general become tractable.Definition 1. An ordered tree decomposition of G is a pair D = T , T , where T = N , E is an ordered tree, and T : N → 2 V is a function such that:for each edge e ∈ E, there exists a node n ∈ N such that e ∈ T(n) 2 ; for each vertex v ∈ V , the set N v = {n ∈ N | v ∈ T(n)} is non-empty, and the restriction of T to N v is connected. The width of D is defined as the integer max{|T(n)| | n ∈ N } − 1, and the tree-width of G is the least width of all tree decompositions of G.Tree automatonThe notion of (deterministic, bottom-up) tree automaton is a powerful tool for expressing and checking properties of finite trees.Definition 2.A tree automaton is a tuple A = Q, Σ, ι, Q end , δ where Q is a finite set of states, Σ is a finite input alphabet, ι ∈ Q is the initial state, Q end ⊆ Q is the set of accepting states and δ : Q 2 × Σ → Q is the transition function.Let T = N , E be a binary ordered labeled tree, with label set Σ. The run of A over T is the function ρ : N → Q such that:for every leaf n of T with label λ, we have ρ(n) = δ(ι, ι, λ); for every internal node n of T with label λ and with children m 1 and m 2 , we have ρ(n) = δ 2 (ρ(m 1 ), ρ(m 2 ), λ). If, furthermore, the run ρ maps the root of T to an accepting state q ∈ Q end , then we say that ρ is accepting, and that the automaton A accepts the tree T .
Dynamic complexity theory
In this paper, we adapt Courcelle's theorem to a dynamic-complexity framework. We briefly introduce the formalisms of descriptive-and dynamic complexity here, and refer to [19,15,13,26] for more details.
Descriptive complexity aims at characterizing positive instances of a problem using logical formulas: the input is then given as a logical structure described by a set of k-ary predicates (the vocabulary) over its universe. For example, a directed graph can be represented as a binary predicate representing its edges, with the set of vertices (usually identified with {1, . . . , n} for some n) as the universe. Whether each vertex has at most one outgoing edge is expressed by the first-order formula ∀x, y, z.(E(x, y) ∧ E(x, z)) ⇒ (y = z). The complexity class FO contains all problems that can be characterized by such first-order formulas. This class corresponds to the circuit-complexity class AC 0 (under adequate uniformity assumptions) [1].
Dynamic complexity aims at developing algorithms that can efficiently update the output of a problem (e.g. reachability of a given vertex in a graph) when the input is slightly changed. In this setting, algorithms may take advantage of previous computations in order to very quickly recompute the solution for the modified input.
Formally, following [26], a decision problem S is a subset of the set of τ -structures Struct(τ ) built on a vocabulary τ . In order to turn S into a dynamic problem DynS, we need to define a finite set of initial inputs and a finite set of allowed updates. For instance, we might use an arbitrary graph as initial input, then use a 2-ary operator ins(x, y) that would insert an edge between vertices x and y.
Hence, we associate the decision problem S with a set Updates of update functions up : Struct(τ ) → Struct(τ ). We identify every non-empty word in Struct(τ ) · Updates * with the τ -structure obtained by applying a sequence of update operations to an initial structure. Denoting by Struct n (τ ) and by Updates n the set of τ -structures and of updates restricted to a universe of size n, we define the dynamic language DynS n as the set of those words in Struct n (τ ) · Updates * n that correspond to structures of S. The dynamic language DynS is then defined as the union (over all n) of all such languages.
Given two complexity classes C and C , a dynamic problem DynS with set of updates Updates belongs to the class Dyn(C, C ) if, and only if, there exists an auxiliary vocabulary τ aux , a C-computable initialisation function f init : Struct(τ ) → Struct(τ aux ), a C -computable update function f up : Struct(τ aux ) × Updates → Struct(τ aux ), and a C -computable decision function f dec : Struct(τ aux ) → {0, 1} such that:
for every structure A ∈ Struct(τ ) and every upate up ∈ Updates, we have
f init (up(A)) = f up (f init (A), up); for every structure A ∈ Struct(τ ), we have A ∈ S ⇔ f dec (f init (A)) = 1.
If, furthermore, f init maps the empty structure of Struct(τ ) to the empty structure of Struct(τ aux ), then we say that DynS belongs to the class DynC .
Informally, DynS belongs to Dyn(C, C ) if, by maintaining an auxiliary structure (which may have an initial cost in C), an algorithm can tackle every update on the input structure with a cost in C . If the initial cost is reduced to zero when the initial input is the empty structure, then DynS belongs to DynC .
In this paper, we consider the case where C = LOGSPACE and C = FO, meaning that precomputations will be carried out in LOGSPACE and that first-order formulas will be used to describe how predicates are updated along transitions.
Main result
We are now in a position to formally define our problem and state our main result. We fix an MSO formula ϕ. We follow the approach of [9], and represent graphs as tuples V, E . Given a universe V , our initial structure consists of a tuple V, E , E , where E is a maximal set of edges and E ⊆ E is an initial set of edges.
We focus below on the operations of insertion and deletion of edges that belong to E . More precisely, we let Updates E = {ins(e), del(e) | e ∈ E }. The effect of a sequence of update operations, represented as a word w ∈ Updates * E , over a set E ⊆ E of edges, is denoted by w(E), and is defined inductively as:
E if w = E ∪ {e} if w = ins(e) w (a(E)) if w = w · a E \ {e} if w = del(e)
For w ∈ Updates * E and E ⊆ E , we write G w(E) for the graph with vertex set V and edge set w(E). It is to be noted that G w(E) is a subgraph of V, E . Finally, we let
DynSat ϕ = { V, E , E · w | w ∈ Updates * E and G w(E) ϕ}.
As mentioned in the introduction, the above problem is unlikely to be solvable in Dyn(PTIME,FO). We therefore adopt the idea of bounding the tree-width of the maximal graph V, E . We fix a positive integer κ and restrict the set of admissible initial inputs: the graph V, E should be of tree-width at most κ. We thus refine our problem as follows:
DynSat κ,ϕ = { V, E , E · w | V, E has tree-width at most κ} ∩ DynSat ϕ .
Our main contribution is a dynamic algorithm for deciding DynSat κ,ϕ . We give a short overview of the proof here. Our algorithm consists in transforming our MSO model checking problem into an equivalent Dyck reachability problem over a labeled acyclic graph. The latter problem is known to be in DynFO [26], although we had to adapt the algorithm to our setting. Our approach for building this acyclic graph follows from an automata-based construction used for proving Courcelle's theorem: along some linearization of a tree decomposition of the maximal graph, we can inductively compute local information about the possible computations of a bottom-up tree automaton. These computations can be represented as finding a path in an acyclic graph. However, we have to resort to Dyck paths in order to make our acyclic graph efficiently updatable when the input graph is modified.
Courcelle's theorem
Courcelle's theorem is not based on working directly with tree decompositions of graphs, but on labeled ordered trees whose labels are chosen from a finite alphabet, and that represent such tree decompositions. We begin with defining such trees.
we set C = { v, w ∈ E | v, w ∈ T(n) 2 \ A 2 }; if X is a set of vertices, then χ(X) = {χ(v) | v ∈ X},
and if X is a set of edges, then
χ(X) = { χ(v), χ(w) | v, w ∈ X}.
These constructions are illustrated in Figure 1, which displays a graph G, a tree decomposition D = T , T of G, a proper D-coloring of G, and its associated succinct tree decomposition.
Observe that, given a tree decomposition D, there always exist D-colorings χ of G and an associated (χ, D)-succinct tree decomposition. They are typically computed from D in a top-down fashion. Furthermore, note that a graph G may have several tree decompositions D of width κ and, for each of them, several proper D-colorings. Hence, G may have several succint tree decompositions. Yet, from all of them we are able to reconstruct G (up to graph isomorphism), and therefore to check whether G satisfies the formula ϕ. A more precise and powerful version of this statement is the theorem stated below, which is a variant of the versions of Courcelle's theorem of [12,Section 11.4] and [17, Section 3.3]. Making Theorem 5 useful further requires being able to compute succinct tree decompositions efficiently. This is possible thanks to the following result, which is proven in [11]:
v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 Graph G a v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 5 v 4 v 2 b v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 4 v 2 v 1 c v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 7 v 5 v 4 h v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 8 v 7 v 5 d v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 7 v 4 v 3 e v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 v 6 v 3 f v 8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 ROOT a χ(A) = ∅ χ(B) = {0, 1, 2} χ(C) = {(0, 1), (1, 0), (2, 0)} b χ(A) = {1, 2} χ(B) = {0} χ(C) = {(0, 0), (0, 2), (1, 0)} c χ(A) = {0, 2} χ(B) = {1} χ(C) = {(1, 0), (1, 2), (2, 1)} g χ(A) = {0, 1} χ(B) = {2} χ(C) = {(0, 2), (1, 2), (2, 1)} d χ(A) = {1, 2} χ(B) = {0} χ(C) = {(0, 1), (0, 2), (1, 0)} e χ(A) = {0} χ(B) = {2} χ(C) = {(2, 0)} f χ(A) = ∅ χ(B) = ∅ χ(C) = ∅ ROOT Tree decomposition D of G (χ, D)-succinct tree decomposition of G D-proper coloring of G: v 1 → 0 v 2 → 1 v 3 → 0 v 4 → 2 v 5 → 0 v 6 → 2 v 7 → 1 v 8 → 2
Towards a dynamic algorithm
In this section, we focus on making Courcelle's theorem dynamic. We fix an input (bottom-up, deterministic) tree automaton A and an input graph G, and we assume that we have a succinct tree decomposition T of G. We transform the language-theoretic problem of checking whether A accepts T into a Dyck reachability problem. More precisely, we build a graph Γ G and establish a correspondence between some Dyck paths in this graph and the (accepting) runs of A on T .
State progression
A first step towards our goal consists in performing a sequence of transformations on T . Bottom-up progression: Observe that, by construction, we always have S 0 = ∅. Figure 2 presents a binary ordered tree T , post indices labeling, and bottom-up progression. Observe that T is the (succinct) tree decomposition presented in Figure 1.
S 0 = ∅ S 1 = {b} S 2 = {b, e} S 3 = {b, e, f } S 4 = {b, d} S 5 = {b, d, g} S 6 = {b, c} S 7 = {a}
When T is binary and has height h, then the bottom-up progression enjoys some conciseness and smoothness properties, which we state below. Finally, assume that there exist two nodes x ≺ x in S i that are at the same height, and let y be the parent of x. By definition, we know that post(x) < post(x ) i < post(y), hence x belongs to the right subtree of y, i.e., x is the right sibling of x. It follows that S i contains at most two nodes at each level, whence |S i | 2h.
The above notion of bottom-up progression also leads to the notion of state progression. Figure 3 presents a tree automaton A, a labeling of T , the (rejecting) run ρ of A on T and the associated state progression. We omit to represent the restriction ρ S0 since S 0 is empty.
λ 0 λ 1 λ 0 λ 1 λ 1 λ 0 λ 1 a b c g d e f
Labeled ordered tree T : Automaton A:
Q = {q 0 , q 1 } Σ = {λ 0 , λ 1 } ι = q 0 Q end = {q 0 } δ : (q i , q j , λ k ) → q max{i·j,k}
Run ρ:
a → q 1 b → q 1 c → q 1 d → q 1 e → q 0 f → q 1 g → q 1
State progression: Figure 3 Labeled tree, tree automaton, run and associated state progression For all i 1, recall that Lemma 8 states that there exists a unique node n i ∈ S i \ S i−1 , and that either n i is a leaf or both of its children belong to S i−1 . The functions ρ Si can therefore be computed in a step-wise manner once the automaton A = Q, Σ, ι, Q end , δ is fixed. More precisely, and denoting by λ the labeling function of T , we have:
ρ S1 : b → q 1 ρ S2 : b → q 1 e → q 0 ρ S3 : b → q 1 e → q 0 f → q 1 ρ S4 : b → q 1 d → q 1 ρ S5 : b → q 1 d → q 1 g → q 1 ρ S6 : b → q 1 c → q 1 ρ S7 : a → q 1
if i = 0, then S 0 = ∅, hence ρ S0 is the empty-domain function; if 1 i and n i is a leaf, then
ρ Si = Π i (ρ Si−1 , λ(n i )), where Π i (ϕ, γ) : n ∈ S i → ϕ(n) if n ∈ S i−1 ; δ(ι, ι, γ) if n = n i ;
if 1 i and n i is an internal node with children m 1 and m 2 , then ρ Si = Π i (ρ Si−1 , λ(n i )), where
Π i (ϕ, γ) : n ∈ S i → ϕ(n) if n ∈ S i−1 ; δ(ϕ(m 1 ), ϕ(m 2 ), γ) if n = n i .
We will rely on this step-wise computation in the following section.
Reduction to the Dyck reachability problem
In this section, we present the reduction of DynSat κ,ϕ to a Dyck reachability problem on an acyclic labeled graph. Our reduction is such that any update (of the edges) in the input graph corresponds to a simple update of the acyclic graph. As we explain in Section 5, this reduction proves Theorem 3.
The Dyck reachability problem in acyclic graphs
Before presenting our reduction, we first define Dyck reachability problems, then recall briefly some results about their dynamic complexity in the case of acyclic graphs: in such graphs, context-free graph queries, and therefore Dyck reachability problems, belong to the dynamic complexity class DynFO [18,26].
Definition 10.
Let G = V, E, L be a labeled graph, with set of labels L, and with edge set E ⊆ V 2 × L. Let v 1 and v 2 be two marked vertices of G. We assume that L can be partitioned as L = L + L − {•}, where L + and L − are in bijection with each other, and • is a fresh "neutral" label symbol, and that a bijection λ + → λ − from L + to L − is given.
The labeling on edges induces in a direct way a labeling on paths in G. The Dyck reachability problem asks whether there exists a path π (in the graph G) from v 1 to v 2 , such that π is labeled with a string in the language D of Dyck words built over the grammar: S → ε | S · • · S | S · λ + · S · λ − · S, where λ + ranges over the set L + .
While [26] assumed a constant-size label set (which is not the case here), the result of [26] can be generalized, as stated below (see [4] for a proof).
Lemma 11. The Dyck reachability problem in acyclic graphs is solvable in
DynFO (under the assumption that updates consist in adding or deleting individual labeled edges), using as only auxiliary predicate a 4-ary predicate ∆(x 1 , y 1 , x 2 , y 2 ) defined by: "There exists a path 1 from x 1 to y 1 with a label λ 1 and there exists a path 2 from x 2 to y 2 with a label λ 2 such that λ 1 · λ 2 is a Dyck word".
Reduction
We fix an MSO formula ϕ, and let A ϕ,4κ+3 = Q, Σ, ι, Q end , δ , which we simply name A in the rest of this section, be the (deterministic, bottom-up) tree automaton of Theorem 5.
We now describe our transformation of any subgraph G of G = V, E into an acyclic labeled graph Γ G for the Dyck reachability problem. Let D = T , T , where T = N , E be a tree decomposition of G of width 4κ + 3, as defined in Lemma 6, and let N be the set of nodes of T . In addition, let χ be a D -coloring function of V , and let (S i ) 0 i N be the bottom-up progression of T .
Let G = V, E be a subgraph of G . Observe that D is also a tree decomposition of G, since E ⊆ E . Hence, there exists a labeling function Λ G : N → Σ that identifies T with a (χ, D )-succinct binary tree decomposition T G of G. In particular, (S i ) 0 i N is also the bottom-up progression of T G . Let ρ G be the run of A over T G . We want to construct a graph Γ G in order to identify the state progression (ρ G Si ) 0 i N with a (Dyck) path in Γ G .
A naive construction. As a first try, we let the vertices of this graph be all pairs (i, π), where 0 ≤ i ≤ N , and π : S i → Q is intended to represent the state progression at step i. Following the local computation described page 7, we include an edge (i − 1, π) → (i, π ) when π = Π i (π, Λ G (n i )), where n i is the unique node in S i \ S i−1 . Then, obviously, if π init is the unique function from ∅ to Q, the path ρ G is accepting if and only if, in this naive graph, the unique path from (0, π init ) to (N, π) is such that π maps the unique element of S N (namely, the root of the tree) onto Q end .
While this naive construction is correct, it is not suitable in a dynamic complexity perspective: adding or removing an edge in G may affect many edges in the above graph. However, we show below that it can only affect edges at a single level (index i); using Dyck constraints, we then adapt the construction above to have updates of G only impact one edge of our graph.
The idea is illustrated on Figure 4. Assume that both upper (naive) graphs represent a parameterized function f with parameters γ (left) and γ (right): there is an edge α x → β y in the left graph whenever β y = f (α x , γ), and an edge α x → β y in the right graph whenever β y = f (α x , γ ). Replacing the value of γ with γ , i.e., transforming the left graph into the right one, requires many edge deletions and insertions. Naive graph
α 0 α 1 α 2 β 0 β 1 β 2 α 0 α 1 α 2 β 0 β 1 β 2 Dyck graph α 0 α 1 α 2 α + 0 α + 1 α + 2 • • γ γ α − 1 α − 0 α − 2 α − 2 α − 0 α − 1 β 0 β 1 β 2 α 0 α 1 α 2 α + 0 α + 1 α + 2 • • γ γ α − 1 α − 0 α − 2 α − 2 α − 0 α − 1 β 0 β 1 β 2
Before the update After the update Figure 4 Using Dyck paths saves many changes when the input graph is updated
We circumvent this problem by using Dyck paths: the value of f (·, γ) is computed thanks to the Dyck path labeled with α + x · • · α − x , which goes from α x to β y = f (α x , γ). Hence, changing the value of γ to γ amounts to replacing the edge •
• − → γ with the new edge • • − → γ .
The refined construction. A "nominal" vertex of the graph Γ G is a pair (i, π), where 0 i N and π is a function π : S i → Q, intended to represent the state progression at step i. Labels of Γ G are the pairs (i, π) + and (i, π) − , and the neutral label •. We write π init for the unique function from ∅ to Q. Now, for 1 ≤ i ≤ N , let n i be the unique node in S i \ S i−1 , and recall that ρ G Si = Π i (ρ G Si−1 , Λ G (n i )). We therefore add the following edges and vertices:
vertices (i − 1) and (i − 1, γ) for all γ ∈ Σ;
edges (i − 1, π) (i−1,π) + − −−−−− → (i − 1) and (i − 1, γ) (i−1,π) − − −−−−− → (i, π )
for all γ ∈ Σ and all π : S i−1 → Q, where π : S i → Q is such that π = Π i (π, γ); one neutral edge (i − 1)
• − → (i − 1, γ), where γ = Λ G (n i ).
Finally, observe that n N is the root of T , that S N = {n N }, and that the run ρ G is accepting if, and only if, ρ G (n N ) ∈ Q end . Hence, we complete the construction of Γ G by adding a last state and neutral edges (N, π)
• − → for those functions π : S N → Q such that π(n N ) ∈ Q end . This construction is both sound and complete, and well-behaved under modifications of G, as outlined by the following results.
(i−1,π) + − −−−−− → (i−1) • − → (i−1, γ) (i−1,π) − − −−−−− → (i, π )
with γ = Λ G (n i ) and π = Π i (π, γ). Hence, such a Dyck path exists if, and only if, = ρ G (n N ), where ρ G is the run of A on T G , in which case the intermediate vertices of the form (i, π) are the vertices (i, ρ G Si ). The result follows immediately. Proposition 13. Let e = (v, w) be an edge of the maximal graph G , and let G and G be two subgraphs of G such that G is obtained by adding (resp. deleting) the edge e to G. The graph Γ G is obtained by deleting an edge e 1 from Γ G and inserting another edge e 2 instead. Furthermore, both edges e 1 and e 2 are FO-definable in terms of e, of Γ G , and of some auxiliary precomputed predicates.
Proof. We only deal here with insertion of an edge. We associate with the edge e = v, w of G a mapping add e : Σ → Σ defined by add e :
χ(A), χ(B), χ(C) → χ(A), χ(B), χ(C) ∪ { χ(v), χ(w) } .
Then, let n be the top-most node of T such that both v and w belong to T(n), and let i = post(n). Lemma 8 states that n ∈ S i \ S i−1 . Hence, the labeling functions Λ G and Λ G coincide on all nodes m = n, and we have Λ G (n) = add e (Λ G (n)). Consequently, the graph Γ G is obtained from Γ G in two consecutive steps:
1. we delete the only outgoing edge, of the vertex (i − 1), which is a neutral edge of the form (i − 1)
• − → (i − 1, γ); 2. we add the new edge (i − 1) • − → (i − 1, add e (γ)).
The case of deletion is analogous, and requires using a mapping del e similar to add e . Since the mappings e → i, (e, γ) → add e (γ) and (e, γ) → del e (γ) can be precomputed, it follows that both the edge e 1 that we deleted from Γ G and the edge e 2 that we inserted instead can be computed with FO formulas.
Overall complexity analysis
In this section, we analyze the complexity of our dynamic algorithm. While adequate notions of reduction do exist in dynamic complexity (see e.g. [19,14]), our reduction does not satisfy all criteria, so we need to compute the complexity of our algorithm by hand. First, denoting by V and L the vertex set and the label set of Γ G , Lemma 11 states that the Dyck reachability problem in Γ G can be solved by using FO update formulas over the universe V L. However, we need FO formulas over the universe V of our MSO model checking problem, i.e., V is the vertex set of the input graph. Hence, we must embed V L into a set of tuples of elements of V of finite arity.
Lemma 6 states that T is of height at most c(κ) · (log 2 (N ) + 1), where N = |V |, and Lemma 8 proves that |S i | 2c(κ) · (log 2 (N ) + 1) for all i N . It follows that |V| = 1 + N i=0 |Q| |Si| = O(N 2c(κ) log 2 (|Q|)+1 ), which is polynomial in |V |. Likewise, |L| is polynomial in |V |, and therefore V L can be embedded into some set V k , where k is a large enough integer (which depends only on κ and on the MSO formula ϕ).
In the end, during the precomputation phase, the algorithm successively computes:
1. a binary rooted tree decomposition D = T , T of the maximal graph G = V, E , of width 4κ + 3, such as described in Lemma 6; 2. a (bottom-up, deterministic) tree automaton A ϕ,4κ+3 such as defined in Courcelle's theorem; 3. a D -coloring function χ, a (χ, D )-succinct tree decomposition of G , and a bottom-up progression (S i ) 0 i N of T ; 4. the vertices, labels and edges of the graph Γ G E , where G E is the initial input graph; 5. an embedding V L → V k ; 6. mappings e → i, (e, γ) → add e (γ) and (e, γ) → del e (γ) mentioned in the proof of Proposition 13; 7. the value of the auxiliary predicate ∆ (mentioned in Lemma 11) on Γ G E .
Lemma 14. Each of these 7 steps can be performed in LOGSPACE.
Proof. The formula ϕ and the integer κ are fixed. Hence, Lemma 6 proves that the step 1 can be performed in LOGSPACE, and the step 2 is completed in constant time. Since performing the steps 3-6 in LOGSPACE is straightforward, it remains to deal with the step 7.
Let be a path with label λ in Γ G . We say that is a Dyck prefix path if λ is a prefix of a Dyck word (which may be λ itself) and if its proper prefixes are not Dyck words; that is a Dyck suffix path if λ is a suffix of a Dyck word and if its proper suffixes are not Dyck words; that is a minimal Dyck path if is both a Dyck prefix and a Dyck suffix path. Minimal non-empty Dyck paths are paths of the form (i − 1)
We sum up the above results as follows. First, we perform a LOGSPACE precomputation, and construct a graph Γ G whose vertex, label and edge sets can be represented as predicates of finite arity on the universe V . Then, during the update phases, whenever introducing or deleting an edge e in G, we replace one edge of Γ G by another one, and these edges are identified by precomputed FO formulas taking the edge e into account, as stated in Proposition 13. Consequently, and since the Dyck reachability problem is in DynFO, updating the edge-membership predicate of Γ G and the auxiliary predicate ∆, which is useful for solving the Dyck reachability problem in Γ G , can be done with FO formulas. Finally, deciding whether G satisfies the formula ϕ, i.e., whether there exists a suitable Dyck path in Γ G , can be done using directly the auxiliary predicate ∆, which completes the proof of Theorem 3.
Conclusion
We developed a dynamic algorithm for checking a (fixed) MSO formula over (evolving) subgraphs of a given graph of bounded tree-width. A natural extension of this work would consist in getting rid of the hypothesis that there exists a maximal graph G of which the graphs under scrutiny are subgraphs. There are two main obstacles for this to be achieved in our approach: first, we would need to be able to dynamically compute tree decompositions of "moderate" width of our dynamic graphs; then, we would have to adapt the structure of our graph Γ G to take into account these evolving tree decompositions. Another direction of research, which was successfully put into practice in [4] when dealing with the particular case of parity games, would consist, given an input formula ϕ = ∃X. ϕ (X) (starting with an existential quantifier), to compute a witness X of the satisfiability of ϕ .
Theorem 3 .
3Fix a positive integer κ and an MSO formula ϕ. The problem DynSat κ,ϕ can be solved in Dyn(LOGSPACE, FO).
Definition 4 .
4Let G = V, E be a graph, and let D = T , T be a binary ordered tree decomposition of G of width κ. Let Σ = 2 {0,...,κ} × 2 {0,...,κ} × 2 {0,...,κ} 2 be a (finite) set of labels. We call proper D-coloring of G a function χ : V → {0, . . . , κ} such that, for all nodes n of T , the restriction of χ to T(n) is injective. We then call (χ, D)-succinct tree decomposition of G (we may omit to mention χ and D if it is clear from the context) the rooted tree obtained by labeling every node n of T with a label λ(n) = χ(A), χ(B), χ(C) ∈ Σ as follows: we set A = T(n) ∩ T(m) if n has a parent m in T , and A = ∅ if n is the root of T ; we set B = T(n) \ A;
Theorem 5 .
5Fix a positive integer κ and an MSO formula ϕ. There exists a (bottom-up, deterministic) tree automaton A ϕ,κ such that, for all graphs G = V, E and all succinct tree decompositions T of G of width κ, G satisfies ϕ if, and only if, A ϕ,κ accepts T .
Figure 1
1Graph, tree decomposition, proper coloring and succinct tree decomposition Lemma 6. Let G be a graph of size N and tree-width κ. We can construct in space O(c(κ) log 2 N ) an ordered binary tree decomposition of G of size at most 2N , width at most 4κ + 3, and height at most c(κ) · (log 2 (N ) + 1), where c(κ) only depends on κ.
Definition 7 .
7Let T be an ordered tree with N nodes. The post order on T is defined as the linear order ≺ such that:if m is a strict ancestor of n, then n ≺ m;
Figure 2
2Ordered tree, depth-first traversal and bottom-up progression if an internal node n has a left child m 1 and a right child m 2 , then m 1 and its descendants are all smaller than m 2 and its descendants (for the order ≺). There exists a unique labeling post : T → {1, . . . , N }, which we call post index, such that n m ⇔ post(n) post(m). We also commonly denote by n i the unique node of T such that i = post(n i ). We further call bottom-up progression of T the sequence S 0 , . . . , S N of subsets of vertices of T defined by S i = {n | post(n) i and post(m) > i for all strict ancestors m of n}.
Lemma 8 .
8Let T be a binary tree with N nodes, of height h, and let (S i ) 0 i N be the bottom-up progression of T . For all i 1, it holds that: the set S i is of cardinality 2h or less; if the node n i is a leaf, then S i = S i−1 ∪ {n i }; if the node n i is an internal node, with children m 1 and m 2 , then both m 1 and m 2 belong to S i−1 , and S i = S i−1 \ {m 1 , m 2 } ∪ {n i }. Proof. First, since i < post(m) for all strict ancestors m of n i , it comes at once that S i \ S i−1 = {n i }. Furthermore, a node n belongs to S i−1 \ S i if, and only if, post(n) < i, n i is an ancestor of n, and post(m) i for all strict ancestors m of n. Since we have post(x) < i < post(y) for all strict descendants x and all strict ancestors y of n i , it follows that S i−1 \ S i consists of the children of n i only (if they exist).
Definition 9 .
9Let T be a labeled binary ordered tree with N nodes, let (S i ) 0 i N be its bottom-up progression, and let A be a (deterministic, bottom-up) automaton. Let ρ be the (unique) run of A over T . We call state progression of A on T the sequence (ρ Si ) 0 i N , where ρ Si denotes the restriction of ρ to the domain S i .
Figure 2
2presented a tree T and a bottom-up progression of T .
Proposition 12 .
12The automaton A accepts the labeled tree T G if, and only if, there exists a Dyck path in Γ G from the vertex (0, π init ) to the vertex . Proof. A path from (0, π init ) to a vertex (N, ), where is a function S N → Q, is Dyck if, and only if, it uses only sub-paths of the form (i−1, π)
and 2 are of length at most 2, and λ 1 · λ 2 is a Dyck word; there exists Dyck paths from x 1 to z 1 and from z 2 to y 2 . Finally, note that every vertex of Γ G is the source of at most one minimal Dyck path. Consequently, for any two vertices x and y of Γ G , checking if there exists a Dyck path from x to y can be done in LOGSPACE, and ∆ can be computed in LOGSPACE too.
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| [] |
[
"Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets",
"Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets"
] | [
"M Holtschneider \nInstitut für Theoretische Physik\nRWTH Aachen\n52056AachenGermany\n",
"S Wessel \nInstitut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany\n",
"W Selke \nInstitut für Theoretische Physik\nRWTH Aachen\n52056AachenGermany\n"
] | [
"Institut für Theoretische Physik\nRWTH Aachen\n52056AachenGermany",
"Institut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany",
"Institut für Theoretische Physik\nRWTH Aachen\n52056AachenGermany"
] | [] | The classical and the quantum, spin S = 1 2 , versions of the uniaxially anisotropic Heisenberg antiferromagnet on a square lattice in a field parallel to the easy axis are studied using Monte Carlo techniques. For the classical version, attention is drawn to biconical structures and fluctuations at low temperatures in the transition region between the antiferromagnetic and spin-flop phases. For the quantum version, the previously proposed scenario of a first-order transition between the antiferromagnetic and spin-flop phases with a critical endpoint and a tricritical point is scrutinized. | 10.1103/physrevb.75.224417 | [
"https://arxiv.org/pdf/cond-mat/0703135v2.pdf"
] | 119,391,445 | cond-mat/0703135 | 9d8c94ce6cb9d2a61f6a4646a81edddf828be00b |
Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets
May 2007
M Holtschneider
Institut für Theoretische Physik
RWTH Aachen
52056AachenGermany
S Wessel
Institut für Theoretische Physik III
Universität Stuttgart
70550StuttgartGermany
W Selke
Institut für Theoretische Physik
RWTH Aachen
52056AachenGermany
Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets
May 2007numbers: 7510Hk7510Jm7540Mg0510Ln
The classical and the quantum, spin S = 1 2 , versions of the uniaxially anisotropic Heisenberg antiferromagnet on a square lattice in a field parallel to the easy axis are studied using Monte Carlo techniques. For the classical version, attention is drawn to biconical structures and fluctuations at low temperatures in the transition region between the antiferromagnetic and spin-flop phases. For the quantum version, the previously proposed scenario of a first-order transition between the antiferromagnetic and spin-flop phases with a critical endpoint and a tricritical point is scrutinized.
The classical and the quantum, spin S = 1 2 , versions of the uniaxially anisotropic Heisenberg antiferromagnet on a square lattice in a field parallel to the easy axis are studied using Monte Carlo techniques. For the classical version, attention is drawn to biconical structures and fluctuations at low temperatures in the transition region between the antiferromagnetic and spin-flop phases. For the quantum version, the previously proposed scenario of a first-order transition between the antiferromagnetic and spin-flop phases with a critical endpoint and a tricritical point is scrutinized.
I. INTRODUCTION
Uniaxially anisotropic Heisenberg antiferromagnets in an external field along the easy axis have attracted much interest, both theoretically and experimentally, due to their interesting structural and critical properties. In particular, they display a spin-flop phase, and multicritical behavior occurs at the triple point of the antiferromagnetic (AF), spin-flop (SF) and paramagnetic phases. 1,2,3,4,5,6,7,8,9,10 A prototypical model for such antiferromagnets is the XXZ model, with the Hamiltonian
H = J (i,j) ∆(S x i S x j + S y i S y j ) + S z i S z j − H i S z i ,(1)
where the sum runs over neighboring spins of a cubic, dimension d = 3, or square lattice, d = 2. The coupling constant J and the field H are positive; the anisotropy parameter ∆ may range from zero to one. Furthermore, S x i , S y i , and S z i denote the spin components at lattice site i.
For the three-dimensional case, early renormalization group arguments 1 and Monte Carlo simulations 3 suggested that the triple point is a bicritical point with O(3) symmetry. Only a few years ago, this scenario has been questioned, based on high-order perturbative renormalization group calculations. 11 It has been predicted that there may be either a first order transition, or that the 'tetracritical biconical' 1 fixed point, due to an intervening 'mixed' or 'biconical' phase in between the AF and SF phases 12,13,14 , may be stable.
In two dimensions, conflicting predictions on the nature of the triple point have been put forward recently 15,16,17,18 , when analyzing the classical version of the above model with spin vectors of unit length, and the quantum version with spin S = 1 2 . Indeed, in the classical case, simulational evidence for a narrow (disordered) phase between the AF and SF phases has been presented 15 , extending presumably down to zero temperature. 16 On the other hand, in the quantum case, based on simulations as well, a direct transition of first order between the AF and SF phases has been argued to occur at low temperatures. 18,19,20 Obviously, experimental data have to be viewed with care because deviations from the XXZ Hamiltonian, Eq. (1), such as crystal field anisotropies or longer-range interactions, may affect relevantly the critical behavior of the triple point. 5,12,13,14,21,22 In the following, we present results from large-scale Monte Carlo simulations of the XXZ model on a square lattice for both the classical and the quantum variant. In the quantum Monte Carlo simulations, the method of the stochastic series expansion (SSE) 23 is used, and the standard Metropolis algorithm is applied for the classical case. The simulations are augmented by a ground-state analysis of the classical model, showing the significance of biconical structures. The outline of the paper is as follows: First we shall discuss our findings on the classical model, followed by a section on the quantum version of the XXZ model. A summary concludes the paper.
II. CLASSICAL MODEL
The ground states of the classical model on a square lattice, see Hamiltonian (1), can be determined exactly. The AF structure is stable for magnetic fields below the critical value
H c1 = 4J 1 − ∆ 2 ,(2)
while for larger fields the SF state is energetically favorable. At H c1 , the tilt angle θ SF of the SF structures, see
θ SF = arccos 1 − ∆ 1 + ∆ .(3)
Increasing the field beyond H c2 = 4J(1 + ∆), all spins perfectly align in the z-direction. At the critical field H c1 , see Eq.
(2), the ground state is degenerate in the AF, the SF, and biconical structures, as illustrated in Fig. 1. This degeneracy in the biconical configurations, following from straightforward energy considerations, seems to have been overlooked in the previous analyses. The structures may be described by the tilt angles, θ 1 and θ 2 , formed between the directions of the spins on the two sublattices of the antiferromagnet and the easy axis. For a given value of θ 1 , the other angle θ 2 is fixed by
θ 2 = arccos √ 1 − ∆ 2 − cos θ 1 1 − √ 1 − ∆ 2 cos θ 1 .(4)
Obviously, the biconical configurations transform the AF into the SF state: The spins on the "up-sublattice" of the AF structure, with the spins pointing into the direction of the field, may be thought of to vary from θ 1 = 0 to ±θ SF , while the spins on the "down-sublattice" vary simultaneously from θ 2 = π to ∓θ SF . Accordingly, θ 1 determines uniquely θ 2 and vice versa. Apart from this continuous degeneracy in θ 1 (or θ 2 ), there is an additional rotational degeneracy of the biconical configurations in the spin components perpendicular to the easy axis, the xycomponents, as for the SF structure, see Fig. (1). These components are, of course, antiferromagnetically aligned at neighboring sites.
To study the possible thermal relevance of the biconical structures at T > 0, we performed Monte Carlo simulations analyzing the joint probability distribution P (θ m , θ n ) for having tilt angles θ m and θ n at neighboring sites, m and n. For comparison with the previous studies 3,15,16 we set ∆ = 4 5 , leading to the phase diagram depicted in Fig. 2. For example, fixing the field at H = 2.41J, we observed at k B T /J ≈ 0.255 an Ising-type transition on approach from higher temperatures and a Kosterlitz-Thouless-type transition on approach from the low-temperature side, extending our corresponding previous findings 15 to even lower temperatures, and in agreement with recent results. 16 Indeed, as depicted in Fig. 3, in that part of the phase diagram, being in the vicinity of the very narrow intervening, supposedly disordered phase, the joint probability P (θ m , θ n ) exhibits a line of local maxima following closely Eq. (4), obtained for the ground state. That behavior is largely independent of the size of the lattices we studied. Similar signatures of the biconical structures are observed in Accordingly, we tend to conclude that biconical fluctuations are dominating in the narrow intervening phase. Whether that phase exists as a disordered phase down to the ground state or whether there is a stable biconical phase in two dimensions, remain open questions, being beyond the scope of this article.
Note that our additional Monte Carlo simulations for the anisotropic XY antiferromagnet in a field on a square lattice show that the analogues of 'biconical' structures (the orientation of the spins being now given by the two tilt angles only) and fluctuations play an important role near the transition regime between the AF and SF phases in that case as well. In fact, Eq. (4) provides an excellent guidance for interpreting our simulational data similar to the ones presented in Fig. 3.
III. QUANTUM XXZ MODEL
The aim of the study on the quantum version, S = 1 2 , of the XXZ model, Eq. (1), has been to check the previously suggested scenario of a first-order phase transition between the AF and SF phases extending up to a critical endpoint and with a tricritical point on the AF phase boundary, see Fig. 4.
We performed quantum Monte Carlo (QMC) simulations in the framework of the stochastic series expansion (SSE) 23 using directed loop updates 24 . We consider square lattices of L × L sites with the linear dimension L ranging from 2 to 150, employing full periodic boundary conditions. Defining, as usual, 23 a single QMC step as one diagonal update followed by the construction of several operator-loops, each individual run typically consists of 10 6 steps and is preceded by at least 2 · 10 5 steps for thermal equilibration. Averages and error bars are obtained by taking into account results of several, ranging from 8 to 32, Monte Carlo runs, choosing different initial configurations and random numbers. Especially for large systems and low temperatures we additionally utilize the technique of parallel tempering (or exchange Monte Carlo) 25,26 to enable the simulated systems to overcome the large energy barriers between configurations related to different phases more frequently. We typically work with a chain of 16 to 32 configurations in parallel which are simulated at different equally spaced temperature or magnetic field values allowing for an exchange of neighboring configurations after a constant number of QMC steps. The achieved reduction of the autocorrelation times, e.g. of the different magnetizations discussed below, amounts up to several orders of magnitude and therefore results in significantly smaller correlations between subsequent measurements which, in turn, allows for shorter simulation times.
To determine the phase diagram and to check against previous work 18 , we calculated various physical quantities including the z-component of the total magnetization,
M z = 1 L 2 i S z i ,(5)
and the square of the z-component of the staggered magnetization,
(M z st ) 2 = 1 L 2 ia S z ia − i b S z i b 2 ,(6)
summing over all sites, i a and i b , of the two sublattices of the antiferromagnet. A useful quantity in studying the SF phase is the spin-stiffness ρ s which is related to the change of the free-energy on imposing an infinitesimal twist on all bonds in one direction of the lattice. In QMC simulations the spin-stiffness can conveniently be measured by the fluctuations of the winding numbers W x and W y , 23
ρ s = k B T 2 W 2 x + W 2 y .(7)
The winding numbers themselves are given by where N + α and N − α denote the number of operators S + i S − j and S − i S + j in the SSE operator sequence with a bond i, j in the α-direction, α ∈ {x, y}.
W α = 1 L N + α − N − α ,(8)
All data for the quantum model presented here are obtained at an anisotropy parameter of ∆ = 2 3 to allow for comparison with previous findings 15, 18 . The phase diagram in the region of interest, where all three phases, the AF, the SF, and the paramagnetic phase occur, is displayed in Fig. 4.
The earlier study 18 asserted a phase diagram with a tricritical point at k B T t /J ≈ 0.141 and a direct first-order transition between the SF and AF phases below the critical endpoint at k B T ce /J ≈ 0.118, see Fig. 4. In detail the authors identified a first-order AF to paramagnetic transition at k B T /J = 0.13 by means of an analysis of the magnetization histograms p(M z ). We studied that case, improving the statistics and considering even larger lattice sizes. Indeed, as expected for a discontinuous change of the magnetization, the histograms of finite systems are confirmed to display two distinct maxima corresponding to the ordered and the disordered phase in the vicinity of the AF phase boundary (see inset of Fig. 5). Note however, that such a two-peak structure can also be found for small systems at a continuous transition, with a single peak in the thermodynamic limit. Thence, a careful finite-size analysis is needed to, possibly, discriminate the two different scenarios. We simulated lattice sizes with up to 150 × 150 spins adjusting the magnetic field such that coexistence of the phases, i.e. equal weight of the two peaks, is provided. As depicted in Fig. 5 the positions of the maxima as a function of the inverse system size exhibit a curvature, which becomes more pronounced for larger lattices. In contrast, in the previous analysis 18 at the same temperature, linear dependences of the peak positions as a function of 1/L had been presumed, leading to distinct two peaks in the thermodynamic limit. We conclude, that the previous claims of a first-order transition at k B T /J = 0.13 and of the existence of a tricritical point at k B T t /J ≈ 0.141 needs to be viewed with care. Indeed, the tricritical point seems, if it exists at all, to be shifted towards lower temperatures.
In the previous work 18 a direct transition of first order between the AF and SF phases has been suggested to take place at lower temperatures, k B T /J ≤ k B T ce /J ≈ 0.118. To check this suggestion we studied the system at constant field H/J = 1.225, where such a direct transition would occur, see Fig. 4. Calculating the expectation values of the different magnetizations as well as the corresponding histograms we obtain an estimate of the critical temperature of the AF phase, k B T AF = 0.09625 ± 0.0005.
Surprisingly, approaching the transition from the AF phase, the finite-size behavior of the squared staggered magnetization (M z st ) 2 , being the AF order parameter, is still consistent with a continuous transition in the Ising universality class: As illustrated in Fig. 6 the asymptotic region is very narrow, similar to the observations in the classical model. 15,16 The dependence on the system size seems to obey (M z st ) 2 ∝ L 1/4 right at the transition, as expected for the Ising universality class. 27 Furthermore, approaching the transition from the SF phase, an analysis of the spin-stiffness ρ s at the same field value of H/J = 1.225 results in about the same transition temperature, k B T SF /J = 0.09625 ± 0.001. Thence, there may be either a unique transition between the SF and AF phases, or, as observed in the classical case, an extremely narrow intervening phase, with phase boundaries of Ising and Kosterlitz-Thouless (KT) type.
To determine, whether a KT transition describes the disordering of the SF phase, we check the theoretical prediction 28,29 that for the infinite system the spinstiffness is finite within the SF phase, takes on a universal value at the KT transition related to T KT by
ρ s (T = T KT , L = ∞) = 2 π k B T KT ,(9)
and discontinuously vanishes in the disordered phase. As depicted in Fig. 7, the spin-stiffness ρ s at T = T SF seems to be, at first sight, significantly larger than the KTcritical value given by Eq. (9). Indeed, in the earlier study 18 it has been argued, based on similar observations, that there is a direct first order AF to SF transition. However, the finite-size effects close to the transition deserve a careful analysis: For the KT scenario, renormalization group calculations 30,31 predict the asymptotic size dependence at T = T KT to obey
ρ s (T = T KT , L) = ρ s (T = T KT , L = ∞) 1 + 1 2 ln L − C 0 ,(10)
where C 0 denotes an apriorily unknown, non-universal, parameter. By studying the quantity 32
C(L) = −2 πρ s k B T − 2 −1 − ln L ,(11)
which, according to Eqs. 9 and 10, converges for L → ∞ and T = T KT to the value C 0 at a KT transition, we obtain a rough estimate of C 0 ≈ 5. A prediction of the finite-size behavior at T SF is obtained by inserting this value, C 0 = 5, into Eq. (10). Comparing the data of the spin-stiffness ρ s in the direct vicinity of the boundary of the SF phase with the prediction according to the KT theory, see Fig. 7 b), one may conclude that the lattice sizes accessible by simulations, L ≤ 64, seem to be too small to capture the asymptotic finite-size behavior. In any event, in case of a KT transition, the spin-stiffness ρ s drops asymptotically very rapidly to its universal critical value as a function of system size, being consistent with the relatively large values for the simulated finite lattices. Thus, a scenario with a KT transition between the SF and a narrow intervening disordered phase cannot be ruled out by the present large-scale simulations down to temperatures as low as k B T /J = 0.09625 ± 0.001. Of course, it is desirable to quantify the role of biconical fluctuations in the quantum case as well. However, accessing the probability distributions of the tilt angles studied in Sect. II for the quantum case is beyond the scope of the present numerical analysis.
IV. SUMMARY
We studied the classical and quantum, S = 1 2 , versions of the XXZ Heisenberg antiferromagnet on the square lattice in an external field along the easy axis. The model is known to display ordered AF and SF as well as disordered, paramagnetic phases. Here we focused attention to the region of the phase diagram near and below the temperature where the two boundary lines between the AF and the SF phases and the disordered phase approach each other, meeting eventually at a triple point. We performed Monte Carlo simulations, augmented, in the classical case, by a ground state analysis.
In the classical version, we presented first direct evidence for the importance of biconical structures in the XXZ model. Indeed, such configurations do exist already as ground states at the critical field H c1 , separating the AF and SF phases. The interdependence of the two tilt angles, characterizing the biconical ground states, persists at finite temperatures, in the region where the narrow phase between the AF and SF phases is expected to occur. Indeed, the joint probability distribution of the tilt angles at neighboring sites demonstrates the thermal significance of those configurations. Previous arguments on O(3) symmetry in that region and down to zero temperature thus have to be viewed with care. The results of the present simulations suggest that, if the biconical configurations do not lead to a stable biconical phase in two dimensions, the narrow intervening phase is a disordered phase characterized by biconical fluctuations. In this sense the "hidden bicritical point" at T = 0 may then be coined into a "hidden tetracritical point."
In the quantum version, previous simulations sug-gested, on lowering the temperature, the existence of a tricritical point on the boundary line between the AF and disordered phases, followed by a critical endpoint being the triple point of the AF, SF and disordered phases, and eventually by a first-order transition between the AF and SF phases at sufficiently low temperatures. The present simulations, considering larger system sizes and improved statistics, provide evidence that this scenario, if it exists at all, has to be shifted to lower temperatures than proposed before. Of course, simulations on even larger lattices and lower temperatures would be desirable, but are extremely time consuming. A clue on possible distinct phase diagrams for the classical and quantum versions may be obtained from an analysis of biconical fluctuations in the quantum case. Experimental studies on signatures of those fluctuations are also encouraged.
PACS numbers: 75.10.Hk, 75.10.Jm, 75.40.Mg, 05.10.Ln
FIG. 1 :
1Ground state configurations of the classical model sketched by the directions of spins on the two sublattices (i. e. at neighboring sites), from left to right: AF, SF, and biconical state. The circles denote the trivial degeneracy in the xyplane.
FIG. 2 :
2Detail of the phase diagram of the XXZ model on a square lattice with ∆ = 4 5 , see Ref. 15. Squares refer to the boundary of the SF, circles to that of the AF phase. The solid line refers to the magnetic field H/J = 2.41, where the probability distribution P (θm, θn), depicted in Fig. 3, has been obtained. Here and in the following figures error bars are shown only if the errors are larger than the symbol size and dotted lines are guides to the eye.
Fig. 1 ,
1is given by
FIG. 3 :
3Joint probability distribution P (θm, θn) showing the correlations between the tilt angles θm and θn on neighboring sites m and n for a system of size L = 80 at H/J = 2.41, kBT /J = 0.255, and ∆ = 4 5 . P (θm, θn) is proportional to the gray scale. The superimposed black line depicts the relation between the two angles in the biconical ground state, see Eq. (4).
FIG. 4 :
4Phase diagram of the XXZ Heisenberg antiferromagnet with spin-1 2 and ∆ = 2 3 . The straight solid lines denote the choices of parameters where our very extensive simulations, discussed in the text, have been performed. The arrows mark the previously 18 suggested locations of the tricritical point (Tt) and the critical endpoint (Tce).the simulations at nearby temperatures, when fixing the field at H = 2.41J, as well as in the vicinity of the entire transition region between the AF and SF phases, seeFig. 2, at higher fields and temperatures.
FIG. 5 :
5Positions of the maxima of the magnetization histograms as a function of the inverse system size. The inset exemplifies two histograms for systems of size L = 32 (circles) and L = 150 (squares) at kBT /J = 0.13 and the coexistence fields H/J = 1.23075 and H/J = 1.232245.
FIG. 6 :
6Doubly logarithmic plot of the staggered magnetization (M z st ) 2 vs. the system size L at H/J = 1.225 for the temperatures kBT /J = 0.095, 0.0955, 0.096, 0.0965, 0.097, and 0.0975 (from bottom to top). The straight line proportional to L 1 4 illustrates the expected finite-size behavior close to a continuous transition of Ising type.
FIG. 7
7: a) Spin stiffness ρs/J vs. temperature kBT /J for the different system sizes L = 8 (circles), 16 (squares), 32 (diamonds), and 64 (triangles). The straight line denotes the critical value of the spin-stiffness according to the formula of Nelson and Kosterlitz 29 , see Eq. (9). b) Finite-size behavior of the spin-stiffness ρs/J at H/J = 1.225 as a function of the inverse system size, 1/L for the temperatures kBT /J = 0.0955, 0.096, 0.0965, 0.097, and 0.0975 (from top to bottom). The dashed curve illustrates the estimated asymptotic behavior according to Eq. (10) with kBTKT/J = 0.09625 and C0 = 5, the corresponding critical value ρs(TKT, L = ∞) ≈ 0.0613 is marked by the filled circle.
AcknowledgmentsFinancial support by the Deutsche Forschungsgemeinschaft under grant No. SE 324/4 is gratefully acknowledged. We thank A. Honecker, B. Kastening, R. Leidl, A. Pelissetto, M. Troyer, and E. Vicari for useful discussions and information.
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| [] |
[
"Reinforcement learning and decision making via single-photon quantum walks",
"Reinforcement learning and decision making via single-photon quantum walks"
] | [
"Fulvio Flamini \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria\n",
"Marius Krumm \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria\n",
"Lukas J Fiderer \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria\n",
"Thomas Müller \nDepartment of Philosophy\nUniversity of Konstanz\nUniversitätsstraße 1078464KonstanzGermany\n",
"Hans J Briegel \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria\n"
] | [
"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria",
"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria",
"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria",
"Department of Philosophy\nUniversity of Konstanz\nUniversitätsstraße 1078464KonstanzGermany",
"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 21aA-6020InnsbruckAustria"
] | [] | Variational quantum algorithms represent a promising approach to quantum machine learning where classical neural networks are replaced by parametrized quantum circuits. Here, we present a variational approach to quantize projective simulation (PS), a reinforcement learning model aimed at interpretable artificial intelligence. Decision making in PS is modeled as a random walk on a graph describing the agent's memory. To implement the quantized model, we consider quantum walks of single photons in a lattice of tunable Mach-Zehnder interferometers. We propose variational algorithms tailored to reinforcement learning tasks, and we show, using an example from transfer learning, that the quantized PS learning model can outperform its classical counterpart. Finally, we discuss the role of quantum interference for training and decision making, paving the way for realizations of interpretable quantum learning agents. * These authors contributed equally to this work. | 10.48550/arxiv.2301.13669 | [
"https://export.arxiv.org/pdf/2301.13669v1.pdf"
] | 256,416,304 | 2301.13669 | d366fac0f044f7f3afdf998364d5973a00c7814e |
Reinforcement learning and decision making via single-photon quantum walks
Fulvio Flamini
Institut für Theoretische Physik
Universität Innsbruck
Technikerstraße 21aA-6020InnsbruckAustria
Marius Krumm
Institut für Theoretische Physik
Universität Innsbruck
Technikerstraße 21aA-6020InnsbruckAustria
Lukas J Fiderer
Institut für Theoretische Physik
Universität Innsbruck
Technikerstraße 21aA-6020InnsbruckAustria
Thomas Müller
Department of Philosophy
University of Konstanz
Universitätsstraße 1078464KonstanzGermany
Hans J Briegel
Institut für Theoretische Physik
Universität Innsbruck
Technikerstraße 21aA-6020InnsbruckAustria
Reinforcement learning and decision making via single-photon quantum walks
Variational quantum algorithms represent a promising approach to quantum machine learning where classical neural networks are replaced by parametrized quantum circuits. Here, we present a variational approach to quantize projective simulation (PS), a reinforcement learning model aimed at interpretable artificial intelligence. Decision making in PS is modeled as a random walk on a graph describing the agent's memory. To implement the quantized model, we consider quantum walks of single photons in a lattice of tunable Mach-Zehnder interferometers. We propose variational algorithms tailored to reinforcement learning tasks, and we show, using an example from transfer learning, that the quantized PS learning model can outperform its classical counterpart. Finally, we discuss the role of quantum interference for training and decision making, paving the way for realizations of interpretable quantum learning agents. * These authors contributed equally to this work.
INTRODUCTION
Machine learning methods are revolutionizing science and technology, with applications ranging from drug discovery [1] to the study of quantum phases of matter [2]. Within this toolbox, reinforcement learning (RL) has demonstrated the capacity to solve tasks better than the human state of the art [3]. Currently, the leading method in machine learning is given by artificial neural networks, with performance beyond theoretical expectations. The main near-term approach to quantum machine learning replaces these artificial neural networks with variational quantum circuits, typically consisting of fixed two-qubit gates and parametrized Pauli rotations. Popular platforms include, e.g., superconducting circuits [4] and ion traps [5]. Another promising platform is offered by photonic circuits, consisting of a fixed, regular mesh of tunable Mach-Zehnder interferometers [6,7].
In this paper, we propose a variational approach to quantum RL based on linear-optical interferometers and on a classical learning model called projective simulation (PS) [8,9]. In PS, the agent's decision making process is modeled as a random walk of an excitation in an episodic memory. Following an approach already investigated by some of the authors [10], we replace this classical random walk with a single-photon quantum walk in an optical interferometer. On the one hand, we are able to preserve the core features of classical PS, so that the main ideas developed in the field can be transferred to the quantum domain. On the other hand, this approach can be tailored to end applications, with actual constraints and needs in mind. These two points link to the main motivation of this work, that is, the development of a learning agent that operates in distributed quantum optical networks [11], and that can interact with other agents by exchanging quantum information encoded in light.
The article is structured as follows: In Sec. I, we review the standard PS model that provides the background of our work. In Sec. II, we present our approach to quantizing PS and, based on that, to designing a photonic implementation.
In Sec. III, we introduce two variational training algorithms tailored to the proposed framework, whose applicability goes beyond the scope of this work. In Sec. IV, we provide numerical evidence that such quantum PS agents can generalize better than their classical counterparts. Most of our analyses can be directly tested with a package we make available online [12] (see App. A), and can be further extended in various directions (using the same package; see Apps. B-D).
I. PROJECTIVE SIMULATION
In this section, we review the basics of RL and PS. RL is a paradigm of machine learning that involves an agent interacting with an environment (see Fig. 1). The agent receives from the environment an input s ∈ S, called state or percept, and decides what action a ∈ A to take next. RL typically considers Markov decision problems, for which the agent's policy can be described by a conditional probability distribution π(a|s). The agent receives immediate feedback on its actions through a reward R; its goal is to find the policy that maximizes the long-term expected reward.
In this work we focus on PS, a physically motivated RL model that has been applied to the study of collective behavior [13], robotics [14], automated experiment design [15] and quantum error correction [16]. PS relies on a memory structure called episodic and compositional memory (ECM), described by a directed graph (see Fig. 1). The vertices of this graph, called clips, are associated with memories of the agent such as remembered percepts, actions or intermediate, more complex steps in the decision making process. Importantly, this means that a clip carries semantic information [17]. When the environment delivers a percept s, the corresponding clip is excited, and a random walk of a single excitation is initiated in its clip network (see Fig. 2), until the excitation hits an action clip a. Each directed edge from clip c i to clip c j is associated with an unnormalized transition probability h ij called h-value. The transition probability p(c j |c i ) is computed by normalizing over all clips that can be reached from c i : . This work uses PS as a framework for reinforcement learning and its quantization. a) When a PS agent receives a percept s from the environment, it initiates a random walk in the graph that describes its memory, until it hits a vertex representing an action a. The performed action may be rewarded, leading to an update of the parameters (h-values) that control its decision making process. b) In classical PS, the vertices of the graph represent clips (c k ). Directed edges are associated with transition probabilities that depend on the h-values. c) In the quantized PS, clips are represented by mode operatorsĉ † k describing a single excitation, while the connectivity is governed by mode-mixing transformations U k . Each U k acts on the mode operators associated with the parent clips of c k (here c 1 , c 2 ), and its complex elements are the transition amplitudes that replace the classical probabilities of transitioning to c k . All panels: all edges that contributed to the decision making process are in green.
p ij ≡ p(c j |c i ) = h ij j h ij .(1
Learning occurs by tuning the h-values, which are initially set to 1. To this end, each edge with h-value h ij is additionally equipped with a so-called glow factor g ij , which contributes to the standard PS update rule [18]:
h (t+1) ij = 1 + (1 − γ)(h (t) ij − 1) + g (t) ij R.(2)
Here, the role of the glow factor g ij is to remember transitions (c i → c j ) taken in past interactions with the environment, so that they are reinforced when a reward R is received. Further, γ ∈ [0, 1] is a forgetting factor: its main purpose is to gradually dampen previously learned transition probabilities, to let the agent adapt to changing environments. γ is also used as a soft cut-off to prevent the h-values to overflow. Each g ij is updated as:
g (t+1) ij = 1 if c i → c j was taken at t, (1 − η) g (t) ij otherwise,(3)
where g
ij = 0 and η ∈ [0, 1] is a discount factor for the actions taken further in the past. We refer to the literature on PS for a more detailed description and additional mechanisms, such as association [19] and reflection [20,21].
II. QUANTUM PROJECTIVE SIMULATION
Since decision making in PS is generated via a random walk of an excitation, it is possible to quantize it by giving learning agents access to a quantum walk in a quantum memory. In the literature, several variants of this idea have been investigated.
In Ref. [20], an algorithm based on discrete quantum walks was presented which provides a quadratic speedup. However, this algorithm relies on sophisticated oracles that likely will not be available in the near future. Furthermore, since the algorithm is described as a quantum circuit, it is not clear how it can be interpreted as a random walk of a physical excitation. On the other hand, in Ref. [8], it was proposed to quantize PS by considering a quantum walk of an excitation in a quantum many-body system. Since Hamiltonians are hermitian, in such systems the probability amplitude of a transition is as large as the amplitude of the backwards transition. To break this symmetry, the authors proposed to use dissipative effects ('quantum jumps'), at the cost of coherence.
In this section, we propose a quantum extension of PS that allows for an optical implementation. Our guiding principle is to design a PS agent natively as a quantum walk of a physical excitation. In particular, our proposal does not rely on the availability of quantum oracles. Our scheme uses an optical interferometer that induces a natural directionality of transitions, without the need for dissipation. We will first describe how to implement the main elements of PS in the quantum domain in general and then, in Sec. II B, on an optical platform.
A. From classical to quantum projective simulation
In order to quantize PS and facilitate an optical implementation, we need to quantize its main elements: the episodic memory, realized by the clip network, and the decisionmaking process, described by the propagation of an excitation through the clip network. A physical realization of a clip network could be a system of harmonic oscillators (modes) with non-trivial many-body interactions between sets of oscillators [8]. In case of an optical implementation, clips c i ∈ C (including percept clips, action clips and intermediate clips) then correspond to modes in an optical interferometric network [10] (see Sec. II B). For the quantization, we associate with each clip c i a pair of creation and annihilation operators of the corresponding (optical) mode, {ĉ † i ,ĉ i }. An excitation of memory clip c i at time τ 0 is then described by the quantum stateĉ † i (τ 0 ) |vac , where |vac denotes the vacuum state, describing the inactive state of the agent's memory. Here we use the Heisenberg picture where the operators carry the agent's internal time dependence. Note that we index RL steps with t, while physical time within a step is denoted by τ k .
A decision-making process is initiated by the excitation of some clip c i at time τ 0 . The subsequent evolution of the agent's memory will be described by a unitary of the form
c † j (τ n ) = U † (τ n , τ 0 )ĉ † j (τ 0 )U(τ n , τ 0 ) = |C| i=1 U ECM jiĉ † i (τ 0 ) ,(4)
which relates the excitation operators of the clip network at different times. The Heisenberg time evolution of the ECM U(τ n , τ 0 ) is thus realized by the interferometric scattering matrix U ECM connecting the modes, which, in turn, is composed . . , c |C| } that describes the agent's episodic and compositional memory (ECM) [17]. When the agent receives a percept s (blue), an excitation triggers and performs a random walk over the network of intermediate clips {c 1 , . . . , c |I| } (green) until it hits one of the action clips {a 1 , . . . , a |A| } (red), which is immediately coupled out. b) In the proposed quantized PS, these random walks become quantum walks of single excitations through layers k = (1, . . . , n) implementing the transformations U k , which mix sets of clips in a certain order (see App. H). Here, clip-to-clip transition probabilities, classically described by the h-values, are replaced by transition amplitudes described by the matrix elements of U k . c) The evolution can be seen as hopping between eigenstates (of the modes) in the Hilbert space (empty disks: in/out modes; filled disks: output modes). This panel reproduces the quantized evolution in the classical ECM in panel (a). Clips are now visited in superposition (green).
of a sequence of n intermediate transformations U k ,
U ECM ( θ) = n k=1 U k ( θ) .(5)
The U k are updated during the learning process via the set of parameters θ (see Fig. 2). The precise role of the U k ( θ) will be discussed in more detail in the following. For convenience and to facilitate an optical implementation, in this work we will consider ECMs that are described by a directed acyclic graph (DAG; see App. H).
Percept encoding -When the agent receives a classical percept at time τ 0 , this leads to the excitation of a percept clip s. The quantum state of the memory is then described by c † s (τ 0 ) |vac . Subsequently, a quantum walk through the clip network starts [8].
Quantum walk in the ECM -To connect the random walk in the ECM to the quantum walk considered in this work, we refer to the examples in Figs. 1 and 2. As shown in panel Fig. 1b, a clip effectively acts as a node that routes incoming edges from sets of clips to other clips in the next step of the random walk. In the quantum ECM sketched in Fig. 1c and Fig. 2b-c, this mechanism translates into the application of a transformation U k , which mixes creation operators of incoming clips withĉ † k . A general clip-to-clip transition c i → c j is then modeled by transformations U k that mix subsets of creation operators, so that the stochastic nature of PS manifests itself at the level of quantum amplitudes.
A crucial departure from classical PS is that clips are now visited in parallel (in superposition, see Fig. 1c), and all regions of the ECM that are reachable from a percept via a sequence of U k contribute to the decision making process. As a consequence, the probability to find an excitation in action clip a at time τ n (described by the quantum state c † a (τ n ) |vac ) when starting with an excitation of percept clip s at time τ 0 (described by c † s (τ 0 ) |vac ) is given by the modulus square of the corresponding transition amplitude
p sa = | a|s | 2 ≡ | a; τ n |s; τ 0 | 2 = U ECM as ( θ) 2 ,(6)
where |c i ; τ k =ĉ † i (τ k ) |vac are the time-dependent eigenstates of the number operatorsĉ † i (τ k )ĉ i (τ k ) corresponding to an excitation being present in clip c i at time τ k . Eq. (6) entails a quantum interference of all paths. While in classical PS the excitation has a well-defined trajectory in the ECM, now the excitation can be delocalized over all the allowed modes (i.e., the quantum amplitude is in general non-zero for more than one mode at the same time). This aspect bears an interesting potential for the learning model, which has already been discussed in the literature [8,22] and that is further discussed in Secs. III-IV and in the Appendix.
Action decoding -The decision-making process ends when U n is applied and the output states are measured. When the excitation is detected in an output state associated with an action, this action is coupled out. In all other cases, the excitation can be (i) routed back to other regions of the reachable ECM; (ii) discarded, exciting again the same percept. Solution (ii) can be interpreted as the agent's capacity to not take action during this decision-making process (see also the notion of suspension of judgment [23]).
Practically, the agent takes action by post-selecting over the outcomes that correspond to actions. In general, one can associate multiple output states with a single action. This makes it possible to avoid post-selection in settings with an action space smaller than the percept space, or to implement quantum channels that are more general than unitaries.
Training the quantum PS agent -To train the agent, we consider a variational approach based on a loss function L that is inspired by classical PS and aims to reproduce its learning mechanism. L depends on the probabilities p sa ( θ) to take action a given a percept s (that is, the policy) and the variational parameters θ. These probabilities can be estimated either numerically (by simulating the quantum walk) or experimentally (via multiple measurements with fixed input). In Sec. III A we introduce and motivate our choice for L (see Eq. 8), along with several modifications and training methods.
B. Quantum optical projective simulation
In this section, we describe our approach to implement quantum PS on a photonic platform. Table I summarizes the main connections with classical PS.
Related works -Photonic circuits provide a promising platform in both classical and quantum domains [6,7], with results of increasing complexity being continuously reported [24,25], in particular for (un)supervised learning [26][27][28][29][30][31][32] and RL [10,22] based on path encoding.
In this work, we focus on RL and aim to design a quantum PS agent that operates with a native quantum walk of physical excitations. Hence, we will replace the single excitation with a single photon propagating in a linear-optical circuit. This approach, while retracing some of the intuitions in Ref. [10] where the p sa were reproduced by optical binary trees, promises the following advantages: (1) it requires only a single (possibly larger) circuit for all percepts, instead of one binary tree per percept and layer, with no explicit need for fast active switching; (2) it allows to take advantage of quantum coherence and interference in the decision making process;
(3) it allows for more advanced training algorithms, which are applicable beyond the scope of this work (see Sec. III); (4) it suggests concrete directions for further developments (see Apps. B-D), such as a framework for learning with quantum data. In the following, we outline how the quantized version of PS can be realized on a photonic circuit.
Photonic architecture -For the implementation of PS on quantum hardware, we want to identify quantum extensions of the Markov chain process that govern the evolution of a classical excitation. These extensions, usually referred to as quantum channels (completely positive trace-preserving maps), would have to reproduce the dynamics in classical PS as a limiting case. A direct implementation of classical PS would require the {U k } to reproduce the stochastic maps that form the random walk in the ECM graph described in Sec. I. Specifically, the real-valued matrices P associated with these transformations need to (i) reproduce the probabilities p ij induced by the h-values, and (ii) fulfill the connectivity in the ECM (P can be seen as a weighted adjacency matrix). Such a class of transformations can in principle be realized on a photonic platform by recalling that any real-valued matrix P can be decomposed as P = V 1 ΣV † 2 via singular value decomposition, where Σ is a diagonal matrix with non-negative real entries (i.e., optical attenuators) and V 1,2 are unitary transformations [26]. Hence, the architecture we use for the variational circuit is the canonical square one [33], since it can implement any unitary transformation by suitably tuning its phase shifters. Indeed, we recall that any such U k can be implemented by a homogeneous mesh of Mach-Zehnder interferometers, each comprising two 50:50 beam splitters and two phase shifters (see Fig. 3 and App. F) [33,34]. The variational parameters θ correspond to the phase settings that control U k .
In the fully coherent case, without additional attenuators, one can set Σ = I in the singular value decomposition, meaning that the U k become unitary transformations. In this case, due to unitarity constraints and since each U k acts as an adjacency matrix, additional edges between clips are created that are not present in the original ECM. This limitation represents an additional departure from classical PS, which can open up interesting opportunities in the context of learning, for instance leveraging the constraints to improve generalization.
Percept encoding -We associate each percept with a photon input in a spatial mode of the circuit. The size of the Hilbert space grows linearly in the number of percepts.
ECM structure -We associate each clip with one optical mode of the architecture. The unnormalized transition probabilities h ij that describe the transitions c i → c j are now replaced by transition amplitudes between optical modes, described by a complex amplitude U k ji . Our chosen architecture is especially useful for reasons that go beyond its universality, noise resilience and compactness [33], since it also promises to support key features of PS:
• Clip creation/deletion: new clips (i.e., optical modes) can be added during the learning process, while clips that are rarely traversed can be removed. • Edge creation/deletion: the random walk can be manipulated by creating or deleting links between clips. • Clip composition: fictitious clips can also be created, by random variation or merging of existing clips.
We observe that this mesh allows one to carve substructures that are in turn universal for unitary transformations, by adjusting the surrounding phases (without the need to modify the hardware; see Fig. 3d). This means that the Hilbert space describing the ECM can be dynamically enlarged to accommodate new clips, and the U k ji can be controlled by intervening on the regions of the circuit that implement U k .
Action decoding -In PS, the decision-making process ends when an action clip is hit. In the proposed implementation, this corresponds to detecting photons from one of the output modes that correspond to an action clip. Whenever a photon is detected in a mode that does not correspond to an action, that event is discarded. In the future, these photons could be routed back into the circuit, or associated with mechanisms of suspension of belief [23].
III. TRAINING THE ARCHITECTURE
In this section, we discuss two algorithms to train the quantum optical PS agent. To this end, we first recall that the stan-dard update rule in PS has no straightforward quantum analogue: while only one path is taken by the classical excitation and rewarded by the update rule (technically, the h-values associated with its edges), all paths from percept to action are visited by the quantum excitation, and the very notion of path loses its meaning. Hence, also motivated by the advances in quantum machine learning, here we take a different, variational approach to update the parameters θ.
A. Variational approach to mimic classical PS
The first algorithm to train the quantum PS agent is based on a variational optimization of a loss function L inspired by classical standard PS.
Since a quantum circuit does not have access to the intermediate states of the computation, the update rule we propose adapts that of a classical 2-layer ECM. This does not mean that the quantum agent is necessarily 2-layered (in fact, the architecture supports multi-layer ECMs, as shown in Fig. 2); rather, it means that we update the observable transition probabilities p sa ( θ). In turn, also the intermediate transition amplitudes in the circuit are affected. However, the update rule of PS requires the use of unnormalized h-values instead of (normalized) probabilities. Thus, let us first introduce the classical normalization factor of the h-values h s , i.e. h s := a h sa , from which we get h sa = p sa h s and the classical 2-layer PS update rule
p (t+1) sa ( θ)h (t+1) s = 1 + (1 − γ) p (t) sa h (t) s − 1 + g (t) sa R. (7)
A loss function L that implements Eq. (7) is
L( θ (t+1) , h (t+1) s ) = s D p (t+1) sa ( θ)h (t+1) s − 1, (1 − γ) p (t) sa h (t) s − 1 + g (t) sa R(8)
Here, θ (t+1) and h (t+1) s are variational parameters, while D is a suitable distance measure such as the Kullback-Leibler divergence (after rearranging the arguments of D), R is the reward, and g sa is the glow factor described in Eq. (3). The expression in Eq. (3) is still valid in the proposed quantum extension, since the update rule underlying the loss function is taken from a classical two-layer PS, and both input (s) and output (a) are classical. However, we emphasize again that the quantum ECM itself is multi-layered.
Note that only the first argument of D in Eq. (8) contains tunable parameters, while the right entry contains the target values calculated from the previous state of the agent, as a reference, similar to deep-Q learning. Similarly, one can use experience replay [35] to collect rewards and glows for a batch of percept-action pairs, and add up or merge the loss functions in appropriate ways. That is because continuous online learning (i.e. updating after each action) has the disadvantage that the unitarity constraints will also affect the transition probabilities of undetected percept-action pairs, without taking their rewards into account.
In the following, we discuss suitable heuristic simplifications of this loss function, which make it more amenable to an actual implementation. First, by approximating h (t) s ≈ h s and introducing an effective reward r s := R hs in the exact update rule in Eq. (7), we obtain
p (t+1) sa ( θ) − 1 h s ≈ (1 − γ) p (t) sa − 1 h s + g (t) sa r s .(9)
Without a reward (i.e. r s = 0), the forgetting mechanism after the approximation seeks to reset p (t) sa to 1 hs . However, in information theory, the no-information distribution is the uniform distribution. This motivates us to replace 1 hs with a uniform distribution 1 |A| ,
p (t+1) sa ( θ) = 1 |A| + (1 − γ) p (t) sa − 1 |A| + g (t) sa r s . (10)
So far, we did not make any choices concerning the source of the rewards R. For our simplifications, we now take the point of view that the reward R is designed by a programmer to achieve an intended agent behavior. Then, a further simplification can be made by directly modelling the effective reward r, rather than R. Therefore, we replace r s with r. With this change in mind, Eq. (10) now looks like an analogue of the exact update rule in Eq. (7), but for probabilities instead of h-values. To ensure that the agent updates get smaller over time, it is recommended to use an "annealing schedule" for the reward, i.e. to put an additional time dependence into the reward of the form r := f (t) · r 0 . Here, f (t) is a function that monotonically decreases from 1 to 0. This allows the training algorithm to make finer and finer updates to the probabilities. In order to stay as close as possible to classical PS, the update rule would have to be applied to all percept-action pairs. This would require implementing the forgetting mechanism (γ > 0) for unvisited percept-action pairs, or enforcing the transition probability of unvisited pairs to remain unperturbed (γ = 0), leading to a complicated loss function. To avoid this overhead, we include only the actually observed perceptaction pairs in the loss function, while the transition probabilities of unobserved pairs are allowed to change according to the unitarity constraints. Combining all of the above simplifications, we obtain the loss function
L( θ) = (s,a) | g (t) sa >0 D p (t+1) sa ( θ) ,(11)C 1 |A| + (1 − γ) p (t) sa − 1 |A| + g (t) sa r
where C is a (smooth) cutoff function to enforce that probabilities are in [0, 1], and only (s, a)-pairs with non-zero glow are considered.
B. Variational approach based on causal diamonds
In the previous section, we implicitly assume that at each step one tunes all phase shifters (i.e., parameters θ) in the circuit. Indeed, this is possible and can be beneficial to train the agent faster and with a better performance. At the same time, updating all phases at each step might not be necessary for the following reason.
In a square architecture with |C| input/output modes, the number of phase shifters is quadratic in |C|. However, for a fixed transition probability p sa ( θ), one can see that only some phases play a role: Those at the intersection of the future and past light cones of s and a, respectively (see Fig. 4a). In the literature on relativity, this intersection is called causal diamond. A number of considerations follow:
i. To update p sa ( θ), one could simply tune the phase shifters inside the causal diamond. ii. For light to reach a from s, what matters is that light does not leak out of their causal diamond. Hence, one can also just tune the O(|C|) phases that lie on the surface of the causal diamond. iii. Since leakage only occurs on the surface of the past light cone, it is sufficient to focus on its intersection with the causal diamond (see Fig. 4). We call these phase shifters leaking nodes.
Overall, in terms of reconfiguration time, stability and power consumption, the above observations allow for progressively simpler experimental requirements. The price to pay is that the surfaces of different causal diamonds (for different (s, a) pairs) intersect and, for this reason and due to unitarity constraints, updating one transition probability also affects the other transitions. Also, the fewer phase shifters are tuned, the smaller the set of values that solve the optimization problem when multiple (s, a) pairs are rewarded. The desired trade-off can be found with the experimental conditions in mind.
We consider two main strategies to tune the sets of relevant phases described by points (i-iii) above: (a) phases are independently updated by gradient descent, and (b) phases are updated sequentially (from left to right, each phase in a layer is set to a new value before the phases in the next layers are adjusted). The latter strategy is possible because we assume that light propagates only forward, and it leads to a faster and smoother training. A quantitative comparison between the two strategies, and a study of the most suitable one with multiple rewarded percept/action pairs, can be done with our shared package, keeping experimental requirements in mind [12]. An illustrative analysis for strategy (b) is shown in Fig. 4c. Additional information can be found in App. E 2.
Importantly, we point out that this approach is not restricted to the regular mesh considered in this work. Rather, similar considerations also apply to circuits with arbitrary mode connectivity and even higher-level structures in the ECM [12]. a) Only the phases inside the causal diamond of (s, a) influence psa (colored nodes indicate the surface of the causal diamond). Since light can only leave the causal diamond from the purple nodes (leaking nodes), it is sufficient to tune their phases. b) The approach works also with multiple transitions, by iteratively tuning different sets of leaking nodes (here shown with five different colors, including gray for the overlaps, and representing pairs of adjacent phase shifters and beam splitters as a single node), even if the updates compete with each other (some leaking nodes belong to multiple causal diamonds). c) The agent learns to maximize psa for the four (s, a) pairs in panel (b), by tuning only the leaking nodes. In some cases, it may be necessary to tune more nodes to maximize all probabilities. Each percept/action is encoded in a pair of adjacent input/output modes.
C. Direct updating via a Gram-Schmidt process
The previous sections consider a variational approach to the optimization problem, where phases are gradually adjusted to produce a new transformation that behaves better under a given figure of merit. We now consider a complementary approach, where we first update the unitary evolution U of the ECM, and then use a well-known decomposition [33] to retrieve the θ that produce the desired U . In this case, one directly adjusts the matrix elements of U to implement an update rule such as Eqs. (7) or (11): the new matrix is no longer unitary; however, a Gram-Schmidt orthonormalization (GSO) process yields a new U that approximates the intended U as .
We add a few remarks on this approach (see also App. E 1). First, if U as is close to the old value U as , one step of GSO only causes a small update of the other entries of U . In this sense, this approach allows for a smooth, flexible and controlled update of U (see Fig. 5). Also, both the GSO and the unitary decomposition are efficient and relatively fast subroutines, which could even be faster than collecting statistics for other variational approaches. In other words, the overhead is outsourced to the classical CPU. The limitation of this approach is that it assumes perfect knowledge of U and a precise control of the experimental settings. The impact of imperfections can be estimated using the shared package [12].
IV. QUANTUM VERSUS CLASSICAL PROJECTIVE SIMULATION IN A TRANSFER-LEARNING SCENARIO
In this section, we apply the proposed quantum PS to an actual learning problem. The PS update rule of Eqs. (7) and (8) has already been used in various applications [13][14][15][16]. Here, we consider a scenario which pronounces the differences between classical and quantum PS. We consider a transfer-learning scenario [36]: a machine learning setting where a model trained for one task can solve another task. The scenario we choose is that of Eva et al. [37] (see Fig. 6). the unitary transformation U that describes the ECM using a Gram-Schmidt orthonormalization. Here, a learning curve (purple) is shown for a single rewarded (s, a) pair in a 2-layer PS described by a 10 × 10 Haar-random U . All other transition probabilities (other colors), from s and to a, decrease to 0. Curves correspond to a single agent, without average. Inset: (|U ij | 2 ) ij after training (arrows indicate (s, a)).
In this scenario, one considers a list of particles characterized by discretized observables, and asks whether two observables have two particular values. To answer these questions, or tasks, in our adapted version of this thought experiment the agent attempts to build a meaningful representation of the particles in two stages (corresponding to the middle layer and a task layer, see Fig. 6a): In stage 1, the agent learns a representation from the percept layer to the middle layer, where each node corresponds to one observable and one value of that observable. While Eva et al. intended that this representation is learned implicitly, in our case we directly train the middle layer to achieve this representation. In stage 2, the connections from percept to middle layer are kept fixed. The task layer has two nodes, which correspond to yes/no-answers. For each question (task), we train a separate set of connections from the middle layer to the corresponding task layer. (i) In the first stage, the PS agent is set up to learn a middle layer of nodes, in which each node corresponds to one observable and one value of that observable. (ii) In the second stage, this middle layer is kept fixed. The task layer is exchangeable and it represents one yes/no question about pairs of observables. Since the excitation of the classical ECM is localized, in the middle layer the classical agent can only use the value of one observable. Interference allows the quantum PS agent to combine knowledge of several observables at once. c) Layout of the quantum PS agent. To fulfill the unitarity constraints, the connection from percept to middle layer is an interferometer tree with 9 outputs (one tree per percept). The connection from the middle layer to the task layer is a 9 × 9 square architecture, shared by all percepts but different for each task layer. Two output modes correspond to the yes/no answers, the others are discarded. d) Prediction accuracy A Q of the quantum PS agent for each of the 27 experiments (yes/no question). Inset: Learning curve for a single experiment (dark orange). A classical agent can achieve at most an accuracy A C = 8/9 (horizontal line, for both plots), always lower than A Q .
The contrast between classical and quantum learning is rooted in the way the decision-making process handles the structured representation of the PS agents: Since the excitation of the classical PS agent is localized, it can only use knowledge about one observable and, therefore, it cannot reliably answer questions involving two observables. The quantum PS agent, however, can use interference to combine knowledge about two observables.
To be more specific, Eva et al. [37] consider three observables, each of which can take three values. We label these observables and their values 0, 1, 2. Each percept can be written as (v 0 , v 1 , v 2 ) with v j ∈ {0, 1, 2}, and v j is the value of observable O j . In total, there are 3 3 = 27 percepts. In addition to the percept layer, there is a middle layer and a final layer (see Fig. 6). The purpose of the middle layer is to explic-itly represent the value of each observable. More specifically, this layer contains 3 2 = 9 nodes, each corresponding to one observable-value pair. For each percept (v 0 , v 1 , v 2 ), the idea is that the agent learns to visit the middle clips (O j , v j ) with uniform probability 1 3 , while not visiting the other middle clips (O j , v) corresponding to wrong values v of the observables.
For the proposed quantum PS, we represent the correct middle-layer state of percept
(v 0 , v 1 , v 2 ) as 1 √ 3 (|v 0 + |3 + v 1 + |6 + v 2 ).
For example, (2, 1, 2) is represented by 1 √ 3 (|2 + |4 + |8 ). Since these states can be non-orthogonal for different percepts, for each percept we use a separate binary tree (see Fig. 6b) with 9 output leaves (27 such trees in total), instead of the unitaries U k shown in Figs. 1 and 2.
To make a fair comparison to a classical PS agent, we consider the following training. Each of the 27 percepts is sampled with uniform probability. Then the middle layer is measured to output one (observable, value) pair. Depending on whether the pair is predicted right or wrong, a fixed reward ±r is given. We use the simplified loss function of Eq. (11) without forgetting-mechanism (γ = 0), and only reward the current percept-action pair (η = 1). To enforce that all relative complex phases are close to zero, we add the 1 norm of these phases to the loss function. Since we use separate binary trees for each percept, and the goal of the tree is just to prepare one simple state, the middle layer can be trained to achieve the intended representation essentially perfectly.
After the middle layer is trained, we fix its weights, meaning that we will not adjust the binary trees for anything that follows. Now, we introduce the final layer, which we call the task layer. The task layer represents yes-no-questions about the values of observables. Our transfer learning scenario requires that the task layer be exchangeable, while we keep the middle layer fixed and independent of the task layer. The task layers we consider here pick two observables O, O and ask whether these observables have specific values v O and v O . There are 27 = 3 2 · 3 2 such task layers. The quantum PS agent uses a 9x9 square architecture for each task layer, as shown in Fig. 6c). We emphasize that the same task unitary is used for each percept. Only two output modes are used, which correspond to yes/no-answers. The 27 percepts are sampled uniformly. Fig. 6d) shows the achieved (weighted) accuracies for all two-observable task layers. Our agent consistently achieved over 97% (weighted) accuracy for all task layers. The analysis was implemented with PyTorch; details are provided in App. G.
While the proposed quantum PS agent can achieve near perfect transfer learning, one can use analytical reasoning to see that the classical PS agent cannot achieve near perfect accuracy: Since the classical excitation is always localized in one node, in the middle layer the classical PS agent can at most know the value of one relevant observable. Therefore, the best thing the agent can do is to guess about the other observable. Since two out of three possible values of that other observable lead to a right no-answer, the best policy the classical PS agent can represent is to always answer no. Hence, the performance of the classical PS agent is upper bounded by 8 9 .
DISCUSSION
In this work, we introduce a variational approach to quantum reinforcement learning that can be implemented with integrated photonic quantum technologies. To this end, we started from preliminary results by some of the authors [10], based on the projective simulation (PS) learning model [8]. This allowed us to establish a map between the key elements of classical PS and its counterparts in the quantum domain. At the core of this analogy lies the random walk of a physical excitation, which becomes a quantum walk in the agent's memory. We then discussed an approach to implement the decision making process and training of the quantum PS agent using photonic circuits.
To this end, in Sec. III we introduced several variational methods to train the agent, all compatible with, or tailored to, a quantum optical implementation. The variational parameters are represented by the phases in the optical architecture, given by the canonical mesh of Mach-Zehnder interferometers. More specifically, we proposed a variational method that is general and can accommodate custom loss functions inside the update rule. In particular, it allows to draw a close connection to the update rule of classical PS. We proposed a modification of the variational training that exploits features of the interferometer implementation to ease the experimental requirements (by reducing the number of parameters that need to be updated during the learning process), which makes training the architecture more efficient and flexible. Furthermore, we proposed an alternative method for updating that consists of a deterministic and exact algorithm to update the overall unitary transformation that describes the quantum PS agent's memory. All algorithms can also be combined to refine the update or to address more complex structures in the memory.
As a specific numerical experiment that demonstrates im-portant differences between classical and quantum PS, we considered a transfer learning scenario. We show how quantum interference allows the quantum PS agent to combine information from several clips, while the classical PS agent only has access to the information of one clip at a time. This work also sets the stage for a number of follow-up investigations, which we outline in the Appendix with additional numerical analyses. In particular, Apps. B and C discuss the potential offered by multi-photon (i.e., multi-percept) and multi-frequency quantum information processing, respectively. Finally, App. D takes a first step towards a notion of interpretability in quantum artificial intelligence, an issue that has attracted more and more attention in the last few years and that is especially relevant within PS. All results can be tested with a package we release online [12].
C 4 C 2 C 3 C 5 C 6 C 7 C 8 C 9 C 1 C 4 C 2 C 3 C 5 C 6 C 7 C 8 C 9 C 1 C 4 C 3 C 7 C 5 C 6 C 8 C 9 C 2 C 4 C 5 C 6 C 7 C 9 C 8
APPENDIX Appendix A: Package overview
The above considerations can be tested and explored with a Mathematica package we make available online [12,38]. Below we describe its main functionalities.
• Function 1 allows to study the structure of the quantum ECM proposed in this work, starting from the directed acyclic graph (DAG) that describes the connectivity of the ECM. It can be used to test the learning model and to develop extensions. This function operates at an abstract level and, specifically, does not presume an optical implementation. Functions 2-4 below can interact with its output by manipulating the transformations associated with each vertex of the DAG. Simple examples and details can be found in Fig. 7.
Function 1 -InitializeQuantumECM
Input: The directed acyclic graph (DAG) G describing an ECM. Output: An equivalent G with explicit layers; the vertices of G that correspond to percepts and actions; all subgraphs of G that are reachable from each percept (where quantum walks take place); a description of the clip-to-clip connectivity in G; extra information.
• Functions 2-4 have been adapted from a package from Ref. [39]. They make it possible to manipulate unitary transformations on a linear-optical circuit, with a focus on the canonical decompositions. These routines serve to instantiate the abstract representation of a quantum ECM, and al- Gram-Schmidt orthonormalization can be applied to the U k associated with an arbitrary path in the ECM. In this example, we reinforce the transition probability psa by updating the U k along the longest path (1,4,5,20) that connects a percept (s: green, k = 1) to an action (a: red, k = 20) for the ECM shown in the inset. Here, we assume that at each step the same (s, a) gets rewarded, for three different factors α (see Sec. E 1). Inset: The method based on causal diamonds can be extended to the ECM graph. While in Sec. III B the analyses zoomed in the structure of each U k , one can also zoom out and reinforce psa by tuning the leaking nodes ("leaking U k ") of (s, a) along the quantum walk (black vertices, k ∈ {5, 6, 14}). Other percepts (actions) are colored in light green (light red).
low to study arbitrary circuits as graphs. They handle the θ parametrization in two formats, one for canonical architectures and one for arbitrary connectivities.
Function 2 -UnitaryDecomposition
Input: A unitary U and the choice of an algorithm for its decomposition. Output: For canonical architectures [33,34], it decomposes U into phases and transmissivities. If beam splitters are implemented as Mach-Zehnder interferometers, it returns the appropriate phase settings.
Function 3 -GetUfromParameters
Input: Parameters in the format handled by Function 2, or in a layer-wise format that allows arbitrary connectivity.
Output: Unitary generated by any of three methods with the input parameters (see Function 2). Optional noise and/or losses can be set.
Function 4 -GetGraph
Input: Information on the mode connectivity in the circuit, in the format handled by Functions 2 and 3.
Output: Graph representation of the input circuit.
• Functions 5-6 enable the update based on the causal diamond for a 2-layer PS (see Sec. III B and Fig. 4) and multilayer PS (see Fig. 8). They make use of the graph representations given by Functions 1 and 4. Function 7 implements the update based on the Gram-Schmidt process (see Sec. III C). Output: Vertices of G in the causal diamond and leaking nodes of (s, a).
Function 6 -UpdateCausalDiamond
Input: Layered parametrization of the circuit (partially pre-computed for a faster evaluation); a percept/action pair (s, a); phase shift δθ.
Output: New layered parametrization and the corresponding unitary.
Function 7 -UpdateGramSchmidt
Input: A unitary U ; a percept/action pair (s, a); a rescaling factor α. Output: Updated U after one Gram-Schmidt process around Uas.
Appendix B: Multi-photon quantum walks
In this work, we replaced the excitation at the core of the decision making process of PS with a single-photon excitation in a quantum walk. In this picture, a photon is always injected into the mode that corresponds to a percept, and the new action is given by the output mode the photon is measured in. An interesting extension of this framework considers the simultaneous injection of multi-photon states. A quantum PS agent equipped with this feature could be able to process multiple percepts at the same time, or a percept with a more complex structure. Furthermore, information encoded in n indistinguishable photons injected in n different input modes would interfere at the level of quantum amplitudes, potentially opening up advantages with respect to the classical scheme, where such a feature is currently and independently under investigation. Foreseen potential advantages in this sense are: (i) the advantages inherited by a multi-excitation classical PS; (ii) the exponentially larger (in n) percept/action (Hilbert) space; (iii) quantum computational advantages related to the hardness of simulating the dynamics [40].
Appendix C: Quantum walks and frequency encoding An interesting possibility offered by an optical implementation is to use additional degrees of freedom of photons, besides the spatial modes, to encode information and/or to train multiple agents in parallel. In particular, frequency appears to be a convenient candidate since the operation of integrated photonic circuits is wavelength-dependent. In fact, both beam splitters' and phase shifters' transformations depend on the wavelength as follows [41]:
U bs λ ∝ cos π 4λ − sin π 4λ sin π 4λ cos π 4λ , U ps λ (P ) ∝ 1 0 0 e i P λ (C1)
where P is the (suitably rescaled) power setting that controls the phase shifter. With these simplified expressions, we can already get an idea about the average impact of frequency encoding on unitary transformations U of increasing size. Results for this analysis are shown in Fig. 9a. We see that, as expected since the architecture consists of hundreds of optical elements, a tiny variation in wavelength λ leads to significantly different effective U s. This means that photons injected with these deviations would undergo different U s, possibly implementing quantum walks in different quantum ECMs. Finally, we investigate whether a variational approach can work also while additional information is encoded in the frequency degree of freedom. To do so, we simulate quantum learning agents that attempt to solve the same RL task (same rewarded percept/action pairs) using different λs. Incidentally, this analysis is similar to, but more sophisticated than, the one presented in Fig. 4d. Results are shown in Fig. 9b, using the approach based on the causal diamond in a 2-layer PS (see Sec. III B). We see that the agent learns to map each of five percepts to the corresponding rewarded action, at the same time for three different λs. Importantly, all phase settings are the same for all λ, that is, the phase configuration found by the algorithm operates well (as shown) when input with different λ.
The above considerations suggest that there is a large potential in the use of additional degrees of freedom that influence the quantum walk. Foreseen potential advantages of this aspect are: (i) extra room for more complex RL mechanism; (ii) parallelism in the agent's decision making process. ) and (c) after training, respectively. We observe that a notion of partial trace emerges inside the circuit, which might be leveraged to study interpretability in quantum learning agents. Analogous results (not shown) are found for the numerical simulations in Fig. 8.
we proposed to consider the square canonical layout [33] for several reasons, above all because it supports the implementation of any unitary transformation U , with no modifications at the hardware level. This is possible because any m × m U can be decomposed as U = D b U b , i.e., the product of a diagonal matrix D (whose elements are complex and have modulo 1) and m(m − 1)/2 unitaries U b implementing complex rotations in the plane spanned by two adjacent modes:
U b = 1 . . . ,(F1)
where the u ij (i, j ∈ {1, 2}) describe the transformation acting on the two modes. This 2 × 2 transformation is implemented by means of a pair of phase shifter (θ 1 ) and imbalanced beam splitter, whose transformation U BS is implemented as a Mach-Zehnder interferometer with tunable phase shifter (θ 2 ):
U BS (θ 2 ) = 1 2 1 i i 1 e iθ2 0 0 1 1 i i 1 .(F2)
Overall, the proposed ansatz can be implemented using well-established, integrated photonic components that only involve tunable phase shifters and fixed, balanced beam splitters (directional couplers). ables in the loss function:
L Shannon = log(3) + 2 j=0 p sOj log p sOj (G2)
where p sOj is the probability of the j-th observable O j , given a percept s. To enforce that the relative phases among the modes of the middle layer vanish, we also add to the loss function the 1 norm of the complex phases, i.e. L phase := 8 m=0 |Φ sm |. Here, Φ sm is the phase in the middle-layer mode m when percept s is considered. The full loss function for the middle layer is then:
L full = 1.0 L PS + 10.0 L Shannon + 1.0 L phase (G3)
We use experience replay, and add up the loss functions of a batch of 600 sampled percepts before running 10 optimizer steps (Adam optimizer [44]) and wiping the batch. The learning rate is set to 0.01, decreased to 0.001 once the accuracy of the last batch was at least 0.95. Training stops when the accuracy of the middle layer exceeds 0.99 for 10 consecutive rounds and, at the same time, the 1 -norm of the phases weighted by p sm is smaller than 0.1.
Once the middle layer is trained, we fix its phase shifters so that the trees do not change anymore. Now, we introduce the task layer, which is exchangeable and independent of the middle layer. While Eva et al. only considered single-observable experiments, our task layers pick two observables (O 0 , O 1 ) and ask whether they have specific values v O0 and v O1 .
The quantum PS agent uses one 9x9 interferometer for each task layer. Again, we use experience replay, with a batch size of 500 (much larger than the number of percepts) and train the agent for 40000 rounds. Since our tasks ask for specific values of two observables simultaneously, only 1 9 of the percepts will correspond to a yes-answer. Hence, in addition to the raw accuracy, we use a weighted accuracy where yes-answers are given an additional weight 8 (otherwise the agent could just always answer no, which would yield a seemingly good accuracy of 8 9 ). The loss function is the one in Eq. (8), with a reward equal to ±0.1 for a right or wrong answer, and an additional factor 8 if the right answer is yes, the Kullback-Leibler divergence for {p sa , 1 − p sa }, ReLU functions to cut off the target values and neither glow nor forgetting. We normalize the probability of the yesand no-mode of the task layer to one, i.e. we post-select on getting one of these two modes. For the learning rate and number of optimizer steps we use the same settings as before.
Overall, the quantum PS agent consistently achieves over 0.97 accuracy in all two-observable tasks (see Fig. 6).
Appendix H: Classical-quantum equivalence of ECM graphs
In this section we formalize the connection between the directed graph of a classical ECM and the quantum walks that derive from it. We refer to this set of quantum walks (from input to output) as unitary routing. We begin by introducing some basic elements from graph theory. Definition 1 (Topological ordering). A topological ordering o G of a directed graph G = (V, E) is a map, o : V → {1, . . . , |V |}, which defines a total ordering of vertices of G such that for every directed edge (v j , v k ) from vertex v j to vertex v k , v j comes before v k in the ordering, i.e., o(v j ) < o(v k ). Let O G be the set of all topological orderings of G.
An example of topological ordering is presented in Fig. 11.
Directed acyclic graphs (DAGs) are directed graphs that do not contain loops. They can be defined as follows [45]:
Definition 2 (Directed acyclic graph). A directed graph is acyclic if and only if it admits a topological ordering.
Thus, the graph in Fig. 11 is an example of a DAG. It is worth noticing that the topological ordering of a DAG such as in Fig. 11 is not unique, because swapping consecutive vertices (in the topological ordering) which are not connected by an edge yields another valid topological ordering.
Below we recall two additional, useful definitions:
G (o) = {o(v i ) | v i ∈ V } i ,(H1){(o(v j ), o(v k )) | (v j , v k ) ∈ E} .
Let ocDAG be the set of all ordered and connected DAGs. Definition 4 (Parents and children). Let G = (V, E) be a graph. For any vertex v i ∈ V , let P G (v i ) be the set of parents of v i , i.e., P G (v i ) = {v j |v j ∈ V, ∃(v j , v i ) ∈ E}. For any vertex v i ∈ V G , let C G (v i ) be the set of children of v i , i.e.,
C G (v i ) = {v j |v j ∈ V, ∃(v i , v j ) ∈ E}.
Equipped with the above definitions, we proceed by defining an ECM graph. Here we focus on ECMs described by connected DAGs.
Definition 5 (ECM graph). An ECM graph is a connected DAG G = (V, E) whose vertices are called either percepts, S = {v j |v j ∈ V, P G (v j ) = ∅}, actions A = {v j |v j ∈ V, C G (v j ) = ∅}, or intermediate clips C = V \(S ∪ A).
The quantum version of a classical ECM graph is defined via a unitary routing. We refer to Table II for an overview of the terminology introduced in this section. (H2) whose matrix elements are given by
U i jk = u i jk if j, k ∈ {i} ∪ P G (o) (i) δ jk else,(H3)
where the u i jk ∈ C are only subject to unitarity constraints. For example, with reference to the example in Fig. 11 and the topological ordering (o(s 1 ), ..., o(a 2 )) = (1, ..., 9), P G (c 2 ) = (s 1 , s 2 ) and P G (o) (4) = (1, 2), hence Given a unitary route route( G (o) ), the unitary matrix describing the whole quantum ECM is then given by
U 4 = U = U i ∈route( G (o) ) U i ,(H6)
where the product is ordered from left to right in descending order of the superscripts. Note that here the superscript i refers to the position of the vertices in the ordering o, while in the main text the superscript refers to an unspecified ordering, and that in general the U i do not commute.
If an ECM graph G admits more than one topological ordering, G can have different unitary routings. Nevertheless, there is the following one-to-one quantum-classical correspondence:
Lemma 9 (Quantum-classical correspondence). The function router is bijective.
Proof. To show bijection we will first show injectivity, that is, we must show that router( G Surjectivity is trivial because the codomain of router is defined as its range, which completes the proof that router is bijective.
While a one-to-one correspondence exists on the level of ordered DAGs G (o) and unitary routings, there is an interesting freedom on the level of ECM graphs and/or unitary routes. In the latter case, a unitary route is not sufficient to uniquely identify a G (o) because if u i jk = u i kj = 0 for some j = k, the corresponding directed edge (j, k) in G (o) cannot be recovered just from the unitary route. Instead, one would recover a G (o) without that edge. Classically, this corresponds to the case of a vanishing h-value, h jk = 0. In the former case, it is worth noticing that ECM graphs are defined without ordering. However, ordering matters for unitary routes since in general the U i s do not commute. This represents an interesting departure from classical ECMs: different orderings can give rise to different unitary routings.
FIG. 1 :
1Projective Simulation (PS)
FIG. 2 :
2Decision making in classical and quantum projective simulation (PS). a) In PS, decision making occurs via a classical random walk over a network of clips {c 1 , .
FIG. 3 :
3Optical implementation of PS. To enable PS on photonic circuits, we use as variational ansatz the square architecture[33]. a) The update of the agent's memory translates into an update of the phase shifters θ. b) Any unitary U can be implemented with a regular mesh of Mach-Zehnder interferometers (MZI), consisting of two 50:50 beam splitters (BS, blue) and two phase shifters (θ 1 , θ 2 )[33,34]. c) When θ = π all light entering a MZI is reflected (bar state), while when θ = 0 it is transmitted (cross state). d) Square, universal architectures can be carved inside larger ones by tuning the phases around them. In this panel, a universal 4 × 4 circuit (in/out modes are labeled) is carved in a larger mesh. Tilted elements represent tunable phase shifters (those inside the carved structure are in fuchsia). BSs with arbitrary transmissivities (orange) are implemented using tunable MZIs. Light can be confined to the pink waveguides by setting the green optical elements.
FIG. 4 :
4Update via causal diamonds. To reinforce a transition probability psa in a 2-layer PS, it is not necessary to update all phase shifters at each step.
FIG. 5 :
5Update via a Gram-Schmidt process. We can directly update
FIG. 6 :
6A transfer-learning scenario for PS agents. a) Structure of the ideal, classical ECM considered in the transfer-learning scenario in Sec. IV. Panel adapted from Ref.[37]. b) Percepts s consist of the values of three observables which characterize particles. The experiment consists of two stages.
ACKNOWLEDGMENTS
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement, Grants No. 801110 and No. 885567. It reflects only the author's view, the EU Agency is not responsible for any use that may be made of the information it contains. ESQ has received funding from the Austrian Federal Ministry of Education, Science and Research (BMBWF). This work was supported in part by the Austrian Science Fund (FWF) through the SFB BeyondC F7102 and the Volkswagen Foundation (Az:97721). This work was also co-funded by the European Research Council (ERC) under Project No. 101055129. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
FIG. 7 :
7Simulating a quantum ECM. a) Directed acyclic graph describing the ECM shown inFig. 2, where each vertex corresponds to a clip c k (here including percepts and actions). b) Vertices, labeled as in panel (a), are rearranged in a layered structure and recolored depending on the order in which the corresponding unitary acts on the excitation (see App. H for a discussion on the ordering of the unitaries). Some unitaries that appear consecutively in the ordering can be seen to commute trivially and are thus grouped in the same layer with the same color. c) Subgraphs corresponding to percepts c 1 and c 2 . Vertices are colored as in panel (b).
FIG. 8 :
8Combining causal diamonds and Gram-Schmidt process. The
Function 5 -
5FindLeakingNodes(AnyDAG)Input: Graph G describing an optical circuit / a quantum ECM (see Function 4); a percept/action pair (s, a).
FIG. 9 :
9Wavelength-dependent quantum walks in the agent's memory. The wavelength dependence of beam splitters and phase shifters turn a limitation of optical technologies into a potential advantage (see Sec. C). a) Fidelity of random effective unitary transformations U of increasing size, as perceived by photons with different separation in wavelength λ (see legend), averaged over 100 pairs of random unitaries. Each U is numerically generated by uniformly sampling power settings in the phase shifters (see Eq. C1) of a square architecture where beam splitters are implemented as tunable Mach-Zehnder interferometers. Standard deviations are smaller than the point size. b) Four learning curves for a 2-layer PS agent based on a 10×10 square architecture (four rewarded percept/action pairs; each percept/action is encoded in a pair of adjacent input/output optical modes). The figure of merit optimized by the variational approach based on causal diamonds (see Sec. III B) is the geometric mean of the transition probability p (λ) sa for three values of λ.
FIG. 10 :
10Partial traceability in the decision making process of quantum PS agents. a-c) Three quantum walks, associated with as many 2-layer ECMs, are numerically simulated with a square optical architecture. Each panel shows the probability P to measure a photon in one of the output modes before training. Each column (label: Layers) displays the squared column (for a fixed percept) of the unitary matrix obtained by adding a new layer of optical elements to the square architecture, from left (input layer) to right (full circuit). The rightmost column of each plot corresponds to the output probabilities of a quantum walk over the full ECM. d-f) Same analysis repeated after training with the variational approach based on the causal diamond (see Sec. III B). Panels (d), (e) and (f) correspond to the evolutions in panels (a), (b
0 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . u 11 u 12 . . . 0 0 . . . u 21 u 22 . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 0 . . . 1
Definition 3 (
3Ordered DAG). An ordered DAG G is a DAG (V, E) with V = {1, . . . , |V |} such that o : V → {1, . . . , |V |}, o(j) = j is a topological ordering of (V, E).For any DAG G = (V, E) with topological ordering o we denote the induced ordered DAG by
. 11: a) Directed acyclic graph shown inFig. 2(top) and one of its topological orderings (bottom). Notice that, for the sake of simplicity, all vertices had already been labeled with subscripts in ascending order. b) Fulfilling unitarity constraints in the mode-mixing transformations U k in Eq. (H3) may generate edges (blue) that are not present in the classical ECM.
Definition 6 (
6Unitary route). Let G = (V, E) be an ECM graph and o a topological ordering of G. A unitary route of the induced ordered DAG G (o) is a sequence of |V | × |V | unitaries, ordered in ascending order of their superscripts, route( G (o) ) = U o(vi) vi∈C∪A ,
Definition 7 (Unitary routing). Let routing( G (o) ) be the set of all unitary routes of G (o) , generated by the freedom in choosing the coefficients u i jk within the constraints of unitarity. Let also ocROUTING = {routing( G (o) )} G (o) ∈ocDAG . Definition 8 (Unitary router). We define a unitary router as the function router : ocDAG → ocROUTING, (H5) G (o) → routing( G (o) ).
(o) ) = router( H (o ) ) ⇒ G (o) = H (o ) . According to the definition of routing, a G (o) has routing( G (o) ) only if (i) it has vertices V = {1, . . . , n}, where n is the number of rows (or columns) of each U i in any of the routes in routing( G (o) ), and (ii) it has directed edges E = {(j, k) | ∃ route( G (o) ) ∈ routing( G (o) ) and ∃ U k ∈ route( G (o) ) such that U k jk = 0}. This completely determines G (o) as G (o) = (V, E). Thus router( G (o) ) = router( H (o ) ) can only be true if G (o) = H (o ) .
TABLE I :
ILinks between the main elements of projective simulation (PS) and its proposed optical quantization.
Object Intuition
IntuitionRoutingSequence of parametrized unitary transformations, constructed from a graph G.RouteA specific sequence of the routing, where all parameters have been fixed.RouterThe map from G o to its routing
TABLE II :
IISummary of the main quantities introduced in App. H.
Appendix D: Towards interpretable quantum artificial intelligenceWhile neural networks show excellent performance, their black-box-like nature can be problematic for some purposes. For example, in delicate decisions that directly impact humans, one may wish to have a justification before following the advice of an enigmatic machine. Similar concerns apply in science where, rather than developing an oracle, one might aim to acquire a deeper understanding of a given phenomenon.These issues have recently sparked interest in interpretable and explainable artificial intelligence[42]. PS offers one viable framework towards this goal. Differently from neural networks[17], in classical PS an interpretation can be found by studying the path taken from percept to action. However, this is no longer true in the quantum domain, where we lose the notion of observable path. Since in a quantum walk all allowed paths contribute to the decision making process, a quantum PS agent loses interpretability in the classical sense (there is no longer a unique path that can be retraced).A notion of interpretability that is applicable to quantum agents is currently being explored. Here we make a first step in this direction by observing that a notion of partial trace can still be recovered, if one studies the likelihood that a photon (or more photons, as in App. B) would be measured in any given mode of the circuit (seeFig. 10). This information can be retrieved by classically simulating the single-photon quantum walk, since all phase settings are known. This possibility still holds for multi-photon quantum walks despite the computational complexity of the task, which becomes practically unfeasible only from n ∼ 30[43].Appendix E: Variational algorithmsIn this section, we provide additional information on the two variational algorithms presented in Sec. III.The following algorithms naturally take into account the unitarity constraint inherent to the architecture. We point out that this constraint is not necessarily a limitation and, rather, it could be advantageous to improve the agent's performance. In fact, since a machine learning ansatz with more freedom is also more prone to overfitting, it is reasonable to believe that a specialized ansatz with restrictions and symmetries might generalize better. A reduced number of degrees of freedom might also help during the training stage.Update via Gram-Schmidt processHere we outline the update via a Gram-Schmidt process. The algorithm consists of four steps (i-iv):as according to a desired update rule. Here, s 0 and a 0 are the observed percept and action of the current round, respectively.In the examples in Sec. III C, we considered a constant rescaling factor α, i.e. U (t+1) The main advantage of this approach is that it allows one to control any theoretical aspect of the learning process, from the update rule to the learning speed. One limitation is that it assumes perfect knowledge and control of the experimental settings. Appending to it one step of variational fine-tuning, as the one we describe in the following section, can help mitigate this issue and speed up the learning process.Update via causal diamondsHere we provide additional considerations on the update based on causal diamonds. The advantage of this approach is that it is efficient (only few phase shifters are adjusted at each step) and that it is variational (it takes into account the imperfect control over the experimental settings).We consider two main approaches (a,b) to update a given subset Θ = (θ 1 , . . . , θ N ) of N phase shifters at each training step, albeit more sophisticated strategies can be devised and tailored to the hardware. These phases can correspond to the set of leaking nodes described in Sec. III B, or to a larger set selected via other criteria.a. Gradient ascent -Each phase θ ∈ Θ is independently tuned to maximize a predefined figure of merit F. b. Sequential -Each phase θ ∈ Θ, in the order given by the light propagation, is tuned to independently maximize a figure of merit F, and is immediately set to the new value before the next phase is probed.The latter approach (b) (seeFig. 4andFig. 8) is usually faster than (a), since each phase update starts from a progressively higher value of F, and the individual updates within a training step do not sabotage each other. However, an update based on gradient ascent as in (a) can be beneficial to escape local minima, which are more likely to cause problems in (b).Appendix F: Optical architectureIn this section we review the basics behind the operation of the optical architecture discussed in the main text. In Sec. II B, Appendix G: Numerical details of the transfer learning scenarioIn this section, we report additional details on the numerical analysis described in Sec. IV, performed with Pytorch.Both classical and quantum PS agents have three layers: percept, middle and task layer. While in classical PS each percept s is represented by a vertex in the ECM graph, in this analysis the quantum PS agent associates with it a (linearoptical) binary tree[10], to overcome the limitation due to the unitarity constraints in the middle layer. All its phases are initialized to π 4 to yield a balanced output distribution. The classical PS agent is expected to pick an observable uniformly at random, and then send the excitation to the middle clip with the right value. For the quantum PS agent, we wish that the percept (v 0 , v 1 , v 2 ) is represented by the stateTo this end, we first train the binary trees, considering a RL scenario where percepts are sampled i.i.d. uniformly. We use the simplified loss function L PS of Eq. (11), with the Kullback-Leibler divergence applied to the probability distribution {p sa , 1 − p sa }. Furthermore, we use ReLU functions to cut off the target values in Eq.(11)such that they are in [0, 1]:As a reward, we assign r := ±0.1 depending on whether or not the agent predicts the right value. To ensure that each observable is picked uniformly, we implement a curiosity mechanism by including the Shannon neg-entropy of the observ-
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| [] |
[
"UNIVERSAL QUADRATIC FORMS AND INDECOMPOSABLES IN NUMBER FIELDS: A SURVEY",
"UNIVERSAL QUADRATIC FORMS AND INDECOMPOSABLES IN NUMBER FIELDS: A SURVEY"
] | [
"Vítězslav Kala "
] | [] | [] | We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main tools. | null | [
"https://export.arxiv.org/pdf/2301.13222v2.pdf"
] | 256,416,229 | 2301.13222 | c9fd795eb0611be85ce958d9bb128b4336bab269 |
UNIVERSAL QUADRATIC FORMS AND INDECOMPOSABLES IN NUMBER FIELDS: A SURVEY
17 May 2023
Vítězslav Kala
UNIVERSAL QUADRATIC FORMS AND INDECOMPOSABLES IN NUMBER FIELDS: A SURVEY
17 May 2023
We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main tools.
Introduction
The study of representations of integers by quadratic forms has a long history; let's briefly start here with a few highlights.
One can perhaps argue that they were first considered as Pythagorean triples, i.e., solutions of the Diophantine equation x 2 + y 2 = z 2 , or, equivalently, representations of 0 by the indefinite ternary form x 2 + y 2 − z 2 . A list of 15 such triples occurs already on the Babylonian clay tablet Plimpton 322 [Rob] from around 1800 BC! The Pell equation, i.e., representation of 1 and other small integers by the binary form x 2 − dy 2 (for some d ∈ Z >0 that is not a square), was considered as early as 400 BC by Greek mathematicians in connection with approximating √ 2, √ 3 by rational numbers. Later it was studied, e.g., by Archimedes (3rd century BC) and Diophantus (3rd century AD) and in India by Brahmagupta (7th century AD) and Bhaskara (12th century AD).
The modern European history starts with giants such as Fermat, Euler, and Gauss, who considered representations of primes by binary definite forms x 2 + dy 2 (for d ∈ Z >0 ) [Cox] and obtained results such as a prime number p is of the form x 2 + y 2 if and only if p = 2 or p ≡ 1 (mod 4).
In 1770 Lagrange proved the Four-Square Theorem stating that every positive integer n is of the form x 2 + y 2 + z 2 + w 2 ; Jacobi then in 1834 gave a formula for the number of representations of n in this form. In a similar vein, Legendre in 1790 proved the Three-Square Theorem that characterizes the integers of the form x 2 + y 2 + z 2 (for much more information on the history, see, e.g., [Di2]). These results eventually led, e.g., to the still active Waring problem, and to using modular forms for studying the representations of integers by quadratic forms.
A quadratic form representing all positive integers is called universal. Lagrange's Theorem thus says that x 2 + y 2 + z 2 + w 2 is a universal quadratic form. The early 20th century saw the characterization of all universal quaternary diagonal positive forms ax 2 + by 2 + cz 2 + dw 2 by Ramanujan [Ra], and an extension of this work to non-diagonal forms by Dickson [Di], who also introduced the term "universal quadratic form". Among such forms are, for example, x 2 +2y 2 +4z 2 +dw 2 for 1 ≤ d ≤ 14. These results were further expanded by Willerding [Wi] to cover also the cases of classical quaternary forms (although her list contains a number of errors, it was nevertheless a big step towards the full classification).
While no ternary positive definite quadratic form is universal (for local reasons), indefinite quadratic forms tend to be universal more easily. For example, any quadratic form x 2 − y 2 − dz 2 is universal as long as 4 does not divide d because x 2 − y 2 = (x + y)(x − y) represents all odd numbers. They form a separate area of study, and we will return to them later only very briefly.
The 15-and 290-Theorems
Let's first discuss in some detail the case of quadratic forms over the ring of integers Z. Recall that a quadratic form of rank r over Z is a polynomial (1)
Q(x 1 , . . . , x r ) = 1≤i≤j≤r a ij x i x j , a ij ∈ Z.
Typically we require the form to be positive definite, meaning that Q(v) > 0 for all v ∈ Z r , v = 0. We attach the Gram matrix to Q, given by
(2) M = M Q = a 11
1 2 a 12 · · · 1 2 a 1r 1 2 a 12 a 22 · · · 1 2 a 2r . . . . . . . . . . . . Taking v ∈ Z r to be a column vector (x 1 , . . . , x r ) t we have
Q(v) = v t M v.
The quadratic form Q is called classical if all the entries of M are integers, i.e., if a ij are even for all i = j.
Each quadratic form has an associated bilinear form B defined by
Q(v + w) = Q(v) + Q(w) + 2B(v, w), v, w ∈ Z r .
A positive definite form satisfies the Cauchy-Schwarz inequality: For all v and w,
Q(v)Q(w) ≥ B(v, w) 2 .
In the 90's, Conway, Miller, Schneeberger, and Simons, and then Bhargava and Hanke [Bh, BH] came up with the following fascinating criteria for universality.
Theorem 2.1. Let Q be a positive definite quadratic form over Z. Then: (a) (15-Theorem, Conway-Schneeberger, ∼ 1995) If Q is classical and represents the integers 1, 2, 3, 5, 6, 7, 10, 14, and 15, then it is universal. (b) (290-Theorem, Bhargava-Hanke, ∼ 2005 [BH]) If Q represents the integers 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, and 290, then it is universal. (c) Both of these lists of integers are minimal in the sense that for each integer n in the list, there exists a corresponding quadratic form that represents all of Z >0 \ {n}, but does not represent n.
While the 15-Theorem in part (a) is not too hard to prove, the 290-Theorem in part (b) is very challenging, not only because of the large amount of computations needed.
There have been a number of further exciting developments related to universal quadratic forms over Z. For example, the conjectural 451-Theorem by Rouse [Ro] says that if a positive definite form represents the integers 1, 3, 5, . . . , 451, then it represents all odd positive integers. This result has been proved only under the assumption that each of the ternary forms x 2 + 2y 2 + 5z 2 + xz, x 2 + 3y 2 + 6z 2 + xy + 2yz, x 2 + 3y 2 + 7z 2 + xy + xz represents all odd positive integers (that seems very hard to establish).
Escalations. We give a sketch of Bhargava's proof of the 15-Theorem. The idea is to "build up" a universal quadratic form Q by gradually adding variables.
In order for Q to be universal, it must represent 1, and so it must contain x 2 . Slightly more precisely, a linear change of variables does not change universality and gives us x 2 (this will be made more precise soon, once we discuss quadratic lattices). Now x 2 is clearly not universal as it does not represent 2, hence Q must contain 2y 2 (again after a change of variables). We get the form x 2 + 2axy + 2y 2 , where the coefficient of xy is 2a because we require the form to be classical, and so the corresponding Gram matrix is 1 a a 2 .
What are the possible values for a? By the Cauchy-Schwarz inequality 1 · 2 ≥ a 2 , which leaves the possibilities a = 0, 1, −1 with the corresponding Gram matrices
1 0 0 2 , 1 1 1 2 , 1 −1 −1 2 .
The quadratic forms x 2 + 2xy + 2y 2 and x 2 − 2xy + 2y 2 are equivalent by the change of variables y → −y so we can forget about the third matrix. As for the second matrix, we can reduce the quadratic form by changing variables:
x 2 + 2xy + 2y 2 = (x + y) 2 + y 2 = X 2 + Y 2 .
Note that, in terms of matrices, the Gram matrix of the resulting form is C t M C for an invertible matrix C. It can be obtained from M by successively applying the same row and column operations:
1 1 1 2 ∼ 1 0 1 1 ∼ 1 0 0 1 .
We see that after two steps of escalations, we have two candidate forms x 2 + 2y 2 and x 2 + y 2 . Since they do not represent 5, respectively 3, we pass to the matrices
1 0 b 1 0 2 c 1 b 1 c 1 5 , 1 0 b 2 0 1 c 2 b 2 c 2 3 .
We again determine all possible values for the coefficients and reduce the forms, which leads to the following possibilities (this can be done as an exercise by the reader):
1 1 d 1 , d 1 = 1, 2, 3 1 2 d 2 , d 2 = 2, 3, 4, 5 1 2 1 1 d 3 , d 3 = 4, 5.
Continuing this process for rank 4, we get 207 forms, 201 of which are universal. This can be proved by local methods and genus theory (i.e., by a suitable use of the local-global principle). The remaining 6 forms represent all but one integers. After adding one more variable, we get 1630 universal forms of rank 5. This procedure showed that if Q is universal, then it contains one of the rank 4 or 5 forms obtained above. These are all universal, and so the converse implication also holds, i.e., any quadratic form that contains one of these forms is universal.
But in the process of escalations, we only considered representations of small integers: rank 1 1 2 1, 2 3 1, 2, 3, 5 4 & 5 1, 2, 3, . . . , 15 Thus if Q represents the integers 1, 2, 3, . . . , 15, then it is universal, proving the 15-Theorem.
Proof of 290-Theorem. The proof of the 290-Theorem, although similar, is much more complicated. First, there are more cases to be considered (we have to continue the escalations up to rank 7, which leaves us with approximately 20 000 cases). Second, proving universality is sometimes very non-trivial and uses tools such as theta series (modular forms). For more information, see the original papers [Bh, BH] or the surveys [Hah, Moo].
Quadratic Lattices
Talking about changes of variables and adding a variable in each step of the escalation process is unpleasant. A more efficient approach is to work with quadratic lattices. Their theory is extensive, but we will keep it to the necessary minimum and refer the reader to the book by O'Meara [OM] for more details.
Abstract lattices. Let K denote a number field of degree d over Q, O K its ring of integers, and
V a finite dimensional K-vector space. A subset L ⊂ V is an O K -lattice if it is a finitely generated O K -module. Example 3.1. If v 1 , v 2 , . . . , v r is a basis of V , then L = O K v 1 + · · · + O K v r is the free O K -lattice of rank r.
We can wonder about the converse statement: Is every O K -lattice of this form? The answer is no, as the following theorem shows.
Theorem 3.2 ( [OM,81:5]). Let L ⊂ V be an O K -lattice. Then there exist linearly independent vectors v 1 , . . . , v r in V and a fractional ideal A in K such that
L = O K v 1 + O K v 2 + · · · + O K v r−1 + Av r .
In particular, the lattice L is not free when A is not a principal ideal.
Of course, when K has class number h K = 1, then all fractional ideals are principal and all O Klattices are free.
When L is as in theorem above, then r is its rank.
Quadratic lattices. Recall that V denotes a finite dimensional K-vector space. We moreover assume that V is a quadratic space, i.e., we a have quadratic form Q : V → K and the attached bilinear form
B(v, w) = 1 2 (Q(v + w) − Q(v) − Q(w)) .
If L ⊂ V is an O K -lattice, then the pair (L, Q) is called a quadratic O K -lattice (more precisely, we should take the restriction Q| L instead of Q).
A
quadratic O K -lattice (L, Q) is called integral if Q(v) ∈ O K for all v ∈ L. In this case B(v, w) ∈ 1 2 O K for all v, w ∈ L. If B(v, w) ∈ O K for all v, w ∈ L,
we say that the quadratic lattice is classical. The language of quadratic lattices lets us make some of the arguments of the preceding section on escalations more formal. Let (L, Q) be a Z-lattice. Instead of "Q contains x 2 " we can say "there exists v 1 ∈ L such that Q(v 1 ) = 1", instead of "Q must represent 2" we would say "there exists v 2 ∈ L such that Q(v 2 ) = 2. What are now the possibilities for B(v 1 , v 2 )?" and so on.
Nevertheless, we will sometimes still just talk about quadratic forms with the understanding that the discussion can be made more precise in the language of lattices.
Specifically, when we have a free lattice O r K , then the corresponding quadratic form looks like (1), except that now a ij ∈ O K , and we again have the Gram matrix given by (2).
As before, we want to study positive definite quadratic forms, but now over a number field K.
First, note that if [K : Q] = d, then there are d embeddings σ : K ֒→ C. We say that K is totally real if σ(K) ⊂ R for every σ. Concretely, let's take K = Q(γ) for an algebraic integer γ whose minimal polynomial is f (X) = X d + a 1 X d−1 + · · · + a d−1 X + a d , a i ∈ Z. Let γ 1 , . . . , γ d denote the complex roots of f (X). Each embedding σ i : K ֒→ C is completely determined by the image of γ, which is sent to one of its conjugates, so that after relabeling σ i (γ) = γ i . We see that K is totally real if and only if γ i ∈ R for all i.
From now on let's always assume that the number field K is totally real.
An element α ∈ K is totally positive if σ(α) > 0 for all the embeddings σ. We write α ≻ β for α, β ∈ K if σ(α) > σ(β) for all σ. In particular α ≻ 0 means that α is totally positive. The totally positive elements α ∈ O K form a semiring which we
denote O + K . The quadratic lattice (L, Q) is totally positive definite if Q(v) ≻ 0 for all v ∈ L \ {0}.
Definition. Let (L, Q) be an integral totally positive definite lattice. We say that Q is universal if it represents all totally positive integers, i.e., if ∀α ∈ O + K ∃v ∈ L : Q(v) = α. Sums of lattices. If V is a finite dimensional K-vector space and L 1 , L 2 ⊂ V are two O K -lattices, we define their sum as
L 1 + L 2 = {v + w | v ∈ L 1 , w ∈ L 2 }. It is a direct sum if L 1 ∩ L 2 = 0, in which case we write L 1 ⊕ L 2 = L 1 + L 2 .
Assuming that V is a quadratic space with a quadratic form Q, the sum of L 1 and L 2 is orthogonal, denoted L 1 ⊥ L 2 = L 1 + L 2 , if B(v, w) = 0 for any v ∈ L 1 and w ∈ L 2 .
We further use the following notation for diagonal quadratic lattices: a is the rank one lattice
O K v with v ∈ V such that Q(v) = a. Then we define a 1 , a 2 , . . . , a r = a 1 ⊥ · · · ⊥ a r = O K v 1 ⊥ O K v 2 ⊥ · · · ⊥ O K v r , where v i ∈ V satisfies Q(v i ) = a i (and B(v i , v j ) = 0 for i = j).
We next state a useful proposition on the "splitting off units" that is proved quite easily using Gram-Schmidt orthogonalization.
Proposition 3.3. Let (L, Q) be a classical O K -lattice, and let v ∈ L be such that Q(v) = ε is a unit in O K . Then L = ε ⊥ L ′ for some lattice L ′ ⊂ L.
Proof. Let L ′ = (O K v) ⊥ = {w ∈ L | B(w, v) = 0}. Clearly O K v = ε is orthogonal to L ′ and we must show that L = O K v + L ′ .
Take any z ∈ L and set w :
= z − B(z, v)ε −1 v. The vector B(z, v)ε −1 v belongs to O K v, because ε is invertible and B(z, v) ∈ O K as L is classical. And we have w ∈ L ′ because B(w, v) = B(z − B(z, v)ε −1 v, v) = B(z, v) − B(z, v)ε −1 B(v, v) = 0, where we used B(v, v) = Q(v) = ε. Thus z = B(z, v)ε −1 v + w ∈ O K v + L ′ .
Before turning to the recent developments on universal forms that constitute our main topic, let's briefly comment on three closely related fields of interest:
Indefinite quadratic forms (and forms over number fields that are not totally real) behave quite differently from our case. Let's only briefly remark that, for example, [Si2] and [EH] characterized general number fields with universal sums of five and three squares, and then [HHX, HSX, XZ] considered more general universal forms. Another interesting topic is the study of universal Hermitian quadratic forms over imaginary quadratic fields, e.g., [EK2,KP1].
Regular quadratic forms are forms that represent all elements that are not ruled out by local obstructions [CI, Ea], and thus present a natural generalization of universal forms. Their theory in many aspects parallels the theory of universal forms; in fact, tools such as Watson's transformations [CE+, CEO, Wa] sometimes allow one to convert regular forms into universal ones (although special care must be paid to what happens dyadically, as well as over number fields with non-trivial narrow ideal class group).
In connection to this, let's also briefly mention the recent computational results by Kirschmer and Lorch [Kir, LK] that classify 1-class genera of quadratic lattices over number fields.
Further, let's note that besides from studying representations of integers by quadratic forms, there have been numerous works considering representations of quadratic forms by quadratic forms and, in particular, by the sum of squares, e.g., [BI, BC+, Ic Oh,Sa1]. Most of them deal with forms over Z, but it is another exciting direction of future research to consider the situation over number fields in detail.
, JKO, KO1, KO2, KO3, Ko, KrY, Mo1, Mo2,
Real Quadratic Fields
In this section, let's consider universal forms over a real quadratic field K = Q( √ D) with squarefree D > 1. For simplicity, we always assume D ≡ 2, 3 (mod 4) so that O K = Z[ √ D] (but everything that we discuss here also generalizes to the case D ≡ 1 (mod 4)).
There are two embeddings K ֒→ R, the identity and
α = a + b √ D → α ′ = a − b √ D.
Thus α is totally positive if and only if a
+ b √ D > 0 and a − b √ D > 0. The norm of α is N (α) = αα ′ = a 2 − b 2 D, and its trace is Tr(α) = α + α ′ = 2a.
Diagonal forms and indecomposables. An easy example of a quadratic lattice is
(O r K , Q), where Q(x 1 , . . . , x r ) = a 1 x 2 1 + · · · + a r x 2 r is a diagonal form.
The lattice is integral and totally positive if and only if all the coefficients a i ∈ O + K . A key tool for working with (diagonal) universal forms is the notion of an indecomposable element:
We say that α ∈ O + K is indecomposable if α = β + γ for β, γ ∈ O + K .
To explain the connection, let's assume that a diagonal quadratic form Q is universal. Then we can express any indecomposable α as α = a 1 v 2 1 + · · · + a r v 2 r , and hence α = a i v 2 i for some i thanks to indecomposability. Thus each indecomposable essentially appears as a coefficient in Q, and we can conclude that the number of variables r of a universal quadratic form is bounded from below by the number of square classes of indecomposables.
The key question that remains to be answered is: Are there any indecomposables? Luckily, yes:
Lemma 4.1. If ε is a totally positive unit, then it is indecomposable.
Proof. Suppose for contradiction that ε = β + γ for some β, γ ∈ O + K . Then
1 = N (ε) = (β + γ)(β ′ + γ ′ ) = N (β) + N (γ) + βγ ′ + β ′ γ > N (β) + N (γ) ≥ 2.
Essentially the same proof also works in a totally real field of a general degree d and shows that each element of norm < 2 d is indecomposable.
Brunotte [Br1,Br2] gave a general upper bound on the norm of an indecomposable integer, and so in each K, there are finitely many indecomposables up to multiplication by totally positive units. Unfortunately, the bound is exponential in the regulator of the number field, and so it is not very useful. While this can be significantly improved (see Theorem 6.1 below), it is still important to obtain more information about indecomposables, ideally in the form of an explicit construction. In real quadratic fields, this is possible using continued fractions, as we shall see next.
Continued fractions. The fundamental unit of a real quadratic field can be given in terms of the continued fraction of
√ D. It is periodic √ D = [u 0 , u 1 , . . . , u s ] = [u 0 , u 1 , . . . , u s , u 1 , . . . , u s , u 1 , . . . ] = u 0 + 1 u 1 + 1 u 2 +··· , and we know that u 0 = ⌊ √ D⌋ and u s = 2⌊ √ D⌋. Let p i q i = [u 0 , . . . , u i ]
be the convergents of the continued fraction. These give the good approximations to
√ D since p i q i − √ D < 1 u i+1 q 2 i .
By an abuse of terminology, the quadratic integers α i = p i + q i √ D will also be called convergents. The element α s−1 is the fundamental unit. In other words, it generates the group of units, which can be described as
O × K = ±α k s−1 | k ∈ Z .
When are the convergents α i totally positive? We always have α i > 0, and it turns out that α ′ i > 0 if and only if i is odd.
Consequently, the fundamental unit α s−1 is totally positive if and only if s is even. Thus for s even, the Pell equation x 2 −Dy 2 = −1 has no solutions. For s odd, it has a solution, namely α s−1 = x+y √ D. When s is odd, the smallest totally positive unit (greater than 1) is then α 2s−1 = α 2 s−1 . The convergents α i satisfy the recurrence
α i+1 = u i+1 α i + α i−1 , i ≥ 0, with α −1 = 1, α 0 = ⌊ √ D⌋+ √ D.
We observe that multiplication by α s−1 shifts indices: α i α s−1 = α i+s . The convergents of a continued fraction are characterized by their best approximation property. One could say that being indecomposable is a form of "totally positive best approximation property", and so the following classical theorem [DS] should not be too surprising.
α = α i,t = α i + tα i+1 , i ≥ −1 odd, 0 ≤ t < u i+2 ,
and their conjugates. Moreover, N (α) ≤ D for every indecomposable α.
Note that α i,u i+2 = α i + u i+2 α i+1 = α i+2
, so indecomposables with a fixed i form an arithmetic progression going from α i to α i+2 .
There have been several improvements on the upper bound for the norm [JK,Ka2,TV] and on the additive structure of indecomposables [HK].
α −1 = 1 α 0 = 2 + √ 6, p 0 q 0 = 2 α 1 = 5 + 2 √ 6, p 1 q 1 = 2 + 1 2 = 5 2 .
The element α 1 is a totally positive unit. Thus all indecomposables up to multiplication by α 1 are α −1,t for 0 ≤ t < u 1 = 2. We have
α −1,0 = 1, α −1,1 = 1 + (2 + √ 6) = 3 + √ 6
with N (3 + √ 6) = 3, and
α 1,0 = α 1 = 5 + 2 √ 6, α 1,1 = α 1 · α −1,1 = 27 + 11 √ 6.
The semiconvergents α 1,t = α 1 · α −1,t , α 3,t = α 2 1 · α −1,t etc. differ from α −1,t by a multiple of a unit but in the case of α 1,t not by a multiple of a square of a unit.
In conclusion, there are four square classes of indecomposables represented by 1, 3 + √ 6, 5 + 2 √ 6, and 27 + 11 √ 6. Now we apply our findings to the problem of universality. Let Q be a diagonal universal quadratic form. Since it must represent 1 and 3 + √ 6, it contains
(3) 1 · x 2 + (3 + √ 6) · y 2 .
The totally positive unit 5 + 2 √ 6 is represented by Q, but it is not a square and N (5 + 2 √ 6) = 1, so it is not represented by (3). Hence Q must contain
(4) 1 · x 2 + (3 + √ 6) · y 2 + (5 + 2 √ 6) · z 2 .
The indecomposable 27 + 11 √ 6 has norm 3 but its square class is not represented by 3 + √ 6, and therefore Q contains (5) 1 · x 2 + (3 + √ 6) · y 2 + (5 + 2 √ 6) · z 2 + (27 + 11
√ 6) · w 2 .
This shows that each diagonal universal quadratic form must have rank r ≥ 4. (Of course, the preceding argument could be more formally stated in the language of quadratic lattices.)
Construction of a universal form.
Observation 4.4. Every α ∈ O + K is a sum of indecomposables. Proof. If α is not indecomposable, then α = β + γ. But then Tr(α) = Tr(β) + Tr(γ) and the traces are positive integers (because the elements are totally positive and integral). Therefore Tr(β), Tr(γ) < Tr(α). The result follows by induction.
Let ε be the totally positive fundamental unit; in other words, a generator of the group of all totally positive units
O ×,+ K = {ε ℓ | ℓ ∈ Z}.
We have seen that ε equals α s−1 or α 2s−1 depending on whether s is even or odd, respectively. Further, let S be the set of representatives of indecomposables up to multiplication by O ×,+ K . We can take
S = {α i,t i | i = −1, 1, 3, . . . , k, 0 ≤ t i < u i+2 },
where k = s − 3 if s is even and k = 2s − 3 if s is odd. In particular, the number of elements in S is
#S = u 1 + u 3 + u 5 + · · · = u 1 + u 3 + · · · + u s−3 + u s−1 , s even, u 1 + u 3 + · · · + u 2s−3 + u 2s−1 = u 1 + u 2 + u 3 + · · · + u s−1 + u s , s odd.
Observation 4.4 tells us that any totally positive α is a sum of indecomposables. We group the indecomposables according to their class in S to express α as
(6) α = σ∈S σe σ ,
where each e σ is a linear combination of totally positive units with non-negative coefficients.
Lemma 4.5. For each e σ , there exist j ∈ Z and integers c, d ≥ 0 such that
e σ = cε j + dε j+1 .
The idea of the proof is to rewrite the linear combination e σ using the minimal polynomial ε 2 − nε + 1 = 0, one just needs to arrange things so that c, d are indeed non-negative. Proof. Let α be a totally positive integer and write it as in (6). By the preceding lemma, each coefficient e σ is of the form e σ = cε j + dε j+1 . All it remains to show is that e σ is represented by the form 1, 1, 1, 1, ε, ε, ε, ε = x 2 1 + x 2 2 + x 2 3 + x 2 4 + ε · (x 2 5 + x 2 6 + x 2 7 + x 2 8 ). When j is even, then c is represented by x 2 1 + x 2 2 + x 2 3 + x 2 4 by Lagrange's Four Square Theorem, and cε j also, and the second term ε · dε j is represented by ε · (x 2 5 + x 2 6 + x 2 7 + x 2 8 ). The case of odd j is very similar.
Sums of continued fraction coefficients. How big is S? We would like to bound its size in terms of D, ε, and the class number h D .
In the case of odd s we have the trivial bound
u 1 + u 2 + · · · + u s ≥ u s = 2⌊ √ D⌋ > √ D.
On the other hand, we know the following:
Theorem 4.7 ([BK2, Corollary 18]). There is a positive constant c such that
u 1 + u 2 + · · · + u s ≤ c √ D(log D) 2 .
Roughly speaking, the preceding estimate (which can be somewhat improved) comes from the class number formula
h D = √ D log ε L(1, χ).
It is know that L(1, χ) ≪ log D, and since clearly h D ≥ 1,
log ε ≪ √ D log D.
Here we use the usual analytic notations that f ≪ g (and g ≫ f ) if there is a constant C > 0 such that f (x) < Cg(x) for all x (that lie in the domains of f, g). We use f ≪ P g or g ≫ P f to stress that the constant C depends on the specified parameter(s) P .
Next we relate log ε to the sum s i=1 u i . We have
ε = α s−1 = u s−1 α s−2 + α s−3 ≥ u s−1 α s−2 ≥ u s−1 u s−2 α s−3 ≥ · · · ≥ u s−1 u s−2 · · · u 0 , hence log ε ≥ s i=1 log u i .
This at least proves that s ≪ √ D log D (if u i = 1 for some i, we have to proceed more carefully). See also [KM] for a more detailed discussion.
In the case when s is even and ε = α s−1 , it is quite subtle trying to estimate u 1 + u 3 + · · · + u s−1 because it can be small: E.g., for the continued fraction
n 2 − 1 = [n − 1, 1, 2(n − 1)],
S contains only one element even though D = n 2 − 1 grows to infinity.
(Non-)Existence of Universal Forms
Let's now turn our attention more generally to the questions of existence of universal forms and of their properties, such as possible ranks. (Again, our discussion always applies to quadratic lattices, even when we talk about quadratic forms.)
We will still (mostly) treat the case of real quadratic fields K = Q( √ D) in this section. Above, we constructed a universal form over every such K (with D ≡ 2, 3 (mod 4), although this assumption is not necessary). In fact, a universal form exists in every number field, and there are (at least) two ways of proving this: a) Proceed similarly as in the case of Q( √ D) (see Corollary 6.2 below). b) Hsia-Kitaoka-Kneser [HKK,Theorem 3] showed a local-global principle for representations of elements with sufficiently large norm by Q, provided that the rank of Q is at least 5. So one can:
• Find a form Q 0 that represents everything locally over all the finite places. For example, Q 0 = 1, 1, 1, α where α ≻ 0 has additive valuation 1 at each dyadic place works, for already 1, 1, 1 is locally universal at all non-dyadic places [OM,92:1b], and at the dyadic places, one can use Beli's theorem [Bel,Theorem 2.1]. Alternatively, one can use Riehm's (much older) theorem [Ri,Theorem 7.4] thanks to which it suffices to make sure that all classes mod 2 are represented -which is easily arranged by adding extra variables. • If necessary, add variables to Q 0 to obtain Q of rank ≥ 5, for which one can use the asymptotic local-global principle [HKK,Theorem 3]. • Finally, add extra variables to cover the (finitely many) square classes of elements of small norms that are not represented by Q. It is easy to see that there is never a universal form of rank r = 1 or 2 (for local reasons). Moreover, when the degree d of K is odd, it quickly follows from Hilbert's reciprocity law that there is no ternary universal form [EK1, Lemma 3].
The most natural candidate for a universal form would be the sum of squares. Unfortunately, it is almost never universal, for Siegel [Si2] showed that a sum of squares is universal over O K only for • K = Q (when 4 squares suffice) and • K = Q( √ 5) (when 3 squares suffice [Ma]). The proof considers representations of units and indecomposables and is sketched below as Theorem 8.1.
One thus has to consider more general quadratic forms and aim at various classification results. This has been the most successful in the quadratic case.
Theorem 5.1 ([CKR, Theorem 1.1]). If K = Q( √ D) has a ternary classical universal form, then D = 2, 3, or 5. In total, there are 11 such forms; examples in the three cases are • x 2 + y 2 + (2 + √ 2)z 2 for D = 2, • x 2 + y 2 + (2 + √ 3)z 2 for D = 3, • x 2 + y 2 + 5+ √ 5 2 z 2 for D = 5.
The best available result in this direction is:
Theorem 5.2 ([KP2, Theorem 3.2]). If K = Q( √ D)
has a universal lattice of rank ≤ 7 (and D is squarefree), then D < (576283867731072000000005) 2 .
This result builds on [Ki1]; in fact, give more precise results, also separately for classical lattices. The proof is based on considering the sublattice representing 1, 2, . . . , 290 (it must have rank at least 4 when D is large thanks to the 290-Theorem), and the sublattice representing ⌈1 · √ D⌉ + 1 · √ D, ⌈2 · √ D⌉ + 2 · √ D, . . . , ⌈290 · √ D⌉ + 290 · √ D (that also must have rank ≥ 4). Note that there is an 8-ary universal form over each Q( √ n 2 − 1) (when n 2 − 1 is squarefree) [Ki2] that is constructed precisely as in Theorem 4.6.
Such results on determining the small possible ranks of universal lattices are motivated by Kitaoka's conjecture.
Conjecture 5.3 (Kitaoka).
There are only finitely many totally real number fields K having a ternary universal form.
The conjecture still remains open. However, B. M. proved at least a weak version of the conjecture saying that when the degree d of K is fixed, then there are only finitely many such fields K.
Some further interesting results are [CL+,De1,De2,Le,KTZ,Sa2].
Lower bounds on ranks. Surprisingly, it turns out that universal lattices can require arbitrarily large ranks. The broad idea behind the proof is the following. In a universal lattice (L, Q), construct a sublattice that must have rank ≥ r, for example by arranging it to contain pairwise orthogonal vectors v i representing suitable convergents α i .
A more precise result in this direction was obtained by Kala-Tinková [KT], with inspiration by earlier results of Yatsyna [Ya]. Let Q be a universal quadratic form over Q( √ D) of rank r. a) If Q is classical, then r ≥ U/2. b) In general, r ≥ √ U /2 (assuming U ≥ 240).
To prove the theorem, we want to use minimal vectors in a quadratic lattice (L, Q), i.e., nonzero vectors v such that Q(v) is minimal. This approach works best over Z, so we need to obtain a Zlattice. In general, if [K : Q] = d and L is an O K -lattice of rank r, then L can be naturally viewed as a Z-lattice of rank rd. Indeed,
L = O K v 1 + · · · + O K v r−1 + Av r
for some fractional ideal A. Now O K and A are isomorphic to Z d as Z-modules and hence we can identify L ≃ Z dr as a Z-module.
We will consider the quadratic form Tr(δQ) for a suitable δ. We choose δ to satisfy that • δ is a totally positive element (for then Tr(δQ) is positive definite), and • Tr(δQ(v)) ∈ Z for any v ∈ L.
This naturally leads us to looking at the codifferent
O ∨ K = {δ ∈ K | ∀α ∈ O K : Tr(δα) ∈ Z}. If O K = Z[ √ D], then O ∨ K = 1 2 √ D · O K .
The inclusion ⊃ is easy to prove as
Tr 1 2 √ D (a + b √ D) = Tr b 2 + a 2D √ D = b
for a, b ∈ Z (and the other inclusion is not too hard either). We next make the following observation: Let α ∈ O + K . If there exists δ ∈ O ∨,+ K such that Tr(δα) = 1, then α is indecomposable. For if α = β + γ for β, γ ∈ O + K , then 1 = Tr(δα) = Tr(δβ) + Tr(δγ) ≥ 2.
Now we have what we need to prove Theorem 5.5.
Sketch of proof of Theorem 5.5.
Step 1. Let U = u i+2 for some odd i and consider the indecomposables α i,t , 0 ≤ t < U . We define
δ = − 1 2 √ D α ′ i+1 .
It can be checked directly that δ ∈ O ∨ K and δ is totally positive. Next we compute the trace of
δα i,t = − 1 2 √ D (p i+1 − q i+1 √ D) · (p i + q i √ D) − t √ D 2D α ′ i+1 α i+1 . Since α ′ i+1 α i+1 = N (α i+1 ) ∈ Z, we have Tr(δα i,t ) = p i q i+1 − p i+1 q i = (−1) i+1 = 1.
Step 2. Take a quadratic O K -lattice (L, Q) representing all the indecomposables α i,t , 0 ≤ t < U , so that Q(v t ) = α i,t for some v t ∈ L. Then (Z 2r , Tr(δQ)) is a Z-lattice containing 2U vectors of length 1, namely ±v t , as Tr(δQ(±v t )) = Tr(δα i,t ) = 1.
Observe that if Q is classical, then Tr(δQ) is also classical. Therefore by repeatedly splitting off 1 (see Proposition 3.3) we get
Z 2r = 1 ⊥ 1 ⊥ · · · ⊥ 1 ⊥ L ′ ,
where 1 is repeated U times in the diagonal part. Thus 2r ≥ U .
Step 3. If Q is non-classical, we use known bound on the number of length-one vectors: There are ≤ max(r 2 , 240) of them in a non-classical Z-lattice of rank r.
Summary. Denote m(K) the minimal rank of a universal O K -lattice over K and m class (K) the minimal rank of a classical universal O K -lattice.
We saw that for
K = Q( √ D) with √ D = [u 0 , u 1 , . . . , u s ], we have 1 2 max(u i ) 1/2 ≤ m(K) ≤ 8 s i=1 u i ≪ √ D(log D) 2 1 2 max(u i ) ≤ m class (K) ≤ 8 s i=1 u i ≪ √ D(log D) 2
If the fundamental unit is not totally positive (i.e., s odd), this is not too bad: the lower bound is D 1/4 and D 1/2 for m(K) and m class (K), respectively. In the case when the fundamental unit is totally positive (i.e., s even), there are arbitrarily large differences between the lower and upper bounds, e.g., for √ D = [u 0 , 1, 1, . . . , 1, 2u 0 ], we get 1/2 ≤ m class (K) ≤ 4s. Obtaining better lower bounds would require including all the indecomposables, not just α i,t for a fixed i.
However, Kala-Yatsyna-Żmija recently expanded on these results by showing that
m class (Q( √ D)) ≫ ε D 1 12 −ε and m(Q( √ D)) ≫ ε D 1 24 −ε .
By "almost all" we mean that the set of such D has (natural) density 1 among the set of all squarefree D > 0.
Finally, let's mention an open problem. We know thanks to Chan-Oh [CO] that there exist analogues of the 15-and 290-Theorems over any number field. However, the corresponding bounds are explicitly known only for classical forms over Q( √ 5) [Le], and determining them more generally seems to be quite hard.
General Results
Let's now turn to fields of higher degree, and to several possible approaches for studying universal forms over them. Throughout this section, K thus denotes a totally real number field of degree d = [K : Q].
Using units. Let (L, Q) be a classical universal quadratic lattice. We saw in Proposition 3.3 that each unit splits off, and in particular L = 1 ⊥ L ′ for some lattice L ′ ⊂ L. Now any square of a unit is represented by 1 , so it need not be represented by L ′ . But if ε ∈ O ×,+ K is a unit which is not a square, then L ′ must represent ε and hence L = 1, ε ⊥ L ′′ for some lattice L ′′ ⊂ L. Continuing like this leads to the following observation: The rank of a classical universal lattice is always greater than or equal to #O ×,
+ K /O ×2 K . Since K is totally real, there are d−1 fundamental units ε 1 , ε 2 , . . . , ε d−1 , which implies O ×,+ K ≃ Z d−1 .
We can distinguish the two extreme cases:
• No fundamental unit is totally positive. Then each totally positive unit is a square, i.e.,
O ×,+ K = O ×2 K . • All fundamental units are totally positive. Then O ×2 K ≃ (2Z) d−1 and #O ×,+ K /O ×2 K = 2 d−1 .
In the general situation when k fundamental units are totally positive, the rank of a classical universal lattice is ≥ 2 k . If k ≥ 2, this proves a special case of Kitaoka's conjecture, i.e., that K has no ternary classical form. Not much is thus missing to prove the full conjecture, it would suffice to show the existence of a few indecomposables! As we already mentioned, recall that Hilbert's reciprocity law implies a theorem of Earnest-Khosravani [EK1]: If d is odd, then there is no ternary universal lattice (for local reasons).
Bounds on indecomposables.
It is not hard to show a general upper bound on the norm of an indecomposable (although surprisingly, this bound was discovered only very recently, even though this question is, e.g., formulated as [Nar,Problem 53]).
Theorem 6.1 ([ KY3,Theorem 5]). Each indecomposable has norm ≤ disc K/Q . In fact, if N (α) > disc K/Q , then α ≻ β 2 for some β ∈ O K .
Proof. Let's just sketch the proof. Let α be an element of norm N α > disc K/Q , and let σ 1 (α), . . . , σ d (α) be its conjugates. By Minkowski's theorem, for a sufficiently small ε > 0 the box
x ∈ R d : |x i | ≤ σ i (α) − ε, i = 1, . . . , d
in the Minkowski space contains a non-zero element β ∈ O K . Then α ≻ β 2 , and α is therefore decomposable.
Before proceeding further, note that we have already seen [Si2] that typically not all totally positive integers are sums of squares, but we can ask: What is the smallest integer P such that if an element is the sum of squares, then it is the sum of at most P squares? This integer P is called the Pythagoras number of the ring O K and is known to be always finite, but can be arbitrarily large [Sc2] (cf. also [Po]). However, there is an upper bound for Pythagoras numbers of orders in number fields that depends only on the degree of the number field [KY2,Corollary 3.3].
In the case of real quadratic number fields K = Q( √ D) the Pythagoras number is always ≤ 5, and this bound is sharp [Pe]. In fact, one can show that P(O K ) = 3 for D = 2, 3, 5 [Co,Sc1] and determine all D for which P(O K ) = 4 (as in [CP]). For some further recent results, see [KY1,Kr,KRS,Ras,Ti2].
Thanks to Theorem 6.1 above, if we have an element of large norm, we can successively subtract squares from it until we are left with something of norm ≤ disc K/Q . If we then rewrite the sum of squares as the sum of P squares, we obtain the following result:
Corollary 6.2 ([KY3, Theorem 6]). The quadratic form αx 2 α + y 2 1 + · · · + y 2 P ,
where we sum over all square classes of elements α ∈ O + K with norm N α ≤ disc K/Q , is universal and has rank ≪ disc K/Q · (log disc K/Q ) d−1 .
It is also sometimes useful to know that there is a partial converse of Theorem 6.1.
Theorem 6.3. Assume that K is primitive, i.e., there is no field Q F K. If α ∈ O + K has norm N α ≤ disc 1/(d 2 −d) K/Q
and is not divisible by any n ∈ Z ≥2 , then it is indecomposable.
We again only sketch the proof. As usual, σ 1 , . . . , σ d denote the d embeddings of K into R. Suppose that α = β + γ for some totally positive integers β and γ. Then
N α = (σ 1 β + σ 1 γ)(σ 2 β + σ 2 γ) · · · (σ d β + σ d γ) ≥ Tr(σ 1 β · σ 2 γ · σ 3 γ · · · σ d γ) ≫ disc 1/(d 2 −d) K/Q
by the Stieltjes-Schur Theorem [Sch,§3]: If µ ≻ 0 and K = Q(µ), then Tr(µ) ≫ disc
1/(d 2 −d) K/Q , cf. [Ka4, Proposition 2].
Elements of trace 1. In Section 4 we saw lower bounds for ranks of universal quadratic lattices in terms of elements of trace 1 in the codifferent. Exactly the same result holds in general.
Theorem 6.4 ( [Ya], [KT,Section 7.1]). Assume that there are β 1 , . . . ,
β u ∈ O + K , δ ∈ O ∨,+ K such that Tr(β i δ) = 1 for all i. Then m(K) ≥ u d , m class (K) ≥ √ u d .
How to find such elements? There is no general way (after all, there may be no totally positive elements in the codifferent that have trace 1), so one may have to rely on explicit constructions (such as in the proof of Theorem 5.5 or 7.1). However, let's also briefly discuss a method based on interlacing polynomials and another one which uses the Dedekind zeta function. The proof uses interlacing polynomials. Let
f (x) = (x − α 1 )(x − α 2 ) · · · (x − α d )
be the minimal polynomial for α, and assume that α 1 < α 2 < · · · < α d . We say that a polynomial
g(x) = (x − β 1 )(x − β 2 ) · · · (x − β d−1 ) interlaces f is α 1 < β 1 < α 2 < β 2 < · · · < β d−1 < α d .
The key fact is that there is a bijection between such polynomials g and the set {γ ∈ O ∨,+ K | Tr γ = 1}.
The Dedekind zeta function is defined as
ζ K (s) = A<O K 1 (N A) s ,
Re s > 1.
The series converges absolutely for Re s > 1 and ζ K has a meromorphic continuation to the entire complex plane with a simple pole at s = 1. It satisfies a functional equation which relates ζ K (s) to ζ K (1 − s). For us, the important important fact is that Siegel related the value ζ K (−1) to elements of small trace [Si1], [Za,§1]: Theorem 6.6 (Siegel's formula for ζ K (−1) and functional equation). Assume that K is a totally real field of degree d = 2, 3, 5, 7. Then
α∈O ∨,+ K Tr α=1 σ (α)(O ∨ K ) −1 = 1 b d disc K/Q 3/2 −1 4π d ζ K (2)
for a suitable b d ∈ Q (e.g., b 2 = 1 240 , b 3 = − 1 504 , . . . ). Here
σ(B) = A|B N (A).
A similar formula holds in each degree d, but as the degree grows, it will involve elements of large traces (roughly, of traces up to d/6).
As a sample application, let's mention the following result on the lifting problem (that will discussed in detail in Section 8).
Theorem 6.7 ([KY2, Theorem 1.2]). If K is a totally real number field of degree d = 2, 3, 4, 5, 7 which has
• principal codifferent ideal, and • a universal quadratic form with coefficients in Z,
then K = Q( √ 5) or K = Q(ζ 7 + ζ −1 7 )
, where ζ 7 = e 2πi/7 . The form x 2 + y 2 + z 2 is universal over Q( 5), and x 2 + y 2 + z 2 + w 2 + xy + xz + xw is universal over Q(ζ 7 + ζ −1 7 ). Large ranks. Let's also summarize here the known results on the existence of number fields with large minimal rank m(K). For quadratic fields, we have already seen this in Section 5, and in the cubic case, this is originally due to Yatsyna [Ya,Theorem 5], and we will establish this in Section 7 below.
A natural idea for extending these results to higher degrees is to start with a field K with large m(K) and to consider overfields L ⊃ K. This was first carried out for multiquadratic fields [KS], and then extended to all fields of degrees divisible by 2 and 3 [Ka4]. Finally, Doležálek [Do] generalized this method to make it readily applicable to quite general extensions L ⊃ K. So it would (mostly) suffice to prove the existence of large ranks m(K) over fields of prime degrees ≥ 5 -which, however, remains open so far.
Families of Cubic Fields
Over a fixed field, one can compute everything explicitly, e.g., there are finitely many totally positive elements α with norm N α ≤ disc (up to multiplication by units), and we can check which ones are indecomposable. We can also compute the codifferent and check which elements have trace 1.
For all fields of a given degree d, the problem is much harder. We used continued fractions to deal with real quadratic fields Q( √ D). We might attempt to use generalized continued fractions [Ber, Sch] for fields of a higher degree -they are much worse behaved but there are some ongoing works in connection with the Jacobi-Perron algorithm [KRSt, KST]. Geometric generalized continued fractions [Kar] are also promising, since there is a close connection to indecomposables.
Rather than working with all fields, it is however typically easier to focus on a suitable family of fields that share some relevant properties (such as the structure of units and indecomposables).
The simplest cubic fields. We describe first the family of totally real cubic fields introduced by Shanks [Sh].
Let K = Q(ρ) where ρ is a root of the polynomial
f (x) = x 3 − ax 2 − (a + 3)x − 1, a ≥ −1.
If we order the three roots ρ, ρ ′ , ρ ′′ as ρ > ρ ′′ > ρ ′ , then they are of approximate sizes ρ ≈ a+ 1, ρ ′′ ≈ 0 and ρ ′ ≈ −1. It is a useful fact that all the roots are units, and are permuted under the mapping α → −1 1+α . We thus see that the other two conjugates ρ ′ and ρ ′′ also belong to K, K is the splitting field of f , and the Galois group Gal(K/Q) ≃ Z/3 is cyclic.
Another consequence is that K has units of all signatures. The discriminant of the polynomial f equals disc f = (a 2 + 3a + 9) 2 . If a 2 + 3a + 9 is squarefree (which happens for a positive density of a),
then O K = Z[ρ].
The units are small, hence the regulator is also small, and the class number formula implies that the class number is large, roughly ≈ a 2 (up to a logarithmic factor).
When we search for indecomposables in a totally real number field K, it is natural to consider K in the Minkowski space by the mapping σ : K ֒→ R d , α → (σ 1 (α), σ 2 (α), . . . , σ d (α)).
For example, consider the situation in a real quadratic field K = Q( √ D) with a fundamental totally positive unit ε. We can multiply every totally positive element by a suitable unit to move it into the cone R ≥0 · 1 + R >0 · ε spanned by 1 and ε. If β ≻ 1 or β ≻ ε, then it is not indecomposable, so we can further restrict our attention to the parallelogram
[0, 1) · 1 + [0, 1) · ε = {t 1 · 1 + t 2 · ε | t 1 , t 2 ∈ [0, 1)}.
The situation in totally real cubic fields is similar. The totally positive units form a discrete set located on the hyperboloid {(x, y, z) ∈ R 3 | xyz = 1} in the Minkowski space. Up to multiplication by units, each element is contained in the polyhedral cone
C = R ≥0 · 1 + R ≥0 · ε 1 + R ≥0 · ε 2 + R ≥0 · ε 1 ε 2 ,
where ε 1 and ε 2 generate the totally positive unit group. This is essentially the content of Shintani's unit theorem [Ne,Thm (9.3)]. The cone C is the union of two "triangular" cones spanned by 1, ε 1 , ε 2 and ε 1 , ε 2 , ε 1 ε 2 , respectively. Again, we can restrict our search for indecomposables to the parallelepipeds [0, 1) · 1 + [0, 1) · ε 1 + [0, 1) · ε 2 and [0, 1) · ε 1 + [0, 1) · ε 2 + [0, 1) · ε 1 ε 2 .
In the simplest cubic fields, this approach (described in more detail in [KT,Section 4]) is explicit enough that we can determine all indecomposables. • there exists a diagonal universal form of rank ∼ 3a 2 , • any classical universal lattice has rank ≥ a 2 6 , • any universal lattice has rank ≥ a 3 √ 2 . Gil Muñoz and Tinková [GMT] recently extended these results to also cover some non-monogenic simplest cubic fields.
• 1 • 1 + ρ + ρ 2 • −v − wρ + (v + 1)ρ 2 , 0 ≤ v ≤ a, v(a + 2) + 1 ≤ w ≤ (v + 1)(a + 1
Other cubic families. Tinková [Ti1] obtained similar results in other families of cubic fields. The fields in such families are again generated by a root ρ of a cubic polynomial depending on some parameters. More specifically,
• Ennola's cubic fields, generated by a root of x 3 + (a − 1)x 2 − ax − 1, a ≥ 3, • a family investigated by Thomas, generated by a root of x 3 − (a + b)x 2 + abx − 1.
Tinková considered indecomposables and quadratic forms over the order Z [ρ]. She showed that, surprisingly, the minimum of Tr(δα) taken over δ in the codifferent can be arbitrarily large for an indecomposable element α. She also applied these results to determine the Pythagoras number (for the simplest cubic fields, it is typically equal to 6) [Ti2].
Continued fraction families of real quadratic fields. For comparison, let's discuss similar families in degree two. The two most well-known examples are:
• Yokoi's family Q( √ m 2 + 4). If m = 2n + 1 is odd, then the relevant continued fraction is 1 + (2n + 1) 2 + 4 2 = [n + 1, 2n + 1].
• Chowla's family Q( √ 4m 2 + 1).
We have also already seen that √ n 2 − 1 = [n − 1, 1, 2(n − 1)]. The idea is to generalize this by considering families Q(
√ D) where √ D = [u 0 , u 1 , u 2 , . . . , u s−1 , 2u 0 ]
with s and u 1 , . . . , u s−1 fixed. A necessary condition is that the sequence u 1 , . . . , u s−1 must be symmetric, i.e., u i = u s−i . It turns out that this condition is almost sufficient for the existence of D.
Theorem 7.3 ( [Fr, Theorem]). Let u 1 , . . . , u s−1 be symmetric, and define the numbers q i via:
q i+1 = u i+1 q i + q i−1 , q −1 = 0, q 0 = 1.
(This will be the sequence of denominators of the convergents p i q i , and it does not depend on u 0 .) There are infinitely many squarefree D ≡ 2, 3 (mod 4) such that √ D = [k, u 1 , . . . , u s−1 , 2k] if and only if q s−2 or q 2 s−2 −(−1) s q s−1 is even (otherwise, there is no such D, even when we drop the condition "squarefree").
In such a case, all D and k are given by
D = D(t) = at 2 + bt + c, k = k(t) = et + f, t ≥ 1
for fixed integers a, b, c, e, f that can be explicitly given in terms of u i .
There is a similar characterization for D ≡ 1 (mod 4) and the continued fraction expansion of 1+
√ D 2 [H-K].
These families have a number of advantageous properties:
• The fundamental unit ε depends linearly on t.
• The class number is large, essentially t/ log t by the class number formula (see [CF+, DK, DL]).
• Indecomposables behave nicely (as in the simplest cubic fields).
Lifting Problem for Universal Forms
When can a quadratic form with coefficients in Z be universal over a larger number field K? The answer to this question, which we call the lifting problem, seems to be "very rarely", at least for number fields of small degrees.
We have already mentioned the result of Siegel on non-universality of sums of squares -let's now sketch its proof.
Theorem 8.1 ([Si2, Theorem I]). If K is a totally real number field such that every element of O + K is a sum of squares, then K = Q or Q( √ 5).
Sketch of proof. Assume that every element of O + K is a sum of squares.
Step 1. We show first that all totally positive units are squares and that we have units of all signatures. By our assumption, each totally positive unit is expressible as ε = α 2 1 + · · · + α 2 t for some α i ∈ O K , and therefore
1 = N (ε) ≥ t i=1 N (α i ) 2 ≥ t,
which implies ε = α 2 1 . Next, let ε 1 , . . . , ε d−1 be a system of fundamental units, and consider the units
ε = (−1) a 0 ε a 1 1 · · · ε a d−1 d−1 , a i ∈ {0, 1}.
There are 2 d of them and they have distinct signatures (otherwise there would exist two of these units whose product is totally positive and hence a square). As there are 2 d signatures in total, we have units of all signatures.
Step 2. We show next that every indecomposable is equal to ε 2 for some unit ε. If β = t i=1 α 2 i is an indecomposable, we must have t = 1, thus β = α 2 for α = α 1 . Let ε be a unit with the same signature as α, or equivalently, εα ≻ 0. If εα ≻ γ for some totally positive γ, then ε 2 β ≻ γ 2 , and ε 2 β is not indecomposable. Thus εα is indecomposable. But if N (β) > 1, then N (β) > N (εα) > 1 and we found an indecomposable with a strictly smaller norm greater than 1. We can continue this infinite descent until we reach a contradiction.
Step 3. Suppose finally that M = O K \ Q( √ 5) is non-empty. Then it contains totally positive elements (because if α ∈ M , then N + α ∈ M is totally positive for a large enough integer N ). Choose a totally positive element λ ∈ M with minimal trace. The rest of the proof runs as follows:
• The fact that λ has minimal trace shows that it is indecomposable. By what we showed earlier, λ = ε 2 for a unit ε. Without loss of generality, we can assume that Tr ε < 0 (substitute −ε for ε if necessary). • Show Tr λ < 3d, where d is the degree of K.
• Consider the decomposition of ε 2 + ε + 1 ≻ 0, and after some estimates get a contradiction.
The general question we are asking is: Over which number fields K is there a universal Z-form, i.e., a positive definite quadratic form with coefficients in Z? For K different from Q and Q( √ 5), the sum of squares is not universal by Siegel's theorem. In particular, there is no universal diagonal Z-form (for each diagonal Z-form is represented by the sum of sufficiently many squares).
We can extend this to general forms over real quadratic fields, as conjectured by Deutsch [De2].
Theorem 8.2 ([KY2, Theorem 1.1]). If K = Q( √ 5) is a real quadratic field, then there is no universal Z-form over K.
For the proof, we again want to work with "minimal vectors" of the corresponding quadratic Olattice (L, Q), as in the proof of Theorem 5.5. For a Z-lattice, they are the vectors v such that Q(v) is the smallest represented positive integer. This does not make a good sense over O K , so we will consider Tr K/Q (Q(v)) (which is a positive integer).
Recall that the codifferent is defined as O ∨ K = {δ ∈ K | Tr(δα) ∈ Z ∀α ∈ O K } . A little more generally, we can look at Tr(δQ(v)) for δ ∈ O ∨,+ K , which is still a non-negative integer. We will be interested in the vectors minimizing this.
As another tool, let's introduce the tensor product (see [KY2,Section 4] for details for the following discussion) (L 1 ⊗ L 2 , Q 1 ⊗ Q 2 ) of two quadratic Z-lattices (L 1 , Q 1 ), (L 2 , Q 2 ). If we fix Z-bases v i and w j for L 1 and L 2 , then L 1 ⊗ L 2 is the Z-lattice whose Z-basis consists of formal elements v i ⊗ w j . The quadratic form is then defined so that (Q 1 ⊗ Q 2 )(v ⊗ w) = Q 1 (v)Q 2 (w) for v ∈ L 1 , w ∈ L 2 .
This gives an integral Z-lattice if at least one of the Q i is classical.
An important observation is that if Q is a Z-form of rank r, then (O r K , Tr(δQ)) is isomorphic to the tensor product (O K ⊗ Z r , T δ ⊗ Q), where T δ (x) = Tr(δx 2 ) (to be more precise, we need to identify O K with Z d by choosing an integral basis).
A Z-lattice L is of E-type if for each Z-lattice M , all the minimal vectors of L ⊗ M are split, i.e., of the form v ⊗ w for v ∈ L, w ∈ M . A theorem of Kitaoka [Kt,Theorems 7.1.1,7.1.2,7.1.3] states that each lattice of rank ≤ 43 is of E-type. The form T δ has rank d, so we are fine when d ≤ 43.
Let Q be a universal Z-form (over a number field K of degree d ≤ 43), and let's us introduce the following condition for α ∈ O + K :
∃δ ∈ O ∨,+ K : Tr(δα) = min β∈O + K Tr(δβ).
One uses minimal vectors to show:
• If α satisfies (7), then α is a square (and indecomposable).
• Every totally positive unit is a square, and so there are units of all signatures in O K .
• If α satisfies (7), then α is a unit.
Sketch of proof of Theorem 8.2. Let K = Q( √ D), so that O K = Z[ω D ], where ω D = √ D, D ≡ 2, 3 (mod 4),
1+
√ D 2 , D ≡ 1 (mod 4). We know that O ∨ K = δO K for some totally positive δ (because we already established that we have units of all signatures). The elements α ∈ O + K such that Tr(δα) = 1 form a convex set in Z d−1 = Z (this set can be described using interlacing polynomials). Hence they form an arithmetic progression of length at least √ D − 1. By the remarks preceding the proof, they are all units. But we cannot have more than 4 units in an arithmetic progression in a real quadratic field by a result of Newman [New]. Now only finitely many values of D need to be examined to finish the proof. Theorem 6.1 is a partial extension of Theorem 8.2 to higher degrees. How to proceed further? There may be some clever approaches using geometry of numbers, elements of trace 2 in the codifferent, and perhaps even different generators of O ∨ K , but at this point we do not know much more than Theorem 6.7 above (although we have a promising work in progress in this direction).
As a few final results, let's mention:
• Let K = Q, Q( √ 5) be a totally real number field. Then there is no classical universal Z-form over K of rank 3, 4, or 5 [KY2,Corollary 3.4].
• Let F be a totally real number field, L an O F -lattice, and d, m ∈ Z >0 . There are at most finitely many totally real number fields K ⊃ F of degree d = [K : Q] such that L ⊗ O K represents all elements of mO + K [KY3, Theorem 2].
Theorem 4.2 ([DS, Theorems 2 and 3]). The indecomposables α are precisely the semiconvergents, i.e., elements of the form
Example 4 . 3 .
43Let's find all indecomposables for D = 6. The convergents of the continued fraction expansion √ 6 = [2, 2, 4] are
Theorem 4.6 ([Ki2, Theorem 1],[BK2, Theorem 10]). The quadratic form ⊥ σ∈S σ, σ, σ, σ, εσ, εσ, εσ, εσ is universal and has 8 · #S variables. (Here ⊥ denotes an orthogonal sum of the diagonal lattices.)
Theorem 5. 4
4([BK1, Theorem 1], [Ka1, Theorem 1.1]). For any positive integer r, there are infinitely many quadratic fields Q( √ D) that do not have a universal lattice of rank ≤ r.
Theorem 5.5 ([KT, Sections 7.1 and 7.3]). Let √ D = [u 0 , u 1 , . . . , u s ] and U = max(u 1 , u 3 , . . . , u s−1 ), if s is even, √ D, if s is odd.
Theorem 5.6 ([KYZ, Theorem 1.1]). Let ε > 0. For almost all squarefree D > 0, we have that
Theorem 6.5 ([Ya, Theorem 4]). Let d and r be positive integers. There are only finitely many totally real number fields K of degree d such that• K is primitive (it has no proper subfields),• K is monogenic (O K = Z[α]for some algebraic integer α),• K has units of all signatures (equivalently, O ×,+ K = O ×2 K ), • K has a universal lattice of rank r.
Theorem 7.1 ([KT, Theorem 1.2]). Let K = Q(ρ) be a simplest cubic field such that O K = Z[ρ]. Up to multiplication by units, all indecomposables are
For the indecomposable 1 + ρ + ρ 2 , we have min Tr(δ(1 + ρ + ρ 2 )) | δ ∈ O ∨+ K = 2.For the indecomposables α = −v − wρ + (v + 1)ρ 2 in the triangle, min Tr(δα) | δ ∈ O ∨+ K = 1.Corollary 7.2 ([KT, Theorem 1.1]). Let K = Q(ρ) be a simplest cubic field such that O K = Z[ρ]. Then
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"Steven R Wilson [email protected] \nUniversity of Michigan\n\n",
"Rada Mihalcea [email protected] \nUniversity of Michigan\n\n"
] | [
"University of Michigan\n",
"University of Michigan\n"
] | [
"28 Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics General rights Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics"
] | The activities we do are linked to our interests, personality, political preferences, and decisions we make about the future. In this paper, we explore the task of predicting human activities from user-generated content. We collect a dataset containing instances of social media users writing about a range of everyday activities. We then use a state-of-the-art sentence embedding framework tailored to recognize the semantics of human activities and perform an automatic clustering of these activities. We train a neural network model to make predictions about which clusters contain activities that were performed by a given user based on the text of their previous posts and selfdescription. Additionally, we explore the degree to which incorporating inferred user traits into our model helps with this prediction task. | 10.18653/v1/p19-1245 | [
"https://web.archive.org/web/20200305192124/https:/www.research.ed.ac.uk/portal/files/117783817/Predicting_Human_Activities_WILSON_DOA13052019_VOR_CC_BY.pdf"
] | 196,174,064 | 1907.08540 | 021d0c5f5e7946f260c72412bf749ff35233531b |
Predicting Human Activities from User-Generated Content Predicting Human Activities from User-Generated Content
Association for Computational LinguisticsCopyright Association for Computational LinguisticsJuly 28 -August 2, 2019. 2019
Steven R Wilson [email protected]
University of Michigan
Rada Mihalcea [email protected]
University of Michigan
Predicting Human Activities from User-Generated Content Predicting Human Activities from User-Generated Content
28 Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics General rights Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
Florence, Italy; Florence, Italy; Florence, ItalyAssociation for Computational LinguisticsJuly 28 -August 2, 2019. 201910.18653/v1/P19-1245Download date: 09. Mar. 2020Edinburgh Research Explorer Citation for published version: Wilson, S & Mihalcea, R 2019, Predicting Human Activities from User-Generated Content. in/07/19. https://doi.org/10.18653/v1/P19-1245 Digital Object Identifier (DOI): 10.18653/v1/P19-1245 Link: Link to publication record in Edinburgh Research Explorer Document Version: Publisher's PDF, also known as Version of record Published In:
The activities we do are linked to our interests, personality, political preferences, and decisions we make about the future. In this paper, we explore the task of predicting human activities from user-generated content. We collect a dataset containing instances of social media users writing about a range of everyday activities. We then use a state-of-the-art sentence embedding framework tailored to recognize the semantics of human activities and perform an automatic clustering of these activities. We train a neural network model to make predictions about which clusters contain activities that were performed by a given user based on the text of their previous posts and selfdescription. Additionally, we explore the degree to which incorporating inferred user traits into our model helps with this prediction task.
Introduction
What a person does says a lot about who they are. Information about the types of activities that a person engages in can provide insights about their interests (Goecks and Shavlik, 2000), personality (Ajzen, 1987), physical health (Bouchard et al., 2018), the activities that they are likely to do in the future (Ouellette and Wood, 1998), and other psychological phenomena like personal values (Rokeach, 1973). For example, it has been shown that university students who exhibit traits of interpersonal affect and self-esteem are more likely to attend parties (Paunonen and Ashton, 2001), and those that value stimulation are likely to watch movies that can be categorized as thrillers (Bardi and Schwartz, 2003).
Several studies have applied computational approaches to the understanding and modeling of human behavior at scale (Yin et al., 2014) and in real time (Wang et al., 2015). However, this previous work has mainly relied on specific devices or platforms that require structured definitions of behaviors to be measured. While this leads to an accurate understanding of the types of activities being done by the involved users, these methods capture a relatively narrow set of behaviors compared to the huge range of things that people do on a day-to-day basis. On the other hand, publicly available social media data provide us with information about an extremely rich and diverse set of human activities, but the data are rarely structured or categorized, and they mostly exist in the form of natural language. Recently, however, natural language processing research has provided several examples of methodologies for extracting and representing human activities from text (Fast et al., 2016;Wilson and Mihalcea, 2017) and even multimodal data (Agrawal et al., 2016).
In this paper, we explore the task of predicting human activities from user-generated text data, which will allow us to gain a deeper understanding of the kinds of everyday activities that people discuss online with one another. Throughout the paper, we use the word "activity" to refer to what an individual user does or has done in their daily life. Unlike the typical use of this term in the computer vision community (Cheng et al., 2015;Zhang et al., 2017), in this paper we use it in a broad sense, to also encompass non-visual activities such as "make vacation plans" or "have a dream" We do not focus on fine-grained sequences actions such as "pick up a camera", "hold a camera to one's face", "press the shutter release button", and others. Rather, we focus on the highlevel activity as a person would report to others: "take a picture". Additionally, we specifically focus on everyday human activities done by the users themselves, rather than larger-scale events (Atefeh and Khreich, 2015), which are typically characterized by the involvement or interest of many users, often at a specific time and location. Given that the space of possible phrases describ-ing human activities is nearly limitless, we propose a set of human activity clusters that summarize a large set of several hundred-thousand selfreported activities. We then construct predictive models that are able to estimate the likelihood that a user has reported that they have performed an activity from any cluster.
The paper makes the following main contributions. First, starting with a set of nearly 30,000 human activity patterns, we compile a very large dataset of more than 200,000 users undertaking one of the human activities matching these patterns, along with over 500 million total tweets from these users. Second, we use a state-of-theart sentence embedding framework tailored to recognize the semantics of human activities and create a set of activity clusters of variable granularity. Third, we explore a neural model that can predict human activities based on natural language data, and in the process also investigate the relationships between everyday human activities and other social variables such as personal values.
Data
While we do not expect to know exactly what a person is doing at any given time, it is fairly common for people to publicly share the types of activities that they are doing by making posts, written in natural language, on social media platforms like Twitter. However, when taking a randomly sampled stream of tweets, we find that only a small fraction of the content was directly related to activities that the users were doing in the real world -instead, most instances are more conversational in nature, or contain the sharing of opinions about the world or links to websites or images. Using such a random sample would require us to filter out a large percentage of the total data collected, making the data collection process inefficient.
Therefore, in order to target only those tweets that are rich in human activity content, we formulate a set of queries that allows us to use the Twitter Search API to find instances of users tweeting about common human activities. Each query contains a first-person, past-tense verb within a phrase that describes a common activity that people do. Using this approach, we are able to retrieve a set of tweets that contains a high concentration of human activity content, and we also find that users who wrote these tweets are much more likely to have written other tweets that describe human activities (Table 1). We build our set of human ac-Sampled tweets w/valid activities 2% Queried tweets w/valid activities 81% Addtl. user tweets w/valid activities 15% tivity queries from two sources: the Event2Mind dataset (Rashkin et al., 2018) and a set of short activity surveys, which we collect ourselves, to obtain nearly 30K queries (Table 2) .
Event2Mind Activities
The Event2Mind dataset contains a large number of event phrases which are annotated for intent and reaction. The events themselves come from four sources of phrasal events (stories, common n-grams found in web data, blogs, and English idioms), and many of them fall under our classification of human activities, making Event2Mind a great resource in our search for concrete examples of human activities. We consider events for which a person is the subject (e.g, "PersonX listens to PersonX's music") to be human activities, and remove the rest (e.g., "It is Christmas morning"). We then use several simple rules to convert the Event2Mind instances into first-person past-tense activities. Since all events were already filtered so that they begin with "PersonX", we replace the first occurrence of "PersonX" in each event with "I" and all subsequent occurrences with "me". All occurrences of "PersonX's" become "my", and the main verb in each phrase is conjugated to its pasttense form using the Pattern python module. 1 For example, the event "PersonX teaches PersonX's son" becomes the query "I taught my son". Since Event2Mind also contains wildcard placeholders that can match any span of text within the same phrase (e.g., "PersonX buys at the store") 2 but the Twitter API doesn't provide a mechanism for wildcard search, we split the event on the string and generate a query that requires all substrings to appear in the tweet. We then check all candidate tweets after retrieval and remove any for which the substrings do not appear in the same order as the original pattern.
Short Survey Activities
In order to get an even richer set of human activities, we also ask a set of 1,000 people across the United States to list any five activities that they had done in the past week. We collect our responses using Amazon Mechanical Turk, 3 and manually verify that all responses are reasonable. We remove any duplicate strings and automatically convert them into first-person and past-tense (if they were not in that form already). For this set of queries, there are no wildcards and we only search for exact matches. Example queries obtained using this approach include "I went to the gym" and "I watched a documentary".
Query Results
Using our combined set of unique human activity queries, we use the Twitter Search API 4 to collect the most recent 100 matches per query (the maximum allowed by the API per request), as available, and we refer to these tweets as our set of queried tweets. We then filter the queried tweets as follows: first, we verify that for any tweets requiring the match of multiple substrings (due to wildcards in the original activity phrase), the substrings appear in the correct order and do not span multiple sentences. Next, we remove activity phrases that are preceded with indications that the author of the tweet did not actually perform the activity, such as "I wish" or "should I . . . ?". We refer to the set of tweets left after this filtering as valid queried tweets (see Table 3 for more details).
In order to gather other potentially useful information about the users who wrote at least one valid queried tweet, we collect both their self-written profile and their previously written tweets (up to 3,200 past tweets per user, as allowed by the Twitter API), and we refer to these as our set of additional tweets. We ensure that there is no overlap between the sets of queried tweets and additional tweets, so in the unlikely case that a user has posted the same tweet multiple times, it cannot be included in both sets.
Further, we use a simple pattern-matching approach to extract additional activities from these additional tweets. We search for strings that match I <VBD> . * <EOS> where <VBD> is any past-tense verb, . * matches any string (nongreedy), and <EOS> matches the end of a sentence. We then perform the same filtering as before for indications that the person did not actually do the activity, and we refer to these filtered matches as our set of additional activities (see Table 4 for more information). Note that since these additional activities can contain any range of verbs, they are naturally noisier than our set of valid query tweets, and we therefore do not treat them as a reliable "ground truth" source of selfreported human activities, but as a potentially useful signal of activity-related information that can be associated with users in our dataset.
For our final dataset, we also filter our set of users. From the set of users who posted at least one valid queried tweet, we remove those who had empty user profiles, those with less than 25 addi-tional tweets, and those with less than 5 additional activities (Table 5).
Creating Human Activity Clusters
Given that the set of possible human activity phrases is extremely large and it is unlikely that the same phrase will appear multiple times, we make this space more manageable by first performing a clustering over the set of activity phrase instances that we extract from all valid queried tweets. We define an activity phrase instance as the set of words matching an activity query, plus all following words through the end of the sentence in which the match appears. By doing this clustering, our models will be able to make a prediction about the likelihood that a user has mentioned activities from each cluster, rather than only making predictions about a single point in the semantic space of human activities.
In order to cluster our activity phrase instances, we need to define a notion of distance between any pair of instances. For this, we turn to prior work on models to determine semantic similarity between human activity phrases (Zhang et al., 2018) in which the authors utilized transfer learning in order to fine-tune the Infersent (Conneau et al., 2017) sentence similarity model to specifically capture relationships between human activity phrases. We use the authors' BiLSTM-max sentence encoder trained to capture the relatedness dimension of human activity phrases 5 to obtain vector representations of each of our activity phrases. The measure of distance between vectors produced by this model was shown to be strongly correlated with human judgments of general activity relatedness (Spearman's ρ = .722 between the model and human ratings, while inter-annotator agreement is .768).
While the relationship between two activity phrases can be defined in a number of ways (Wilson and Mihalcea, 2017), we we chose a model that was optimized to capture relatedness so that our clusters would contain groups of related activities without enforcing that they are strictly the same activity. Since the model that we employed was trained on activity phrases in the infinitive form, we again use the Pattern python library, this time to convert all of our past-tense activities to this form. We also omit the leading first person pronoun from each phrase, and remove user mentions (@<user>), hashtags, and URLs. We then 5 Shared by the first author of the referenced paper.
"Cooking" make cauliflower stir-fry for dinner make garlic and olive oil vermicelli for lunch start cooking bacon in the oven (on foil in a sheet) burn the turkey make perfect swordfish steaks tonight "Pet/Animal related" get a new pet spider today cuddle 4 dogs get a pet sitter feel so happy being able to pet kitties today spend some time with cats "Spectating" watch football italia watch a football game in the pub watch basketball today watch sports watch fireworks today in the theatre "Passing Examinations" ace the exam pass one's exam thank god get a perfect score on one's exam get a c on one's french exam pass another exam omg define the distance between any two vectors using cosine distance, i.e., 1 − A·B ||A||||B|| , for vectors A and B.
We use K-means clustering in order to find a set of k act clusters that can be used to represent the semantic space in which the activity vectors lie. We experiment with k act = 2 n with n ∈ Z ∩ [3, 13] and evaluate the clustering results using several metrics that do not require supervision: within-cluster variance, silhouette coefficient (Rousseeuw, 1987), Calinski-Harabaz criterion (Caliński and Harabasz, 1974), and Davies-Bouldin criterion (Davies and Bouldin, 1979). In practice, however, we find that these metrics are strongly correlated (either positively or negatively) with the k act , making it difficult to quantitatively compare the results of using a different number of clusters, and we therefore make a decision based on a qualitative analysis of the clusters. 6 For the purpose of making these kinds of Distance to "Cooking": 0.11 cook breakfast cook the spaghetti start cooking cook something simple start cooking a lot more Distance to "Cooking": 0.52 feed one's ducks bread all the time give one's dog some chicken stop eating meat eat hot dogs and fries get one's dog addicted to marshmellows Distance to "Cooking": 0.99 take a picture with her post a photo of one bring something like 1000 rolls of film draw a picture of us holding hands capture every magical moment to give to the bride predictions about clusters, it is beneficial to have a smaller number of larger clusters, but clusters that are too large are no longer meaningful since they contain sets of activities that are less strongly related to one another. In the end, we find that using 2 10 = 1024 clusters leads to a good balance between cluster size and specificity, and we use this configuration for our prediction experiments moving forward. Examples of activities that were assigned the same cluster label are shown in Table 6, and Table 7 illustrates the notion of distance within our newly defined semantic space of human activities. For example, two cooking-related clusters are near to one another, while a photography-related cluster is very distant from both.
Methodology
Given a set of activity clusters and knowledge about the users who have reported to have participated in these activities, we explore the ability of machine learning models to make inferences about which activities are likely to be next performed by a user. Here we describe the supervised learning setup, evaluation, and neural architecture used for the prediction task.
Problem Statement
We formulate our prediction problem as follows: for a given user, we would like to produce a probability distribution over all activity clusters such that: argmax
c i ∈C P (c i |h, p, a) = c t ,
where C is a set of activity clusters, h, p, and a are vectors that represent the user's history, profile, and attributes, respectively, and c t is the target cluster. The target cluster is the cluster label of an activity cluster that contains an activity that is known to have been performed by the user. If a model is able to accurately predict the target cluster, then it is able to estimate the general type of activity that the user is likely to write about doing in the future given some set of information about the user and what they have written in the past. By also generating a probability distribution over the clusters, we can assign a likelihood that each user will write about performing each group of activities in the future. For example, such a model could predict the likelihood that a person will claim to engage in a "Cooking" activity or a "Pet/Animal related" activity.
The ability to predict the exact activity cluster correctly is an extremely difficult task, and in fact, achieving that alone would be a less informative result than producing predictions about the likelihood of all clusters. Further, in our setup, we only have knowledge about a sample of activities that people actually have done. In reality, it is very likely that users have participated in activities that belong to a huge variety of clusters, regardless of which activities were actually reported on social media. Therefore, it should be sufficient for a model to give a relatively high probability to any activity that has been reported by a user, even if there is no report of the user having performed an activity from the cluster with the highest probability for that user.
Model Architecture
As input to our activity prediction model, we use three major components: a user's history, profile, and attributes. We represent a history as a sequence of documents, D, written by the user, that contain information about the kinds of activities that they have done. Let t = |D|, and each document in D is represented as a sequence of tokens. We experiment with two sources for D: all additional tweets written by a user, or only the additional activities contained in tweets written by a user, which is a direct subset of the text contained in the full set of tweets.
A user's profile is a single document, also represented as a sequence of tokens. For each user, we populate the profile input using the plain text user description associated with their account, which often contains terms which express selfidentity such as "republican" or "athiest." We represent the tokens in both the user's history and profile with the pretrained 100dimensional GloVe-Twitter word embeddings (Pennington et al., 2014), and preprocess all text with the script included with these embeddings. 7 Finally, our model allows the inclusion of any additional attributes that might be known or inferred in order to aid the prediction task, which can be passed to the model as a dim a dimensional real-valued vector. For instance, we can use personal values as a set of attributes, as described in Section 3.3.
We train a deep neural model, summarized in Figure 1, to take a user's history, profile, and attributes, and output a probability distribution over the set of k act clusters of human activities, indicating the likelihood that the user has reported to have performed an activity in each cluster. There are four major components of our network: Document Encoder This is applied to each of the t documents in the history-either an activity phrase or a full tweet. For document i in D, it takes a sequence of token embeddings as input and produces a dim d dimensional vector, d i as output.
History Encoder This layer takes the sequence 7 nlp.stanford.edu/projects/glove/preprocess-twitter.rb {d 0 , . . . , d t } as input and produces a single dim H dimensional vector, h, as output, intended to represent high-level features extracted from the entire history of the user.
Profile Encoder Takes each token in the user's profile as input and produces a single dim p dimensional vector, p as output.
Classifier As input, this module takes the concatenation a ⊕ h ⊕ p, where a is the predefined attribute vector associated with the user. Then, a prediction is made for each of the k act clusters, first applying softmax in order to obtain a probability distribution. We refer to the dimension of the output as dim o .
For any of the three encoder layers, several layer types can be used, including recurrent, convolutional, or self-attention based (Vaswani et al., 2017) layers. The classifier layer is the only layer that does not take a sequence as input and we implement it using a simple feed-forward multilayer network containing c layers with h c hidden units each. The network is trained with crossentropy loss, which has been shown to perform competitively when optimizing for top-k classification tasks (Berrada et al., 2018).
Incorporating Personal Values
While the attributes vector a can be used to encode any information of interest about a user, we choose to experiment with the use of personal values because of their theoretical connection to human activities (Bardi and Schwartz, 2003). In order to get a representation of a user's values, we turn to the hierarchical personal values lexicon from . In this lexicon, there are 50 value dimensions, represented as sets of words and phrases that characterize that value. Since users' profiles often contain value-related content, we use the Distributed Dictionary Representations (DDR) method (Garten et al., 2018) to compute a score, s v for each value dimension, v, using cosine similarity as follows:
s v = R(prof ile) · R(lexicon v ) ||R(prof ile)||||R(lexicon v )|| ,
where R(·) is a representation of a set of vectors, which, for the DDR method, is defined as the mean vector of the set; prof ile is a set of word embeddings, one for each token in the user's profile; and lexicon v is another set of word embeddings, one for each token in the lexicon for value dimension v. Finally, we set a = (s 0 , . . . , s dim L ) where dim L = 50, the number of value dimensions in the lexicon. Examples of profiles with high scores for sample value dimensions are shown in Table 8.
Category Top Scoring Profile Family a mother to my son Nature
Environment & nat resource economist tweeting about climate change/risk, energy, environmental protection, green finance, commodities, data science, politics Work-Ethic Football is like life -it requires perseverance, self-denial, hard work, sacrifice, dedication and respect for authority Religion /Galatians 2:20/ I love our Lord Jesus Christ. Further, we explore the types of activity clusters that contain activities reported by users with high scores for various value dimensions. For a given value, we compute a score for each cluster s C v by taking the average s v of all users who tweeted about doing activities in the cluster. For each value v, we can then rank all clusters by their s C v score. Examples of those with the highest scores are presented in Table 9. We observe that users whose profiles had high scores for Family were likely to report doing activities including family members, those with high scores for Nature tweeted about travel, and those with high Work-Ethic scores reported performing writing related tasks.
Category Activities in High Scoring Cluster
give one's daughter a number of plants Family take one's family to the park work in the garden with mom visit another castle Nature visit france go on a fishing trip add another footnote to the dissertation Work-Ethic file a complaint with the fcc write one's first novel by hand follow the rules Religion study really hard do a good deed
Evaluation
We evaluate our activity prediction models using a number of metrics that consider not only the most likely cluster, but also the set of k eval most likely clusters. First, we evaluate the average per-class accuracy of the model's ability to rank c t , the target cluster, within the top k eval clusters. These scores tell us how well the model is able to make predictions about the kinds of activities that each user is likely to do. Second, we test how well the model is able to sort users by their likelihood of having reported to do an activity from a cluster. This average comparison rank (ACR) score is computed as follows: for each user in the test set, we sample n other users who do not have the same activity label. Then, we use the probabilities assigned by the model to rank all n + 1 users 8 by their likelihood of being assigned c t , and the comparison rank score is the percentage of users who were ranked ahead of the target user (lower is better). We then average this comparison rank across all users in the test set to get the ACR. The ACR score tells us how well the model is able to find a rank users based on their likelihood of writing about doing a given activity, which could be useful for finding, e.g., the users who are most likely to claim that they "purchased some pants" or least likely to mention that they "went to the gym" in the future.
Experiments and Results
We split our data at the user-level, and from our set of valid users we use 200,000 instances for training data, 10,000 as test data, and the rest as our validation set.
For the document encoder and profile encoder we use Bi-LSTMs with max pooling (Conneau et al., 2017), with dim d = 128 and dim p = 128. For the history encoder, we empirically found that single mean pooling layer over the set of all document embeddings outperformed other more complicated architectures, and so that is what we use in our experiments. Finally, the classifier is a 3-layer feed-forward network with and dim c = 512 for the hidden layers, followed by a softmax over the dim o -dimensional output. We use Adam (Kingma and Ba, 2014) as our optimizer, set the maximum number of epochs to 100, and shuffle the order of the training data at each epoch. During each train- ing step, we represent each user's history as a new random sample of max sample docs = 100 documents 9 if there are more than max sample docs documents available for the user, and we use a batch size of 32 users. Since there is a class imbalance in our data, we use sample weighting in order to prevent the model from converging to a solution that simply predicts the most common classes present in the training data. Each sample is weighted according to its class, c, using the following formula:
w c = N count(c) * dim o
where count(c) is the number of training instances belonging to class c. We evaluate our model on the development data after each epoch and save the model with the highest per-class accuracy. Finally, we compute the results on the test data using this model, and report these results.
We test several configurations of our model. We use the complete model described in section 3.2 using either the set of additional tweets written by a user as their history (full T ), or only the set of additional activities contained in those tweets (full A ). Then, to test the effect of the various model components, we systematically ablate the attributes vector input a, the profile text (and subsequently, the Profile Encoder layer) p, and the set of documents, D, comprising the history along with the Document and History Encoders, thereby removing the h vector as input to the classifier. We also explore removing pairs of these inputs at the same time. To contextualize the results, we also 9 We empirically found that increasing this value beyond 100 had little effect on the development accuracy.
include the theoretical scores achieved by random guessing, labeled as rand. 10 We consider two variations on our dataset: the first is a simplified, 50-class classification problem. We choose the 50 most common clusters out of our full set of k act = 1024 and only make predictions about users who have reportedly performed an activity in one of these clusters. The second variation uses the entire dataset, but rather than making predictions about all k act classes, we only make fine-grained predictions about those classes for which count(c) ≥ minCount. We do this under the assumption that training an adequate classifier for a given class requires at least minCount examples. All classes for which count(c) < minCount are assigned an "other" label. In this way, we still make a prediction for every instance in the dataset, but we avoid allowing the model to try to fit to a huge landscape of outputs when the training data for some of these outputs is insufficient. By setting minCount to 100, we are left with 805 out of 1024 classes, and an 806th "other" class for our 806-class setup. Note that this version includes all activities from all 1024 clusters, it is just that the smallest clusters are grouped together with the "other" label.
While our models are able to make predictions indicating that learning has taken place, it is clear that this prediction task is difficult. In the 50-class setup, the full T − a, p model consistently had the strongest average per-class accuracy for all values of k eval and the lowest (best) ACR score (Table 10). The full A − a, p model performed nearly as well, showing that using only the human-activity Table 11: Per-class accuracy (%) @ k eval and ACR scores for the 806-class prediction task. Note that removing h from either full T or full A gives the same model. For ACR only, lower is better. relevant content from a user's history gives similar results to using the full set of content available. When including the attributes and profile for a user, the model typically overfits quickly and generalization deteriorates.
In the 806-class version of the task, we observe the effects of including a larger range of activities, including many that do not appear as often as others in the training data (Table 11). This version of the task also simulates a more realistic scenario, since predictions can be made for the "other" class when the model does to expect the user to claim to do an activity from any of the known clusters. In this setting, we see that the full T − p model works well for k eval ≤ 25, suggesting that the use of the attribute vectors helps, especially when predicting the correct cluster within the top 25 is important. For k eval ≥ 50, the same full T − a, p model that worked best in the 50-class setup again outperforms the others. Here, in contrast to the 50-class setting, using the full set of tweets usually performs better than focusing only on the human activity content. Interestingly, the best ACR scores are even lower in the 806-class setup, showing that it is just as easy to rank users by their likelihood of writing about an activity, even when considering many more activity clusters.
Conclusions
In this paper, we addressed the task of predicting human activities from user-generated content. We collected a large Twitter dataset consisting of posts from more than 200,000 users mentioning at least one of the nearly 30,000 everyday activities that we explored. Using sentence embedding models, we projected activity instances into a vec-tor space and perform clustering in order to learn about the high-level groups of behaviors that are commonly mentioned online. We trained predictive models to make inferences about the likelihood that a user had reported to have done activities across the range of clusters that we discovered, and found that these models were able to achieve results significantly higher than random guessing baselines for the metrics that we consider. While the overall prediction scores are not very high, the models that we trained do show that they are able to generalize findings from one set of users to another. This is evidence that the task is feasible, but very difficult, and it could benefit from further investigation.
We make the activity clusters, models, and code for the prediction task available at http://lit.eecs.umich.edu/downloads.html
1 :
1Predictive model architecture.
Table 1 :
1Effect of targeted query approach on activity frequency in tweets. "Valid activities" are defined as first-person verb phrases that clearly indicate that the author of the text has actually performed the concrete activity being described. For each set of tweets, a random subset of 100 was chosen and manually annotated for validity.count
unique
Event2Mind activities 24,537 24,537
Survey activities
5,000
4,957
Total
29,537 29,494
Table 2 :
2Number of human activity queries from mul-
tiple sources.
Table 3 :
3Summary of query results.
Table 4 :
4Summary of additional data.Initial number unique users
358,091
Users with non-empty profiles
96.9%
Users with ≥ 1 addtl. tweets
94.9%
Users with ≥ 25 addtl. tweets
93.1%
Users with ≥ 1 addtl. activities
93.5%
Users with ≥ 5 addtl. activities
87.1%
Final number unique valid users 214,708
Table 5 :
5Summary valid user filtering.
Table 6 :
6Examples of clustered activities (with manually provided labels, for reference purposes only).
Table 7 :
7Three sample clusters and their distances from the first cluster inTable 6, showing the closest cluster, a somewhat distant cluster, and a very distant cluster.
Table 8 :
8Profiles scoring the highest for various values categories when measured with the values lexicon.
Table 9 :
9Activity clusters associated with the highest
scoring users for various values categories when mea-
sured with the values lexicon.
Table 10 :
10Per-class accuracy (%) @ k eval and ACR scores for the 50-class prediction task. Note that removing h from either full T or full A gives the same model. For ACR only, lower is better.
ACR full T 0.15 0.36 0.61 0.97 1.91 4.65 8.66 12.24 16.15 30.69 43.96 44.10 −a 0.32 0.61 0.98 1.39 2.96 5.99 10.21 14.61 18.95 35.19 49.26 42.61 −p 0.45 1.02 1.37 1.96 3.38 7.41 12.71 17.17 21.60 37.53 51.11 41.14 −a, p 0.41 0.70 1.10 1.66 3.03 6.88 12.89 17.86 22.76 38.61 52.38 40.82 full A 0.29 0.41 0.72 1.04 2.05 4.50 8.50 12.14 15.48 30.04 44.24 45.98 −a, p 0.26 0.47 0.83 1.35 2.24 4.61 8.90 13.24 16.80 31.29 45.11 44.56 −h 0.10 0.28 0.44 0.73 1.37 4.08 7.60 10.96 14.28 27.60 40.77 47.94 −a, h 0.10 0.36 0.53 1.00 1.85 4.64 8.58 12.57 16.23 29.31 41.57 46.94 −p, h 0.10 0.23 0.41 0.68 1.49 3.72 7.12 10.46 13.65 26.90 39.93 48.15 rand 0.12 0.25 0.37 0.62 1.24 2.98 6.34 9.19 12.54 26.21 36.77 50.00k eval
1
2
3
5
10
25
50
75
100
200
300
−a
0.24 0.44 0.75 1.02 2.02 4.62 8.70 12.19 15.56 30.18 43.34 45.99
−p
0.23 0.46 0.66 1.13 2.29 5.27 9.66 14.33 18.75 34.00 47.71 42.64
www.clips.uantwerpen.be/pattern
We also treat instance of "PersonY" as a wildcard since this could be any name or even a user (@) mention on Twitter.3 www.mturk.com 4 developer.twitter.com/en/docs/tweets/search/apireference/get-search-tweets.html
We acknowledge that similar experiments could be run with different cluster assignments, and our preliminary experiments showed comparable results. It is important to note that we do not treat these clusters as the definitive organization of human activities, but as an approximation of the full activity space in order to reduce the complexity of making predictions about activities in that space.
We set n = 999 in this study to achieve comparison samples of size 1000.
For the evaluation metrics considered in this paper, random guessing is as strong or stronger than a "most frequent class" baseline, so we do not report it.
AcknowledgmentsThis research was supported in part through computational resources and services provided by the Advanced Research Computing at the University of Michigan. This material is based in part upon work supported by the Michigan Institute for Data Science, by the National Science Foundation (grant #1815291), by the John Templeton Foundation (grant #61156), and by DARPA (grant #HR001117S0026-AIDA-FP-045). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the Michigan Institute for Data Science, the National Science Foundation, the John Templeton Foundation, or DARPA. Many thanks to the anonymous reviewers who provided helpful feedback.
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| [] |
[
"Study on the Electronic Structure and Stability of Some OPE(oligo- phenylene-ethynylene derivative)-RE 3 N@C 80 Dyads by PM7",
"Study on the Electronic Structure and Stability of Some OPE(oligo- phenylene-ethynylene derivative)-RE 3 N@C 80 Dyads by PM7"
] | [
"Kye-Ryong Sin \nDepartment of Chemistry\n\n",
"Sun-Gyong Ko \nDepartment of Chemistry\n\n",
"Hong-Gol O ",
"Song-Jin Im \nDepartment of Chemistry\n\n\nDepartment of Physics\nKim Il Sung University\nDaesong districtPyongyangDPR Korea\n"
] | [
"Department of Chemistry\n",
"Department of Chemistry\n",
"Department of Chemistry\n",
"Department of Physics\nKim Il Sung University\nDaesong districtPyongyangDPR Korea"
] | [] | In this paper, we investigated the electronic structure and stability of some mesomorphic OPE-RE 3 N@C 80 dyads from the oligo-phenylene-ethynylene derivatives (OPE) and the trimetallic nitride template endohedral fullerenes (TNT-EMFs) -RE 3 N@C 80 (RE=Sc,Y,La) by using PM7, the updated version of the semi-empirical Hartree-Fock method.In OPE-RE 3 N@C 80 , the fullerene cages were modified to have the opened cage (fulleroid) structure by addition of OPE on the [6,6] position of the fullerene cages. There was no considerable charge transfer between OPE and fullerene cage, but the fullerene cages had the remarkable minus charges mainly due to the electron transfer from RE 3 N to the cage. The calculated electronic spectra showed that light absorption bands of OPE-C 80 were more redshifted than that of OPE-RE 3 N@C 80 and all of OPE-RE 3 N@C 80 seem to have a couple of Vis-NIR absorption peaks. | null | [
"https://export.arxiv.org/pdf/1407.4828v1.pdf"
] | 100,618,428 | 1407.4828 | 59cc3d1f9b2baee36fa3cac62380ef493f965071 |
Study on the Electronic Structure and Stability of Some OPE(oligo- phenylene-ethynylene derivative)-RE 3 N@C 80 Dyads by PM7
Kye-Ryong Sin
Department of Chemistry
Sun-Gyong Ko
Department of Chemistry
Hong-Gol O
Song-Jin Im
Department of Chemistry
Department of Physics
Kim Il Sung University
Daesong districtPyongyangDPR Korea
Study on the Electronic Structure and Stability of Some OPE(oligo- phenylene-ethynylene derivative)-RE 3 N@C 80 Dyads by PM7
Key-words: endohedral fullereneoligo-phenylene-ethynylene derivativesTNT-EMFquantum chemistryPM7
In this paper, we investigated the electronic structure and stability of some mesomorphic OPE-RE 3 N@C 80 dyads from the oligo-phenylene-ethynylene derivatives (OPE) and the trimetallic nitride template endohedral fullerenes (TNT-EMFs) -RE 3 N@C 80 (RE=Sc,Y,La) by using PM7, the updated version of the semi-empirical Hartree-Fock method.In OPE-RE 3 N@C 80 , the fullerene cages were modified to have the opened cage (fulleroid) structure by addition of OPE on the [6,6] position of the fullerene cages. There was no considerable charge transfer between OPE and fullerene cage, but the fullerene cages had the remarkable minus charges mainly due to the electron transfer from RE 3 N to the cage. The calculated electronic spectra showed that light absorption bands of OPE-C 80 were more redshifted than that of OPE-RE 3 N@C 80 and all of OPE-RE 3 N@C 80 seem to have a couple of Vis-NIR absorption peaks.
Introduction
Now it is well-known that fullerenes are not chemically inert and undergo various chemical reactions such as nucleophilic addition, Diel's-Alder reaction, 1,3-dipolar cycloaddition, radical addition, oxidation, reduction etc. [1,2,3] In these decades many researches in the fullerenes chemistry have been focused on the synthesis of the endohedral metallofullerenes (EMFs), the so-called "cluster-fullerene", containing metal atoms or clusters therein and on their application in manufacture of the novel nano-materials such as molecular devices, sensors and medical tools [4] . Especially, a variety of the trimetallic nitride template endohedral fullerenes (TNT-EMFs) such as RE 3 N@C n ( RE = Sc, Y, La; n=78, 80, 82, … ) have been synthesized and modified for utilizing their functionalities in molecular electronics and bio-technology. [5,6] One of the most prosperous applications of RE 3 N@C n can be found in organic photovoltaics due to their excellent electron acceptor abilities.
A recent research was carried out for synthesis of some π-conjugated system -fullerene dyads for photovoltaic applications, where the donor units were either oligo-phenylene-ethynylene (OPE) or oligo-phenylene-vinylene (OPV) derivatives and for the acceptor, C 60 or Y 3 N@C 80 were used. [7] The liquid crystalline (LC) behavior, shown by the synthesized dyads was expected to improve the photovoltaic efficiency of the BHJ (block hetero-junction) organic solar cells by ambipolar charge transfer.
In this paper, PM7 in MOPAC2012 [8] , one of the updated semi-empirical Hartree-Fock methods, was applied in the theoretical study on the electronic structure and stability of OPE-C 80 and OPE-RE 3 N@C 80 dyads (RE = Sc, Y, La). There have been some reports on DFT (Density Functional Theory) study on RE 3 N@C n and their derivatives [4,9,10,11] , but still no research has been done on theoretical calculation of the electronic structures of the OPE-RE 3 N@C 80 dyads.
Computational Models and Method
Here the quantum chemical study has been done on four OPE-FD dyads (OPE-C 80 and OPE- Figure 2 shows the chemical structures of the OPE-Y 3 N@C 80 dyad, synthesized in the previous research. [7] According to that experimental research, the models for OPE-C 80 and OPE-RE 3 N@C 80 dyad (RE=Sc, Y, La) were chosen as [6,6] adducts, where OPE was covalently linked to C 80 or RE 3 N@C 80 just on the [6,6] addition site of the fullerene cage (the nearest site to RE atom in case of RE 3 N@C 80 ). To simplify the task and avoid the overload in computation, the long alkyl chain R (-C 12 H 25 ) in the OPE was shortened as -CH 3 in all the models for OPE-FDs.
The OPE-FDs can be separated as two individual subunits, OPE and FD, for comparing their electron donor -acceptor interaction. Here OPE 1 symbolizes a half part of OPE ( Figure 3). To consider the effect of RE 3 N on the electronic structure of the OPE-RE 3 N@C 80 , the empty fullerene cage model without RE 3 N was also calculated as OPE-C 80 .
The geometric and electronic structures of the models were calculated by PM7 from MOPAC2012, the latest version of the semi-empirical MO software package, that has been well-known as one of the most efficient quantum chemical tools with the enhanced accuracy for a wide range of molecules, complexes, polymers, crystals, and TNT-EMFs. [8,12] Especially, it offers good parameter set for the calculation of most of the elements on the periodic table including rare earth elements.
The first step of calculation was the geometry optimization by EF (Eigenvector-Following) routine, which was followed by the configuration interaction (CI) calculation based on the Relative Stability of the OPE-FDs was evaluated by ΔE t , the difference of total energies (E t ) between the resulting model (OPE-FD) and the separated subunits (OPE and FD). It can be seen from the optimized geometric structures of RE 3 N@C 80 in Figure 5 that RE 3 N (RE = Sc, La) has the planar form, but Y 3 N has the pyramidal form, which resembles the previous XRD measurement of Gd 3 N@C 80 -I h and DFT calculation of Y 3 N@ C 78 -D 3h . [13,14] From the electronic structures of RE 3 N@C 80 calculated from their optimized geometry, it was found out that the positive charge of RE atoms was increased and the negative charge of N atom decreased in the cluster fullerene compared with those in free RE 3 N, which shows that in RE 3 N@C 80 more portion of electrons of RE atoms was transferred to the fullerene cage, not to N atom. [12] Figure 5. The optimized geometric structures of RE 3 N@C 80 Figure 6 shows that OPE 1 and OPE have the similar HOMO-LUMO levels, which means there can not be apparent π-conjugation between two phenylethynyl branches in OPE. All FDs (C 80 and RE 3 N@C 80 ) have lower LUMO levels than OPE, therefore they can accept electron from OPE. HOMO levels of RE 3 N@C 80 are lower than that of the empty C 80 . It can be explained as the result of stabilization of the C 80 cage by RE 3 N incorporation. [15]
Results and Discussion
1) The geometric and electronic structures of the separated subunits (OPE and FDs)
2) The geometric and electronic structures of OPE-FDs
Four models of the OPE-FDs discussed in this paper had the similar configurations after geometric optimization (Figure 8). These configurations may be different from those of the real OPE-FD dyads [7] because these models have the shorten alkyl group (-CH 3 ) instead of the long chain (-C 12 H 25 ) and can not express their well-assembled frameworks in the liquid crystalline phase. Figure 9 shows ΔE t of the OPE-fullerenes calculated as the total energy difference of OPE-FD from its separated subunits (OPE and FD). OPE-C 80 became to be unstable after the formation of the dyad and it can be considered as the result of structural deformation of the subunits, especially C 80 due to the formation of the dyad. The most stable one was OPE-La 3 N@C 80 and other OPE-RE 3 N@C 80 were also more stable than OPE-C 80 because all of RE 3 N@C 80 had been stabilized by electron transfer from RE 3 N to C 80 . structures. Table 1 shows the C-C distances at the [6,6] OPE-addition sites of FDs and their increases (△R C-C ) after OPE-addition, where more stable OPE-La 3 N@C 80 had the less △R C-C and less stable OPE-C 80 and OPE-Y 3 N@C 80 had the larger △R C-C . From the electronic spectra of OPE-FDs (Figure 10), it can be found out that light absorption band of OPE-C 80 was more red-shifted than that of OPE-RE 3 N@C 80 , but its maximum absorption intensity was far less than OPE-RE 3 N@C 80 , and all of OPE-FDs seem to have a couple of Vis-NIR absorption peaks. In all of OPE-FDs, the fullerene cages were modified to have the open-up cage (fulleroid) structure by addition of OPE on the [6,6] position of the fullerene cages. The C-C distance at the [6,6] addition site of the cages was less increased in the more stable OPE-FDs.
There was no considerable charge transfer between OPE and FDs, but in OPE-RE 3 N@C 80 the fullerene cages had the remarkable minus charges mainly due to the electron transfer from RE 3 N to the cage.
Light absorption bands of OPE-C 80 were more red-shifted than that of OPE-RE 3 N@C 80 and all of OPE-FDs seem to have a couple of Vis-NIR absorption peaks.
Figure 1 .
1RE 3 N@C 80 ), where FD means C 80 and three kinds of RE 3 N@C 80 (RE=Sc,Y,La).For all the OPE-FDs, the geometry of C 80 -I h , one of the geometric isomers of C 80 , was chosen as the fullerene cage, where I h shows the geometric symmetry of the fullerene cage.(Figure 1) Models for FDs (RE=Sc,Y,La)
Figure 2 .
2Models for OPE-FDs
Figure 3 .
3Models for OPE and OPE 1
single-point MO results. The configurations for the singlet electronic transitions were composed of 20 MOs near HOMO and LUMO (10 occupied MOs and 10 unoccupied MOs). The electronic spectra were drawn by using the Gaussian smoothing function based on the transition energies (mode positions) and the oscillator strengths (mode intensities).
Figure 4
4shows the optimized geometric structures of OPE and its one branch (OPE 1 ), where the phenylethynyl unit (-C 6 H 4 -C≡C-C 6 H 4 -C≡C-C 6 H 4 -) is arranged to forms a straight line, but its three phenyl rings are not on the same plane, which prevents to form larger π-conjugation plane in OPE.
Figure 4 .
4The optimized geometric structures of OPE and its one branch (OPE 1 )
Figure 6 .
6HOMO -LUMO energy levels of subunits of OPE-FDs FromFigure 7, it can be seen that OPE has UV absorption, but FDs can interact with visible light or even with NIR.
Figure 7 .
7The electronic spectra of OPE and FDs ( red: mode positions, green: mode intensities )
Figure 8 .
8The different views of the optimized structures of OPE-FDs
Figure 9 .
9Relative stability (ΔE t ) of OPE-FDsIn all of OPE-FDs, the fullerene cages were modified to have the open-up cage (fulleroid)
Figure 10 .
10The electronic spectra of OPE-FDs ( red: mode positions, green: mode intensities )4. ConclusionPM7 calculations were carried out on four OPE-fullerene dyads ( OPE-FDs ) such as OPE-C 80 and OPE-RE 3 N@C 80 ( RE = Sc, Y, La ).
Table 1
1there was no considerable charge transfer between OPE and FDs, but in OPE-RE 3 N@C 80 the fullerene cages had the remarkable minus charges mainly due to the electron transfer from RE 3 N to the cage. It seems that the more electrons were transferred from RE 3 N to C 80 , the more stable OPE-RE 3 N@C 80 was formed..
C-C distance at the [6,6] addition site of FDs (nm)
model
OPE-C 80
OPE-Sc 3 N@C 80 OPE-Y 3 N@C 80 OPE-La 3 N@C 80
free FD
0.142
0.148
0.153
0.149
OPE-FD
0.235
0.239
0.247
0.234
△R C-C
0.093
0.091
0.094
0.085
Table 2 shows the local charges of the subunits (OPE, C 80 cage, RE 3 N) in OPE-FDs, where
Table 2.
Local charges of the subunits in OPE-FDs (e)
model
OPE-C 80
OPE-Sc 3 N@C 80 OPE-Y 3 N@C 80 OPE-La 3 N@C 80
OPE
0.012
0.071
0.037
0.069
C 80 cage
-0.012
-4.454
-3.976
-4.471
RE 3 N
-
4.383
3.939
4.402
AcknowledgementThe authors thank Stewart Computational Chemistry for its efficient MOPAC2012.
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| [] |
[
"A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models",
"A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models"
] | [
"Rei Aso \nTokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan\n",
"Tatsuya Chuman \nTokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan\n",
"Hitoshi Kiya \nTokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan\n"
] | [
"Tokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan",
"Tokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan",
"Tokyo Metropolitan University\n6-6 Asahigaoka, Hino-shi191-0065TokyoJapan"
] | [] | In this paper, a privacy preserving image classification method is proposed under the use of ConvMixer models. To protect the visual information of test images, a test image is divided into blocks, and then every block is encrypted by using a random orthogonal matrix. Moreover, a ConvMixer model trained with plain images is transformed by the random orthogonal matrix used for encrypting test images, on the basis of the embedding structure of ConvMixer. The proposed method allows us not only to use the same classification accuracy as that of ConvMixer models without considering privacy protection but to also enhance robustness against various attacks compared to conventional privacy-preserving learning. | 10.48550/arxiv.2301.03843 | [
"https://export.arxiv.org/pdf/2301.03843v2.pdf"
] | 255,570,006 | 2301.03843 | e29f5b7c55dc655414d66d23308fd7ac3710399a |
A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models
Rei Aso
Tokyo Metropolitan University
6-6 Asahigaoka, Hino-shi191-0065TokyoJapan
Tatsuya Chuman
Tokyo Metropolitan University
6-6 Asahigaoka, Hino-shi191-0065TokyoJapan
Hitoshi Kiya
Tokyo Metropolitan University
6-6 Asahigaoka, Hino-shi191-0065TokyoJapan
A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models
In this paper, a privacy preserving image classification method is proposed under the use of ConvMixer models. To protect the visual information of test images, a test image is divided into blocks, and then every block is encrypted by using a random orthogonal matrix. Moreover, a ConvMixer model trained with plain images is transformed by the random orthogonal matrix used for encrypting test images, on the basis of the embedding structure of ConvMixer. The proposed method allows us not only to use the same classification accuracy as that of ConvMixer models without considering privacy protection but to also enhance robustness against various attacks compared to conventional privacy-preserving learning.
Introduction
Deep learning has been deployed in many applications including security-critical ones. Generally, data contains sensitive information such as personal informational, so privacypreserving methods for deep learning have become an urgent problem [1]. To achieve privacy-preserving learning, various methods have been proposed. One of them is Federated Learning (FL) [2], which is a type of distributed learning. FL allows us to train a model over multiple participants without directly sharing their raw data. However, FL have not considered the protection of test data in cloud environments so far. In this paper, we propose a novel method for protecting visual information on test images.
To protect visual information on plain images in untrusted cloud environments, many learnable encryption methods have been studied so far [3]- [13]. Learnable encryption has to satisfy three requirements in general: (a) having a high accuracy that is almost the same as that of plain models, (b) being robust enough against various attacks, and (c) easily updating a secret key. However, most of existing methods [3]- [11] degrade the accuracy of models due to the use of encrypted images, and moreover, need to retrain models to update the key. In contrast, the similarity between block-wise encryption and the architecture of isotropic networks has been pointed out to enable us to perfectly stratify the two requirements that the existing methods cannot [12] [13]. Information on embeddings in isotropic networks such as the vision transformer [14] and ConvMixer [15] is encrypted by random matrixes generated with secret keys for privacy-preserving learning. However, in the conventional methods [12] [13], simple permutation matrixes are used for image and model encryption, so encrypted images are not robust enough against various attacks. Accordingly, we propose the use of a novel random matrix, which is called a random orthogonal one generated by using the Gram-Schmidt orthonormalization. The proposed method allows us to enhance the visual protection of images, while maintaining the same as that of plain models and the easy update of a secret key.
ConvMixer
Before discussing the proposed method, we summarize ConvMixer and its properties briefly. ConvMixer is mainly used for image classification tasks and is known for its high classification performance [15]. The structure of ConvMixer is inspired by the Vision Transformer (ViT) [14]. ViT consists of two Embedding processes (Patch Embedding and Position Embedding) and a Transformer structure. On the other hand, ConvMixer consists of a Patch Embedding and a CNN structure. Figure 2 shows the structure of ConvMixer, which consists of two main structures: Patch Embedding and Con-vMixer Layer. In this paper, we focus on Patch Embedding. In Patch Embedding, an input image ∈ R × × of height , width , and number of channels is divided into patches of size × . Each patch is then transformed into a vector ∈ R 2 , multiplied by the learnable filter and linearly transform it into a vector of -dimensions by taking the prod-
uct of ∈ R 2 as = [ 1 , ..., , ..., ](1)∈ R × , ∈ R ( 2 )×
In previous studies [12][13], it is known that it is possible to protect the privacy of test images by transforming the filter with a secret key. In this paper, we propose a method to achieve stronger privacy preserving of test images by using random orthogonal matrices. The proposed method aims to protect visual information on test images. To achieve this aim, we encrypt test images and a transform model by using an random orthogonal matrix. The framework is summarized as below.
Proposed Method
• A third party (trusted) generates random numbers with a secret key (seed), and prepares a random orthogonal matrix from the random numbers and an inverse random orthogonal matrix −1 . • The third party trains a ConvMixer model with plane images. The trained model is transformed into an encrypted model by using −1 .
• The third party provides the random orthogonal matrix to a client (trusted) and model to a provider (untrusted). • The client transforms a test image into an encrypted imageˆby using . After that, the client sendsˆto the provider.
• The provider inputsˆinto model , and sends back a prediction result to the client.
Even if the provider is not trusted, the client does not give visual information of test images and matrix used for image encryption to the provider. Thus, the client can receive prediction results while maintaining the privacy preserving of test images.
Test Image Encryption
A test image ∈ R × × is transformed into an encrypted imageˆ∈ R × × as below.
1. Divide into blocks with a size of × such that = { 1 , ..., }, where × is the same size as the patch size used in a ConvMixer model, and is ( × )/ 2 .
Flatten each block
∈ R × × into a vector ∈ R 2 as = [ (1), ..., ( 2 )].(2)
3. Generate a encrypted vectorˆ∈ R 2 by multiplying vector by matrix ∈ R ( 2 )×( 2 ) aŝ = .
(
4. Rebuild vectorˆinto blockˆin the reverse order of step 2. 5. Concatenateˆ= {ˆ1, ...,ˆ} into an encrypted test imageˆ.
Model Encryption
To avoid the performance degradation caused by encryption of test images, in Eq.(1) is transformed by using −1 as
= −1 .(4)
When replacing and with andˆ, respectively, vector z in Eq.(1) is reduced to as
= [ˆ1 , ...,ˆ , ...,ˆ ].(5)
Thus, by substituting Eqs.
From Eq.(5), encrypted model allows us to have the same performance as that of the model trained with plane images, under the use of encrypted images.
Generation of Random Orthogonal Matrices
A random orthogonal matrix can be generated by using the Gram-Schmidt orthonormalization. The procedure for generating with a size of × is given as follows.
1. Generate an real matrix with a size of × by using a random number generator with a seed.
Calculate
( ), and proceed to 3 if ( ) ≠ 0. Otherwise, return to 1.
3. Compute a random orthogonal matrix from by using the Gram-Schmidt orthogonalization.
In this framework, any regular matrix can be used as A for image encryption. Several conventional methods for privacypreserving image classification use permutation matrices of pixel values, in which many elements have zero values in matrices as
= 0 1 0 0 0 1 1 0 0 .(7)
In contrast, the proposed random orthogonal matrices include no the zero values as elements. The use of such matrices allows us not only to more strongly protect visual information on plain images but to also enhance robustness against various attacks, while maintaining the same performance as that of models trained with plain images. In addition, −1 can easily be calculated as the transposed matrix of A.
Experiment Results
To verify the effectiveness of the proposed method, we ran a number of experiments on the CIFAR-10 dataset.
Setup
We used the CIFAR-10 dataset, which consists of 60,000 color images (dimension of 32 × 32 × 3) with 10 classes (6000 images for each class) where 50,000 images are for training and 10,000 for testing. ConvMixer was trained and tested on the CIFAR-10 dataset. In the model setting, we set the patch size to 4, the number of channels after patch embedding to 256, the kernel size of depth-wise convolution to 7 and the number of ConvMixer layers to 8. models were trained for 200 epochs with the Adam optimizer, where the learning rate was 0.001. We also used a random orthogonal matrix with a size of 48 × 48 for the encryption of test images and models. Figure 3 shows an example of images encrypted with a conventional encryption method [12] [13], in which pixel shuffling and negative-positive transformation are carried out for image encryption, and an example of images encrypted with the proposed method, where the images had × × = 512×512×3 as an image size, and the block sizes used for encryption were = 8 and = 16. When using an orthogonal matrix for encryption, transformed pixel values are real values, so (b) in Fig. 3 were displayed after normalizing the pixel values to the range of [0.1]. From the figures, the selection of a larger the block size gave smaller visual information. The use of random orthogonal matrices was also demonstrated to have a stronger visual protection performance than that of the conventional method. In addition to visual protection, encrypted images have to be robust enough against various attacks, which aim to restore visual information from encrypted images. We already confirmed that images encrypted with the proposed method are more robust against attacks including jigsaw puzzle solver attacks [16]. In particular, unlike ViT, ConvMixer models do have position embedding, so the position of patches cannot be changed. Therefore, privacy-preserving ConvMixer needs a stronger encryption method than ViT.
Visual Protection Performance
Classification Performance
We evaluated the classification performance of the proposed method as shown in Table 1, where plain and encrypted indicate plane test images and an encrypted test images, respectively, and plain model and encrypted model are models trained with plain images, and models trained with encrypted images. Table 1 shows the classification results for each combination. From the table, the proposed method (the combined use of Encrypted models and encrypted images) had the same classification accuracy as that of the baseline without privacy protection (plain models and plain images). Accordingly, the proposed method can not only protect the visual information of test image, but also classify the encrypted image without any degradation of classification accuracy.
Conclusion
In this paper, we proposed a novel method for protecting visual informational on test images under the use of ConvMixer models. The proposed method allows us to use a random orthogonal matrix for image encryption, and it was demonstrated not only to enhance the visual protection of images but to also maintain the same accuracy as that of models trained with plain images.
Figure 1 :Figure 2 :
12Architecture Framework of proposed method Figure 2 illustrates the framework of the proposed method.
=
[ 1 , ..., , ..., ] = .
Figure 3 :
3Example of encrypted images with (a) conventional method[12][13] and (b) proposed method
Table 1 :
1Classification accuracy (%)model\test image plane encypted
plane model
90.38
13.2
encypted model 10.27
90.38
AckowledgmentThis study was partially supported by JSPS KAKENHI (Grant Number JP21H01327).
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| [] |
[
"It Takes Two Flints to Make a Fire: Multitask Learning of Neural Relation and Explanation Classifiers",
"It Takes Two Flints to Make a Fire: Multitask Learning of Neural Relation and Explanation Classifiers"
] | [
"Zheng Tang \nUniversity of Arizona\nUniversity of Arizona\n\n",
"Mihai Surdeanu \nUniversity of Arizona\nUniversity of Arizona\n\n"
] | [
"University of Arizona\nUniversity of Arizona\n",
"University of Arizona\nUniversity of Arizona\n"
] | [] | We propose an explainable approach for relation extraction that mitigates the tension between generalization and explainability by jointly training for the two goals. Our approach uses a multi-task learning architecture, which jointly trains a classifier for relation extraction, and a sequence model that labels words in the context of the relation that explain the decisions of the relation classifier. We also convert the model outputs to rules to bring global explanations to this approach. This sequence model is trained using a hybrid strategy: supervised, when supervision from pre-existing patterns is available, and semi-supervised otherwise. In the latter situation, we treat the sequence model's labels as latent variables, and learn the best assignment that maximizes the performance of the relation classifier. We evaluate the proposed approach on the two datasets and show that the sequence model provides labels that serve as accurate explanations for the relation classifier's decisions, and, importantly, that the joint training generally improves the performance of the relation classifier. We also evaluate the performance of the generated rules and show that the new rules are great add-on to the manual rules and bring the rule-based system much closer to the neural models. | 10.1162/coli_a_00463 | [
"https://export.arxiv.org/pdf/2204.11424v5.pdf"
] | 248,377,455 | 2204.11424 | b91a9accfad959a1b3a0cfd7e9f6e6d5a967463e |
It Takes Two Flints to Make a Fire: Multitask Learning of Neural Relation and Explanation Classifiers
Zheng Tang
University of Arizona
University of Arizona
Mihai Surdeanu
University of Arizona
University of Arizona
It Takes Two Flints to Make a Fire: Multitask Learning of Neural Relation and Explanation Classifiers
We propose an explainable approach for relation extraction that mitigates the tension between generalization and explainability by jointly training for the two goals. Our approach uses a multi-task learning architecture, which jointly trains a classifier for relation extraction, and a sequence model that labels words in the context of the relation that explain the decisions of the relation classifier. We also convert the model outputs to rules to bring global explanations to this approach. This sequence model is trained using a hybrid strategy: supervised, when supervision from pre-existing patterns is available, and semi-supervised otherwise. In the latter situation, we treat the sequence model's labels as latent variables, and learn the best assignment that maximizes the performance of the relation classifier. We evaluate the proposed approach on the two datasets and show that the sequence model provides labels that serve as accurate explanations for the relation classifier's decisions, and, importantly, that the joint training generally improves the performance of the relation classifier. We also evaluate the performance of the generated rules and show that the new rules are great add-on to the manual rules and bring the rule-based system much closer to the neural models.
correcting the underlying model because "changing one thing changes everything" in a neural network (Sculley et al. 2015).
Our article focuses on addressing the limitations of these local and post-hoc explainability approaches by providing a self-explanatory neural architecture (i.e., explanations are part of classification) that can provide both local and global explanations. In particular, we propose an approach for relation extraction (RE) that jointly learns how to explain and predict. Intuitively, our approach trains two classifiers: an explainability classifier (EC), which labels words in the textual context where the relation is expressed as important or not for the relation to be extracted, and a relation classifier (RC), which predicts the relation that holds between two given entities using only the words deemed as important. As such, our approach is self-explanatory because of inter-dependency between RC and EC, and generates faithful explanations that correctly depict how the relation classifier makes a decision (Vafa et al. 2021).
The contributions of this article are the following:
(1) We introduce a hybrid strategy to jointly train the EC and RC. Our method trains the EC as a supervised classifier when information about which words are important for a relation exists. For example, in this article we use a small set of linguistic rules to identify the important words in the relation's context. For example, in the sentence "John was born in France," such a rule may identify the words born and in as important. Importantly, our approach requires minimal supervision for explanations, e.g., we report results when using an average of 7 rules per relation type on one dataset and fewer on another dataset. For the more common situation where training examples are not associated with such rules, we train using a semi-supervised strategy: we treat EC's labels as latent variables, and learn the best assignment that maximizes the performance of the RC.
(2) We evaluate our approach on two datasets: TACRED (Zhang et al. 2017) and CoNLL04 (Roth and Yih 2004). For (partial) explainability information, we select from the surface rules provided with the dataset (Zhang et al. 2017;Chang and Manning 2014) as well as from a small set of syntactic rules developed in-house using the Odin framework . Our evaluation demonstrates that jointly training for prediction and explainability improves the performance of the relation classifier considerably on CoNLL04, and maintains the same level of performance on TACRED when compared with a state-of-the-art neural relation classifier. Importantly, our method achieves its best performance when using an average of 7 rules per relation type on TACRED and 4 rules per relation type for CoNLL04, which indicates that only minimal guidance from such rules is needed.
(3) More relevant for the goals of this work, we also evaluate our method for explainability using two strategies. The first strategy is automated and focuses on the capacity of our method to identify the same words in the context as the ones identified by rules, to verify that our approach indeed encodes the proper linguistic knowledge. Thus, this evaluation looks at examples associated with rules. In this situation, we measure the overlap between the words identified by the EC as important and the words used by rules using standard precision, recall, and F1 scores. The second strategy relies on plausability, i.e., can the machine explanations be understood and interpreted by humans (Wiegreffe and Pinter 2019a;Vafa et al. 2021)? To this end, we compare the tokens identified by the EC against human annotations of the context words marked as important for the relation. In both evaluations, our approach achieves considerably higher overlap with rules/human annotations than other strong baselines such as saliency mapping (Simonyan, Vedaldi, and Zisserman 2013), LIME (Ribeiro, Singh, and Guestrin 2016), SHAP (Lundberg and Lee 2017), CXPlain (Schwab and Karlen 2019), and greedy rationales (Vafa et al. 2021).
(4) We also explore the feasibility of transforming the local explanations into global ones. That is, instead of using the EC to explain individual predictions, we introduce a simple algorithm that converts the tokens marked as important into a set of rules that becomes a new, fully-explainable model that approximates the behavior of the neural RC. We compare the performance of this rule-based model with the performance of the rules written by domain experts, as well as with the neural RC model. The results show that our rule-based model has a considerably higher performance that the manually-written rules, approaching the performance of the neural classifier within a reasonable gap. In some real-world scenarios, this gap may be an acceptable cost, as the generated rulebased model provides actionable explainability. That is, when a rule is incorrect, a domain expert can improve it without impacting other parts of the models ).
Related Work
Our work lies at the intersection of relation extraction and explainability. We summarize these two research areas next.
Relation Extraction
Information extraction (IE), i.e., extracting structured information from text such as events and their participants, is one of the fundamental tasks in NLP that was shown to be useful for many end-user applications such as question answering Li 1999, 2000), and summarization (Rau, Jacobs, and Zernik 1989;Zechner 1997). Our work focuses on a subtask of IE: relation extraction (RE), which addresses the extraction of (mostly) binary relations between entities such as place_of_birth, which connects a person named entity with a location. RE has received tremendous attention in the past several decades. We group the works on RE into two categories: before the "deep learning tsunami" (Manning 2015), and after.
2.1.1 Relation extraction before deep learning. The first approaches for RE were rulebased. For example, Hearst (1992) proposed a method to learn hyponymy relations using hand-written patterns. Riloff (1996) introduced a pattern acquisition method that alternates between learning paterns and extracting relation mentions. Brin (1998) proposed a dual iterative pattern/relation expansion, which exploited the duality between patterns and relations. Hassan, Awadallah, and Emam (2006) used Hyperlink-Induced Topic Search (HITS) (Kleinberg 1999) to jointly learn patterns and relations in an unsupervised manner. In general, these rule-based methods usually obtain high precision but suffer from low recall. While our explanations can be interpreted as rules, our work differs from these directions in two significant ways. First, most of these directions are iterative, alternating between learning patterns (or rules) and relations. In contrast, our approach trains relation and explanation classifiers jointly. Second, and probably more importantly, we show that our explanations often focus on parts of speech that are necessary for plausability (according to the human annotators) but are semantically-ambiguous such as prepositions and determiners. On the other hand, most pattern acquisition methods usually focus on clear syntactic structures such as subject-verb-object and words with more clear semantics such as nominals and verbs.
Statistical methods that followed the above rule-based approaches address the limited generality of rules. In terms of supervision, "traditional" machine learning approaches for RE include fully supervised methods (Zelenko, Aone, and Richardella 2003;Bunescu and Mooney 2005), or methods that rely on distant supervision, where training data is generated automatically by (noisily) aligning existing knowledge bases with texts (Mintz et al. 2009;Riedel, Yao, and McCallum 2010;Hoffmann et al. 2011;Surdeanu et al. 2012). Most of these approaches used explicit features such as lexical, syntactic, and semantic. For example, Kambhatla (2004) proposed a maximum entropy classifier using these features. Zhou et al. (2005) found that additional features such as syntactic chunks further help the classification performance. Jiang and Zhai (2007) evaluate the effectiveness of different feature spaces for RE. Similarly, Chan and Roth (2011) expanded feature representations to include syntactico-semantic structures that improve RE.
Our work is conceptually similar to the method of Chan and Roth (2011). Similar to them, we extract relations only from the smaller context identified by a distinct component (the explainability classifier in our case). However, there are several important differences between these two efforts. First, the method of Chan and Roth (2011) operates as a pipeline: they start by matching syntactico-semantic structures potentially indicative of relations, and then they apply a relation classifier only on the texts that match them. In contrast, our method jointly trains the relation and explainability classifiers. Second, the syntactico-semantic structures in (Chan and Roth 2011) were manually extracted and categorized, whereas our explanations are learned in a semi-supervised way from data and a small number of rules. Last but not least, the patterns of Chan and Roth (2011) are non-lexicalized. In contrast, the explanations produced by our explainability classifier are lexicalized, which is critical for human understanding.
Kernel methods were also a popular direction for relation extraction due to their advantage of avoiding feature engineering. To this end, Miller et al. (2000) introduced a sequence kernel for relation extraction. Several researchers proposed kernels designed around constituent parse trees to capture sentence grammatical structure (Miller et al. 2000;Zelenko, Aone, and Richardella 2003;Moschitti 2006). Bunescu and Mooney (2005) ;Nguyen, Moschitti, and Riccardi (2009) introduced kernels based on syntactic dependencies, a simpler representation that flattens constituent trees while preserving most syntactic information. To combine the information captured by individual kernels that model different representations, Zhao and Grishman (2005) presented a composite kernel which combines multiple such individual kernels.
Deep learning methods for relation extraction.
Deep learning approaches for RE that rely on sequence models range from using CNNs or RNNs (Zeng et al. 2014;Zhang and Wang 2015), to augmenting RNNs with different components (Xu et al. 2015;Zhou et al. 2016), or to combining RNNs and CNNs (Vu et al. 2016;Wang et al. 2016). Other approaches take advantage of graph neural networks (Zhang, Qi, and Manning 2018) or attention mechanisms (Zhang et al. 2017).
More recently, transformer-based (Vaswani et al. 2017) approaches have shown considerable improvements on many natural language tasks including RE. For example, Wu and He (2019) applied BERT (Devlin et al. 2018) to the TACRED RE task. Devlin et al. (2018); Yamada et al. (2020) showed that further improvements are possible with a better representation for the pre-trained language model. Our approach also fits in this space. We deploy a transformer-based classifier to capture relation mentions, but we also include a novel component dedicated to explainability, which tags the words important for the relation at hand. Importantly, our direction has the relation classifier operate directly on top of the words deemed important for the relation by the explainability classifier, which guarantees that our explanations are faithful, i.e., our explanations correctly depict how the relation classifier makes a decision (Vafa et al. 2021). Further, we propose an efficient semi-supervised strategy to jointly train the relation and explainability classifiers using a small amount of linguistic supervision for explainability.
Explainability
Explainable artificial intelligence (XAI) has recently experienced a resurgence in the context of deep learning (Adadi and Berrada 2018; Gunning and Aha 2019; Arrieta et al. 2020;Danilevsky et al. 2020).
A taxonomy of explanations.
Explanations can be categorized along two main aspects: whether they explain a complete model (global) or individual predictions (local); and whether they are built in the classification model itself (self-explaining) or are generated through a post-processing step (post-hoc).
Global vs. Local. Rule-based approaches (Hearst 1992;Brin 1998) or decision trees (Béchet, Nasr, and Genet 2000;Boros, Dumitrescu, and Pipa 2017) provide global explainability by constructing transparent models that people can understand. However, these directions were slowly replaced by deep learning, which tends to yield better classifiers (at least with respect to accuracy). Several efforts aimed at bringing back global explainability into deep learning. For example, in the non-NLP context of high-stakes decision-making at population level, Rawal and Lakkaraju (2020) proposed a model-agnostic framework that constructs global counterfactual explanations that provide an interpretable and accurate summary of recourses for an entire population affected by a certain problem such as bad financial credit. Closer to our work, Craven and Shavlik (1996); Frosst and Hinton (2017) both proposed distilling a neural network into a globally-interpretable model such as a decision tree.
However, most recent approaches focus on local model explainability, which preserves the underlying neural classifier and interprets its individual predictions. In this category, Hendricks et al. (2016) produced natural language explanations of individual model outputs. Han, Wallace, and Tsvetkov (2020) used influence-based training-point ranking to study spurious training artifacts in NLP settings. Wachter, Mittelstadt, and Russell (2018); Karimi et al. (2020) used counterfactual explanations to understand model decisions.
Self-explaining vs. Post-hoc. Self-explaining strategies make explanations an integral part of model predictions. For example, Tang, Hahn-Powell, and Surdeanu (2020) proposed an encoder-decoder method for relation extraction, which jointly classifies relations and decodes rules that explain the relation classifier's decisions. Rajani et al. (2019) proposed a framework that provide both answer and explanation for a commonsense QA task. In contrast, post-hoc explanations include an additional component that generates explanations after the main model produces its decisions. In this space, Liu et al. (2018) learn a taxonomy post-hoc to better interpret network embeddings. As mentioned above, Craven and Shavlik (1996); Frosst and Hinton (2017) both proposed post-hoc strategies to distill neural network into decision trees. ;Fong, Patrick, and Vedaldi (2019) ;Hoover, Strobelt, and Gehrmann (2020) provided post-hoc visualizations as model explanations. Belinkov et al. (2017); Peters, Ruder, and Smith (2019); Zhao and Bethard (2020); Hewitt et al. (2021) introduced probes, i.e., models trained to predict certain linguistic properties in order to verify that the underlying neural models have learned the desired linguistic knowledge.
With respect to this taxonomy, our approach is self-explaining because our relation extractor has access solely to the context identified as important by the explainability classifier, and local because our core method explains individual predictions. However, in the latter part of this article we propose a simple strategy that converts local explainability into global by converting the entire neural model into a set of rules using the words deemed as important in a dataset by the explainability classifier.
Finding rationales.
From a different perspective, our approach can be seen as finding rationales, i.e., subsets of context that explain individual model decisions (Vafa et al. 2021). Although these directions fit under local explainability (and mostly post-hoc) we discuss them separately due to their recent popularity and proximity to our work.
Some efforts in this space used gradient-based saliency mapping to determine the importance of tokens in context (Baehrens et al. 2010;Simonyan, Vedaldi, and Zisserman 2013;Devlin et al. 2018;Voita, Sennrich, and Titov 2021). However, gradients can be saturated, i.e., they may be close to zeros and, thus, lose explanatory signal. Ghorbani, Abid, and Zou (2019); Wang et al. (2020) also warn that gradients are fragile and they can be distorted while keeping the same prediction.
As an alternative, some researchers focused instead on attention weights in transformer networks (Wiegreffe and Pinter 2019b;Mohankumar et al. 2020). However, there is also evidence that attention weights may not be good explanations (Jain and Wallace 2019;Brunner et al. 2019;Kobayashi et al. 2020). Other efforts have used adversarial attacks on inputs to identify their importance. For example, HotFlip (Ebrahimi et al. 2017) used word-level substitutions to impact predictions. CXPlain (Schwab and Karlen 2019) calculates feature importance by masking them and comparing differences in output confidences. Feng et al. (2018); Li, Monroe, and Jurafsky (2016) focused on input reduction to identify the importance of input features. Instead of reducing, Vafa et al. (2021) greedily added input information to locate meaningful rationales. However, other research has showed that input perturbation cannot always guarantee a good explanation (Poerner, Roth, and Schütze 2018).
In a different direction, surrogate approaches (Ribeiro, Singh, and Guestrin 2016; Lundberg and Lee 2017) generated artificial data in the neighborhood of a prediction to be explained, by randomly hiding features from the instance and learning a surrogate model to explain the predictions. AllenNLP ) combined adversarial attacks and gradient-based saliency mapping in their toolkit. Lastly, Lei, Barzilay, and Jaakkola (2016); Situ et al. (2021) trained a generator model to produce feature importance.
Other than the problems we mentioned above, most of these approaches are either passively reflecting the model behavior or learning rationales in an unsupervised way. Because of this, these methods cannot guarantee faithfulness and plausibility. In contrast, our proposed approach provides local explanations (or rationales) that are designed to be faithful. Further, our empirical evaluation shows that our explanations are also more plausible than other rationale finding methods (see Section 4).
All of the approaches discussed above address the task of finding rationales. However, a relatively new direction focuses on the opposite effort: if rationales are provided by a human expert, how can they be integrated in a statistical model? For example, Bao et al. (2018) proposed a method to map discrete rationales to continuous attention, and showed that the performance on low-resource tasks can be improved by transferring these mappings from resource-rich tasks. Hancock et al. (2018) showed that human-provided natural language explanations for labeling decisions can be converted to noisy labels using a semantic parser. They empirically demonstrated that through this process they can train classifiers with comparable F1 scores considerably faster. Incorporating rationales in a classifier is a key part of the our approach. However, our method jointly trains the explanation classifier with the relation classifier, rather than depending on human rationales for the entire training data.
Approach
At a high level, our approach consists of two main components: a neural relation classifier with an integrated explainability classifier, and a rule generation component, which generates a rule-based model from the explainability information, i.e., context words that explain a relation, provided by the neural model.
Walkthrough Example
Before getting into the details of our approach, we highlight its key functionality with the walkthrough example shown in Table 1. Consider the sentence "John's daughter, Emma, likes swimming.". As shown in Table 1 (a), the task input includes: the raw text in the sentence, the entities participating in the relation (denoted as subject and object) and their types (PERSON here), and the syntactic dependency parse tree. Table 1 (b) shows the output of our relation classifier (RC) and explanation classifier (EC): the RC returns the predicted relation per:children, while the EC labels the word daughter as the trigger of the predicted relation.
Step (c) shows the information that is collected for rule generation. This information includes: the two entities, the relation predicted, the tokens identified by the EC as the rationale for the relation, and the shortest syntactic path connecting the two entities with the rationale words. The output rule generated by our approach is shown in step (d). This rule is written in the Odin language (Valenzuela-Escarcega et al. 2015;. The rule captures the relation to be predicted (per:children), its trigger (daughter), the two arguments and their type, e.g., subject with the type SUBJ_Person, and the syntactic paths between each argument and the trigger phrase, e.g., nmod:poss for the subject argument. Note that in this simple example, the trigger consists of a single word, but, in general, an Odin rule can take any arbitrary sequence of words as its trigger.
This example shows that our method can be deployed in two ways. First, one can use the joint RC and EC neural classifiers, which predict relations that hold between pairs of entities, as well as local explanations (or rationales) that explain the prediction. Alternatively, a different class of users may use the output of step (d), which, once applied on large text collections, contains a set of rules that describes multiple relation classes. This usage may be preferred in real-world situations that have to mitigate the "technical debt" of neural methods, i.e., reduce the cost of maintaining these models over time (Sculley et al. 2015). Although not within the scope of this work, other works have shown that rule-based methods for IE can be improved and maintained at a low cost ). Table 1: Walkthrough example of our approach. The task input includes information about the entities participating in the relation (denoted as subject and object) and their types (PERSON here). Our neural architecture, which includes both a relation and explanation classifier, predicts the relation that holds between the two entities (per:children here, i.e., the object is the child of the subject), as well as which words best explain the decision (in red). In step (c), the rule generator collects the necessary information from the annotated sentence, i.e., the shortest syntactic dependency path that connects the two entities with the explanation words (in red in the figure).
Step (d) shows the generated rule in the Odin language.
Joint Relation and Explainability Classifiers
As mentioned, our approach jointly trains an explainability classifier (EC) and a relation classifier (RC). The RC is a multiclass classifier that distinguishes between actual relation labels seen in training. We couple the RC with a binary classifier that first predicts if the current example contains an actual relation or no relation (marked as no_relation).
For conciseness, we call this classifier the no relation classifier (NRC). The EC is a binary Figure 1: Flow of our semi-supervised training procedure for an individual training example. All the "Train . . . " blocks (green background) involve parameter updates of the corresponding classifiers. These updates are shown here for an individual training example, but are batched in the actual implementation.
word-level classifier, which labels words in the sentence that contains the relation with 1, if they are important for the underlying relation, or 0, otherwise. We start this section with the description of the overall training procedure, and follow with details about the individual classifiers.
Training Procedure.
The overall flow of the training procedure is shown in Figure 1. This flow is temporally split in two periods: a burn-in period, which is fully supervised, followed by a period that includes semi-supervised learning (SSL). This distinction is necessary because while all training examples in this task are guaranteed to have RC labels, most examples will not have gold explainability annotations. For example, for the sentence "[CLS] John was born in London.", the training data contains information that there is a per:city_of_birth relation between John and London, but may not contain information about which words are critical for this relation (born and in).
Burn-in period. In this stage, shown in the left-hand side of Figure 1, we only use the training examples that are associated with explainability annotations (see Section 3.2.2 for details on how these annotations are generated). Here we train initial versions of the three classifiers: NRC, EC, and RC (see Section 3.2.3 for details on the three classifiers). The purpose of this stage is to initialize the three classifiers such that they can be successfully used to reduce the search space for explainability annotations in the next SSL stage.
After burn-in. In this stage, the training procedure is exposed to all training examples, including those without annotations for explainability. That is, for such training examples, we simply have annotations for the relation labels (or no_relation), without knowing which context words explain the underlying relation. In such situations, the right-hand side of the flow in Figure 1 is used, which triggers two additional components: one to generate candidates for explainability annotations, and one to choose the best sequence of word labels (i.e., which words are important and which are not).
For the former component, exhaustively generating all possible label assignments is prohibitively expensive (i.e., O(2 N ) for a sequence of length N ). To mitigate this cost, we rely on the prediction scores of the EC classifier to reduce the number of candidates. That is, if the score of the binary EC for a given token is higher than a threshold (t up ), we directly annotate the corresponding token as important (i.e., assign label 1); if this score is lower than a second threshold (t low ), we annotate the token as not important (label 0); and, lastly, if the the score is between the two thresholds, we generate two candidate labels for this token (both 0 and 1). For example, given an input sentence "[CLS] [SUBJ-PER] was born in [OBJ-CITY] .", 1 and these prediction scores from the EC: [0.12, 0.14, 0.19, 0.86, 0.25, 0.15, 0.01], using t up = 0.8 and t low = 0.2, we produce the following candidate label sequences: [0, 0, 0, 1, 0, 0, 0] and [0, 0, 0, 1, 1, 0, 0], because the assignment for the token in is ambiguous according to the two thresholds.
Once these candidates are generated, we loop through all the generated sequences of word labels, and pick the sequenceĉ that yields the highest score for the correct relation label according to the current RC:ĉ
= argmax c p(R|c)(1)
where R is the gold label of the instance, p(R|c) is the score at the gold label R predicted by the RC for a given annotation candidate c. In the previous example, if the RC scores of the two candidates for the correct relation label per:city_of_birth are 0.8 and 0.5, we select the first candidate over the second one. Then this sequence of labels is used as (pseudo) gold data to train the EC on this training example. This guarantees that each training example has annotations (gold, or generated through the above procedure) for both EC and RC.
Because these two components rely on having reasonable predictions from the EC and RC classifiers, we found it beneficial to include the previous burn-in period, where these classifiers are trained using the (small) amount of supervision available. relying on manual annotations, which are expensive, we repurpose rules that extract the same relation. The intuition behind our approach is that if a rule exists that extracts the same relation label as the gold label in a training example, then this rule (and, specifically, its lexical elements) can be seen as an explanation of the extraction. In particular, in this article we focus on the TACRED dataset (Zhang et al. 2017), and select explanations from two sets of rules:
Explainability
(1) Surface rules: The TACRED project generated a set of high-precision rules for the task, implemented in the Tokensregex language (Chang and Manning 2014). For example, the rule SUBJ-PER was born in * OBJ-CITY 2 extracts a per:city_of_birth relation between a person named entity (the subject) and a city named entity (the object) if the sequence was born in occurs somewhere between the two entities. For such rules, we label all tokens contained in the rule (e.g., was, born, in) with the label 1 (i.e., they are important for explainability), and all other tokens in the sentence with 0.
(2) Syntactic rules: In initial experiments, we observed that the TACRED surface rules have high precision but low recall. To improve generalization, we also wrote 38 syntaxbased rules using the Odin language (Valenzuela-Escárcega, Hahn-Powell, and Surdeanu 2016). 3 Figure 2 shows an example of such a rule. For these syntactic rules, we marked all their lexical elements (typically the trigger predicates such as work or write in the figure) as important (label 1), and all other words as not important (label 0). Figure 3: Neural architecture of the proposed multitask learning approach. The entity tokens (subject in blue and object in orange) are masked with their named entity labels, e.g., SUBJ-Person, in the actual implementation.
Classifiers.
As mentioned, the building blocks of our approach consist of three classifiers: the no-relation classifier (NRC), the relation classifier (RC), and the explainability classifier (EC). These are jointly trained using the schema previously described in this section. Below we describe their individual details, which are also visualized in Figure 3.
SpanBERT Encoder and NRC:. We follow the entity masking schema from (Zhang et al. 2017) and replace the subject and object entities with their provided named entity (NE) labels, e.g., "
[CLS] [SUBJ-PER] was born in [OBJ-CITY] . . . ". We feed this input to a SpanBERT-based (Joshi et al. 2020) encoder:
[h h h 0 , . . . , h h h n ] = Encoder([w 0 , . . . , w n ])(2)
where w n is the id of the word at position n, and h n is the hidden representation generated by the encoder. We add the special masking tokens for SUBJ-* and OBJ-* to the vocabulary so that the encoder can handle them properly. We implement the NRC using a feedforward layer with a sigmoid function on top of the encoder's [CLS] token.
Explainability Classifier (EC):. We implement the EC as a binary token-level classifier, where the positive label indicates that the corresponding token is important for the underlying relation. Section 3.2.2 discusses how these annotations are generated from rules; Section 3.2.1 explains the SSL training procedure when these annotations are not available.
Relation Classifier (RC):. Crucially, the RC relies only on words that are marked as important by the EC, or are part of the subject/object entity. This is an important distinction between our approach and other relation extraction methods, which typically rely on the [CLS] representation for classification. In the next section, we empirically show that this latter strategy is considerably less explainable than ours. This is because the [CLS] representation aggregates information from all tokens in the sentence, whereas our method focuses only on the important ones. We build the aggregated representation of the important context words, subject and object as follows:
h h h f inal = f (h h h ctx 1 :ctx n ) • f (h h h subj 1 :subj n ) • f (h h h obj 1 :obj n )(3)
where h h h denotes the hidden representations produced by the encoder, f : R d×n → R d is the average pooling function that maps from n output vectors into one; and • is the concatenation operator. Importantly, h h h ctx iterates only over words marked as important by the EC.
The concatenated representation h h h f inal is fed to a feedforward layer with a softmax function to produce a probability distribution p p p over relation types.
The three classifiers are trained using the following joint loss function:
loss = loss nrc + loss ec + loss rc (4) loss nrc = −(t n * log(y n ) + (1 − t n ) * log(1 − y n )) (5) loss ec = −(t e * log(y e ) + (1 − t e ) * log(1 − y e ))(6)loss rc = −log(p(R))(7)
where the losses of the NRC and EC (loss nrc and loss ec , respectively) are implemented using binary cross entropy. For both, t indicates the corresponding gold label, and y is the respective sigmoid's activation. The loss of the RC (loss rc ) is implemented using categorical cross entropy, where p(R) is the likelihood predicted by the model for the correct relation R.
Aggregating Local Explanations into a Global, Rule-based Model
As mentioned, the last component of our approach aggregates all local RC and EC predictions into a single rule-based model that explains the overall behavior of the RC and EC models. As such, the produced rule-based model brings global explainability to the task. We will show in Section 4 that this transformation comes with a cost in performance, but this cost might be acceptable in scenarios where such RE extraction must be deployed, maintained, and improved over a long period of time. Table 1 (c), our relation and explanation classifiers produce all the information necessary to generate an Odin rule. At a high-level, the Odin rules we employ here follow a predicate (or trigger in the Odin language) and argument template, where all arguments are connected to the trigger using a syntactic dependency path. This information is either provided by our classifiers, e.g., we use the rationale tokens identified by the EC as triggers, or can be automatically extracted from the sentence, e.g., we represent the syntactic connections between predicate and arguments using the shortest path that connects them in the syntactic dependency tree. Algorithm 1 describes this entire rule generation process:
Rule Generation. As shown in
Algorithm 1 Rule Generator
Input: set of annotated sentences, S Input: model output L (from RC) and T (from EC) 1: R ← ∅ 2: for every sentence s in S do 3:
Get the subject and object entity e s and e o from s 4:
Get the predicted relation label l from L and the rationale words t from T
5:
if s hasn't been extracted by any manual rule then 6:
Find the shortest path p s between t and e s in the dependency tree 7:
Find the shortest path p o between t and e o in the dependency tree 8:
r ← empty Odin rule template 9:
Assign l to r as the label to match 10:
Assign t to r as the relation predicate (or trigger) 11:
Assign p s and p o to r as the argument patterns.
12:
R ← R ∪ {r} 13:
end if 14: end for Output: set of generated rules R
Experimental Results
Data Preparation
We report results on the TACRED dataset (Zhang et al. 2017) and CoNLL04 dataset (Roth and Yih 2004). As discussed in Section 3.2.2, we provided rules for explanation supervision. For the TACRED data, we selected rules from the surface patterns of Angeli et al. (2015), and we combined them with an additional set of 38 syntactic rules in the Odin language (Valenzuela-Escárcega, Hahn-Powell, and Surdeanu 2016) that were manually created by one of the authors from the training data. For CoNLL04 data, we selected from a set of 19 syntactic rules in Odin language, 10 of which are borrowed from the TACRED syntactic rules, since the two datasets shared some overlapping relations.
These rules match 20.7% of positive examples in the TACRED training set and 24.2% of positive examples in the CoNLL04 training set. On average, 7.27 rules are assigned to each TACRED relation, and 3.8 rules are assigned to each CoNLL04 relation.
Importantly, our approach does not use rules at evaluation time. However, we take advantage of all existing rules to automatically evaluate the quality of the explanations generated by our method. In the TACRED dataset, the combined set of rules from (Angeli et al. 2015) and our syntactic rules match 23.9% data points in the development set, and 23.9% examples in the test set; in the CoNLL04 dataset, the syntactic rules match 20.1% of examples and 20.9% of examples respectively. We use only these matches for an automated evaluation of explainability (discussed below).
Baselines 4.2.1 Relation Extraction
Baselines. For the relation extraction task, we compare our approach with three baselines: an extended version of the rule-based approach of Angeli et al. (2015), a neural state-of-the-art RE approach based on Span-BERT (Joshi et al. 2020), and a neural approach with built-in explainability (Lei, Barzilay, and Jaakkola 2016):
• Rule-based Extraction. As mentioned in Section 4.1, we employ two sets of rules. First, we use the tokensregex surface rules from (Angeli et al. 2015), which are executed in the Stanford CoreNLP pipeline (Manning et al. 2014a • Unsupervised Rationale. Lei, Barzilay, and Jaakkola (2016) proposed an approach that combines an unsupervised rationale generator with a task-specific classifier, both of which are trained to operate together (similar to our approach). However, there are several key differences between their method and ours. First, their explanation generator cannot incorporate human input (as we do through rules); instead, it is indirectly guided by the loss of the downstream task. Second, their architecture is more complex, i.e., they use two distinct encoders: one for explanation generation and another for the downstream task (both of which are implemented with recurrent networks). We adapt this method to our RE framework, by replacing our EC with their rationale generation algorithm (which is a token-level binary classifier that produces an output compatible with our EC). For a fair comparison with our method, we kept the other components unchanged.
That is: we encode the input text using the same SpanBERT, then we use their generated rationales and the given entities as pooling mask to construct the final vector to feed into the relation classifier 5 . Originally, Lei, Barzilay, and Jaakkola (2016) proposed their approach to sentiment analysis and text retrieval. Bastings, Aziz, and Titov (2019) extended this method and adapted it to a natural language inference task. To our knowledge, this is the first attempt to apply this explainability strategy to relation extraction.
Note that all baselines as well as our method receive inputs in the standard TACRED format, 6 which contains tokenized sentences, spans of the subject and object mentions, and the types of the two entity mentions. The only difference between the RC baselines and our method is that, as discussed in Section 4.1, our approach receives information on which sentence tokens were matched by rules during the burn-in training period.
Explainability Baselines.
For explainability, we compare our approach against eight baselines, detailed below. These are all popular explanation approaches published in recent years. Most of them provide a feature importance score for each feature 7 and most of them are post-hoc 8 . Here, we labeled the top N positive features identified by the baselines as important. 9 In the first quantitative evaluation of explainability (Section 4.4.2), for all baselines we set N to be equal to the number of words in the gold explanation. Importantly, this means that all baselines have an unfair advantage over our approach, which is non-parametric with respect to N , i.e., it identifies N on the fly for each sentence. In the second, qualitative evaluation of explainability (Section 4.4.3), N is a hyper parameter that we tuned to maximize the baselines' performance. 10 We detail the eight explainability baselines below:
• Attention. Attention weights have been proposed as an explanation mechanism by Bahdanau, Cho, and Bengio (2014). Followup work debated the validity of this strategy (Jain and Wallace 2019; Wiegreffe and Pinter 2019b; Kobayashi et al. 2020). However, because this remains a popular approach, we include attention weights as a baseline in this work. In particular, we use the attention weights from the last layer of a "vanilla" SpanBERT model, i.e., one that is trained on top of the [CLS] representation, without an EC. For this baseline, we label as important the top N tokens with the highest [CLS] attention weights.
• Saliency Mapping. The feature importance score of the token x i is determined by the highest prediction's accumulated gradients in each dimension of the token in the embedding layer. These scores are obtained through a back-propagation of the highest prediction's probability. Although there are different implementations of the gradient saliency mapping approach (Devlin et al. 2018;Voita, Sennrich, and Titov 2021), we use the simple back-propagation approach from (Simonyan, Vedaldi, and Zisserman 2013).
• LIME. Ribeiro, Singh, and Guestrin (2016) proposed the LIME framework, which provides explanations to any black-box classifier. LIME samples the neighbors of the local instance x to be explained, by generating perturbations of the tokens in x. Then, it trains a linear separator from these samples to approximate the local behavior of the model. The coefficients of the separator are later used as the feature importance score. 6 We converted the CoNLL04 data into the same format as TACRED.
7 Except for greedy adding and unsupervised rationale approaches which rely on labeling the features to be included in the rationale, similar to what we do. 8 Except for unsupervised rationale approach which trains a generator together with the rest of the model, similar to what we do. 9 We ignored tokens part of the subject and object entities for a fair comparison. 10 We used N = 3 for TACRED, and N = 1 for CoNLL04.
•
Unsupervised Rationale. As mentioned in the previous sub-section, this baseline replaces our EC with the unsupervised method of Lei, Barzilay, and Jaakkola (2016). Here we use this method as an explainability baseline.
• SHAP. The Shapley value (Shapley 1952) is a cooperative game theory concept that calculates the score of feature x i by taking into account its interactions with all other subsets of features. Similar to what LIME does, Lundberg and Lee (2017) also train a linear model to approximate the local behavior around the sampled neighbors. However, unlike LIME, which uses cosine similarity or L2 distance as its kernel, they propose a SHAP kernel which is determined by the number of permutations of features.
• CXPlain. Schwab and Karlen (2019) proposed an approach called CXPlain that explains the decisions of any machine-learning model by measuring the importance of the model's features. To this end, CXPlain masks each token x i in x, and calculates the score of x i by comparing the output with the masked input x against the output that relies on the original input x. The difference between the two is calculated using a causal objective.
• Greedy Adding. Instead of randomly sampling from perturbations or masking the features, Vafa et al. (2021) proposed a method that greedily adds the features to the input data point. That is, it starts with an empty rationale, and each time it selects and adds the feature that increases the probability of the correct label y t the most. The process repeats as long as the confidence in predicting y t keeps increasing.
• All Words between Subject and Object. We have observed that most of the important words that determine the relation between the entities occur in the span between the two entities. To capture this intuition, we implemented this simple baseline, which simply includes all the words between subject and object in its rationale.
Similarly to the RC settings discussed in the previous sub-section, these baselines and our method rely on the standard TACRED input format. However, our EC is semi-supervised, i.e., during burn-in it receives explainability annotations generated by rules. In contrast, the EC baselines do not rely on rule information.
Implementation and Evaluation Details
Before introducing our results, we discuss key details about our implementation and evaluation.
To avoid the RC classifier overfitting on the names in the sentence (Suntwal et al. 2019), we mask the subject and object entities by replacing the original tokens in these entities with a special token, i.e., SUBJ-<NE> or OBJ-<NE>, where <NE> is the corresponding name entity type provided in the dataset. We use the pre-trained SpanBERT to encode the input sentence. For the TACRED dataset, which is organized to contain a single relation per sentence, we feed the [CLS] token to the final linear layer for relation classification. However, for the CoNLL04 data, which typically contains more than one relation per sentence, we used the concatenation of the [CLS] hidden state and the average pooling of [SUBJ] and [OBJ] hidden state embeddings. This was necessary to distinguish between the different relations that co-occur in the same sentence. We used the AdamW optimizer (Loshchilov and Hutter 2019) for all training processes. We evaluated all RC classifiers using the standard micro precision, recall, and F1 scores. All neural models were trained using 5 different random seeds; we report the average scores and standard deviation over these seeds for RC.
For explainability, we report two evaluations. 11 For the first, automated evaluation, we use only the data points that are associated with a rule that produces the same relation label as the gold data. For these examples, we consider the lexical artifacts of the rule as gold information for explainability (as explained in §3.2.2). We measure the overlap between the important words produced by the analyzed methods and this data using precision, recall, and F1 scores. We also include a second, qualitative evaluation on the plausability of the generated explanations (Vafa et al. 2021), where a more plausible explanation will overlap more with a relation explanation manually generated by domain experts. For this evaluation, we sampled 100 and 60 data points from the test sets of TACRED and CoNLL04, respectively. These are sentences where our model predicted a relation, and where there is no gold annotation from rule-based method (i.e., no rule matched). We split these data points into two sets: a subset where our method predicted the correct relation, and one where it did not. In other words, in the former set, we investigate the capacity of the explainability methods to explain correct predictions, while in the latter we analyze their capacity to explain why the machine was incorrect. Two domain experts 12 manually annotated rationales for these sentences and the provided relation labels. The annotators were asked to identify the minimal set of tokens that explain the provided relation. Or, in other words, identify the tokens that when replaced with other words change the relation to be predicted. For example, in the sentence SUBJ-PER was born in OBJ-CITY., if we replace the words born in with other words (e.g., moved to), the relation between the subject and object changes. Importantly, to avoid any potential bias, the two annotators worked completely independently of each other, and had no access to explanations provided by any algorithm. 13 We evaluate the overlap between the machine and human rationales using the same standard precision, recall, and F1 measures.
Appendix A lists the hyperparameters used to train all RC and EC models. Lastly, we evaluate the quality of the generated rule-based model. To this end, we evaluated two sets of rules: rules generated from the training sentences, 14 and rules generated over the test set. In the latter scenario, we do not use any gold data. That is, we rely on the predicted relation labels (from the RC) and rationales (from the EC) to generate rules. Thus, the latter setting is akin to transductive learning, i.e., where the model has access to the unlabeled data from the testing partition, but no access to any human annotations. We evaluate the performance of these rule-based models using the same micro precision, recall, and F1 scores as the first RC evaluation. 11 We did not include an evaluation of faithfulness, which is typically done by post-hoc explainability approaches (Ribeiro, Singh, and Guestrin 2016;Schwab and Karlen 2019) because our approach is faithful by design, i.e., our RC only relies on the tokens identified by the EC. 12 These were two of the authors. 13 To encourage reproducibility, we release the annotations at https://github.com/clulab/releases/tree/master/cl2022-twoflints/dataset 14 We filter our training relations which matched a gold rule, since there is already a rule assigned to them Table 3: Relation extraction results on the CoNLL04 test partition. We used the pre-trained SpanBERT-large. Our full model trains on the entire training partition using the SSL method discussed in Section 3.2.1. The "burn-in only" setting trains just on the training subset that has annotations from rules.
Results and Discussion
In this section, we introduce and discuss the results for both relation and explainability classification. We conclude this section with an error analysis that highlights some typical errors in our models. Tables 2 and 3 report the RE performance of all methods discussed on the TACRED and CoNLL04 datasets. The results of all statistical approaches are averaged over three random seeds. For all these models we report average performance and standard deviation in the tables. We draw the following observations from these tables:
Relation Extraction.
• First, the SSL variant of our approach improves considerably over the equivalent burn-in only setting (i.e., training just on the data points that have matching rules). The improvement is 20.91% F1 (absolute) on TACRED, and 21.83% (absolute) on CoNLL04. These results highlight the importance of SSL for this task.
• Second, our approach is slightly better than SpanBERT on TACRED, and yields a statistically-significant improvement of nearly 4% F1 (absolute) on
CoNLL04. 15 This indicates that jointly training for classification and explainability helps the classification task itself (or, in the worst case, does not hurt relation classification). Table 3 also shows that our approach has the highest RE recall on CoNLL04, higher than the vanilla SpanBERT by 5%. All in all, this suggests that explainability also serves as a disambiguator in situations where multiple relations co-occur in the same sentence (the common setting in CoNLL04) by narrowing the text to just the context necessary for the relation at hand. As further evidence that performing RC on top of explanations helps disambiguate the underlying text, the standard deviation of our approach on CoNLL04 is five times smaller than that of SpanBERT.
• Interestingly, the unsupervised rationale method approaches the performance of our full model on both datasets. However, as we will show in the next sub-section, this comes with considerably worse explanations.
• Lastly, our approach nearly doubles the F1 score of the rule-based approach on TACRED, and more than doubles it on CoNLL04. This is caused by large improvements in recall, which highlights the importance of hybrid strategies that combine rules and neural components.
To understand the runtime overhead introduced by the EC, we compared our method's runtimes during training and inference against the runtime of the vanilla SpanBERT. The average training time of our method is 0.37 sec/batch in the burnin period and 0.38 after burn-in. In contrast, the average training time of SpanBERT is 0.06 sec/batch. 16 The inference time for both our model and SpanBERT is 0.10 sec/batch on the same device. The larger overhead in training is caused by: (a) backpropagating through a larger computational graph due to the joint EC and RC loss, and (b) iterating through multiple candidate explanations. We measured the average number of explanation candidates to be 85 in the first training epoch after burn-in period, and 22 after 10 epochs. However, considering that inference time are similar, we believe that the training overhead is justified by the additional explainability functionality included in the framework. Tables 4 and 5 show that our approach generally improves explainability quality considerably. Post-hoc explanation methods do not provide the same explanation quality compared to our method, which actively models explainability. Note that the high performance of annotating all the words between subject and object is caused by the fact that most data points in this evaluation are associated with surface rules, which prefer shorter contexts that are more likely to contain only significant information. Nevertheless, the 20% F1 gap between this strong baseline and our method indicates that our method successfully learns how to generalize beyond these simple scenarios.
Quantitative Evaluation of Explainability. The results of the automated evaluation of explainability in
However, we note that these results are not terribly surprising: our method is trained to generate explanations that mimic lexical artifacts of rules, while the other explainability 15 We performed statistical significance analysis using non-parametric bootstrap resampling with 1000 iterations. 16 All times measured on an NVIDIA RTX 3090 GPU. Table 5: Automated evaluation of explainability on CoNLL04, in which we compare explainability annotations produced by these methods against the lexical artifacts of rules.
baselines have not been exposed to rules during their training. Thus, this evaluation is necessary (to validate that our approach is learning to do what we intended, which is to mimic the lexical artifacts of rules) but not sufficient. In the next sub-section, we will show that our approach overlaps with human explanations much more than all other explainability baselines. Table 6 lists a learning curve for our approach on TACRED, as we vary the amount of rules available per relation. That is, for each relation, we use up to top k rules, where k varies from 1 to 10. In the table we include results for both relation and explainability classification using the same measures as the previous tables. The table shows that even in the "up to top 5 rules" configuration (which means an average of 3.6 rules per relation type in practice), our model obtains a close F1 to the our best model with good explainability. This result indicates that our approach performs well with minimal human supervision for explanation guidance. Note that we do not include the learning curve for CoNLL04 since there are only 19 rules applied to this dataset, which translates into only 3.8 per relation type. . Tables 7 and 8 Table 7: TACRED evaluation of the plausability of explanations, which measures the overlap between machine explanations and human annotations. For each method, we pick the higher scores between the two human annotators.
Qualitative Evaluation of Explainability
annotations of explainability. Similar to evaluations of machine translation, we choose the higher scores between the machine methods and any of the two human annotators. Note that the human annotators had a Kappa agreement (McHugh 2012) of 69.8% on labeling the same tokens as part of an explanation. This is considered moderate (Landis and Koch 1977), which we found encouraging considering the complexity of the task and the fine granularity of the annotations. We investigated the differences between the human annotators and observed that they are caused either by legitimate annotation errors or by the fact that there are multiple valid rationales for a given relation. For example, in the sentence OBJ-PER is the CEO and president of SUBJ-ORG, the relation org:top_membersemployees can be explained either by the tokens CEO or president.
The two tables indicate that our approach generates explanations that have considerably higher overlap with human-generated explanations, even though all data points part of this evaluation were chosen to not have a matching rule. This suggests that our approach generates high-quality explanations of its predictions regardless of whether it has seen the underlying pattern or not. Moreover, the recall of our approach is much higher than that of the other post-hoc explanations, which have not been exposed to rules during training. This shows that with a small amount of supervision, the generated explanations can be better aligned with human intuitions. Table 8: CoNLL04 evaluation of the plausability of explanations, which measures the overlap between machine explanations and human annotations. For each method, we pick the higher scores between the two human annotators.
The fact that our method outperforms considerably the unsupervised rationale approach of Lei, Barzilay, and Jaakkola (2016), which is driven solely by relation classification performance, further emphasizes that a "human-in-the-loop" method such as ours is necessary to yield meaningful explanations. We include several examples of the generated rationales in Figures 4, 5, 6, and 7. These examples indicate that most of the baselines are noisier, i.e., they contain a considerable amount of false positives (words that should not be part of the rationale) and false negatives (words that should be included but are not). In contrast, our method does a better job focusing on the right explanation tokens.
In the example in Figure 4, both our RC model and vanilla BERT predicted the correct relation. However, our method labels only the preposition of and the determiner the as its explanation, while other baselines such as LIME and SHAP completely missed them. Greedy adding and CXPlain label more irrelevant words in the context such as ( and press conference. The attention weights do capture the key words, but we can clearly see additional noise surrounding the entities. In the example in Figure 5, both our model and vanilla model predicted the incorrect relation. Our model labels the preposition for, which provides a strong hint for its (possibly) incorrect prediction (per:countries_of_residence). In contrast, the baselines focus more on the nouns such as defender and champion. Applying the substitution heuristic indicates that the preposition for is necessary for the explanation (e.g., changing it to against changes the relation), while the nouns are not relevant. In this example, the attention weights are almost completely noisy.
In Figure 6, both our model and the vanilla SpanBERT model produce the correct prediction. The words Secretary-General clearly explain the Work_For relation in the explanations generated by our model and greedy adding. The other baselines do not provide meaningful explanations here. In Figure 7, which shows an incorrect prediction, only our model can defend its prediction by its explanation. The baseline approaches cannot provide valid explanations to defend the prediction at all. We also find that with the explanation provided from our model, one can argue that the predicted relation is actually correct, and we should change the gold label instead. Lei, Barzilay, and Jaakkola (2016) state that rationales should be short, coherent, and be sufficient for the correct prediction. However, short does not necessarily mean simple. To highlight this point, Figure 8 compares the distribution of POS tags in the TACRED test partition with the distribution of POS tags that participate in explanations in the same attention weights for BERT's 16 heads (the lighter the color the higher the weight). We shrink the weight range margin to make the color scale distinguishable. For a fair comparison, we masked the subject and object in the attention weights. partition. We draw two observations from this data. First, to extract plausible rationales, our EC has to diverge from the distribution of POS tags in the data in a non-trivial way. For example, the frequency of verbs (VB * ), prepositions (IN), and commas is considerably higher in the explanations than the raw data. Second, the figure indicates that our explanations often focus on parts of speech that are necessary for plausability (according to the human annotators) but are semantically-ambiguous such as prepositions (IN), commas, 17 and determiners (DT). This is different from traditional pattern acquisition "Barack Obama" and "former president", respectively) cover most lexical information relevant to the relation. In these cases, the remaining signal that indicates the apposition is the comma. methods (Riloff 1996), which usually focus on words with more clear semantics such as nominals and verbs. 18
Ablation Study.
To understand the impact of the classifiers employed by our approach (i.e., NRC, RC, and EC), we implemented ablation experiments on both datasets, which are summarized in Tables 9 and 10. Note that the method without both NRC and EC becomes equivalent to the vanilla SpanBERT (as we discussed in Section 4.2.1).
Overall, this experiment re-emphasizes that not only does our approach outperform
Counts of POS tags in the test partition.
Counts of POS tags in the explanations in the test partition.
Interpretability: from Local to Global.
Lastly, we evaluate the performance of our rule-based model that relies solely on rules, some of which were manually written (see Section 4.1), while some were automatically generated by our approach, as described in Section 3.3. The results are summarized in Tables 11 and 12. We draw two observations from these results:
• Automatically-generated rules can outperform manually-written ones. However, in order to approach the performance of the neural RC, our method benefits from being aware of the distribution of words in each testing sentence to be processed (setting [3] in the tables). Importantly, we reiterate that when using the test sentences, our approach does not have access to any gold human annotations for RC and EC. That is, the rules generated from test sentences rely only on predicted relation labels and predicted explanations for each given sentence. The fact that rules need to be exposed to more data before they generalize is not extremely surprising: the rule matching engine we currently use relies on exact lexical matching, which means that the actual tokens to be matched must be present in the rule. However, the fact that the knowledge necessary to encode a relation extraction can be encoded into rules is exciting. The combination of these observations suggests that a future avenue for research that focuses on "soft rule matching" (Zhou et al. 2020), might be the direction that captures the advantages of both rules and neural methods.
• Interestingly, automatically-generated rules tend to be complementary to the manual ones. The combination of all three rule sets ([1], [2], and [3] in the tables) outperforms considerably both the setting that relies solely on manual rules and the configuration that relies only on automatically-generated ones. The combination of all rule sets outperforms the manually-generated rules by 31% F1 and 38% F1 (absolute) in TACRED and CoNLL04, respectively. Furthermore, the TACRED result of the combined rule set approaches the performance of the neural RC within less than 3% F1. The performance gap between the combined rule set and neural Table 13: Typical errors that our explainability classifier commits. These include errors of under prediction (first two rows), misleading prediction (middle two rows), and errors of over prediction (last two rows). This figure follows the same convention as Figure 4.
RC in CoNLL04 is larger (over 14% F1). 20 Nevertheless, all in all, this result suggests that humans and machines can collaborate towards building a fully-explainable model that comes reasonably close to the performance of neural classifiers.
Error Analysis.
We conclude this section with a brief error analysis of our explainability classifier in the TACRED and CoNLL04 datasets. Table 13 summarizes a few typical errors observed in the two datasets. The first two rows in the table show examples where the EC generates explanations that rely solely on the subject and object entities, without including any word in the relations' contexts. Note that the example shown in the first row is potentially correct: it is likely that a location name that immediately precedes an organization name indicates the location of that organization. However, the second example is clearly incorrect: the correct explanation to justify the no_relation label should minimally include not and relative. Further, please note that a hypothetical RC that had access to the unmasked entities could potentially perform even better. For example, in the first case, one could infer that O Globo is based in Rio de Janeiro because the former organization name is Portuguese. However, our RC only sees masked subjects and objects. Nevertheless, we believe that our strategy of masking entities participating in relations is a valuable exercise, as it investigates the capacity of neural methods to identify explicit context necessary for relation extraction.
Rows 3 and 4 in the table show examples when our RC makes incorrect predictions due to incorrect tokens labeled by the EC. For example, the token president in row 4 guides the RC towards the incorrect prediction org:top_members/employees. The situation in the third row is more subtle: one might argue that China here can also be referring to the government, which makes the prediction Work_for correct. In any case, these errors indicate that our explanations can be used for debugging purposes when the RC makes incorrect predictions.
The last two rows in the table show examples where our EC over included words in its explanations. For example, in the last row, a likely interpretation is that the verb is should be part of the correct explanation, but all the other words are unnecessary. This happens because the rule lexical triggers in TACRED tend to contain multiple words, which encouraged the EC to learn to include additional words in its explanation. In contrast, in CoNLL04 (second to last row), most triggers are single-word phrases. This prompted the EC to include one token in its explanation, even though it is unnecessary for the prediction of the relation label in this case.
For a more complete bigger picture, we analyzed the overall frequency of these error types on the same sampled instances we used for the qualitative explanation evaluation (Section 4.4.3). Errors where the EC provided no explanations 21 occurred in 4.12% of examples in TACRED, and 19.41% in CoNLL04. Errors where the explanations caused false positive relations to be predicted appeared 25.95% times in TACRED, and 16.49% in CoNLL04. Nevertheless, as Tables 4, 5, 7, and 8 show, our EC makes considerably fewer errors than all other explainability methods. There is no reason to believe that its current errors cannot be fixed with human feedback that would provide a (hopefully small) number of rules to adjust imperfect explanations.
Conclusion
We introduced an explainable approach for relation extraction that jointly trains for prediction and explainability. Our approach uses a multi-task learning framework with a shared encoder, and jointly trains a classifier for relation extraction with a second explainability classifier that labels which words in the context of the relation explain the underlying relation. Further, our method is semi-supervised, as annotations for the latter classifier are usually not available.
We evaluated the proposed approach on a relation extraction task in two datasets: TACRED and CoNLL04. Our evaluation showed that, even with minimal supervision for explanation guidance, our method generates explanations for the relation classifier's decisions that are considerably more accurate and plausible than other strong baselines such as LIME, or relying on attention weights (Simonyan, Vedaldi, and Zisserman 2013;Bahdanau, Cho, and Bengio 2014;Ribeiro, Singh, and Guestrin 2016;Lundberg and Lee 2017;Schwab and Karlen 2019;Vafa et al. 2021). Further, our results indicated that jointly training for explainability and prediction improves the prediction task itself, i.e., the relation classifier performs better when it is exposed only to the textual context deemed important by the explainability classifier.
We also showed that it is possible to convert these local explanations into global ones. We converted the outputs of our explainability classifier into a set of rules that globally explains the behavior of the neural relation classifier. Our results showed that our strategy for generating a rule-based model pushes the performance of rule-based approaches closer to that of neural methods.
Longer term, we envision our approach being used in an iterative semi-supervised learning scenario akin to co-training (Blum and Mitchell 1998). That is, the newly generated rules can be converted to executable rules that can be applied over large, unannotated texts to generate new training examples for the relation classifier, and vice versa. Further, our method could potentially benefit from traditional pattern bootstrapping approaches (Riloff 1996;Lin and Pantel 2001), which could reduce the amount of human supervision necessary by automatically expanding the set of initial patterns available.
At a higher level, we hope that this work will support meaningful collaborations between NLP researchers and subject matter experts in other domains (e.g., medical, legal), who benefit from the output of NLP systems (e.g., large-scale extraction of biomedical events) but may not understand the intricacies of the neural methods that underlie these NLP approaches.
We release all code and data behind this work at: https://github.com/clulab/ releases/cl2022-twoflints/ This interest has been properly disclosed to the University of Arizona Institutional Review Committee and is managed in accordance with its conflict of interest policies.
Figure 2 :
2An example of a relation extraction rule in the Odin language that extracts the per:employee_of relation relation. The rule is driven by verbal triggers such as work, play or serve. The relation's arguments (the subject and object) are identified through both semantic constraints (subject must be Person), and syntactic ones (subject must be attached to the trigger through a certain syntactic dependency pattern: an optional (?) adnominal clause (acl), followed by a nominal subject (nsubj). This rule would extract a per:employee_of relation from the text ". . . Joe is a research scientist working at IBM. . . ".
Figure 4 :
4Examples of explainability annotations on TACRED for a correct RC prediction. The subject and object entities, which are provided in the task input, are highlighted in blue and orange. The important tokens for explainability identified by the various methods are highlighted in red. The bottom of the figure shows the heatmap of[CLS]
Figure 5 :
5Examples of explainability annotations on TACRED for an incorrect RC prediction. This figure follows the same convention asFigure 4.
Figure 6 :
6U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Gold label: Work_For; predicted label: Work_For Saliency ' ' The U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Predicted label: Work_For LIME ' ' The U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Predicted label: Work_For SHAP ' ' The U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Predicted label: Work_For CXPlain ' ' The U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Predicted label: Work_For Greedy Adding ' ' The U.S. has the biggest stock of chemical arms in the world , and it is trying to obstruct other countries from having their own , ' ' said Arab League Secretary-General Chedli Klibi . Predicted label: Examples of explainability annotations on CoNLL04 for a correct RC prediction. This figure follows the same convention as Figure 4.
Figure 7 :
7Examples of explainability annotations on CoNLL04 for an incorrect RC prediction. This figure follows the same convention asFigure 4.
Figure 8 :
8The distributions of POS tags in the TACRED test partition. The top figure shows how many times each POS tag appears in the test data. The bottom figure shows how many times each POS tag appears in the generated explanations from the same partition.
Annotation. As mentioned, a key part of our approach requires that EC annotations be available for a few of the training examples. To this end, rather thanlabel: per:employee_of
pattern: |
trigger =
[lemma=/work|write|play|consult|serve/]
subject: SUBJ_Person = <acl? nsubj
object: OBJ_Organization = nmod
Label(s) to assign to a match.
Lexical constraints on the relation's predicate.
argName:ArgType, where ArgType indi-
cates the named-entity category expected for
this argument.
full content of the masked span without depending on individual token representations within it. SpanBERT outperforms BERT in many tasks including relation extraction. Further, SpanBERT is currently the best TACRED BERT-based model available in the HuggingFace transformer library(Wolf et al. 2020) that does not use any external resources, or does not rely on complex hybrid architectures.).
Second, we include the Odin syntactic rules we developed in-house, which
are executed in the Odin framework (Valenzuela-Escárcega, Hahn-Powell,
and Surdeanu 2016). 4
•
SpanBERT. SpanBERT (Joshi et al. 2020) is an extension of the original
BERT (Devlin et al. 2018) that: (1) masks continuous random spans instead
of random tokens, and (2) trains the span boundary representations to
predict the
Table 2 :
2Relation extraction results on the TACRED test partition. Joshi et al. 2020) 81.30±4.89 71.01±5.11 75.78±4.79 Unsupervised Rationale 83.91±2.88 74.88±1.44 79.11±1.01 Our Approach Burn-in Only 62.71±2.27 53.32±0.95 57.63 ±1.39 Full Model 83.01±2.16 76.30±3.08 79.46±0.92We used the pre-trained
Table 4 :
4Automated evaluation of explainability on TACRED, in which we compare
explainability annotations produced by these methods against the lexical artifacts of
rules.
Approach Precision Recall
F1
Attention
69.44
69.44 69.44
Saliency Mapping
42.42
42.42 42.42
LIME
62.45
89.39 68.45
Unsupervised Rationale
5.47
86.94
9.84
SHAP
34.85
34.85 34.85
CXPlain
50.00
50.00 50.00
Greedy Adding
23.24
54.55 29.58
All words in between SUBJ & OBJ
72.99
96.59 77.29
Our Approach
99.29
100
99.52
lists the results of our evaluation of the plausability of explanations by comparing them against humanNum of Rules Precision Recall
F1
Relation Classification
Up to top 1 (0.98 rules/relation)
72.48
66.23 69.21
Up to top 5 (3.56 rules/relation)
72.97
69.02 70.94
Up to top 10 (5.02 rules/relation)
69.30
71.64 70.45
All rules (7.27 rules/relation)
71.15
71.13 71.14
Explainability Classification
Up to top 1 (0.98 rules/relation)
74.62
85.35 75.02
Up to top 5 (3.56 rules/relation)
92.19
94.06 91.28
Up to top 10 (5.02 rules/relation)
91.06
95.62 91.22
All rules (7.27 rules/relation)
95.63
97.92 95.76
Table 6 :
6Learning curve of our approach on TACRED based on amount of rules used. In
each experiment, we use up to top k rules per relation type; the number in parentheses is
the actual average number of rules per type.
Approach Precision Recall
F1
Attention
41.39
20.60 26.50
Saliency Mapping
18.73
35.58 23.41
LIME
14.31
26.03 18.09
Unsupervised Rationale
4.73
69.66
8.30
SHAP
13.86
22.85 16.79
CXPlain
28.84
55.06 36.48
Greedy Adding
31.59
33.52 30.16
Our Approach
74.72
61.20 62.05
CXPlain ' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in Idaho , where more than 200 fires continued to burn. Predicted label: Work_For Greedy Adding ' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in Idaho , where more than 200 fires continued to burn. Predicted label: Work_ForOur Approach
' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in
Idaho , where more than 200 fires continued to burn.
Gold label: no_relation; predicted label: Work_For
Saliency ' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in
Idaho , where more than 200 fires continued to burn.
Predicted label: Work_For
LIME
' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in
Idaho , where more than 200 fires continued to burn.
Predicted label: Work_For
SHAP ' ' We 're in for a long haul , ' ' said Dave Olson of the Payette National Forest in
Idaho , where more than 200 fires continued to burn.
Predicted label: Work_For
Attention Weights
the vanilla SpanBERT, but it does so while generating an explanation for its decisions. Removing the NRC drops the relation classification F1 score by approximately 3 points on TACRED, and 2 points on CoNLL04. This impact is explained by that fact the using the NRC avoids the meaningless scenario where the EC (which was trained only on positive examples) is applied to negative examples. Interestingly, removing the EC has no statistical impact on relation classification performance on TACRED, but it reduces the relation classification F1 by approximately 3 points on CoNLL04. As discussed in Section 4.4.1, this is caused by the fact that the EC serves as a useful disambiguator in CoNLL04, where multiple relations co-occur in the same sentence. The EC is not thatRC F1
Quantitative EC F1 Qualitative EC F1
Full Model
70.52±0.54
95.76
62.05
− NRC
67.47 ±0.54
92.95
54.70
− EC
70.62±0.46
N/A
N/A
Vanilla SpanBERT 70.07±0.73
N/A
N/A
Table 9 :
9Ablation results on the TACRED test partition, i.e., "-" indicates that the corresponding component was removed from the full system, and "N/A" indicates that that metric is not applicable.RC F1
Quantitative EC F1 Qualitative EC F1
Full Model
79.46±0.92
99.52
58.97
− NRC
77.34±2.33
99.00
50.12
− EC
76.58±1.52
N/A
N/A
Vanilla SpanBERT 75.78±4.79
N/A
N/A
Table 10 :
10Ablation results on the CoNLL04 test partition, i.e., "-" indicates that the corresponding component was removed from the full system, and "N/A" indicates that that metric is not applicable.Approach Precision Recall
F1
Baseline
Manual Rules [1]
85.93
24.24 37.81
Our Approach
Rules from Training [2]
49.39
30.26 37.52
Rules from Test [3]
59.69
55.04 57.27
Combination of [1] and [2]
54.12
62.95 58.20
Combination of [1] and [3]
65.28
71.64 68.31
Combination of [2] and [3]
56.34
40.90 47.40
Combination of [1], [2] and [3]
57.36
72.00 63.85
Table 11 :
11Performance of the rule-based model on the TACRED test partition.[1] is the set of manually-written surface rules ofAngeli et al. (2015) coupled with our syntactic rules (see Section 4.1).[2] is the set of rules generated from our explainability classifier's outputs with gold labels on the training partition.[3] is the set of rules from the explainability classifier's outputs with predicted labels on the test partition. We also evaluate the performance on combinations of these sets of rules: [2]+[3] contain all rules generated by our approach; [1]+[2]+[3] combine machine-generated rules with the manually-written rules. impactful in TACRED, which has a more artificial setting with much fewer relations per sentence.19 Approach Precision Recall
F1
Baseline
Manual Rules [1]
81.82
17.06 28.24
Our Approach
Rules from Training [2]
66.10
27.73 39.07
Rules from Test [3]
67.95
50.24 57.77
Combination of [1] and [2]
71.06
39.57 50.84
Combination of [1] and [3]
68.48
59.72 63.80
Combination of [2] and [3]
64.01
55.21 59.29
Combination of [1], [2] and [3]
66.67
63.03 64.80
Table 12 :
12Performance of the rule-based model on the CoNLL04 test partition. This table follows the same conventions as Table 11, except, in this case, [1] is the set of manually-written Odin rules we wrote for CoNLL04.
Approach SpanBERT Unsupervised Rationale Our ApproachTable 1: Hyperparameter details for training the neural models for relation classification (for SpanBERT) and both components (Unsupervised Rationale and our approach). The numbers with * are the default values from the SpanBERT implementation available at: https://github.com/facebookresearch/SpanBERT.Number of epochs
10 *
20
20
Learning rate
2e-5 *
1e-5
1e-5
Dropout rate
0.1 *
0.1
0.1
Batch size
32 *
32
32
Max sequence length
128 *
128
128
Scheduler
Linear scheduler with warm up *
The entities participating in a relation are masked with their named entity labels (see Section 3.2.3).
We simplified the Tokensregex syntax for readability. 3 All these rules are included in this submission as supplemental material.
The rule set from(Angeli et al. 2015) also included some syntactic rules, but we found out that they only matched the simpler per:title relation, so we did not use them. 5 We also observed that our architecture that uses a single, shared transformer encoder performs better than their original architecture with two distinct encoders.
Commas are necessary to capture appositive constructs, which are often indicative of relations, e.g., "Barack Obama, the former president." In cases such as these, the subject and object of the relation (e.g.,
Note that traditional patterns may include prepositions and particles, e.g., in verb constructs such as SUBJECT was born in OBJECT. However, these patterns are usually semantically headed by verb phrases or nominalized predicates, e.g., born, and seldom by prepositions.
The average number of relations per sentence in TACRED is approximately 2 in training, and 1 in development and test.
We conjecture that the cause for this larger gap is the lower quality of the rules used for the CoNLL04 dataset. That is, the TACRED rules were developed by a larger team over a longer period of time, whereas the CoNLL04 rules were developed by one of the authors in only a few hours.
We included in this category the situations where the explanation was completely empty or it included only the subject and/or object entity mentions.
https://huggingface.co/SpanBERT/spanbert-large-cased
AcknowledgmentsWe thank the reviewers and action editor for their thoughtful comments and suggestions. This work was partially supported by the Defense Advanced Research Projects Agency (DARPA) under the World Modelers program, grant #W911NF1810014, and by the National Science Foundation (NSF) under grant #2006583. Mihai Surdeanu declares a financial interest in lum.ai.Appendix A: Experimental DetailsWe use the dependency parse trees, POS tags and NER labels as included in the original release of the TACRED dataset. All these were generated with Stanford CoreNLP(Manning et al. 2014b).We use the pretrained SpanBERT model(Joshi et al. 2020)available in the HuggingFace transformer library(Wolf et al. 2020) as our encoder. 22Table 1shows the hyperparameter details for training the neural models for relation classification (SpanBERT) and both relation and explainability classification (Unsupervised Rationale and our approach). Note that we relied mostly on the default hyperparameter values from SpanBERT, but used a larger number of epochs with a smaller learning rate to fine-tune the additional explainability component. The Unsupervised Rationale method was tuned for relation classification, which boosted its RC performance(Tables 2 and 3), but negatively impacted its explainability power(Tables 4 and 5).Some of the explainability baselines do not have hyper parameters, including: attention, saliency mapping, greedy adding, and all words in between. For SHAP, we use all default settings from the API provided by the authors at: https://shap. readthedocs.io/en/latest/index.html. For LIME, the number of samples we used is 2000. And for CXPlain, the explanation model we use is a 2-layers RNN model, with learning rate of 0.001, dropout rate of 0.2, and trained for 2 epochs.
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Exploring various knowledge in relation extraction. Guodong Zhou, Jian Su, Jie Zhang, Min Zhang, Proceedings of the 43rd Annual Meeting of the Association for Computational Linguistics (ACL'05). the 43rd Annual Meeting of the Association for Computational Linguistics (ACL'05)Ann Arbor, MichiganAssociation for Computational LinguisticsZhou, GuoDong, Jian Su, Jie Zhang, and Min Zhang. 2005. Exploring various knowledge in relation extraction. In Proceedings of the 43rd Annual Meeting of the Association for Computational Linguistics (ACL'05), pages 427-434, Association for Computational Linguistics, Ann Arbor, Michigan.
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"Super-Resolving Face Image by Facial Parsing Information",
"Super-Resolving Face Image by Facial Parsing Information"
] | [
"Journal Of L A T E X Class ",
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] | [] | [] | Face super-resolution is a technology that transforms a low-resolution face image into the corresponding highresolution one. In this paper, we build a novel parsing map guided face super-resolution network which extracts the face prior (i.e., parsing map) directly from low-resolution face image for the following utilization. To exploit the extracted prior fully, a parsing map attention fusion block is carefully designed, which can not only effectively explore the information of parsing map, but also combines powerful attention mechanism. Moreover, in light of that high-resolution features contain more precise spatial information while low-resolution features provide strong contextual information, we hope to maintain and utilize these complementary information. To achieve this goal, we develop a multi-scale refine block to maintain spatial and contextual information and take advantage of multi-scale features to refine the feature representations. Experimental results demonstrate that our method outperforms the state-of-the-arts in terms of quantitative metrics and visual quality. The source codes will be available at https://github.com/wcy-cs/FishFSRNet. | 10.1109/tbiom.2023.3264223 | [
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Super-Resolving Face Image by Facial Parsing Information
AUGUST 2015 1
Journal Of L A T E X Class
Files
Super-Resolving Face Image by Facial Parsing Information
148AUGUST 2015 1Index Terms-Face hallucinationface super-resolutionfacial priormulti-scaleparsing map
Face super-resolution is a technology that transforms a low-resolution face image into the corresponding highresolution one. In this paper, we build a novel parsing map guided face super-resolution network which extracts the face prior (i.e., parsing map) directly from low-resolution face image for the following utilization. To exploit the extracted prior fully, a parsing map attention fusion block is carefully designed, which can not only effectively explore the information of parsing map, but also combines powerful attention mechanism. Moreover, in light of that high-resolution features contain more precise spatial information while low-resolution features provide strong contextual information, we hope to maintain and utilize these complementary information. To achieve this goal, we develop a multi-scale refine block to maintain spatial and contextual information and take advantage of multi-scale features to refine the feature representations. Experimental results demonstrate that our method outperforms the state-of-the-arts in terms of quantitative metrics and visual quality. The source codes will be available at https://github.com/wcy-cs/FishFSRNet.
I. INTRODUCTION
F ACE super-resolution (FSR), also known as face hallucination, aims to recover a high-resolution (HR) face from a low-resolution (LR) face. In real-world scenarios, due to the limitations of low-cost cameras and the influence of imaging conditions, the captured face images are always in low resolution, which not only provides the user poor visual perception, but also has adverse effects on face-related tasks, such as face attribute analysis [1], face detection, face recognition [2], etc. Thus, FSR has a wide range of application and drawn increasingly attention in recent years.
In the literature, Baker and Kanada [3] propose face hallucination for the first time and recover HR face by searching local features from training set. After that, more and more representative methods are developed. In the early stage, researchers mainly design shallow learning-based methods by exploring the energy of local linear embedding [4], eigentransformation [5], principal component analysis [6] and others. Due to the insufficient representation capability of these methods, they are hard to generate an outstanding HR face image, especially when the upscale factor is large, such as ×8, ×16.
Recently, motivated by the success of deep learning, a variety of deep learning-based FSR methods are proposed and have achieved great breakthrough. In the early stage, Zhou et al. [7] develop the first convolution neural network-based FSR method. Then, Yu et al. [8] develop URDGN, which is a generative adversarial network-based FSR method. In [9], [10], inspired by human perception, the researchers combine deep learning and reinforcement learning to recover HR progressively. In [11], WSRNet is built to recover faces in wavelet domain. The work of [12] develops a suppression module to encode semantic information for FSR. Chen et al. [13] build a facial attention for learning local details. Recently, SISN [14] designs an internal-feature split attention for capturing internal correlation-ship among intermediate features and then improving the quality of face images.
Human face is a highly structural object, and the face image has its inherited characteristics, such as facial heatmap, facial parsing map, and others, which can improve FSR performance. Then, many methods propose to use the priors to guide the reconstruction process. Zhu et al. [15] design a two-branch network for FSR and face correspondence field estimation, respectively. Super-FAN [16] first recovers a super-resolved face and then estimates the heatmaps of super-resolved faces. To utilize the heatmap, Super-FAN forces the heatmap of super-resolved face to be close to the one of HR face, which is called heatmap loss. However, heatmap loss only works in the training phase, but does not participate in the inference phase of FSR. To solve this problem, some methods have been proposed to insert the heatmap estimation into the superresolution network. The approach of [17] first generates the intermediate features and then estimates the heatmaps from the intermediate features, and then fuses the estimated heatmaps and intermediate features for the following FSR procedure. FSRNet [18] recovers a coarse super-resolved face, then estimates face-specific information which is used to assist the next fine reconstruction. Ma et al. [19] develop a deep FSR model with iterative collaboration (DIC). It iteratively performs FSR and prior estimation to facilitate two tasks each other.
Although these prior-guided FSR methods have achieved impressive performance, the following concerns still need to be carefully considered: i) On the one hand, methods using priorbased loss like [16], [20] Considering the shortcomings mentioned-above, we propose a new parsing map attention fusion network for face hallucination. Our method is a two-step network, which is shown in Fig. 1. To address the first problem, we develop the ParsingNet to extract parsing map directly from LR instead of the intermediate results in [17]- [19]. This change can avoid the accumulation of error influenced by inaccurate intermediate results. Then, we recover the final super-resolved face images with the extracted parsing map and LR input by a fishshape network, FishFSRNet. For capturing and utilizing multiscale features fully, we develop a multi-scale refine block (MSRB) that reserves multi-scale features from the previous layers and uses the reserved features to refine the current features. In addition, we build a parsing map attention fusion block (PAFB) which combines the effectiveness of attention mechanism and parsing map for recovering the important details and face contours.
The main contributions of the proposed method are summarized as follows:
• We propose a two-step deep face hallucination method which first builds ParsingNet to directly estimate facial parsing map from LR to avoid the wrong information caused by the intermediate results, and then feeds the parsing map and LR into FishFSRNet to recover HR face. • We develop a multi-scale refine block (MSRB) to reserve previous features at different scales and refine the current features. In this way, the information of all resolution of face images can be well captured and utilized. • We develop a parsing map attention fusion block (PAFB) that can not only exploit the channel and spatial correlation of features but also make full use of parsing map. • Our method achieves the state-of-the-art performance in multiple upscale factors (e.g., ×4, ×8, ×16). Extensive experiments demonstrate the superiority of our method over existing deep learning-based FSR methods in terms of visual and quantitative results. This work is an extension of our previous work of [21]. The essential improvements over our previous work include: i) We have provided a more comprehensive review about related work; ii) We have developed a much more powerful network architecture by exploiting rich features. Considering that features in different resolutions have different information, we have improved the network architecture and designed a multiscale refine block to maintain and utilize multi-resolution features; iii) We have developed a novel parsing map attention fusion block which can combine the potential of parsing map and attention mechanism and make full use of them; iv) We have conducted more comprehensive experiments and ablation studies to verify the effectiveness of our method.
II. RELATED WORK
In the following, we will introduce the related work of FSR, including shallow learning and deep learning-based methods.
A. Shallow Learning-based FSR
Face hallucination is first proposed by Baker and Kanade [3] in 2000, and they enhance resolution by searching a similar structure from trainset. Since then, face hallucination has attracted increasingly attention, and a series of works have been proposed. Liu et al. [22] design a two-step FSR method that first uses global linear model to recover a coarse result, and then designs a patch-based nonparametric Markov network for compensating high-frequency details. Inspired by locally linear embedding, Chang et al. [4] recover a high-quality face with neighbor embedding. Then, Wang et al. [5] take advantage of eigentransformation to recover high quality faces. The work of [6] utilizes kernel principal component analysis prior model to extract valuable information for boosting face reconstruction. Except that, there are still many methods utilizing convex optimization [23], canonical correlation analysis [24], Bayesian approach [25], local structure prior [26], kernel regression [27], etc. to recover a high quality face. However, these methods fail to restore high-quality faces when the upsampling factor is large (i.e., ×8, ×16).
B. Deep Learning-based FSR 1) General FSR Methods: Recently, with the booming development of deep learning in computer vision [28], [29], deep learning-based face hallucination methods have achieved great breakthroughs [30]. In the early stage, scholars mainly design efficient network structures for FSR without considering face specificity and these methods are called general FSR methods. Zhou et al. [7] make the first attempt to super-resolve face images with convolution neural network. In detail, URDGN [8] constructs network based on generative adversarial network. Considering that LR can be unaligned, TDN [31], [32] inserts spatial transformer networks [33] into the network to perform the alignment. To enhance robustness of the model, Yu et al. [34] build transformative discriminative autoencoders for noisy and unaligned face repair. Considering that the LR face images may be non-frontal, the work of [35] proposes to jointly perform FSR and face frontalization. [2] and [36] develop effective networks to recover face images degraded by atmospheric turbulence. After that, inspired by that human perception process in which human starts from whole images and then explore a sequence of regions with an attention shifting mechanism, Cao et al. [9], [10] combine reinforcement learning to recover faces gradually. Jiang et al. [37] first recovers a smooth noise-free intermediate result, then takes advantage of high-quality trainset and facial landmarks to compensate high-frequency details for the smooth intermediate results. Different from the aforementioned methods that recover faces in image domain, Huang et al. [11], [38] find that wavelet transformation can depict the textural and contextual information of the images, thus they super-resolve faces in the wavelet domain and propose WSRNet. Jiang et al. [39] utilizes ensemble learning to recover HR faces. Recently, the work in [40] develops a supervised pixel-wise GAN to improve FSR performance. Dou et al. [41] apply PCA to recover face images progressively. [42] develops a compact network and introduces global learning strategy for face hallucination. Chen et al. [13] design a facial attention to recover local details well while Lu et al. [14] develop internal-feature split feature for capturing facial semantic information.
2) Prior-guided FSR Methods: Instead of focusing on network architecture design, prior-guided FSR methods always utilize facial prior (i.e., geometric prior, reference prior, dictionary prior, codebook prior, generative prior, etc.) to improve FSR performance. We introduce them in detail as follows.
Geometric prior-based methods: Since face image is a highly structured object and has face-specific geometric prior (including facial heatmaps, facial landmarks, facial parsing map, etc.) which is helpful for facial detail reconstruction, geometric prior-based FSR methods are proposed. At first, CBN [15] recovers face images and estimates face-specific information (dense correspondence field) progressively. However, CBN needs complex and extensive preprocess. Later on, Super-FAN [16] constrains the facial heatmaps of superresolved results should be close to the ones of HR faces. FSRNet [18] builds a coarse FSR network to produce a coarse super-resolved result which is used to estimate facial prior information and extract features, finally the extracted prior and features are fed into fine FSR network, generating the final results. Yu et al. [17] embed facial heatmap estimation into FSR network. Ma et al. [19] iteratively perform FSR and estimate heatmaps which are used for performing FSR in the next iteration (DIC). In addition, DIC develops an attention fusion module to utilize facial prior. Different from these methods, JASRNet [43] has a share encoder for FSR and facial prior estimation to extract features with maximum information. After that, Li et al. [44] design a structure enhancement network to estimate and utilize facial boundaries. HCRF [45] combines random forest to recover different facial components effectively. Yu et al. [46] build a semantic-driven face hallucination network and develop an improved residual block to combine facial prior. Wang et al. [47] design a prior distillation framework to recover face images. Instead of using 2D prior, Hu et al. [48] focus on the utilization of 3D prior. Among these works, FSRNet [18] is the most related to our method. Here we analyze the difference between them. Firstly, FSRNet extracts prior from coarse FSR result and utilizes it by concatenation while our method extracts prior from LR face, designs PAFB to finely fuse prior at multiple scale and builds MSRB to utilize complementary multi-scale information.
Reference prior-based methods: In addition to explore geometric prior, some researchers focus on reference prior extracted from high-quality reference face image. Originally, researchers use high-quality face images with the same identity with LR face as reference. The works of [49], [50] first align the reference with LR and then extract identity-aware features from reference to improve face image quality. In case of having many reference images with the same identity, ASFFNet [51] designs guidance selection module to select the best reference which has the most similar pose, and then develops adaptive feature fusion block to combine reference information.
Dictionary prior-based methods: Considering that different people may have similar facial components, facial component dictionary constructed from a high-quality face dataset is used to guide FSR. Li et al. [52] first extract feature from the entire dataset and then crop the facial component and then clustering K classes to form dictionary. After that, it designs dictionary transfer module to transfer high-frequency features from dictionary to LR features.
Codebook prior-based methods: Apart from facial component dictionary which can represent facial structure explicitly, discrete codebook prior presenting facial information implicitly is also utilized [53]- [56]. To be specific, the codebook P !"
×8
Conv
ResBlock ResBlock Conv is constructed by VQVAE [57] with high-quality face dataset. Zhao et al. [56] map LR face into code space and then find the most similar code in code book as reference code, then replace the LR code with reference code to reconstruct high-quality face by reconstruction branch while Gu et al. [55] introduce an additional texture branch to capture texture information. Instead of replacing the LR code with reference code, Wang et al. [54] use multi-head self-attention to fuse LR code and reference code. However, codebook is constructed by high-quality face image while LR code is from LR face image, directly finding the most similar code between them is unreasonable. In view of this, the work of [53] designs a transformer module to predict code sequence.
Generative prior-based methods: Rather than learning a codebook, the works of [58]- [62] pretrain a generative model with a high-quality face dataset and use the generator as generative prior. Specially, PULSE [62] directly uses the generator to generate SR result while [58] builds an encoder to obtain latent code from LR and feeds the latent code to the generator to generate SR result. Different from them, method in [60], [61] only use the synthesis network of the generator to assist FSR. Zhu et al. [59] combine generative prior and geometric prior to perform blind face restoration.
III. METHODS
Face super-resolution (FSR) aims to recover a highresolution (HR) face from the corresponding low-resolution (LR) one, in which face-specific prior knowledge is expected to boost the performance. To achieve this goal, we develop a two-stage network as illustrated in Fig. 1. Overall, our method first estimates face-specific information i.e., a parsing map, from LR by our ParsingNet, and then feeds the LR and prior into the super-resolution network (FishFSRNet) to recover the HR one. In FishFSRNet, we develop a multi-scale refine block (MSRB) which reserves and utilizes multi-scale features to refines the current features. Moreover, we build a novel parsing map attention fusion block (PAFB) to explore and combine the potential of parsing map and attention mechanism.
A. ParsingNet
Different from general images which have various objects and landscapes, one aligned face image always has a face in the center and has face-specific information. Thus, FSR methods tend to explore the utilization of face-specific information, including facial landmarks, facial parsing map, facial heatmaps and others. In this paper, we explore the potential of facial parsing map in FSR. In this section, we propose a parsing map prediction network named ParsingNet, which predicts a facial parsing map directly from LR. Considering that the LR face images are low-quality and extracting accurate facial parsing map from LR is very difficult, we simplify the task and decrease the difficulty of the task. Specifically, a simple parsing map which is a 0-1 matrix where the skin is 1 and the facial components and others are 0, is used as the facial prior. Then we feed the LR face images into the ParsingNet, and obtain the parsing maps, which can be formulated as
P = f ParsingNet (I LR ),(1)
where f ParsingNet is the function of our ParsingNet, and P is the parsing map estimated by the ParsingNet. As shown in Fig. 3, our ParsingNet is comprised of two convolutional layers and eight Resblocks [63]. To force our ParsingNet to estimate accurate facial parsing map, we design a parsing map loss, which can be defined as
L ParsingNet = P − Y 1 ,(2)
where Y is the ground truth of facial parsing map.
B. FishFSRNet
After introducing ParsingNet, we come to the superresolution network FishFSRNet. HR features contain spatiallyprecise information while LR features have rich contextual information. Due to their complementarity, they should be all kept and used for refining other features. In view of this, we aim to design a network that can extract and utilize multi-scale features. To achieve this goal, we develop our FishFSRNet which consists of a feature extraction layer, fish head, fish body, fish tail and a reconstruction layer, as shown in Fig. 2. In our FishFSRNet, the LR faces are first progressively upsampled, and then downsampled and finally upsampled to generate the final super-resolved face images. In this pattern, our FishFSRNet can obtain multi-scale feature at different layers and utilize these features for boosting FSR.
Specifically, given the LR face image I LR , the FishFSRNet extracts features from it by the feature extraction layer which is a 3 × 3 convolutional layer,
F 0 = f Feature Extraction (I LR ),(3)
where f Feature Extraction is the function of the feature extraction layer, and F 0 is output feature. Then, extracted feature is fed into the fish head. To extract multi-scale features and increase the receptive field, our fish head upsamples the F 0 three times. Moreover, to promote the utilization of the multiscale features, the result of each upsampling is preserved and passed to the following layers. Thus, the process of fish head can be formulated as:
F 1 , F 2 , F 3 , F 4 = f Fish Head (F 0 , P ),(4)
where f Fish Head is the function of fish head, and F 1 , F 2 , F 3 are the multi-scale features that are preserved, and F 4 is the final output of fish head.
• To be specific, F 0 is first upsampled to obtain F 1 which is preserved for the following layers,
F 1 = f Up (F 0 ),(5)
where f Up denotes an upsampling module which is comprised of a 3×3 convolutional layer and a ×2 pixelshuffle [64] layer. Then, F 1 and parsing map P are passed through two cascaded PAFBs to explore facial prior information, and the results are upsampled again to acquire the feature in ×4 resolution,
F 2 = f Up (f PAFBs (F 1 , P )),(6)
where F 2 is the output by ×4 upsampling and it is preserved for the following utilization, and f PAFBs corresponds to two cascaded PAFBs. Then, with the parsing map, F 2 is processed and upsampled again,
F 3 = f Up (f PAFBs (F 2 , P )),(7)
where F 3 is the output upscaled by ×8. Then, the last PAFBs in the fish head are applied,
F 4 = f PAFBs (F 3 , P ),(8)
where F 4 is the final output of the fish head. Then, come to the fish body. In contrast to the fish head, fish body downsamples the features three times to generate multi-scale features, and inserts a multi-scale refine block (MSRB) before the first downsampling module, and every downsampling module is followed by two PAFBs. The MSRB aims to utilize multi-scale features (F 1 , F 2 , F 3 ) passed from the previous fish head for refining the current features. The process of the fish body can be formulated as:
F 5 , F 7 , F 9 , F 10 = f Fish Body (F 4 , F 1 , F 2 , F 3 , P ),(9)
where f Fish Body is the function of fish body, and F 5 , F 7 , F 9 are the multi-scale features in different resolutions and F 10 is the final output. The formulation of fish body is introduced in detail in the following.
• Firstly, MSRB in fish body utilizes the previous multiscale features to refine the current features for exploring the complementary information in high-and low-resolution features, and downsamples the refinement result,
F 5 = f Down (f MSRB (F 4 , F 1 , F 2 , F 3 )),(10)
where f Down is a downsampling module (comprised of a 3×3 convolutional layer and a inverse pixelshuffle [65] layer to perform ×2 downsampling), f MSRB denotes the MSRB module, and F 5 is the output feature refined and downsampled by the MSRB and the downsampling module. Then, two cascaded PAFBs are applied to F 5 ,
F 6 = f PAFBs (F 5 , P ),(11)
obtaining F 6 . After that, the MSRB exploits the multiscale features F 1 , F 2 , F 3 passed from the previous layers to refine the current feature F 6 again,
F 7 = f Down (f MSRB (F 6 , F 1 , F 2 , F 3 )),(12)
where F 7 is feature refined by MSRB. Similar to F 5 , it is also processed by two PAFBs to combine facial prior,
F 8 = f PAFBs (F 7 , P ),(13)
generating F 8 . Then, F 8 is passed to incorporate with previous multi-scale features and be downsampled,
F 9 = f Down (f MSRB (F 8 , F 1 , F 2 , F 3 )),(14)
where F 9 is the refined result. Until now, F 5 , F 7 , F 9 in three different resolutions are obtained and preserved. Finally, F 10 is acquired by PAFBs,
F 10 = f PAFBs (F 9 , P ).(15)
Similar to the fish head, the fish body also preserves the multiscale features generated by the downsampling modules for the utilization in the following layers. These features are fed into the fish tail by skip connection. After that, with features from the fish body, our fish tail progressively upsamples features, generating features in the same resolution with the HR face. Similar to the fish head, our fish tail is also comprised of three upsampling modules, PAFBs and MSRBs. The utilization of previous features in fish tail is also finished by MSRBs. The process can be defined as
F t = f Fish Tail (F 10 , F 9 , F 7 , F 5 , P ),(16)
where f Fish Tail denotes our fish tail and F t is its output.
• In detail, the process of fish tail can be formulated into three sequential MSRB-Down-PAFBs operations,
F 11 = f PAFBs (f Down (f MSRB (F 10 , F 9 , F 7 , F 5 )), P ),(17)F 12 = f PAFBs (f Down (f MSRB (F 11 , F 9 , F 7 , F 5 )), P ),(18)F t = f PAFBs (f Down (f MSRB (F 12 , F 9 , F 7 , F 5 )), P ),(19)
where F 11 and F 12 are the intermediate features. Finally, we generate the recovered HR face through the reconstruction layer:
I SR = f Reconstruction (F t ),(20)
where f Reconstruction denotes the function of the reconstruction layer (a 3×3 convolutional layer), and I SR is recovered face.
To measure the pixel consistency between the superresolved results and the ground truth, we use a pixel loss that measures the pixel distance, which can be expressed as
L FishFSRNet = I SR − I HR 1 ,(21)
where I HR is the ground truth. To supervise the model, we choose L1 loss as our loss function.
C. Multi-scale Refine Block (MSRB)
HR features tend to contain more spatially-precise information while LR features have rich contextual information. Since the information contained by HR and LR features is complementary, features in different resolutions should be all utilized. In light of this, we first pass all previous features at different scales to every downsampling module in the fish body and every upsampling module in the fish tail. Then, we develop MSRB and insert it before every downsampling module in fish body as well as the upsampling module in fish tail to refine the current features with previous feature of different resolutions. In the following, we will introduce our MSRB and show the architecture of MSRB in Fig. 4. Taking the upscale factor ×8 as an example. The input of MSRB includes four parts: the current feature F , previous features F i , F j and F k . First, we feed F with every previous feature into three refine blocks respectively to refine F . After that, we concatenate the refined outputs of refine blocks. Considering that features in LR contain rich contextual information and features in HR provide spatially-precise information, refined features should be treated discriminatively. Thus, we further feed the concatenation result into a channel attention mechanism to capture information along channel dimension. Finally, the sum of F and the output of the channel attention is calculated, which is the final fused and refined feature.
Refine
Projection Function
Refine block: Inspired by the back-projection algorithm [66], we develop a refine block which is used to utilize the multi-scale features from the previous stage and remedy the missing information of the current feature. In essence, our refine block is designed to enhance and refine the current features with an error feedback mechanism. As shown in Fig. 4, for every previous feature (here we take F k as an example), our refine block first resizes (implemented by the nearest interpolation or a convolution layer) it to generate F kc that shares the same resolution with F . Then, the error feedback mechanism can be implemented.
• The first step is to compute the difference F ke between the current feature F and the previous feature F kc ,
F ke = F − F kc .(22)
• The second step is to update the current feature F with the projected error features:
F ko = f P (F ke ) + F,(23)
where f P is the function of projection implemented by two convolution layers and F ko is a refined feature.
D. Parsing Map Attention Fusion Block (PAFB)
Recently, attention mechanism, mainly including channel attention mechanism or spatial attention mechanism, is widely used in the image super-resolution field and has achieved a great breakthrough, such as the work in [67], [68]. It is generally accepted that channel and spatial attention mechanisms can extract effective features and boost image super-resolution. We assume that the face image is a special case of general image, thus the technique that can promote general image super-resolution can also benefit face restoration. Depending on this assumption, we introduce channel and spatial attention mechanism into FSR. In addition, face is a high-structured object which has its own face-specific information. Thus, we also take advantage of face-specific prior knowledge (i.e., parsing map) to improve face restoration. Combining facespecific information and an effective attention mechanism, we develop a parsing map attention fusion block (PAFB) which consists of an attention branch and a parsing map branch as shown in Fig. 5. Our proposed PAFB can not only explore the information dependency along the spatial and attention dimensions, but also make full use of the parsing map. Attention Branch: Motivated by the powerful representative ability of attention mechanism in general image superresolution, we develop an attention branch to explore interchannel and spatial relationship. To be specific, our attention branch first employs two convolution layers to extract features, and then feed extracted features into parallel channel and spatial attentions to capture inter-channel and spatial dependencies. To effectively fuse the channel and spatial mechanisms, the attention branch directly concatenates the outputs of them and applies a fusion convolution layer to further combine them, generating F A .
Parsing Map Branch: Given that a face has its own face-specific prior knowledge that can boost FSR, we further develop a parsing map branch. First, we interpolate P into the same size with F by nearest interpolation. Since F and parsing map P are in different domains and can provide different information, two convolution layers are applied on them to project them into a similar domain. After that, concatenation followed by a fusion convolution layer is applied on the projected features to simply fuse them, generating F P .
Since the attention branch pays more attention to the interchannel and spatial relationship while the parsing map branch lays stress on the exploration of prior, we further combine them to boost the cooperation between them and promote the performance of FSR. Specifically, concatenation followed by a convolution layer is used to fuse them adaptively. The effectiveness of the two branches is verified in Section IV.
E. Architecture Novelty of FishFSRNet
The architecture design novelty of FishFSRNet is mainly reflected in MSRB and PAFB which have their specific characteristics. i) MSRB: In light of that high-resolution features contain more precise spatial information while low-resolution features provide strong contextual information, we hope to maintain and utilize these complementary information. Thus, MSRB is designed. It not only fuses features from previous layers, but also exploits the potential of features in different resolutions. In addition, MSRB has a refinement mechanism to utilize the multi-scale features to refine the current features. ii) PAFB: in PAFB we not only introduce attention mechanism to FSR but also fuse them with the face-specific information (i.e., paring map). Specifically, on the one hand, we use the spatial and channel attention to capture spatial-and channelwise information implicitly. On the other hand, we embed the face parsing map to explore facial structure information explicitly. By the collaboration between them, the proposed method can achieve a very good performance.
IV. EXPERIMENTS
A. Datasets and Metrics
We conduct extensive experiments on two widely used datasets: CelebA [69] and Helen [70]. In our ablation study, we adopt CelebA to verify the effectiveness of every component. In quantitative and qualitative comparisons with state-of-theart methods, both CelebA and Helen are used.
CelebA: CelebA is a a large-scale face dataset, and contains more than 200,000 faces in large pose diversity and background clutter, providing 5 facial landmarks, 40 attributes and identity information. For CelebA dataset, we use 168,854 faces as training set, and 100 faces as validation set, and 1,000 faces as testing set, following [19].
Helen: Helen is a face dataset that contains 2,330 faces under a broad range of appearance variation, providing 194 landmarks. For Helen dataset, we following the setting of [19] and use 2,005 faces for training (wherein 1,955 faces as the training set and 50 faces as validation set), and the remaining 50 faces as the testing set.
Peak Signal-to-Noise Ratio (PSNR) and structural similarity (SSIM) [71] indices are introduced as metrics for evaluation. They are computed on the Y channel of YCbCr space.
B. Implementation details
Training setting Following DIC [19], we first preprocess the face images. For both datasets, we use 68 landmarks extracted by OpenFace [72]- [74]. Depending on the face region, we first crop the faces and resize them into 128×128 as ground truth HR faces, and then resize the ground truth into 64×64, 32×32, and 16×16 as corresponding LR faces with upscaling factor ×4, ×8, and ×16 respectively. For facial parsing maps, we adopt pretrained BiSeNet [75] to extract parsing map of different facial components from HR and fuse. For ParsingNet with different scales, we downsamples the extracted parsing map with the corresponding scale to obtain the ground truth parsing map of ParsingNet. Our experiments are implemented with the popular toolbox Pytorch [76]. We first train the ParsingNet, and then with the estimated parsing map, we train the super-resolution network FishFSRNet. Both ParsingNet and FishFSRNet are optimized by ADAM with β 1 = 0.9, β 2 = 0.99 and = 1e − 8. The learning rate is set as 1e − 4 in the training phase, and the mini-batch size of the model is set as 8. Note that there are 2, 3, 4 upsampling modules in fish head and fish tail, 2, 3, 4 downsampling modules in fish body for ×4, ×8, and ×16 FSR respectively, and every upsampling or downsampling module is followed by two cascaded PAFBs.
C. Comparisons with State-of-the-Art 1) Qualitative and Quantitative Comparisons on CelebA [69] and Helen [70]: To illustrate the superiority of our method in terms of quantitative results, we choose several state-of-the-art methods for comparisons and conduct experiments on CelebA [69] and Helen [70] with multiple upscale factors (×4, ×8 and ×16). The comparison methods include two representative image super-resolution methods SRCNN [78] and VDSR [79], three general FSR methods URDGN [8], WSRNet [11] and recently proposed SISN [14] without using any face-specific prior knowledge, and three recently proposed face-specific prior-guided FSR methods Super-FAN [16], FSRNet [18] and DIC [19]. We also introduce the Bicubic interpolation as a baseline comparison method. For a fair comparison, we retrain all the models on our datasets. The quantitative evaluation in terms of PSNR and SSIM metrics are depicted in Table I, and the visual results of different methods on ×4, ×8 and ×16 are presented in Fig. 6, Fig. 7 and Fig. 8, respectively. In these three figures, we list the LR and ground truth HR in the first and last columns, respectively, and the middle columns are the results of different methods. Here, we present the detailed comparison results.
SRCNN [78] is the first deep learning-based image superresolution method, which is too shallow to recover face images well. The quantitative results of SRCNN are only better than these of Bicubic. URDGN [8] is a generative adversarial network-based FSR method, which pays more attention to recover realistic face images. Due to the property of the generative adversarial network, URDGN is inferior in quantitative metrics and always generates face images with unreal artifacts and missing facial details in visual quality, as shown in Fig. 7. VDSR [79] is more powerful than SRCNN with a much deeper network. However, it fails to recover face images well. WSRNet [11] is a wavelet-based FSR method that super-resolves face images in the wavelet domain instead of image domain. Since the wavelet domain can depict the contextual information of the images, WSRNet can recover some high-frequency details. However, WSRNet ignores the utilization of face-specific prior knowledge, resulting in the loss of important facial details. SISN [14] builds internalfeature split attention to capture inter-feature information for recovering facial semantic texture. As shown in Fig. 7, SISN can recover global facial structure well but cannot reconstruct local facial details, especially on facial components, such as the mouth and eyes. This phenomenon is due to the fact that SISN does not exploit the function of the facial prior. Super-FAN [16] measures the distance between the heatmaps of SR and HR to maintain the structural consistency, named heatmap loss. However, the heatmap loss is only used in the training phase, and face-specific prior knowledge is not used in the testing phase, leading to unfavorable faces, as shown in Fig. 7. As shown in Fig. 7, FSRNet generates clear faces for ×8 FSR, but there are still many artifacts existing, especially on the location of eyes and mouth. When applied for ×16 FSR on CelebA [69], the faces generated by FSRNet are too smooth in Fig. 8. However, as shown in Fig. 9 Our method In this paper, we propose a two-stage framework, which first estimates the facial parsing map and then recovers faces under the guidance of the parsing map. Our proposed framework consists of two subnetworks, i.e., Pars-ingNet and FishFSRNet. The ParsingNet attempts to estimate a facial parsing map directly from LR. This pattern can avoid the bad influence caused by the wrong intermediate results. Then, FishFSRNet enhances the quality of LR with the guidance of the parsing map. Based on the previous FishFSRNet, we further develop a multi-scale refine block to reserve previous multi-scale features and refine the current features. In addition, we build a parsing map attention fusion block which can not only explore inter-channel and interspatial relationships, but also make the best use of face-specific prior knowledge. Towards ×4 FSR, our methods can repair a clear face image with sharp and realistic details on facial components. As shown in Fig. 7, our method can recover rich facial details and holistic facial structure. Although ×16 FSR is difficult, our method can still recover visually pleasing face images on CelebA [69], as shown in Fig. 8. Compared to FSRNet [18], DIC [19] and SISN [14] that fail to recover facial components, our method can recover much clearer facial outline and components even in Helen [70] with upscale factor ×16, as shown in Fig. 9. In terms of quantitative metrics, as indicated in Table I, our method performs better than other methods on two datasets and all upscale factors. For example, our method outperforms the second-best methods with a large margin of 0.25 dB on Helen [70] with upscale factor ×16. Compared to these general FSR methods [8], [11], [14], [78], our method takes face-specific prior knowledge into account and achieves better results. Compared to previous prior-guided FSR methods [16], [18], [19], [77], our method extracts the parsing map from LR and inserts the parsing map into every proposed PAFB to make full use of the parsing map, resulting in superior performance.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)(a) (b) (c) (d) (e)(a) (b) (c) (d) (e)
2) Qualitative Comparisons on Real-world Images: The above experiments are based on simulated datasets, which are significantly different from real-world LR face images. Thus, we apply our method to some real-world faces with unknown degradation. The results are shown in Fig. 10. Since the ground truth is not available, we only present the super-resolution results. For comparison, we present the generated results of FSRNet [18] and SISN [14] in Fig. 10. Faces recovered by FSRNet [18] have artificial rings and are smooth and blurred. The faces generated by SISN [14] are visually pleasing, but have blurred details on the facial components. In contrast to these methods, our method can recover visually pleasing faces and generate clear and realistic high-frequency details and textures, especially on facial components.
3) Comparison on Face Recognition:
To further verify the superiority of the proposed method, we compare the performance of different FSR methods on face recognition. To be specific, we first obtain the representative vector of the SR and HR face images by pretrained DeepFace [80], and then calculate the cosine similarity of these vectors. The cosine similarity is used to evaluate the face recognition performance of different methods. The results are presented in Table II. As shown in Table II, our method achieves the highest cosine similarity in most cases except on ×16 Helen [70]. On ×16 Helen, DIC performs best and our method achieves secondbest performance. In general, our method can not only improve FSR performance, but also boost face recognition.
D. Ablation Study
In this section, we analyze and verify the effectiveness of our proposed MSRB and PAFB. To have a clear comparison, we first replace our proposed MSRB and PAFB with concatenation and ResBlock, respectively, and we name this model Model 1. Based on the Model 1, we further conduct a set of experiments and present the quantitative results in Table III.
The Effectiveness of MSRB: Compared with our previous work [21], we develop a MSRB to boost the feature representation ability. To verify the effectiveness of it, we directly replace the concatenation in Model 1 with our proposed MSRB, and name it Model 2. Then, we compare the performance of Model 1 and Model 2. From the Table III, it is obvious that the performance of Model 2 is much better than that of Model 1, which proves that our MSRB can boost the FSR effectively. From the prospective of visual quality, we present the results of Model 1 and Model 2 in Fig. 11. The visual FSR results of Model 2 is globally clearer than the ones of Model 1. However, our proposed MSRB is not tailored for faces and leaves out face-specific prior knowledge. Thus, the super-resolved results on facial components is not sharp enough. The Effectiveness of PAFB: In Section III-D, we have introduced our PAFB which contains two branches (i.e., an attention branch and a parsing map branch). Here, we conduct another series of experiments to validate the effectiveness of our PAFB. First, we verify the effectiveness of two branches respectively. We only reserve the attention branch and remove the parsing map branch (it should be noted that the concatenation followed by a convolution layer at the end of PAFB is also discarded), and this model is called Model 3. At this time, Model 3 only explores the inter-channel and inter-spatial relationship without considering face-specific prior knowledge. In contrast to Model 3, we remove the attention branch and only reverse the parsing map branch in PAFB, which is Model 4. As shown in Table III, with the parsing map providing facial structure information, Model 4 can improve the performance of FSR, verifying the effectiveness of face-specific knowledge. In addition, we conduct experiments to analyze the function of different attention mechanisms in FSR when the parsing map branch is used. Specially, we remove the channel attention (CA) or spatial attention (SA) from the PAFB, and introduce the edited PAFB to Model 1, generating Model 5 and Model 6 respectively. From Table III, we can find that both SA and CA can promote FSR under the condition that the parsing map is explored. We analyze that facial parsing map provides facial structure information which is a semantic-level guidance, while the attention mechanism aims to explore relationship along spatial and channel dimensions, which can be viewed as pixel-level information. Thus, they are complementary, and the collaboration between them can enhance the representation ability of the network, promoting FSR. Finally, we add MSRB to Model 7, generating Model 8. Clearly, Model 8 achieves the best performance in terms of PSNR and SSIM. In terms of visual quality, the results of Model 1, Model 2 and Model 8 are all presented in Fig. 11. Due to the consideration of facespecific prior knowledge, Model 8 not only recovers more visually pleasing face images than Model 1 and Model 2 globally, but also reconstructs much sharper details on the facial components such as eyes and mouths. In summary, the effectiveness of every proposed component is verified by the above experiments.
(a) (b) (c) (d) (e)(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
The Effectiveness of Parsing Map: In addition, we also compare the visual results of different parsing maps to present the influence of the parsing maps. As shown in Fig. 12, given an LR face, we feed the LR with a right parsing map P shown in Fig. 12(c), and then our model generates I SR (shown in Fig. 12(d)) which is very close to the ground truth. When we give the model a wrong or mismatched parsing map P shown in Fig. 12(a), we obtain the I SR 1 (shown in Fig. 12(b)) which has many artifacts and distorted facial components. Clearly, a wrong parsing map would lead to distorted face images, while a correct parsing map can boost the FSR. Except that, the results also verify that our model makes full use of face-specific prior knowledge. Considering the huge differences between the wrong and right parsing maps, we give some disturbance to the right parsing and conduct another experiment. To be specific, we also rotate the right parsing map by 15, 30, 45, 60 degrees, and then obtain the super-resolved results with these disturbed parsing maps. As shown in Fig. 13, the greater the disturbance, the worse the visual quality.
In addition, we present the face parsing accuracy of our ParsingNet in Table IV and present a visual comparison between our final model and our model without the parsing map in Fig. 14 to further verify the effectiveness of the parsing map. To be specific, we remove the parsing map in the parsing map branch and keep parameter similar, and train the revised model. Then, we show the visual comparison between these two models in Fig. 14. We can see that our model with the parsing map can recover more visually pleasing face images than the model without the parsing map. Especially in the regions of facial components, the face images hallucinated by our method have much clearer texture and details. By these experiments, we can conclude that the facial parsing map plays an important role in FSR.
V. CONCLUSION This paper proposes a parsing map attention fusion network for face hallucination. In particular, it directly estimates a facial parsing map from LR, and then recovers the final face images with the extracted parsing map. Based on this, we further develop a novel multi-scale refine block to reserve the previous features of different resolutions and refine the current features, which can significantly improve the performance. Except that, we build a parsing map attention fusion block which can not only explore the relationship among channel and spatial dimensions but also exploit the contribution of face-specific prior (parsing map). Finally, the paper conducts massive comparison experiments to verify the effectiveness of every proposed component, and the proposed method can achieve favorable performance against existing methods.
Fig. 1 .
1Corresponding author: Junjun Jiang C. Wang, J. Jiang, Z. Zhong, D. Zhai and X. Liu are with the School of Computer Science and Technology, Harbin Institute of Technology, Harbin 150001, China E-mail: {wangchy02, jiangjunjun, zhwzhong, zhaideming, csxm}@hit.edu.cn. The overall framework of our method.
Fig. 3 .
3The architecture of our proposed ParsingNet.
Fig. 4 .
4The architecture design of our proposed multi-scale refine block.
Fig. 5 .
5The architecture of our parsing map attention fusion block.
Fig. 6 .
6Visual quality comparison of state-of-the-art methods by the scale of ×4. Please zoom in to view the differences. (a): LR; (b): DIC [19]; (c): SISN [14]; (d): Ours; (e): HR.
Fig. 7 .Fig. 8 .
78Visual quality comparison of state-of-the-art methods for several side-face examples selected from Helen [70] (the first four rows) and CelebA [69] (the last four rows) datasets by the scale of ×8. Please zoom in to view the differences. (a): LR; (b): Bicubic; (c): URDGN [8]; (d): WSRNet [11]; (e): Super-FAN [16]; (f): FSRNet [18]; (g): DIC [19]; (h): SISN [14]; (i): Ours; (j): HR. Visual quality comparison of state-of-the-art methods for several face examples selected from CelebA [69] dataset by the scale of ×16. Please zoom in to view the differences. (a): LR; (b): DIC [19]; (c): SISN [14]; (d): PSFRGAN [77]; (e): Ours; (f): HR.
Fig. 9 .
9Visual quality comparison of state-of-the-art methods for several sideface examples selected from Helen [70] dataset by the scale of ×16. Please zoom in to view the differences. (a): LR; (b): DIC [19]; (c): SISN [14]; (d): Ours; (e): HR.
Fig. 10 .
10Visual quality comparison of state-of-the-art methods on the realworld face images. (a): Real LR face images; (b) FSRNet [18]; (c): SISN [14]; (d): Ours; Please zoom in to view the differences.
Fig. 11 .
11Visual comparison results of different models on CelebA [69] by ×8. (a): LR; (b): The super-resolved results of Model 1; (c): The super-resolved result of Model 7; (d): The super-resolved results of Model 8; (e): HR.
Fig. 12 .
12Visual comparison results of different parsing maps on CelebA [69] by ×8. (a): Wrong parsing map; (b): The super-resolved result with the wrong parsing map; (c): Right parsing map; (d): The super-resolved result with the right parsing map; (e): HR.
Fig. 13 .
13Visual comparison results of turbulent parsing maps on CelebA [69] by ×8. (a): Right parsing map; (b): The parsing map rotated by 15 degrees; (c): The parsing map rotated by 30 degrees; (d): The parsing map rotated by 45 degrees; (e): The parsing map rotated by 60 degrees; (f): LR; (g): The super-resolved result with right parsing map; (h): The super-resolved result with parsing map rotated by 15 degrees; (i): The super-resolved result with the parsing map rotated by 30 degrees; (j): The super-resolved result with the parsing map rotated by 45 degrees; (k): The super-resolved result with the parsing map rotated by 60 degrees; (l): HR.
Fig. 14 .
14Visual quality comparison of the model with the parsing map and the model without the parsing map. Please zoom in to view the differences. (a): LR; (b): The results of the model without the parsing map; (c): The results of the model with the parsing map; (d): HR.
only apply loss in the training phase arXiv:2304.02923v1 [cs.CV] 6 Apr 2023 and the prior information does not participate in the inference phase, which cannot fully capture the potential of the prior information. On the other hand, the prior knowledge derived from the intermediate results is directly affected by the quality of intermediate results, which is usually limited, leading to poor and even wrong prior knowledge learned. ii) Existing deep FSR models always ignore the fusion and utilization of multi-scale (or multi-resolution) features. HR features contain spatially-precise information while LR features have rich contextual information. Due to their complementarity, they should be all kept and used for refining other features. However, most methods are not able to take full advantage of this point.Fish Head
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TABLE I QUANTITATIVE
IEVALUATION OF VARIOUS FACE HALLUCINATION METHODS ON CELEBA [69] AND HELEN[70].Method
Venue
CelebA [69]
Helen [70]
Average
×4
×8
×16
×4
×8
×16
PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM
Bicubic
-
27.48 0.8166 23.68 0.6258 20.34 0.4671 28.22 0.8540 23.88 0.6628 20.65 0.5061 24.04 0.6554
SRCNN [78]
TPAMI'15 28.04 0.8369 23.93 0.6348 20.54 0.4672 28.60 0.8678 23.98 0.6670 20.73 0.5088 24.30 0.6638
URDGN [8]
ECCV'16 30.11 0.8844 25.62 0.7261 22.29 0.5786 30.29 0.9019 25.23 0.7205 21.68 0.5452 25.87 0.7261
VDSR [79]
CVPR'16
31.25 0.9056 26.36 0.7605 22.42 0.5942 30.74 0.9014 25.31 0.7266 21.35 0.5417 26.24 0.7383
WSRNet [11]
ICCV'17
30.92 0.9081 26.83 0.7873 23.13 0.6343 30.95 0.9163 26.02 0.7731 22.00 0.5763 26.64 0.7659
Super-FAN [16]
CVPR'18
31.37 0.9054 27.08 0.7841 23.42 0.6515 30.94 0.9099 26.23 0.7656 22.54 0.5991 26.93 0.7693
FSRNet [18]
CVPR'18
31.46 0.9084 26.66 0.7714 23.04 0.6293 30.76 0.9101 25.89 0.7605 22.05 0.5820 26.64 0.7603
DIC [19]
CVPR'20
31.44 0.9091 27.41 0.8022 23.47 0.6573 31.81 0.9269 26.69 0.7953 22.60 0.6122 27.24 0.7838
SISN [14]
MM'21
31.88 0.9157 27.31 0.7978 23.42 0.6743 31.63 0.9245 26.66 0.7920 22.55 0.6037 27.24 0.7847
Ours
-
31.97 0.9170 27.54 0.8072 23.68 0.6784 32.01 0.9292 26.86 0.7984 22.85 0.6346 27.49 0.7917
TABLE II COSINE
IISIMILARITY OF HR AND SR FACE IMAGES RECOVERED BY DIFFERENT METHODS ON CELEBA [69] AND HELEN [70] DATASETS IN FACE RECOGNITION.TABLE III EFFECTIVENESS STUDY OF DIFFERENT PARTS OF THE PROPOSED NETWORK ON CELEBA WITH SCALE ×8. AB AND PMB DENOTE THE ATTENTION BRANCH AND PARSING MAP BRANCH RESPECTIVELY. SA AND CA ARE SPATIAL ATTENTION AND CHANNEL ATTENTION RESPECTIVELY.Datasets
CelebA [69]
Helen [70]
×4
×8
×16
×4
×8
×16
Bicubic
0.7979 0.3851 0.3057 0.8982 0.5039 0.3363
FSRNet [18]
0.9066 0.7171 0.5701 0.9313 0.7453 0.5461
DIC [19]
0.9074 0.7728 0.4975 0.9462 0.7865 0.5886
SISN [14]
0.9204 0.7709 0.5919 0.9488 0.7853 0.5177
Ours
0.9206 0.7780 0.6358 0.9527 0.7960 0.5692
Models
PAFB-AB
PAFB-PMB MSRB PSNR
SSIM
SA CA
1
27.21
0.7951
2
27.44
0.8030
3
27.31
0.7990
4
27.30
0.7988
5
27.35
0.8013
6
27.41
0.8028
7
27.44
0.8036
8
27.54
0.8072
TABLE IV FACE
IVPARSING ACCURACY ON CELEBA [69] AND HELEN [70]. Accuracy 0.9763 0.9671 0.9564 0.9520 0.9228 0.8672Dataset
CelebA [69]
Helen [70]
×4
×8
×16
×4
×8
×16
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"Diagnosing The Ejecta Properties of Engine-Driven Supernovae from Observables in Their Initial Phase",
"Diagnosing The Ejecta Properties of Engine-Driven Supernovae from Observables in Their Initial Phase"
] | [
"Keiichi Maeda \nDepartment of Astronomy\nKyoto University\nKitashirakawa-Oiwake-cho, Sakyo-ku606-8502KyotoJapan\n",
"Akihiro Suzuki \nResearch Center for the Early Universe\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n",
"Luca Izzo \nDARK\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 1282200CopenhagenDenmark\n"
] | [
"Department of Astronomy\nKyoto University\nKitashirakawa-Oiwake-cho, Sakyo-ku606-8502KyotoJapan",
"Research Center for the Early Universe\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan",
"DARK\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 1282200CopenhagenDenmark"
] | [
"MNRAS"
] | Engine-driven explosions with continuous energy input from the central system have been suggested for supernovae (SNe) associated with a Gamma-Ray Burst (GRB), super-luminous SNe (SLSNe), and at least a fraction of broad-lined SNe Ic (SNe Ic-BL) even without an associated GRB. In the present work, we investigate observational consequences in this scenario, focusing on the case where the energy injection is sufficiently brief, which has been suggested for GRB-SNe. We construct a simplified, spherical ejecta model sequence taking into account the major effects of the central engine; composition mixing, density structure, and the outermost ejecta velocity. Unlike most of the previous works for GRB-SNe, we solve the formation of the photosphere self-consistently, with which we can predict the photometric and spectroscopic observables. We find that these ejecta properties strongly affect their observational appearance in the initial phase ( ∼ < a week since the explosion), highlighted by blended lines suffering from higher-velocity absorptions for the flatter density distribution and/or higher outermost ejeca velocity. This behaviour also affects the multi-band light curves in a non-monotonic way. Prompt follow-up observations starting immediately after the explosion thus provides key diagnostics to unveil the nature of the central engine behind GRB-SNe and SNe Ic-BL. For SN 2017iuk associated with GRB 171205A these diagnosing observational data are available, and we show that the expected structure from the engine-driven explosion, i.e., a flat power-law density structure extending up to ∼ > 100, 000 km s −1 , can explain the observed spectral evolution reasonably well. | 10.1093/mnras/stad1075 | [
"https://export.arxiv.org/pdf/2304.04146v1.pdf"
] | 258,048,998 | 2304.04146 | 3937e76de033779161b1bff51bce642ef29627f7 |
Diagnosing The Ejecta Properties of Engine-Driven Supernovae from Observables in Their Initial Phase
2022
Keiichi Maeda
Department of Astronomy
Kyoto University
Kitashirakawa-Oiwake-cho, Sakyo-ku606-8502KyotoJapan
Akihiro Suzuki
Research Center for the Early Universe
Graduate School of Science
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Luca Izzo
DARK
Niels Bohr Institute
University of Copenhagen
Jagtvej 1282200CopenhagenDenmark
Diagnosing The Ejecta Properties of Engine-Driven Supernovae from Observables in Their Initial Phase
MNRAS
0002022Accepted XXX. Received YYY; in original form ZZZPreprint 11 April 2023 Compiled using MNRAS L A T E X style file v3.0transients: supernovae -gamma
Engine-driven explosions with continuous energy input from the central system have been suggested for supernovae (SNe) associated with a Gamma-Ray Burst (GRB), super-luminous SNe (SLSNe), and at least a fraction of broad-lined SNe Ic (SNe Ic-BL) even without an associated GRB. In the present work, we investigate observational consequences in this scenario, focusing on the case where the energy injection is sufficiently brief, which has been suggested for GRB-SNe. We construct a simplified, spherical ejecta model sequence taking into account the major effects of the central engine; composition mixing, density structure, and the outermost ejecta velocity. Unlike most of the previous works for GRB-SNe, we solve the formation of the photosphere self-consistently, with which we can predict the photometric and spectroscopic observables. We find that these ejecta properties strongly affect their observational appearance in the initial phase ( ∼ < a week since the explosion), highlighted by blended lines suffering from higher-velocity absorptions for the flatter density distribution and/or higher outermost ejeca velocity. This behaviour also affects the multi-band light curves in a non-monotonic way. Prompt follow-up observations starting immediately after the explosion thus provides key diagnostics to unveil the nature of the central engine behind GRB-SNe and SNe Ic-BL. For SN 2017iuk associated with GRB 171205A these diagnosing observational data are available, and we show that the expected structure from the engine-driven explosion, i.e., a flat power-law density structure extending up to ∼ > 100, 000 km s −1 , can explain the observed spectral evolution reasonably well.
INTRODUCTION
Massive stars can lead to various explosive phenomena as their end products, following the collapse of the central core. In most of the cases, the outcome is believed to be a core-collapse supernova (CCSN), which is divided into several subclasses based on spectral line footprints (e.g., Filippenko 1997); type II (H-rich), IIb (H-poor), Ib (H-deficient and He-rich), Ic (H-and He-deficient), forming a sequence of envelope stripping during their pre-SN evolution (a red-supergiant progenitor for SNe II to a Wolf-Rayet-like C+O star progenitor for SNe Ic; e.g., Langer 2012;Maeda 2022). While the CCSN explosion mechanism has not yet been fully understood, it is widely believed that a neutron star (NS) is formed behind the so-called delayed neutrino-heating mechanism in most, if not all, of these canonical CCSNe (e.g., Janka 2012; Burrows & Vartanyan 2021;Sato et al. 2021).
However, more extreme explosions that are probably associated with demise of a massive star have also been found. Gamma-Ray Bursts (GRBs), as a brief burst of gamma-rays (see, e.g., Piran 2004, for a review), were proposed to belong to this category (Paczynski ★ E-mail: [email protected] 1986), and the link to CCSNe was directly confirmed by the discovery of a peculiar SN associated with a weak and nearby GRB 980425 (Galama et al. 1998). The SN, termed SN 1998bw, showed much broader spectral features than canonical SNe. A high velocity at the line forming region of C+O-rich ejecta was introduced, which results in a blend of various spectral lines based on SN Ic spectra (Iwamoto et al. 1998). Combined with a slower evolution than canonical SNe Ic, it has been proposed that the kinetic energy and the ejecta mass are ∼ > 3 × 10 52 erg and ∼ 10 , which are both much larger than those derived for canonical SNe Ic (Iwamoto et al. 1998;Woosley et al. 1999;Nakamura et al. 2001;Maeda et al. 2006b). It has therefore been suggested that SN 1998bw was a hyper-energetic SN Ic following a collapse of a massive star whose main-sequence mass was larger than those triggering canonical CCSNe.
Since then, an increasing number of SNe have been found to be associated with nearby GRBs when such search for the SN component is possible (but see Della Valle et al. 2006;Gal-Yam et al. 2006), forming 'GRB-SNe' (e.g., Woosley & Bloom 2006;Hjorth 2013;Cano et al. 2017). They share the characteristic broad features with the prototypical GRB-SN 1998bw, and thus classified as 'broadlined SNe Ic (SNe Ic-BL)'. Indeed, SNe Ic-BL have been found even without an associated GRB, while the GRB-SNe represent the most extreme cases among SNe Ic-BL (Cano 2013;Modjaz et al. 2016).
Given the distinct nature of the ejecta properties and the association with a GRB, the explosion mechanism of GRB-SNe (and at least a fraction of SNe Ic-BL) is believed to be different from that behind canonical CCSNe. Given its hyper-energetic nature, it is called the central engine. A popular model is the so-called collapser scenario, which involves a formation of a rapidly-rotating black hole (BH) and an accretion disc in the centre of a collapsing star (Woosley 1993;MacFadyen & Woosley 1999;Kumar et al. 2008;Hayakawa & Maeda 2018). Another popular model is a 'magnetar'-driven scenario, i.e., a formation of a highly-magnetized and rapidly-spinning NS following the core collapse (Duncan & Thompson 1992;Usov 1992;Mazzali et al. 2006a;Maeda et al. 2007a;Metzger et al. 2011). The enginedriven explosion has been further extended as a possible scenario of another flavor of exotic SNe, the H-poor class of 'super-luminous SNe (SLSNe-I)' which are characterized by high luminosities and large energy budget in the radiation output (Quimby et al. 2011;Gal-Yam 2012). The central engine, similar to the case for GRB-SNe, has been proposed as a power source; the magnetar scenario (Kasen & Bildsten 2010; see also Maeda et al. 2007b) or the BH accretion scenario (Dexter & Kasen 2013).
The continuous energy input from the central engine may change the ejecta dynamics and the composition structure as compared to the canonical SN ejecta. The mixing is an inevitable consequence, which has been found to lead an overall flatter density distribution throughout the ejecta toward the outermost region (with a general structure of ∝ −5 or −6 where is the ejecta density as a function of the velocity, , as derived analytically and numerically; Suzuki & Maeda 2017, rather than the rapidly steepening structure toward the surface usually assumed for canonical CCSNe (especially for SNe Ic) as a result of a 'point-like and instantaneous' explosion (Matzner & McKee 1999;Tan et al. 2001). The composition structure is also mixed (Suzuki & Maeda 2021). These features are also found at least qualitatively in a jet-like energy input from the central engine, especially along the jet direction Tominaga et al. 2007;Eisenberg et al. 2022;Suzuki & Maeda 2022;Pais et al. 2023). The scenario further suggests a link between GRB-SNe and SLSNe (Metzger et al. 2017); a key may be the difference in the time duration in which the central engine operates, with short-duration and long-duration systems resulting in GRB-SNe and SLSNe, respectively (Suzuki & Maeda 2021). Depending on the engine-operation duration, the outermost ejecta velocity may also be different (Eisenberg et al. 2022;Suzuki & Maeda 2022;Pais et al. 2023), which is further affected by the radiation loss (Suzuki & Maeda 2021).
Deriving the ejecta structure based on observational data of GRB-SNe thus provides powerful diagnostics on the nature of the central engine. It has been performed based on synthetic spectra (for spherical, one-dimensional models) as compared to spectra of individual GRB-SNe (or SNe Ic-BL) (e.g., Iwamoto et al. 1998;Mazzali et al. 2000;Nakamura et al. 2001;Mazzali et al. 2002Mazzali et al. , 2003Mazzali et al. , 2006aAshall et al. 2019;Izzo et al. 2019;Ashall & Mazzali 2020;Kwok et al. 2022). Such study has been showing that observational properties of GRB-SNe and SNe Ic-BL can be generally explained by an energetic explosion of massive C+O-rich ejecta. Further, some degree of the composition mixing (especially of 56 Ni; e.g., Ashall et al. 2019;Izzo et al. 2019) and a flat density structure at a high velocity (e.g., Mazzali et al. 2000;Izzo et al. 2019) have been inferred.
These results are based on the assumption that a photosphere is formed at a higher velocity than in canonical SNe Ic -this is indeed a critical shortcoming in most of the previous works, which parameterized the position (velocity) of the photosphere and its temperature (or luminosity). Surprisingly, the spectral synthesis simulations for GRB-SNe based on the self-consistent calculations for the nature of the photosphere are generally lacking, with only a few exceptions as summarized below. Dessart et al. (2017) performed detailed non-LTE (Local-Thermodynamic Equilibrium) radiation transfer simulations for 1D spherical models, including the self-consistent photosphereformation processes. However, their models did not take into account the expected effects of the central engine as mentioned above, resulting in a relatively poor fit to the observed spectra of SN 1998bw. Multi-dimensional radiation transfer simulations were performed by Rapoport et al. (2012) based on a specific series of parameterized 2D jet-like explosion models of Maeda et al. (2002). Similarly, Barnes et al. (2018) and Shankar et al. (2021) performed multi-dimensional radiation transfer based on a series of parameterized jet-like explosion models. While these multi-dimensional studies are regarded as being more realistic than one-dimensional models, a drawback is that the results are relatively limited to the particular models which do not necessarily cover a wide range of the possible effects of the central engine. Most critically, all these works focused on the maximumphase spectral formation, lacking the spectral-model prediction in the first week since the explosion.
Indeed, the very early-phase spectra potentially provide powerful diagnostics on the nature of the central engine; the central engine manifests its nature in the outermost ejecta, both in the density and composition structures. Given that the ejecta become progressively transparent from the outer region as time goes by, the spectralformation site initially resides in the outermost layer while it eventually recedes to the inner, lower velocity region. The power of the infant spectroscopic data was demonstrated by Izzo et al. (2019) who inferred the existence of extremely high-velocity materials reaching to ∼ 100, 000 km s −1 in SN 2017iuk associated with GRB 171205A, based on spectral-synthesis models for its spectra starting on day 1 since the GRB (see also Ashall et al. 2019, for another example, GRB 161219B/SN 2016jca). Recently added is SN Ic-BL 2020bvc, which showed similarly high-velocity absorption features reaching to 60, 000 − 70, 000 km s −1 on day 1.5, despite its non-association with a GRB (Ho et al. 2020;Izzo et al. 2020;Rho et al. 2021); the development of high-cadence optical surveys now allows such prompt spectroscopic followup observations even without an associated GRB, which has led to the discovery that some of SNe Ic-BL without a GRB can also exhibit extreme observational properties as demonstrated by SN 2020bvc.
In the present work, we predict the photometric and spectroscopic properties of the engine-driven SNe including a self-consistent photosphere-formation processes. Our investigation is based on idealized/simplified 1D ejecta models, which however includes key effects of the engine-driven explosion in the density and composition structure. We especially focus on the initial, rising-phase behaviours and investigate how such observations can be used to diagnose the ejecta structure and thus the nature of the central engine. In Section 2, we describe our method and models. The results are presented in Section 3, where photometric and spectral properties are addressed in Sections 3.1 and 3.2, respectively. The paper is closed in Section 4 with concluding remarks.
METHOD AND MODELS
In the present work, we restrict ourselves to one-dimensional, spherical models, based on the success of such models to reproduce basic light-curve and maximum-phase spectral properties (see references in Section 1). This is mainly to simplify/idealize the problem so that we can extract characteristic and general properties especially associated with the outermost ejecta, but also helps to reduce computational time. This is indeed a good approximation for quasi-spherical energy input from the central engine (e.g., Suzuki & Maeda 2017, but also can capture basic ejecta properties of a jet-like energy input at least along the line-of-sight if the associated hydrodynamics effects (e.g., Suzuki & Maeda 2022) are properly parameterized (see below).
We assume that the ejecta structure is immediately frozen after the shock breakout and possible energy input from the central engine, well before observations. Further, we assume that the thermal energy input is dominated by the 56 Ni/Co radioactive heating; these conditions require that the progenitor radius is sufficiently small (which is usually met for a compact C+O star progenitor) and that the energy input by the central engine is terminated well before the time of observations ( ∼ < 10 4 sec or so; Suzuki & Maeda 2021) 1 . When both of these requirements are satisfied, the energy deposited either by the initial shock or subsequent operation of the central engine, or both, should be quickly converted to the kinetic energy, leading to the freezeout of the ejecta structure and the homologous expansion. This is a scenario appropriate for GRB-SNe (Suzuki & Maeda 2021). The opposite case, where the central engine operates in a long time scale and thus determines the ejecta energy content, will be investigated in the future as a scenario for SLSNe (K. Maeda, in prep.).
The properties of the ejecta model in the present work are thus specified by the (one-dimensional) density structure ( ( , ) = 0 ( ) −3 ) and the distributions of different elements and radioactive isotopes ( i ( )), where the relations are expressed in the velocity space (i.e., the Lagrangian coordinate). For the density structure, our reference model is the CO138 model applied to the prototypical GRB-SN 1998bw, which was constructed by a one-dimensional thermal-bomb hydrodynamic simulation based on a 13.8 C+O star progenitor model (Iwamoto et al. 1998;Nakamura et al. 2001). The ejecta mass and the kinetic energy are 10.6 and 3 × 10 52 ergs, respectively. In addition to the original CO138 density structure, we test a power-law density distribution, e.g., 0 ( ) ∝ −6 ; the energy input by a central engine introduces substantial mixing within the ejecta, and the resulting density structure is well described by a power-law distribution with the index of −5 to −6 as a function of the velocity (Suzuki & Maeda 2017. A similar mixing effect is found for a jet-like energy injection along the jet axis (Eisenberg et al. 2022;Suzuki & Maeda 2022;Pais et al. 2023), and our one-dimensional simulation may also be regarded as a rough approximation along the jet axis which is expected to be aligned with the line-of-sight for GRB-SNe (see also Section 4 for further discussion). The power-law distribution is constructed by hands, keeping the same ejecta mass and kinetic energy as in the original CO138 model.
The operation of the central engine will also introduce the mixing of the chemical compositions (Maeda et al. 2002;Tominaga et al. 2007;Suzuki & Maeda 2021, 2022. To examine the effect of the composition structure in the observables, we examine two cases; one adopting the (original) stratified structure, and the other adopting the homogeneous mixing of all the elements and 56 Ni introduced by hands. Given that the power-law density model already assumes a large-scale mixing, we adopt only the mixing case for the power-law model. In both cases, the solar abundance compositions are added to the heavy elements beyond neon. We however note that adding the solar compositions introduces only a minor modification for the mixing model, since the metal content is dominated by the newly-synthesized elements everywhere in the ejecta.
An additional model parameter in our model sequence is the outermost ejecta velocity ( max ), above which the density is set zero. This quantity depends on the hydrodynamics of the shock propagation in the steeply decreasing progenitor density structure (Matzner & McKee 1999), together with the energy redistribution following the shock breakout due to the radiation loss (Falk & Arnett 1977;Ensman & Burrows 1992;Suzuki et al. 2016). The operation of the central engine further provides an additional factor, again through the combined effect of the hydrodynamic evolution Pais et al. 2023) and radiation loss (Suzuki & Maeda 2021). In other word, our aim here is to clarify what observables keep the information on the outermost ejecta velocity, and how the observations can be used to constrain it; we thus aim at providing possible diagnostics on the nature of the central engine through max .
Introducing max and the power-law distribution is conducted in a way to keep the ejecta mass and the kinetic energy in the original CO138 model (10.6 and 3 × 10 52 ergs), by adjusting/scaling the innermost velocity cut and the density scale, i.e., 0 ( ) for given . The innermost velocity cut does not affect the present conclusions at all, as the innermost part is not exposed in the time window of interest in the present work. The effect of the density scale is also minimal; even for the two extreme choices of max = 120, 000 and 60, 000 km s −1 in the power-law density model sequence, the difference in the density scale is only ∼ 15% since the mass content in the outermost layer are negligible even for the power-law distribution sequence since it is much steeper than 0 ( ) ∝ −3 . The (homogeneous) mass fractions of different elements are computed to keep the masses of each element in the original CO138 model (after adding the solar compositions), and the differences in the mass fractions between models with different values of max are negligible for the same reason. Table 1 summarizes the model sequence examined in the present work. Fig. 1 shows the density structure for Models CO138 and POW, as well as the composition structure for cases 'nomix' and 'mix'. The mass of 56 Ni is set to be 0.4 in all the models. To demonstrate the difference in the ejecta structures of the present models and that adopted for a typical SN Ic, we also show a representative 'SN Ic' ejecta structure in Fig. 1; an explosion of a stripped CO core evolved from a 18 main-sequence star (Fang & Maeda 2023, their CO18 model). The ejecta mass and the kinetic energy are set to be the typical values estimated for SNe Ic (∼ 2.7 and 2 × 10 51 ergs; Taddia (2023) is also shown (grey).
by the density structure and max are demonstrated in Fig. 2, which shows the velocities of different layers as a function of the mass contained above a given layer up to the outermost surface of the ejecta.
A high velocity could be obtained toward the outermost layer even in the canonical SN Ic ejecta due to the acceleration of the shock wave in propagating the surface of a progenitor star with a steep density gradient (Matzner & McKee 1999), depending on the radiative loss at the shock breakout (Falk & Arnett 1977;Ensman & Burrows 1992). However, in the canonical SN Ic ejecta, the density is very low at the outermost layer ( Fig. 1), and thus changing max would not introduce much effect on the optical display; the mass of the highvelocity material is too small to participate in the spectral formation (Fig. 2). The CO138 model has a larger amount of materials at a high velocity (Iwamoto et al. 1998;Nakamura et al. 2001), and thus such high-velocity materials potentially affect the radiation transfer, therefore the result can be dependent on the choice of max . Further, the POW model sequence has even a larger amount of materials at a high velocity due to the flat density distribution (Fig. 2); ∼ > 0.04 and ∼ > 10 −3 at ∼ > 60, 000 and ∼ > 100, 000 km s −1 , respectively, which are larger than those in the CO138 model (∼ 5 × 10 −3 at ∼ > 60, 000 km s −1 and ∼ 5 × 10 −5 at ∼ > 100, 000 km s −1 ), and much larger than in the 'SN Ic' ejecta (∼ 10 −4 at ∼ > 60, 000 km s −1 ).
For each model structure thus constructed, we simulate multiband light curves and spectra using HEIMDALL (Handling Emission In Multi-Dimension for spectrAL and Light curve calculations; Maeda et al. 2006bMaeda et al. , 2014. Despite the multi-dimensional capability of HEIMDALL, the present investigation is restricted to onedimensional spherical configuration.
In the radiation transfer simulations, the -ray packets are created POW_mix_12 Figure 2. The velocity as a function of the mass coordinate as measured from the outermost ejecta. Shown here are the structures of CO138_mix_10 (brown), POW_mix sequence (blue, green, orange, and red for max = 60, 000, 80, 000, 100, 000, and 120, 000 km s −1 , respectively), and the representative SN Ic model (grey).
through the radioactive input following the 56 Ni distribution given as the model input. The propagation of the high-energy photons originally emitted as the decay lines is followed as a random walk process with the Monte-Carlo method, including the Compton scattering, photoelectric absorption, and pair production as the photon-matter interaction processes (Maeda 2006). The energy deposited to the gas by these processes is recorded, together with the energy provided by positrons immediately following the positron-emission decay channel.
Once the thermal energy deposition to each spatial and temporal grid is determined, this information is used to create optical photon packets. The transfer of these optical photons is then performed again using the Monte-Carlo method, where iteration is solved in each time step to determine the ionization status, the electron number density, and the temperature, as obtained by requiring the self-consistency between the radiation field and gas conditions. The LTE is assumed. The radiation transfer calculations involve ∼ 5 × 10 5 bound-bound transitions, as well as free-free and bound-free transitions (Maeda et al. 2006b), using the opacity data taken from Kurucz & Bell (1995).
The number of the radial grids used in the radiation transfer simulations is 300 − 600 for the models with max = 60, 000 − 120, 000 km s −1 . It then provides the resolution of < 200 km s −1 in the input ejecta models, which is sufficient to resolve wavelength-dispersion elements in typical spectroscopic observations. The number of the photon packets is ∼ 10 8 , in all the simulations. At the end of each simulation, the photon packets are sampled into 90 bins in time (with a logarithmic spacing between 0.3 and 160 days) and 3,000 bins in wavelengths (with a logarithmic spacing between 100 and 20,000Å). The number of the photon packets is sufficiently large to achieve the convergence in the thermal condition in each space and time grid, as well as to provide a good Signal-to-Noise ratio in the synthetic , , and bands from bottom to top (shown in violet, blue, green, orange, and red, respectively). The same model sequence with different max (see Table 1) is shown by grey lines. For an illustrative purpose, the light curves of SN 1998bw (Galama et al. 1998;Sollerman et al. 2002;Clocchiatti et al. 2011) are also shown by the filled squares, using the same colour coordinate as that used for the models. The Vega system is used throughout the present paper. spectra (i.e., on average ∼ 300 − 400 packets in each time-step and wavelength bin). Fig. 3 shows the synthetic multi-band light curves of Models CO138_nomix_10, CO138_mix_10, and POW_mix_10. In the same figure, we also plot the multi-band light curves of the prototypical GRB-SN 1998bw, but noting that it is merely for a demonstration purpose since we have not tuned the models to fit to the observational data of SN 1998bw.
RESULTS
Photometric Properties
With the mixing of 56 Ni, the light curves show a quicker rise to the peak, and the peak is reached earlier. Accordingly, the 56 Ni and composition mixing affects the colour evolution, as shown in Fig. 4. These are well-known effects of the extended distribution of 56 Ni (e.g., Nakamura et al. 2001;Dessart et al. 2012;Bersten et al. 2013;Yoon et al. 2019). The need for the chemical mixing, especially of 56 Ni, has indeed been suggested to fit the light curves of SN 1998bw (Nakamura et al. 2001), as is evident from Fig. 3. The 56 Ni mixing is indeed a property shared by SNe Ic-BL in general (Taddia et al. 2019), or even by canonical SNe Ic (Taddia et al. 2015;Yoon et al. 2019).
We find that the outermost ejecta velocity, max , also affects the initial, rising part of the light curves. This is seen in the colour evolution, as shown in Fig. 4. As the photosphere recedes in the velocity space, the models with different values of max converge as the material well above the photosphere starts decoupling from the radiation field. For example, Models POW with max = 120, 000 (red) and 100, 000 km s −1 (orange) show different colour evolution in the first week. Then these two models converge, but keep showing different colours from Model POW with max = 80, 000 km s −1 (green) until ∼ 10 days. After that, these three models become mutually indistinguishable in their colours, but show different colours (especially in the − colour) from Model POW with max = 60, 000 km s −1 (blue). Finally all these models converge to the same colour evolution around the maximum light.
The effect of the outermost velocity on the − colour evolution is monotonic on max but that of the − colour evolution is non-monotonic. These effects are not explained simply by different evolution of the photosphere nor by additional radioactive-decay energy input in the outermost layer for models with larger max . These differences in the colours are indeed driven by the differences in the spectral formation above the photosphere. We will clarify this issue in discussing the spectral formation process in the next section.
The observed colour evolution of SNe 1998bw, 2017iuk, and 2020bvc is shown in Fig. 5 (Clocchiatti et al. 2011;Izzo et al. 2019Izzo et al. , 2020Ho et al. 2020), overplotted with the CO138_nomix model and the POW_mix sequence with different values of max . For SN 1998bw, unfortunately the early evolution is not well sampled; we therefore perform interpolation of the data points to compute the colours (open squares), unless there are data points in the two bands in close proximity in the epochs (filled squares). Fig. 5 is for a demonstration purpose, since the model has not been tuned to individual SNe; as such, direct comparison between the model colour and the observed one should not be conducted, while the colour 'evolution' may still be meaningful. SN 2020bvc shows a change in the colour evolution at ∼ 5 days both in the − and − colours, which is not seen in (or opposite to) the model predictions. Indeed, SN 2020bvc shows the initial rapid decline in the bolometric luminosity, which cannot be attributed to the 56 Ni/Co heating. The present models thus do not apply to the early photometric evolution of SN 2020bvc, and additional powering mechanism must be considered; the issue will be investigated in a forthcoming paper (K. Maeda, in prep.). For the other two SNe, the overall colour evolution is indeed well explained by the present models; especially, it is clear that substantial 56 Ni mixing is necessary to explain the early colour evolution (see also Yoon et al. 2019). The initial rapid evolution toward the red in the − colour, as well as the transition from the red-to-blue evolution then to the blue-to-red evolution in the − colour, are seen in SN 2017iuk as predicted by the models; it is seen that the evolution roughly matches to the POW_mix models with max = 80, 000 to 100, 000 km s −1 . This is indeed consistent with the conclusion from the spectral modeling (Section 3.2; see also Izzo et al. 2019). The photometric data of SN 1998bw are less constraining, due to the sparsely sampled data points; the colours estimated with the interpolation involves large uncertainties and should not be over-interpreted. Still, SN 1998bw shares the qualitative behaviors in the colour evolution with SN 2017iuk and thus with the model predictions. The POW_mix model with max = 80, 000 km s −1 provides a better match to the data than . The colour evolution of the model sequences CO138_nomix (brown) and CO138_mix (cyan and magenta for max = 80, 000 and 100, 000 km s −1 , respectively), and POW_mix sequence (blue, green, orange, and red for max = 60, 000, 80, 000, 100, 000, and 120, 000 km s −1 , respectively). the other models (with the above caveat bared in mind). These investigations demonstrate that the early photometric colour evolution can serve as a powerful diagnostics of the outermost ejecta structure, especially the outermost velocity.
Spectroscopic Properties
Fig . 6 shows the evolution of the synthetic spectra in the initial phase, on days 0.5, 1, 2, and 7, well before the maximum light 2 . While we do not intend to tune the models to fit the observational data of specific objects, we also show the spectra of GRB-SNe 1998bw (Patat et al. (Hiramatsu et al. 2020) for a demonstration purpose 3 . Fig. 7 shows the spectral evolution covering the maximum phase. In the present work, we restrict ourselves up to about one month since the explosion for two reasons; (1) in the post-maximum phase, an additional dense core is required to explain both the light curve ) and spectral properties (Maeda et al. 2006a; see also Dessart et al. 2017), and (2) the deviation from the LTE becomes progressively important toward a late phase.
The chemical mixing manifests itself mainly in three effects. First, the emergence of the SN light becomes earlier for a more extensive 56 Ni mixing, as already been discussed in many previous works (e.g., Nakamura et al. 2001;Dessart et al. 2012;Bersten et al. 2013;Yoon et al. 2019, see also Section 3.1). The 56 Ni mixing also affects the evolution of the photosphere, and therefore the photospheric velocity itself, e.g., at the maximum light (Dessart et al. 2016;Moriya et al. 2020). This is also seen in the spectral line velocities (Fig. 7). The other effect is the additional absorption in the 'mixed' models especially in the blue, which is provided by metals, e.g., Fe II and Co II, in the outer layer (see the feature between 4,000 and 5,000 Å in Fig. 7). The general behaviour in the time evolution, for different choice of the outermost ejecta velocity ( max ), is clearly seen in the spectral evolution. Initially Models POW with different values of max show noticeable differences in the synthetic spectra as can be seen in the spectra on days 0.5, 1, and 2. On day 7, the models with max = 120, 000 and 100, 000 km s −1 converge to show indistinguishable spectra, while the difference is still seen as compared to the other two models ( max = 80, 000 and 60, 000 km s −1 ). The similar behaviour is discerned between Models CO138 with max = 10, 000 and 80, 000 km s −1 , which show clear difference up to day 2, while the difference disappears on day 7. As time goes by, the difference between Models POW with max ≥ 100, 000 and 80, 000 km s −1 becomes smaller on day 11, and these two models become indistinguishable on day 15. Finally, their spectra merge into Model POW with max = 60, 000 km s −1 on day 19, i.e., around the peak.
The main difference between the model sequences CO138 and POW is the density structure at ∼ > 50, 000 km s −1 ; the POW sequence has a flatter density structure and thus a higher density there than the CO138 sequence. It is seen in Fig. 7 that the difference between these two sequences becomes progressively small toward the maximum light. If Models CO138 and POW with the same max are compared, the POW models show a more blue-shifted absorption feature (due to the line blending) than the CO138 models.
The difference in the degree of the blueshift seen in the absorption as a function of max (as complemented by the density slope) explains the early-phase colour evolution for different models (Fig. 4 and Section 3.1). In the following, we focus on the model POW sequence, but essentially the same argument applies to the model CO138 sequence. Fig. 6 shows that the synthetic spectra do not show noticeable difference in the band. The absorption feature in the models with large max starts overlapping the -band bandpass, with the flux suppression in the band especially strong for Model POW_mix_12 in the first week. This is the reason why this model shows very blue − colour in the initial phase, followed by Model POW_mix_10. The -band flux is more easily affected by the absorption. Therefore, models with large max suffered from the flux suppression both in the and bands, while decreasing max leads to the suppression only in the band. This explains the non-monotonic behaviour as a function of max seen in the − colour. Note that the difference here is not due to difference in the photospheric temperature -it is seen in Figs. 6 and 7 that Models POW_mix (and Models CO138_mix) have essentially the same continuum colour irrespective of max 4 ; except for the wavelength ranges that experience the high-velocity absorption associated with the materials well above the photosphere, the synthetic spectra for the models with different values of max are overlapping (e.g., at ∼ < 4, 500Å and ∼ > 7, 000Å) -if the photospheric temperature is different, the overal spectral slope must be different. This effect of max on the colour is thus different from that by the 56 Ni mixing, later of which changes the photospheric velocity and temperature (Yoon et al. 2019).
The above considerations show a potential power of the spectroscopic observations (as well as the less-direct photometric observations) to diagnose the properties of the outermost layer, which is otherwise difficult. The spectra in the first few days provide dramatic difference in the degree of blueshift in the (blended) absorption feature at 5,000-7,000Å, depending on max . This dependence on max is to some extent coupled with the density structure, since the larger density in the outermost layer (i.e., Models POW) tend to show a larger blueshift in the absorption feature than the lower-density models (i.e., Models CO138), for given max (i.e., the cyan line vs. the green line, or the magenta line vs. the orange line, in Fig. 6). The time evolution is however somewhat different (e.g., Fig. 4 for the − colour evolution; Figs. 6 and 7 for the longer appearance of the additional absorption in Models POW than CO138 for same max ), and the detailed time-series modeling could help solve the degeneracy.
To demonstrate the power of the very early-phase spectra to diagnose the ejecta properties, we compare the synthetic model spectra to the observational ones of GRB-SNe 2017iuk and 1998bw, and SN Ic-BL 2020bvc, with the caveat that the present models are not tuned to reproduce observational properties of specific objects. The intensive spectral coverage in the initial, rising phase is available for GRB-SN 2017iuk (Izzo et al. 2019). Fig. 6 shows that Models POW_mix_10 and POW_mix_12 capture key spectral properties of SN 2017iuk. On day ∼ 1, a broad absorption feature is seen in ∼ 5, 200 − 6, 400Å with the minimum at ∼ 5, 800Å in the observation. The models (both POW and CO138) with max ∼ < 80, 000 km s −1 instead show an emission-like feature peaking at ∼ 6, 000Å, contrary to the observational spectrum. With max ∼ 100, 000 km s −1 , this peak starts being suppressed by a blue-shifted absorption component originally in the redder wavelength, creating instead the absorption feature in the same wavelength range. This property persists on day 2, where again the models with max ∼ > 100, 000 km s −1 are favored. The same effect is seen in the model spectra on days 7 and 11; the peak at ∼ 6, 200Å seen in the models with the smaller max starts being overlapped by the absorption for Model POW_mix_8, and it is totally smeared out to produce a featureless continuum-like spectra in ∼ 5, 500 − 8, 000Å in Models POW_mix_10 and POW_mix_12. The latter is consistent with the observational spectrum. Model CO138_mix_10 fails to ex-plain this feature due to the lower density than Model POW_mix_10 in the line-formation region. All the models start showing the peak at ∼ 6, 200Å on day 15, when the corresponding emission feature is discerned in the observed spectrum. In summary, despite various simplifications in the model construction, the power-law density structure (with the index of ∼ −6 in the velocity space) extending to ∼ > 100, 000 km s −1 , with substantial mixing in the composition structure including 56 Ni, reproduces the spectral evolution of SN 2017iuk reasonably well. We thus confirm the main conclusions of Izzo et al. (2019), emphasizing that the formation of the photosphere is treated in a self-consistent (first-principle) manner in the present work, rather than the parameterized treatment of the photosphere adopted in many spectral-synthesis studies including the models presented in Izzo et al. (2019).
As emphasized above, this comparison is for a demonstration purpose, given that we have not tuned the models to individual objects. For example, on day 1, the main absorption feature discussed above has a wide absorption feature up to ∼ 7, 500Å in the models, while the observed spectrum of SN 2017iuk shows a peak at ∼ 6, 500 km s −1 . In addition, the observed spectrum does not show a clear peak at ∼ 4, 800Å, which is seen as a strong peak in the model spectra. These may be affected by the detailed density and composition structures more sensitively than the absorption structures (including the possibility of non-spherical geometry); the absorption mainly probes the information along the line-of-sight toward the photosphere, while the emitting region is generally more extended. Further, in the earliest phase of SN 2017iuk, there is an uncertainty in the subtraction of the GRB afterglow component, which may artificially create or smooth peak structures while the absorption features are less affected by such subtraction. As such, we focus on the absorption features in the present work, but emphasize that more detailed study with finer/wider model grids is required for detailed spectral study of individual objects (e.g., Izzo et al. 2019).
SN Ic-BL 2020bvc was not associated with a GRB, but it was discovered soon after the explosion thanks to the recent development of high-cadence optical surveys. The follow-up observations revealed that it belongs to an extreme end of SNe Ic-BL, with many properties shared with GRB-SNe, including the high-expansion velocity as well as strong radio and X-ray emissions (Ho et al. 2020;Izzo et al. 2020;Rho et al. 2021). SN 2020bvc has been thus suggested to be driven by a powerful central engine creating a sub-relativistic ejecta component, either by a chocked jet (Ho et al. 2020) or an off-axis jet (Izzo et al. 2020). Very high-velocity absorption features reaching to ∼ 60, 000 − 70, 000 km s −1 are seen in its spectrum taken on day 1.5, which shares the property seen in SN 2017iuk at a similar epoch. It then evolved to show similar maximum-light spectra with SN 1998bw. Fig. 6 shows the spectrum on day 1.5. SN 2020bvc showed a blue continuum on day 1.5; to focus on the absorption/emission features, its spectrum shown in Fig. 6 is subtracted by an arbitrarily-defined power-law continuum and then the flux is normalized. The difference to SN 2017iuk is striking, which may be mainly attributed to the difference in the outermost velocities; the spectrum resembles the models with max = 60, 000 or 80, 000 km s −1 (i.e., CO138_mix_8, POW_mix_6, and POW_mix_8), confirming the claim by Ho et al. (2020); Izzo et al. (2020); Rho et al. (2021). SN 2020bvc demonstrates three key issues; (1) the very-early spectra of GRB-SNe and SNe Ic-BL can be used to constrain the outermost ejecta properties, (2) there could indeed be a diversity in the outermost ejecta properties among GRB-SNe and SN Ic-BL, and they might further be linked to their association or non-association with GRBs, and (3) the high-cadence optical surveys now provide an excellent opportunity to perform systematic investigation of the outermost ejecta properties, even without an associated GRB.
A caveat here on the comparison between the present models and the spectrum of SN 2020bvc is that the models presented here take into account only the 56 Ni/Co heating, while SN 2020bvc probably exhibited a cooling emission either from a jet-heated cocoon or a confined circumstellar matter (CSM) hit by the sub-relativistic ejecta (Izzo et al. 2020;Ho et al. 2020;Jin et al. 2021;Rho et al. 2021); we plan to extend the model to include the cooling emission in the early-phase spectral formation (see Suzuki & Maeda 2021, for the light curve simulations). Indeed, the present investigation may already place a potential constraint on the underlying mechanism for the early, bright and rapidly declining emission. Jin et al. (2021) and Rho et al. (2021) showed that the early light curve evolution of SN 2020bvc can be explained by the cooling emission resulting from the interaction between the SN ejecta and a dense and confined CSM, with the mass of ∼ 0.05 − 0.3 . The expectation from this scenario is that the high-velocity ejecta should be decelerated, and thus the formation of the high-velocity blueshifted absorption may indeed be suppressed. Taken the model POW_mix_8 as an example (with max = 80, 000 km s −1 ), Fig. 2 shows that the reverse shock will decelerate the outermost ejecta to ∼ 60, 000, 40, 000, and 30, 000 km s −1 for the CSM masses of ∼ 0.05, 0.1, and 0.3 , respectively, according to the rough estimate that the swept-up mass of the CSM and that of the ejecta are comparable. Given that the high-velocity absorption features reaching to ∼ 60, 000 − 70, 000 km s −1 are seen in its earliest spectrum, the consideration here places a strong constraint on the SN-CSM interaction scenario; the scenario is generally disfavored, with only the least massive CSM case ( ∼ < 0.05 ) still surviving as a possibility. We will investigate the origin of the early emission of SN 2020bvc in a forthcoming paper (K. Maeda, in prep.).
The same level of comparison to SN 1998bw is not possible as the early-phase spectra are observationally missing. We however note that SNe 2017iuk and 1998bw show overall similarities in the maximum-phase spectra when spectra of both SNe are available, despite the difference of ∼ 1 magnitude at the peak. SN 1998bw shows larger degree of blueshift in overall spectral shape than SN 2017iuk, which probably reflects some intrinsic difference in the ejecta kinematics (Izzo et al. 2019). The viewing angle effect is probably not strong in the maximum phase for models constructed for GRB-SNe where the SN component is quasi-spherical, and both SNe are probably viewed on-axis (Maeda et al. 2006b;Tanaka et al. 2007;Rapoport et al. 2012;Barnes et al. 2018;Shankar et al. 2021), and it would not explain the difference between the two SNe. SN 1998bw is bluer than the present models, which might be explained by a low metallicity (Rapoport et al. 2012) or by a less substantial mixing than in the 'mixed' models. In any case, the overall spectral properties are well explained by the present models.
CONCLUDING REMARKS
In the present paper, we demonstrate how the ejecta properties of GRB-SNe and engine-driven SNe Ic can be constrained through the comparison between the synthetic observables (both in photometry and spectroscopy) and the observed data, especially focusing on the initial, rising phase soon after the explosion. Based on insights obtained through a series of the hydrodynamic and radiationhydrodynamic simulations of the engine-driven SNe (Suzuki et al. 2016;Suzuki & Maeda 2017, 2022, we investigate the effects of the composition mixing, density structure, and the maximum ejecta velocity. Out findings can be summarized as follows:
(i) The different degree of the composition mixing, especially of 56 Ni, changes the rising time and the colour evolution substantially. The effect is also seen in synthetic spectra, where the line velocities are also affected, with the mixing leads to the higher velocities.
(ii) The density slope in the outermost layer affects the initial spectral evolution, where the flatter density distribution, as expected from the hydrodynamic behaviour induced by the central engine, leads to high velocities in the spectral features.
(iii) The outermost ejecta velocity can be strongly constrained from the spectroscopic data in the initial phase ( ∼ < a week since the explosion), in a way that the higher velocity leads to broader spectral features resulting in featureless spectra.
(iv) The dependence of the colour evolution on the parameters (e.g., the maximum ejecta velocity) is not necessarily monotonic. The characteristic colour evolution for different models can be understood from the spectral formation processes affecting fluxes in different bandpasses.
(v) The effects of the density structure and the maximum velocity are seen clearly in the infant phase (within a week since the explosion in the model sequence considered in the present work). The synthetic observables however eventually converge toward the maximum phase, leaving the earliest phase observation critical to diagnose the ejecta properties.
We compare the synthetic spectra with the observed spectral sequences of SNe 1998bw, 2017iuk, and 2020bvc. While our models are simplified (e.g., assuming spherically symmetric structure; see below), and not tuned to fit the data (e.g., assuming a complete mixing and investigating only a fixed set of the ejecta mass and kinetic energy), the present models reproduce the observed spectra reasonably well. The characteristic featureless spectra due to the blending of individual broad lines are well explained. The spectra in the first week are not available for SN 1998bw, and not much constraint can be placed on the density structure and maximum ejecta velocity. On the other hand, the intensive spectral sequence in the initial phase available for SN 2017iuk provides powerful diagnostics on the ejecta properties. The power-law density structure as combined with the maximum ejecta velocity of ∼ > 100, 000 km s −1 provides good reproduction of the spectral features and their evolution. The similar analysis is possible for non-GRB (or off-axis GRB) SN Ic BL 2020bvc, which also favors the existence of the high-velocity ejecta but with somewhat lower maximum velocity, i.e., 60, 000 − 80, 000 km s −1 . These analyses highlight the power of the infant-phase spectroscopic observations of GRB-SNe and SNe Ic-BL.
One important caveat in the present investigation is that we have assumed the spherically symmetric ejecta. This is indeed the case for most of the previous works; based on the success of such models and the accumulated experiences obtained there, we believe that the characteristic ejecta properties, i.e., the density and composition structures as well as the maximum velocity, are reasonably well constrained within this framework. Indeed, these 'spherical' properties may be regarded to represent those along the line of sight (thanks to the large optical depth in the initial phase up until the maximum phase). For example, a series of the modeling activities for SN 1998bw using a jet-like explosion model suggests that this argument is largely justified; the required energy may be reduced by a factor of a few as compared to the spherical model, but otherwise many conclusions based on the spherical models would not change (Maeda et al. 2006b;Maeda 2006;Maeda et al. 2006a;Tanaka et al. 2007;Rapoport et al. 2012; Barnes et al. 2018), provided that the key Engine-Driven Supernovae 11 multi-dimensional effects, i.e., the change in the density structure and mixing in the composition structure, are taken into account. GRB observations are triggered by the -ray photons in an unbiased way, and thus a search for an emerging SN component in optical wavelengths is possible starting just after the detection of the GRB. When the SN component is detectable is dependent on the competition between the decreasing contribution of the GRB afterglow and the increasing contribution of the SN component toward the maximum light. Therefore the clear detection of the SN component in the first few days after the explosion requires some specific conditions, e.g., close distance and weak GRB afterglow including possible off-axis events. So far, GRB 161219B/SN 2016jca and GRB 171205A/SN 2017iuk have provided such opportunities, with the SN spectra showing broad features available already 2 days (SN 2016jca) and even 1 day (SN 2017iuk) after the explosion. While further increasing the sample depends on luck, it is important not to miss the chance to obtain such irreplaceable data set for the next (rare) nearby GRB.
Indeed, the strategy may work even better for SNe Ic-BL without an associated GRB. Thanks to the recent advance of the new-generation, unbiased and wide-field optical surveys, e.g., Pan-STARRS (Panoramic Survey Telescope And Rapid Response System; Kaiser et al. 2002), ATLAS (Asteroid Terrestrial-impact Last Alert System; Tonry et al. 2018) and ZTF (Zwicky Transient Facility; Bellm et al. 2019;Masci et al. 2019), nearby SNe are routinely discovered in the infant phase just after the explosion. For a survey limiting magnitude of ∼ 20 mag, SNe within the distance of ∼ 100 Mpc can be discovered at the absolute magnitude of ∼ −15 mag; taking the light curve of SN 1998bw as a template, this corresponds to the magnitude within a day of the explosion, which allows a prompt spectroscopic observations to obtain the spectra within a few days, or even within a day, of the explosion. Within this distance, ∼ 100 − 200 CCSNe are expected to be discovered per year. Given the relative frequency of SNe Ic-BL being ∼ 1 − 2% among CCSNe (Graur et al. 2017), we may expect that infant spectra can be obtained for a few SNe Ic-BL per year. The model predictions in the present work can be applied to such data, which will hopefully form essential background to investigate whether and how the ejecta properties are different between GRB-SNe and non-GRB SNe Ic-BL. While being rare, such investigation will particularly important for non-GRB 'engine-driven' SNe Ic-BL, such as SNe 2009bb (Soderberg et al. 2010;Pignata et al. 2011), 2012ap (Margutti et al. 2014Milisavljevic et al. 2015), 2014ad (Sahu et al. 2018;Kwok et al. 2022), and 2020bvc (Ho et al. 2020;Izzo et al. 2020;Rho et al. 2021).
Figure 1 .
1et al. 2015; Lyman et al. 2016). The density structure of Models CO138 (black-dashed) and POW (with the power-law index of −6; black-solid). The outermost velocity, max , is set as 100, 000 km s −1 in this figure. The density is scaled at 1 day since the explosion. The mass fractions of 56 Ni (red), Si (green), and O (blue) are shown, for Models CO138_nomix_10 (dashed) and POW_mix_10 (solid). The 'representative' SN Ic structure taken from Fang & Maeda
Figure 3 .
3The light curves of Models CO138_nomix_10 (left), CO138_mix_10 (middle), and POW_mix_10 (right). The light curves are shown for the , ,
Figure 4
4Figure 4. The colour evolution of the model sequences CO138_nomix (brown) and CO138_mix (cyan and magenta for max = 80, 000 and 100, 000 km s −1 , respectively), and POW_mix sequence (blue, green, orange, and red for max = 60, 000, 80, 000, 100, 000, and 120, 000 km s −1 , respectively).
Figure 5 .
5The colour evolution of SNe 1998bw (red), 2017iuk (blue), and 2020bvc (black), overplotted with the CO138_nomix model and the POW_mix sequence ( max = 60, 000, 80, 000, 100, 000, and 120, 000 km s −1 ; seeFig. 4) as shown by grey lines. For SN 1998bw, the colours computed by the magnitudes in the two bands in the close proximity in the observed epochs are shown in the filled squares, while the colours that involve interpolation are shown by open squares. For SNe 2017iuk and 2020bvc, the -and -band magnitudes are obtained with the SWIFT UVOT. For SN 2017iuk, the − colour is shown using the data obtained with the GROND -band filter. For SN 2020bvc, the -band magnitude is replaced by the ZTF -band magnitude.
Figure 6 .
6The synthetic spectra in the infant to early phases, for Models CO138_nomix (brown), CO138_mix with max = 80, 000 km s −1 (cyan) and 100, 000 km s −1 (magenta), POW_mix with max = 60, 000 km s −1 (blue), 80, 000 km s −1 (green), 100, 000 km s −1 (orange), and 120, 000 km s −1 (red). Shown here for a demonstration purpose are the spectra of GRB-SN 2017iuk(black; Izzo et al. 2019), GRB-SN 1998bw (grey; Patat et al. 2001), and SN Ic-BL 2020bvc (grey: Hiramatsu et al. 2020) at similar epochs (when available) on the bottom of each panel. The spectrum of SN 2020bvc is subtracted by an arbitrary power-law continuum to highlight the spectral features. 2001), 2007iuk (Izzo et al. 2019), and SN Ic-BL 2020bvc
Figure 7 .
7The same asFigure 6, but for the maximum-light phases.
Table 1 .
1Models: For the density structure, 'CO138' adopts the original 1D hydrodynamic model, while 'POW' adopts the power-law distribution ( ∝ −6 ). For the composition structure, 'nomix' adopts the original stratified structure, while 'mix' adopts the homogeneous mixing.Model
Density
Mixing
max (10 9 cm s −1 )
CO138_nomix_10 Original
No
10
CO138_mix_8
Original
Yes
8
CO138_mix_10
Original
Yes
10
POW_mix_6
Power
Yes
6
POW_mix_8
Power
Yes
8
POW_mix_10
Power
Yes
10
POW_mix_12
Power
Yes
12
Note that the present model is applicable both to the 'jet-driven explosion' (e.g.,Suzuki & Maeda 2022) and the 'quasi-spherical explosion followed by an additional energy input by the central engine' (e.g.,Suzuki & Maeda 2017).
MNRAS 000, 1-12(2022)
The synthetic spectra for the model CO138_nomix are shown only on day 7 and thereafter. The model has essentially zero flux before day 7(Fig. 3).
The spectra of SNe 1998bw and 2020bvc are downloaded from the WIS-eRep (Yaron & Gal-Yam 2012); https://www.wiserep.org/. The spectra of SN 2017iuk are available on GRBSpec (de Ugarte Postigo et al. 2014); http://grbspec.eu/?index.php.
Note that the photospheric velocities derived for SN 2017iuk were 59, 000 and 53, 000 km s −1 on days 1 and 2, respectively(Izzo et al. 2019). These are already below the minimum value of max (60,000 km s −1 ) examined in the present work.
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-12(2022)
ACKNOWLEDGEMENTSThe authors thank the anonymous referee for her/his constructive and insightful comments. K.M. acknowledges support from the Japan Society for the Promotion of Science (JSPS) KAKENHI grant JP18H05223, JP20H00174, and JP20H04737. A.S. acknowledges support from the JSPS KAKENHI grant JP19K14770 and JP22K03690. Numerical computations were carried out on Cray XC50 at Center for Computational Astrophysics, National Astronomical Observatory of Japan. We also used the Yukawa Institute Computer Facility. Some data presented in this work are obtained from WISeREP (https://www.wiserep.org) and GRBspec database (http://grbspec.eu/?index.php).DATA AVAILABILITYThe simulated light curves and spectra are available upon request to K.M.
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| [] |
[
"EXPLICIT SMOOTH REAL ALGEBRAIC FUNCTIONS WHICH MAY HAVE BOTH COMPACT AND NON-COMPACT PREIMAGES ON NON-SINGULAR REAL ALGEBRAIC MANIFOLDS",
"EXPLICIT SMOOTH REAL ALGEBRAIC FUNCTIONS WHICH MAY HAVE BOTH COMPACT AND NON-COMPACT PREIMAGES ON NON-SINGULAR REAL ALGEBRAIC MANIFOLDS"
] | [
"Naoki Kitazawa "
] | [] | [] | In our previous preprint, which is withdrawn due to an improvement, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on non-singular real algebraic manifolds. This paper presents its variant. Our result is new in obtaining nonproper smooth real algebraic functions on non-singular real algebraic manifolds satisfying explicit conditions on (non-)compactness of preimages whereas previously the manifolds are only semi-algebraic.Explicitly, this mainly contributes to two different regions. One is singularity theory of differentiable maps and applications to differential topology. More precisely, construction of nice smooth maps with desired preimages. The other is real algebraic geometry. More precisely, explicit construction of smooth real algebraic functions and maps: we can know the existence and consider approximations of smooth maps by maps of such classes in considerable cases. | null | [
"https://export.arxiv.org/pdf/2304.02372v2.pdf"
] | 257,952,222 | 2304.02372 | 03cf5c25fbecf3980d35fc9d48138b191c354a61 |
EXPLICIT SMOOTH REAL ALGEBRAIC FUNCTIONS WHICH MAY HAVE BOTH COMPACT AND NON-COMPACT PREIMAGES ON NON-SINGULAR REAL ALGEBRAIC MANIFOLDS
24 May 2023
Naoki Kitazawa
EXPLICIT SMOOTH REAL ALGEBRAIC FUNCTIONS WHICH MAY HAVE BOTH COMPACT AND NON-COMPACT PREIMAGES ON NON-SINGULAR REAL ALGEBRAIC MANIFOLDS
24 May 2023
In our previous preprint, which is withdrawn due to an improvement, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on non-singular real algebraic manifolds. This paper presents its variant. Our result is new in obtaining nonproper smooth real algebraic functions on non-singular real algebraic manifolds satisfying explicit conditions on (non-)compactness of preimages whereas previously the manifolds are only semi-algebraic.Explicitly, this mainly contributes to two different regions. One is singularity theory of differentiable maps and applications to differential topology. More precisely, construction of nice smooth maps with desired preimages. The other is real algebraic geometry. More precisely, explicit construction of smooth real algebraic functions and maps: we can know the existence and consider approximations of smooth maps by maps of such classes in considerable cases.
Introduction.
Our paper is on new explicit construction of smooth real algebraic functions and maps on non-singular real algebraic manifolds. This is also a variant [11], which is withdrawn due to an improvement. Presentations of [11] and our present paper are similar in considerable scenes.
1.1. Two interest related to our study: one is from singularity theory of differentiable maps and its applications to differential topology and the other is from real algebraic geometry. Related to our study, we present a study on singularity theory of differentiable maps and applications to differential topology. Previously, the author has constructed smooth functions with prescribed preimages explicitly in [5,6,9]. These are answers to a problem of the author, a revised version of Sharko's problem. Sharko asks whether we can obtain smooth functions whose Reeb graphs are as desired. [16,18] are among explicit answers to Sharko's original question and they have also motivated the author to obtain the results. The Reeb graph of a smooth function is a graph obtained as the space of connected components of preimages of smooth functions of some nice wide class. Such a class is studied in [22] for example. Reeb graphs are classical objects already in [21]. They inherit several important topological information of the manifolds.
We are also interested in real algebraic geometry. According to general theory, pioneered by Nash for example, smooth manifolds are so-called non-singular real algebraic manifolds and we can approximate smooth maps by smooth real algebraic maps and know the existence in considerable cases. [1,2,14,15,19,24,28] explain about this for example. Our interest lies in explicit construction of explicit smooth real algebraic functions or maps of non-positive codimensions. Natural projections of spheres embedded naturally in the one-dimensional higher Euclidean spaces are simplest examples on closed manifolds. As functions regarded as generalized cases, see [17,20,27] for example. They are given by suitable real polynomials.
Under these backgrounds, we consider the following problem.
Problem 1. On the other hands, as natural questions on geometry, explicit global structures of these real algebraic functions and properties are hard to understand. For example, can we know about preimages? As another example, can we know about the real polynomials for our desired zero sets and real algebraic manifolds? [7] is one of our pioneering study. [8] is the abstract of our related talk in a conference. The work is followed by us in [10,11,12]. Our new study is on smooth functions which may be non-proper whereas these studies show explicit construction of real algebraic functions on non-singular real algebraic closed manifolds.
1.2. Manifolds and maps. Let X be a topological space homeomorphic to some cell complex of a finite dimension. We can define its dimension dim X as a unique integer. A topological manifold is well-known to be homeomorphic to some CW complex. A smooth manifold is well-known to be homeomorphic to some polyhedron. We can also define the structure of a certain polyhedron for a smooth manifold in a canonical and unique way. This is a so-called PL manifold. We do not need to understand such theory precisely here.
Let R k denote the k-dimensional Euclidean space, which is a simplest k-dimensional smooth manifold. This is also the Riemannian manifold endowed with the standard Euclidean metric. Let R := R 1 and Z ⊂ R denote the set of all integers. Let N ⊂ Z denote the set of all positive ones. For each point x ∈ R k , let ||x|| ≥ 0 denote the distance between x and the origin 0 where the metric is the standard Euclidean metric. This is also naturally a (non-singular) real algebraic manifold. This is also a so-called (non-singular) Nash manifold and a real analytic manifold. This real algebraic manifold is also the k-dimensional real affine space. Let S k := {x ∈ R k+1 | ||x|| = 1} denote the k-dimensional unit sphere. This is a k-dimensional smooth compact submanifold of R k+1 . It has no boundary. It is connected for any positive integer k ≥ 1. It is a discrete set with exactly two points for k = 0. It is a non-singular real algebraic set, which is the zero set of the real poly-
nomial ||x|| 2 = Σ k+1 j=1 x j 2 with x := (x 1 , · · · , x k+1 ). Let D k := {x ∈ R k | ||x|| ≤ 1} denote the k-dimensional unit disk.
It is a k-dimensional smooth compact and connected submanifold of R k for any non-negative integer k > 0. It is also a semialgebraic set.
Let c : X → Y be a differentiable map from a differentiable manifold X into another differentiable manifold Y . x ∈ X is a singular point of the map if the rank of the differential at x is smaller than the minimum between the dimensions dim X and dim Y . We call c(x) a singular value of c. Let S(c) denote the singular set of c. It is defined as the set of all singular points of c. Unless otherwise stated, here, differentiable maps are smooth maps, which are maps of the class C ∞ .
A canonical projection of the Euclidean space R k is defined as the smooth surjective map mapping each point x = (x 1 , x 2 ) ∈ R k1 × R k2 = R k to the first component x 1 ∈ R k1 with the conditions on the dimensions k 1 , k 2 > 0 and k = k 1 + k 2 . Let π k,k1 : R k → R k1 denote this map. We can define a canonical projection of the unit sphere S k−1 as the restriction of this there.
1.3. Our main results. Let X ⊂ R be a subset. Let a ∈ X. Let X a ⊂ X denote the set of all elements of X smaller than or equal to a.
The real algebraic set X ∈ R k defined by l ≥ 0 real polynomials is the intersection of the zero sets for these l real polynomials. In the case l = 0, it is R k . In short, (non-singular) real algebraic manifolds are based on such sets and (non-singular) Nash manifolds are based on semi-algebraic sets, defined by considering not only the zero sets of real polynomials, but also the inequalities.
Our real algebraic manifolds are unions of connected components of some real algebraic sets defined by finitely many real polynomials unless otherwise stated. Non-singular real algebraic manifolds are defined by these polynomials. More precisely, such notions are defined by the ranks of the maps defined canonically from these polynomials with implicit function theorem.
Main Theorem 1. Let l > 1 be an integer. Let {t j } l j=1 be an increasing sequence of real numbers. Let l N l−1 : N l−1 → {0, 1} be a map. Let m be a sufficiently large integer. Then we have a suitable m-dimensional non-singular real algebraic connected manifold M with no boundary and a smooth real algebraic function f : M → R enjoying the following properties.
(
1) f (M ) = [t 1 , t l ] and the image f (S(f )) of the singular set of f is {t j } l j=1 . (2) f −1 (t) is closed and connected if t ∈ (t j , t j+1 ) and l N l−1 (j) = 0. f −1 (t)
is connected and non-compact and it has no boundary if t ∈ (t j , t j+1 ) and l N l−1 (j) = 1. f −1 (t) is connected and homeomorphic to a CW complex whose dimension is at most m − 1 for any t ∈ f (M ). Furthermore, for a suitable integer n > 1 and another sufficiently large integer m > n, we can suitably have a desired m-dimensional manifold M and a desired function f : M → R as the composition of a nice smooth real algebraic map on M into R n with a canonical projection. The nice smooth map into R n is presented as a very explicit map in Main Theorem 2, later.
Main Theorem 2 shows that the map into R n can be constructed as one having a globally simple structure. Furthermore, this is very explicit and we can know the real polynomial for the definition of the real algebraic manifold (set). This is a variant of our main result of [12], which is withdrawn due to an improvement. There we have constructed the manifolds as Nash ones and the functions and maps as the restrictions of smooth real algebraic functions and maps to semi-algebraic sets. Main Theorems here are also motivated by some differential topological viewpoint or [6] and there we have constructed smooth functions with prescribed preimages on non-compact manifolds with no boundaries. The functions of [6] are non-proper in general. We also construct smooth functions which are not real analytic there.
The next section explains about Main Theorems including our proofs. As our related previous work, we have constructed our functions and maps avoiding existence theory and approximations. The third section is a kind of appendices and related problems are presented.
Conflict of interest.
The author was a member of the project JSPS Grant Number JP17H06128 and the project JSPS KAKENHI Grant Number JP22K18267 "Visualizing twists in data through monodromy" (Principal Investigator: Osamu Saeki). Their support has helped the author to do this study. The author is a researcher at Osaka Central Advanced Mathematical Institute (OCAMI researcher). Note that he is not employed there. This is for our studies and helps our studies.
Data availability.
Data essentially supporting our present study are all in the paper.
On Main Theorems.
2.1. Important subsets in the real affine spaces. We introduce several subsets in the real affine spaces. They are important in proving Main Theorems.
2.1.1. L P,p1,p2 . For distinct two points p 1 , p 2 ∈ R 2 , let L P,p1,p2 ⊂ R 2 denote the straight line, which is uniquely defined and regarded as a copy of the 1-dimensional real affine space embedded in the 2-dimensional real affine space R 2 by the canonically defined smooth real algebraic embedding. This is also an affine subspace in R 2 , defined later in Definition 1.
(x 1 , x 2 ) ∈ R 2 | (x 1 − a)(x 2 − b) = c}.
Let it be denoted by C H,[a,∞),b,c . This is the real algebraic set defined by the single real polynomial and this is a 1-dimensional non-singular real algebraic manifold. It consists of exactly two connected components. These connected components are denoted by
C H+,[a,∞),b,c := {(x 1 , x 2 ) ∈ C H,[a,∞),b,c | x 1 < a, x 2 > b} and C H−,[a,∞),b,c := {(x 1 , x 2 ) ∈ C H,[a,∞),b,c | x 1 > a, x 2 < b}, respectively.
Last, this is also for a hyperbola.
2.1.4. B E,{aj} k j=1 ,{rj } k j=1 . Let {a j } k j=1 be a sequence of real numbers. Let {r j } k j=1
be a sequence of positive numbers. We consider the subset in R k represented as
{(x 1 , · · · x k ) ∈ R k | Σ k j=1 (xj −aj ) 2 rj ≤ 1}. Let it be denoted by B E,{aj } k j=1 ,{rj} k j=1
. This is a k-dimensional smooth manifold diffeomorphic to the unit disk D k . This is also a semi-algebraic set. Its boundary is a (k − 1)-dimensional non-singular real algebraic manifold and the real algebraic set defined by the single real polynomial Σ k j=1 (xj −aj ) 2 rj − 1. This is diffeomorphic to S k−1 of course. This is for a k-dimensional ellipsoid and its boundary.
2.1.5. Affine transformations and some classes of sets. Respecting these subsets for example, we define some important subsets. We also regard R k as the k-dimensional real vector space in the canonical way. A linear transformation on a linear space means a linear map which is also a linear isomorphism there. An affine transformation on the real affine space R k is a map represented as the composition of a linear transformation on R k with an operation considering the sum with a fixed value a ∈ R k or the operation mapping x to x + a.
Definition 1.
(1) An affine subspace in R k is R k itself or a non-singular real algebraic submanifold embedded by the smooth real algebraic embedding mapping x ∈ R k1 to (x, 0) ∈ R k1 × R k2 = R k with the conditions k 1 , k 2 > 0 and k 1 + k 2 = k. It is also an affine subspace in R k if it is mapped onto such a subspace by some affine transformation on the real affine space R k .
(2) A (k − 1)-dimensional cylinder of a hyperbola in R k is a non-singular real algebraic manifold and it is also the real algebraic set defined by a naturally defined single real polynomial and represented as a subset in 2.1.2 or 2.1.3 in the case k = 2 or a form of the product of a subset of 2.1.2 or 2.1.3 in R 2 and R k−2 in the case k ≥ 3. A non-singular real algebraic submanifold mapped onto such a subspace by some affine transformation on the real affine space R k is also called in this way.
(3) A k-dimensional ellipsoid in R k is B E,{aj} k j=1 ,{rj} k j=1
or a semi-algebraic set mapped onto such a subspace by some affine transformation on the real affine space R k .
(4) A k-dimensional cylinder of some ellipsoid in R k is B E,{aj} k ′ j=1 ,{rj } k ′ j=1 × R k−k ′ ⊂ R k ′ × R k−k ′ with
some integer satisfying 1 ≤ k ′ < k or a semialgebraic set mapped onto such a subspace by some affine transformation on the real affine space R k .
2.2.
A proposition for our proof of Main Theorems. For one of our main ingredients in our paper, we review construction of smooth real algebraic maps first presented in [7] and followed by [11,12] for example. For a finite set X, let |X| ∈ N {0} denote its size. Definition 2. Let D be a connected open set in R n enjoying the following properties.
(1) There exists a positive integer l > 0.
(2) There exists a family {f j } l j=1 of l real polynomials with n variables.
(3) D := l j=1 {x ∈ R n | f j (x) > 0} and for the closure D of D, D = l j=1 {x ∈ R n | f j (x) ≥ 0}. They are also semi-algebraic sets. (4) S j is a connected component of the real algebraic set {x ∈ R n | f j (x) = 0}
defined by f j (x) and a non-singular connected real algebraic hypersurface.
(5) ({x ∈ R n | f j (x) = 0} − S j ) D is empty and D is in R n − ({x ∈ R n | f j (x) = 0} − S j ),
which is an open set. (6) Distinct hypersurfaces in the family {S j } intersect satisfying the following conditions. In other words, "transversality" is satisfied.
(a) Each intersection j∈Λ S j (Λ ⊂ N l ) is empty or a non-singular real algebraic hypersurface with no boundary and of dimension n − |Λ| for each non-empty set Λ.
(b) Let e j : S j → R n denote the canonically defined smooth real algebraic embedding. For each intersection j∈Λ S j as before and each point p there, the following two agree.
(i) The image of the differential of the canonically defined embedding j∈Λ e j of j∈Λ S j at p. (ii) The intersection of the images of all differentials of the embeddings in {e j } j∈Λ at p.
Then D is said to be a normal and convenient domain. We also call it an NCD.
Remark 1. "Normal and convenient domain" or "NCD" seems to be identical to our paper, in addition to the withdrawn paper of us [12].
Remark 2. In Definition 2, in this case of non-singular hypersurfaces S j , so-called normal vectors at each point in each intersection can be chosen as ones which are mutually independent. Of course, the number of the mutually independent normal vectors are same as the number of the distinct hypersurfaces intersecting there.
Our transversality can be also discussed in this way.
The following reviews main ingredients of [7,11,12]. (2) For each point p ∈ R n , the preimage f D −1 (p) is empty or diffeomorphic to the product of finitely many unit spheres and at most (m − n)-dimensional.
Proof. This essentially reviews a main result or Main Theorem 1 of [11] and [12].
For example, we abuse the notation such as one from Definition 2. For local coordinates and points for example, we use the notation like x := (x 1 , · · · , x k ) where k > 0 is an arbitrary positive integer.
We define a set S :
= {(x, y) ∈ D × R m−n+l ⊂ R n × R m−n+l = R m+l | f j (x 1 , · · · , x n ) − ||y j || 2 = 0, j ∈ N l }
where y j := (y j,1 , · · · , y j,dj ) ∈ R dj and y := (y 1 , · · · , y l ) with a suitable positive integer d j > 0.
We investigate the function defined canonically from the real polynomial f j (x 1 , · · · , x n )− Σ dj j ′ =1 y j,j ′ 2 . We investigate its partial derivative for variants x j and y j,j ′ . We also show that we can apply implicit function theorem to each point of the set S to show that S is non-singular. Case 1. The case of a point (x 0 , y 0 ) ∈ S such that y 0 is not the origin.
By the assumption f j (x 0 ) > 0 for each j ∈ N l . Here we calculate the partial derivative of the function for each variant y j,j0 , at (x 0 , y 0 ). For our notation on y 0 , we introduce y 0 := (y 0,1 , · · · y 0,l ) and y 0,j := (y 0,j,1 , · · · , y 0,j,dj ) ∈ R dj . As a result, the value is 2y j,j0,0 = 2y 0,j,j0,0 = 0 for some variant y j,j0,0 with j 0 = j 0,0 . We calculate the partial derivative of the function for each variant y j ′ ,j0 ′ satisfying j ′ = j. The value is 2y j ′ ,j0 ′ = 2y 0,j ′ ,j0 ′ = 0 for any variant y j ′ ,j0 ′ satisfying j ′ = j.
Based on this, we explain about the map into R l obtained canonically from the family of the l functions defined from the real polynomials. The differential of the restriction of the function defined canonically from the real polynomial is not of rank 0. This is not a singular point of this function defined from this polynomial. The map into R l obtained canonically from the family of such l functions is of rank l.
Case 2. The case of a point (x O , y O ) ∈ S such that y O is the origin.
For our notation on y O , we introduce y O := (y O,1 , · · · y O,l ) and y O,j := (y O,j,1 , · · · , y O,j,dj ) ∈ R dj . In other words, "0" in "y 0 " of Case 1 is changed into "O". We calculate the partial derivative of the function for each variant y j ′ ,j0 ′ satisfying j ′ = j, at (x O , y O ). The value is 2y j ′ ,j0 ′ = 2y O,j ′ ,j0 ′ = 0 for any variant y j ′ ,j0 ′ satisfying j ′ = j.
We may suppose the existence on some non-empty subset J with the property that x O ∈ S j for each j ∈ J ⊂ N l and that x O / ∈ S j for each j / ∈ J. By the assumption, for the real polynomial f j1 (x), f j1 (x O ) > 0 for each j 1 / ∈ J. Here we calculate the partial derivative of the function defined from the real polynomial f jJ (x) − ||y jJ || 2 for each j J ∈ J and each variant x j (1 ≤ j ≤ n). The function defined canonically from the real polynomial f jJ has no singular points on the hypersurface S jJ for each j J ∈ J. This is since S j is non-singular for all j ∈ N l . From this, the partial derivative of the function defined from the real polynomial
f jJ (x) − ||y jJ || 2 is not 0 for some variant x jJ,O (1 ≤ j J,O ≤ n) with j J ∈ J.
We explain about the map into R |J| obtained canonically from the |J| real polynomials corresponding canonically to the |J| variants. The rank of the differential at the point is |J|. This comes from the assumption on the "transversality" for the hypersurfaces S j in Definition 1.
We consider the canonically obtained map into R l−|J| similarly respecting the remaining real polynomials. The rank of the differential at the point is l − |J|. We can know this from the fact that for each of these l − |J| polynomials the partial derivative by some variant y j1,j1,0 ′ satisfying j 1 / ∈ J is not 0. We can also have such a case for the suitably and uniquely chosen j 1 / ∈ J and some suitably chosen integer 1 ≤ j 1,0 ′ ≤ d j1 which may not be unique in general. We cannot have such a case for any integer j 1 ∈ N l − J and any integer 1 ≤ j 1,0 ′ ≤ d j1 . This also comes from an essential argument in Case 1. This is also presented in the beginning of our argument of Case 2 essentially.
Integrating these arguments here, we have a result as in Case 1.
By our assumption and definition, for each point x ′ in R n − D sufficiently close to D, we can define the set
S x ′ := {(x ′ , y) ∈ R n × R m−n+l ⊂ R n × R m−n+l = R m+l | f j (x 1 , · · · , x n ) − ||y j || 2 = 0, j ∈ N l }.
We can also see that it is empty. This with implicit function theorem yields the fact that the set S is an m-dimensional smooth closed and connected manifold and a non-singular real algebraic manifold. It is also a connected component of the real algebraic set defined by the l real polynomials. We can also put M := S and define f D : M → R n as the restriction of the canonical projection π m+l,n . This completes the proof.
2.3. Some important NCDs. The complementary set of an affine subspace L P,p1,p2 ⊂ R 2 in R 2 consists of exactly two connected components. One of this is regarded as a connected component represented by either of the following two forms. Figure 1. R H,(−∞,a],0,c and some important (sub)sets of R 2 . We can introduce the following subsets in R n as the closures of explicit NCDs. They are important in our arguments.
• {(x 1 , x 2 ) ∈ R 2 | x 2 ≥ a 1 x 1 + a 2 } where a 1 , a 2 ∈ R. → ↓ ↑• {(x 1 , x 2 ) ∈ R 2 | x 1 ≥ a} where a ∈ R.2.3.1. R +,{s1,j } 3
j=1 ,s2,s and R −,{s1,j } 3 j=1 ,s2,s . Let s 1,1 < s 1,2 < s 1,3 and s 2 , s > 0 be real numbers. We also assume the following. We define R +,{s1,j } 3 j=1 ,s2,s as the intersection L P+,(s1,1,0),(s1,1,1) L P+,(s1,1,0),(s1,2,0) L P+,(s1,2,0),(s1,3,s2) R H,(−∞,s1,2],0,s .
As before, let s 1,1 < s 1,2 < s 1,3 and s 2 , s > 0 be real numbers. We also assume the following.
A proof of Main Theorems.
Main Theorem 2. In Main Theorem 1, we can have the function as the composition of a suitable smooth real algebraic map into R n enjoying the following properties with the canonical projection π n,1 : R n → R, mapping (x 1 , · · · x n ) to x 1 .
(1) The image of the map is the closure D of an NCD D where we abuse the notation used in Definition 2 for example. Here let {t i(j) } l ′ j=1 ⊂ {t j } l j=1 be a subsequence of numbers consisting of all numbers i(j) ∈ N l−1 such that l N l−1 (i(j)) = 1. l ′ is a non-negative integer which may be 0. In the case l ′ = 0, the sequence is empty.
We define n and D as follows. We put D 0 as the interior of L P+,(t1,0),(t1,1) L P+,(t1,0),(t l ,0) L P−,(t l ,0),(t l ,1) L P−,(t1,1),(t l ,1) ⊂ R 2 and we can see that D 0 is an NCD. We can define D 1 as an open set of R 2 × R l ′ +1 = R × R l ′ +2 = R n enjoying the following rule. Here we put t j,1 := t i(j) and t j,2 := t i(j)+1 . Let x := (x 1 , · · · , x l ′ +3 ) ∈ R l ′ +3 . x ∈ D 1 if and only if the following three hold.
•
(x 1 , x 2 ) ∈ D 0 . • (x 1 , x j+2 ) is in the interior of R {t j,j ′ } 2 j ′ =1 ,0,1 for each j ∈ N l ′ . • −R < x l ′ +3 < R for some positive real number R.
For each integer 1 ≤ j ≤ l − 3, we choose a suitable and sufficiently small
B E,{a j,j ′ } l ′ +3 j ′ =1 ,{r j,j ′ } l ′+3 j ′ =1
satisfying a j,1 = . We can choose these n-dimensional ellipsoids in R n disjointly in the interior of D 1 and we do. We remove the interiors. Our desired set D is the interior of the resulting set.
Proposition 1 yields our desired map f D : M → R n . We explain about some important remarks. We can know types of each resulting hypersurface S j by the construction. We can consider the natural connections on Euclidean spaces as the natural Riemannian manifolds and consider the notion of (mutually) "parallel" objects such as parallel tangent vectors and parallel subsets for example. The following list shows all of these hypersurfaces S j here. Here let e i ∈ R l ′ +3 denote the tangent vector at the origin 0 ∈ R l ′ +3 represented as the vector whose i-th component is 1 and whose i ′ -th component is 0 for any integer i ′ = i satisfying 1 ≤ i ′ ≤ l ′ + 3.
• Two mutually disjoint affine subspaces parallel to the affine subspace {0} × R l ′ +2 . At each point there, each normal vector is regarded to be parallel to some vector represented by the form te 1 with some number t = 0. • Two mutually disjoint affine subspaces parallel to the affine subspace R × {0} × R l ′ +1 . At each point there, each normal vector is regarded to be parallel to some vector represented by the form te 2 with some number t = 0. • Two mutually disjoint affine subspaces parallel to the affine subspace R l ′ +2 × {0}. At each point there, each normal vector is regarded to be parallel to some vector represented by the form te l ′ +3 with some number t = 0.
• Exactly l ′ affine subspaces. The i-th affine subspace in these l ′ affine subspaces is equal to R i+1 × {0} × R l ′ −i+1 . At each point there, each normal vector is regarded to be parallel to some vector represented by the form te i+2 with some number t = 0. • Exactly 2l ′ cylinders of hyperbolas. For each integer 1 ≤ i ≤ l ′ , the (2i − 1)th hypersurface and the (2i)-th one here are mutually disjoint. These two are represented as subsets of R l ′ +3 obtained in the following steps.
-We choose two subsets C H+,[ti,1,∞),0,1 and C H+,(−∞,ti,2],0,1 . Consider the product of each subset and R l ′ +1 , This new subset is a subset of
R 2 × R l ′ +1 = R l ′ +3 .
-We map the previously obtained subsets by an affine transformation σ : R l ′ +3 → R l ′ +3 defined uniquely in the following way. * For each x ∈ R l ′ +3 , the 1st component of σ(x) does not change under the transformation σ and it is equal to x 1 . * For each x ∈ R l ′ +3 , the j-th component of σ(x) may change under the transformation σ and equal to x j+1 for 2 ≤ j ≤ i + 1. * For each x ∈ R l ′ +3 , the (i+2)-th component of σ(x) may change under the transformation σ and equal to x 2 . * For each x ∈ R l ′ +3 , the j-th component of σ(x) does not change under the transformation σ and equal to x j for i + 3 ≤ j ≤ l ′ + 3. At each point there, each normal vector is regarded to be parallel to some vector represented by the form t i,1 e 1 +t i,2 e i+2 with some numbers t i,1 = 0 and t i,2 = 0. Furthermore, these two are also apart from the i-th affine subspace equal to R i+1 × {0} × R l ′ −i+1 presented before.
• The boundaries of some n-dimensional ellipsoids in R n . The n-dimensional ellipsoids in R n are mutually disjoint and they are also apart from the other hypersurfaces presented here.
We discuss transversality. Observe the list and see the normal vectors explicitly. Recall also Remark 2 for example. We can see that the transversality is satisfied.
We discuss singularities of the function f . For the boundary of each n-dimensional ellipsoid in R n , only points regarded as the two "poles" are regarded as points in this connected component of D−D whose preimages (for our map f D ) contain some singular points of the function f := π n,1 • f D . On singular points of the function f , we can explain similarly about points in the two real affine subspaces in R l ′ +3 parallel to {0} × R l ′ +2 or the set {t 1 , t l } × R l ′ +2 . In the image D ⊂ R n of the map f D , except such points, the preimages contain no singular points of f . This completes the proof.
We can have another similar result.
Main Theorem 3. In Main Theorem 1, let l = 3. Instead of Main Theorem 2, we can have the function as the composition of a suitable smooth real algebraic map into R n enjoying the following properties with the canonical projection π n,1 : R n → R, mapping (x 1 , · · · x n ) to x 1 .
(1) The image of the map is the closure D of an NCD D where we abuse the notation used in Definition 2 for example. (2) D − D is a union of finitely many (n − 1)-dimensional smooth connected submanifolds of S j as in Definition 2. Each S j is a connected component of the real algebraic set defined by a single real polynomial of degree at most 2. More precisely, The family {S j } of hypersurfaces satisfies the following conditions. (a) It is an affine subspace of R n , a connected component of an (n − 1)dimensional cylinder of a hyperbola in R n , the boundary of some ndimensional ellipsoid in R n , or the boundary of some n-dimensional cylinder of some ellipsoid in R n represented as the product of a 2dimensional ellipsoid in R 2 and R n−2 . (b) If l N l−1 (N l−1 ) = {0}, then each S j is the boundary of some n-dimensional ellipsoid and these hypersurfaces are mutually disjoint. (c) If l N l−1 (N l−1 ) = {0, 1}, then each hypersurface S j0 which is the boundary of some n-dimensional ellipsoid and any hypersurface S j satisfying j = j 0 are mutually disjoint.
A proof. We obtain a desired NCD D as follows.
Case 1. The case l = 2.
In the case l N1 (1) = 0, we put n = 2 and define D as the interior of in the interior disjointly for 1 ≤ j ≤ l − 3. We remove their interiors from B E,{a 0,j ′ } 3
B E,{a 0,j ′ } 2 j ′ =1 ,{r 0,j ′ } 2 j ′ =1j ′ =1 ,{r 0,j ′ } 3 j ′ =1 .
We consider the case l N l−1 (N l−1 ) = {0, 1}. Here, as in the proof of Main Theorem 2, let {t i(j) } l ′ j=1 ⊂ {t j } l j=1 be a subsequence of numbers consisting of all numbers i(j) ∈ N l−1 such that l N l−1 (i(j)) = 1. l ′ is a positive integer.
We define n and D as follows. We put D 0 as the interior of
B E,{a 0,j ′ } 2 j ′ =1 ,{r 0,j ′ } 2 j ′ =1
satisfying a 0,1 = t1+t l 2 and r 0,1 = (t l −t1) 2 4 and we can see that D 0 is an NCD. We can define D 1 as an open set of R 2 × R l ′ +1 = R × R l ′ +2 = R n enjoying the following rule. Here we put t j,1 := t i(j) and t j,2 := t i(j)+1 . Let x := (x 1 , · · · , x l ′ +3 ) ∈ R l ′ +3 . x ∈ D 1 if and only if the following three hold.
• (x 1 , x 2 ) ∈ D 0 .
• (x 1 , x j+2 ) is in the interior of R {t j,j ′ } 2 j ′ =1 ,0,1 for each j ∈ N l ′ . • −R < x l ′ +3 < R for some positive real number R.
For each integer 1 ≤ j ≤ l − 3, we choose a suitable and sufficiently small
B E,{a j,j ′ } l ′ +3 j ′ =1 ,{r j,j ′ } l ′+3 j ′ =1
satisfying a j,1 = tj+1+tj+2 2 and r j,1 = (tj+2−tj+1) 2 4 . We can choose these n-dimensional ellipsoids in R n disjointly in the interior of D 1 and we do. We remove the interiors. Our desired set D is the interior of the resulting set.
Proposition 1 yields our desired map f D : M → R n . We explain about some important remarks. We can know types of each resulting hypersurface S j by the construction. We can consider the natural connections on Euclidean spaces as the natural Riemannian manifolds as in the proof of Main Theorem 2. The following list shows all of these hypersurfaces S j here. As in the proof of Main Theorem 2, let e i ∈ R l ′ +3 denote the tangent vector at the origin 0 ∈ R l ′ +3 represented as the vector whose i-th component is 1 and whose i ′ -th component is 0 for any integer i ′ = i satisfying 1 ≤ i ′ ≤ l ′ + 3.
• The boundary of the (l ′ + 3)-dimensional cylinder ∂D 0 × R l ′ +1 ⊂ R l ′ +3 of the ellipsoid D 0 in R l ′ +3 . At each point there, normal vectors are regarded to be parallel to some vector represented by the form t 1 e 1 + t 2 e 2 with some pair (t 1 , t 2 ) = (0, 0) of real numbers. • Two mutually disjoint affine subspaces parallel to the affine subspace R l ′ +2 × {0}. At each point there, normal vectors are regarded to be parallel to some vector represented by the form te l ′ +3 with some number t = 0. • Exactly l ′ affine subspaces. The i-th affine subspace in these l ′ affine subspaces is equal to R i+1 × {0} × R l ′ −i+1 . At each point there, normal vectors are regarded to be parallel to some vector represented by the form te i+2 with some number t = 0. • Exactly 2l ′ cylinders of hyperbolas. For each integer 1 ≤ i ≤ l ′ , the (2i − 1)th hypersurface and the (2i)-th one here are mutually disjoint. These two are represented as subsets of R l ′ +3 obtained in the following steps.
-We choose two subsets C H+,[ti,1,∞),0,1 and C H+,(−∞,ti,2],0,1 . Consider the product of each subset and R l ′ +1 , This new subset is a subset of R 2 × R l ′ +1 = R l ′ +3 . -We map the previously obtained subsets by an affine transformation σ : R l ′ +3 → R l ′ +3 defined in the following way. * For each x ∈ R l ′ +3 , the 1st component of σ(x) does not change under the transformation σ and it is equal to x 1 . * For each x ∈ R l ′ +3 , the j-th component of σ(x) may change under the transformation σ and equal to x j+1 for 2 ≤ j ≤ i + 1. * For each x ∈ R l ′ +3 , the (i+2)-th component of σ(x) may change under the transformation σ and equal to x 2 . * For each x ∈ R l ′ +3 , the j-th component of σ(x) does not change under the transformation σ and equal to x j for i + 3 ≤ j ≤ l ′ + 3. At each point there, each normal vector is regarded to be parallel to some vector represented by the form t i,1 e 1 +t i,2 e i+2 with some numbers t i,1 = 0 and t i,2 = 0. Furthermore, these two are also apart from the i-th affine subspace equal to R i+1 × {0} × R l ′ −i+1 presented before.
• The boundaries of some n-dimensional ellipsoids in R n . The n-dimensional ellipsoids in R n are mutually disjoint and they are also apart from the other hypersurfaces presented here. We discuss transversality. Observe the list and see the normal vectors explicitly. Recall also Remark 2 for example. We can see that the transversality is satisfied.
We discuss singularities of the function f . For the boundary of each n-dimensional ellipsoid in R n , only points regarded as the two "poles" are regarded as points in this connected component of D−D whose preimages (for our map f D ) contain some singular points of the function f := π n,1 • f D . On singular points of the function f , we can explain similarly about points in {(t 1 , a 0,2 ), (t l , a 0,2 )} × R l ′ +1 . This is a (l ′ + 1)-dimensional affine subspace in R l ′ +3 . In the image D ⊂ R n of the map f D , except such points, the preimages contain no singular points of f . This completes the proof.
Remark 3. This paper shows a variant of [12], which is withdrawn due to an improvement. Construction of the real algebraic map into R 3 of Main Theorem 2 of [12] aims to construct smooth real algebraic maps on non-singular real algebraic manifolds whereas we only succeed construction in cases the following two are satisfied "in terms of our Main Theorem 1".
• l N l−1 (j) = 0 for 1 < j < l − 1.
• At least one of l N l−1 (1) = 0 or l N l−1 (1) = 0 holds. More precisely, we can check the version arXiv:2303.14988v2 and "A Proof of Main Theorems". More precisely, in the case either l N l−1 (1) = 0 or l N l−1 (1) = 0 holds, check a case in Case 8 which is presented through Case 3 or Case 4 first.
In constructing nice maps into R 3 like ones in our Main Theorem 2 in more general cases, we need semi-algebraic sets or (non-singular) Nash manifolds there. We also consider construction such that the images of maps are represented as the intersections of the closures of open subsets of specific types of fixed Euclidean spaces in [11]. For main results of [12] and our present paper, it seems that we can have various explicit construction.
For example, if we admit the hypersurfaces S j to be more general ones regarded as some connected components of the real algebraic sets defined by (single) real polynomials of degrees at most 2, what can we know? See Problem 3, presented in the last.
3. Appendices, mainly problems related to our study.
We present some problems related to our study. For example, it may be important to explicitly give definitions of notions on graphs. However, we omit them. We expect that maybe at least we have some elementary knowledge on graphs. See also [5,6,7,9,10,11,12] for example. Problem 2. In Main Theorems, can we construct the function f whose preimages have prescribed topologies?
In the case of smooth functions, [5,6] are important. [5] considers the case of smooth functions on 3-dimensional closed and orientable manifolds. According to this, we can have a smooth function with mild singularities of some natural class whose Reeb graph is an arbitrary finite graph and connected components containing no singular points of whose preimages are prescribed closed and connected surfaces. The singularities are, in short, singularities of Morse functions, fold maps, which are higher dimensional versions of Morse functions, and compositions of such functions and maps. See [4] for fundamental theory of singularities of Morse functions, fold maps, and more general smooth maps. [9] generalizes (some) results of [5] for Morse functions on closed manifolds of general dimensions. [6] considers cases where connected components of preimages containing no singular points for the functions and the manifolds have no boundaries and may not be closed. This also constructs smooth functions which are not real analytic. [22] is also a related paper, which is based on our informal discussions on [5]. It considers very general cases. It also constructs smooth functions which are not real analytic.
[10] studies a very explicit case for smooth real algebraic functions on closed manifolds such that preimages are empty or connected. More precisely, (connected components of) the preimages are empty, spheres, or manifolds represented as connected sums of products of two spheres if they contain no singular points of the function.
Problem 3. The images of our real algebraic maps into R n , constructed in Main Theorem 2 for example, are regarded as the closures of some n-dimensional open sets in R n . They collapse to some cell (CW) complexes whose dimensions are lower. Can we consider suitable classes of graphs, CW complexes or more generally, cell complexes for such phenomena? Can we study about (the closures of) open sets surrounded by non-singular real algebraic hypersurfaces of some nice classes and collapsing to such nice complexes. In other words, can we study about nice arrangements of real algebraic hypersurfaces forming (the closures of) the open sets?
For this, [3] is one of related studies. This is on realization of domains surrounded by non-singular connected real algebraic curves in the affine space R 2 and collapsing to given graphs. In short, they first find desired domains topologically or in the smooth category. Next they apply a kind of approximations by real polynomials. This also has motivated us to obtain our related pioneering result [7]. For related studies, see [13] and see also [25,26] for example. Problem 4. In our case of smooth real algebraic functions and maps on non-singular real algebraic (or more generally, Nash) manifolds, can we have explicit and nice affirmative answers for more general (Reeb) graphs? Here, we also need explicit and nice conditions on (the topologies and the differentiable structures of) connected components of preimages containing no singular points.
2.1.2. C H,(−∞,a],b,c . Let a, b and c > 0 be real numbers. We consider the subset in R 2 represented as {(x 1 , x 2 ) ∈ R 2 | (x 1 − a)(x 2 − b) = c}. Let it be denoted by C H,(−∞,a],b,c . This is the real algebraic set defined by the single real polynomial and this is a 1-dimensional non-singular real algebraic manifold. It consists of exactly two connected components. These connected components are denoted by C H+,(−∞,a],b,c := {(x 1 , x 2 ) ∈ C H,(−∞,a],b,c | x 1 > a, x 2 > b} and C H−,(−∞,a],b,c := {(x 1 , x 2 ) ∈ C H,(−∞,a],b,c | x 1 < a, x 2 < b}, respectively. Last, this is for a hyperbola. 2.1.3. C H,[a,∞),b,c . Let a, b and c < 0 be real numbers. We consider the subset in R 2 represented as {
Proposition 1 .
1For an NCD D ⊂ R n and an arbitrary sufficiently large integer m, we have some m-dimensional non-singular real algebraic connected manifold M being also a connected component of the real algebraic set defined by some l real polynomials and a smooth real algebraic map f D : M → R n enjoying the following properties. (1) The image f D (M ) is the closure D and f D (S(f D )) = D − D.
Let L P+,p1,p2 ⊂ R 2 denote the closure of the connected component. Let L P−,p1,p2 ⊂ R 2 denote the closure of the other connected component. The complementary set of C H,(−∞,a],b,c or C H,[a,∞),b,c in R 2 consists of exactly two connected components. One of them is surrounded by C H+,(−∞,a],b,c := {(x 1 , x 2 ) ∈ C H,(−∞,a],b,c | x 1 > a, x 2 > b} and C H−,(−∞,a],b,c := {(x 1 , x 2 ) ∈ C H,(−∞,a],b,c | x 1 < a, x 2 < b} or C H+,[a,∞),b,c := {(x 1 , x 2 ) ∈ C H,[a,∞),b,c | x 1 < a, x 2 > b} and C H−,[a,∞),b,c := {(x 1 , x 2 ) ∈ C H,[a,∞),b,c | x 1 > a, x 2 < b} according to the chosen hyperbola C H,(−∞,a],b,c or C H,[a,∞),b,c . Let R H,(−∞,a],b,c denote the closure of the connected component for C H,(−∞,a],b,c . Let R H,[a,∞),b,c denote the closure of the connected component for C H,[a,∞),b,c . See FIGURE 1, showing an example for b = 0, for example.
•
(s 1,3 , s 2 ) ∈ C H,(−∞,s1,2],0,s and (s 1,3 , s 2 ) ∈ C H+,(−∞,s1,2],0,s . • There exists a unique negative number s 2 ′ < 0 and (s 1,1 , s 2 ′ ) ∈ C H,(−∞,s1,2],0,s and (s 1,1 , s 2 ′ ) ∈ C H−,(−∞,s1,2],0,s hold.
•
(s 1,1 , s 2 ) ∈ C H,[s1,2,∞),0,−s and (s 1,1 , s 2 ) ∈ C H+,[s1,2,∞),0,−s . R −,{s1,j } 3 j=1 ,s2,s as the intersection L P−,(s1,3,0),(s1,3,1) L P+,(s1,2,0),(s1,3,0) L P+,(s1,2,0),(s1,1,s2) R H,[s1,2,∞),0,−s . FIG-URE 1 of [11] shows a subset very similar to R −,{s1,j } 3 j=1 ,s2,s in an explicit way. In our definition the thick curve in the parabola there is changed into a straight segment passing through the boundary point in the left of the connected component of the hyperbola in the left. 2.3.2. R {s1,j } 2 j=1 ,s2,s . Let s 1,1 < s 1,2 and s 2 , s > 0 be real numbers. We can define R {s1,j } 2 j=1 ,s2,s as the intersection L P+,(s1,1,0),(s1,2,0) R H,[s1,1,∞),0,−s R H,(−∞,s1,2],0,s of the three disjoint non-singular connected curves.
( 2 )
2D − D is a union of finitely many (n − 1)-dimensional smooth connected submanifolds of S j as in Definition 2. Each S j is a connected component of the real algebraic set defined by a single real polynomial of degree at most 2. More precisely, The family {S j } of hypersurfaces satisfies the following conditions. (a) It is an affine subspace of R n , a connected component of an (n − 1)dimensional cylinder of a hyperbola in R n , or the boundary of some n-dimensional ellipsoid in R n . (b) If S j0 is the boundary of some n-dimensional ellipsoid in R n , then for any other S j satisfying j = j 0 , S j S j0 is empty.A proof of Main Theorems. We obtain a desired NCD D as follows.Case 1. The case l = 2. In the case l N1 (1) = 0, we put n = 2 and define D as the interior of L P+,(t1,0),(t1,1) L P+,(t1,0),(t2,0) L P−,(t2,0),(t2,1) L P−,(t1,1),(t2,1) and in the case l N1 (1) = 1, we put n = 2 and define D as the interior of L P+,(t1,0),(t1,1) L P+,(t1,0),(t2,0) L P−,(t2,0),(t2,1)to complete the proof.Case 2. The case l = 3. We put n = 2 and define D as the interior of either of the following four to complete the proof.• L P+,(t1,0),(t1,1) L P+,(t1,0),(t2,0) L P+,(t2,0),(t3,1) L P−,(t1,1),(t3,1) in the case l N2 (j) = 0 for each j. • L P+,(t1,0),(t1,1) L P+,(t1,1),(t2,0) L P+,(t2,0),(t3,1) L P−,(t3,0),(t3,1) in the case l N2 (j) = 1 for each j.• R +,{tj } 3 j=1 ,0,1 in the case l N2 (1) = 1 and l N2 (2) = 0. • R −,{tj } 3 j=1 ,0,1 in the case l N2 (1) = 0 and l N2 (2) = 1.Case 3. Remaining general cases l ≥ 4.
.
case l N1 (1) = 1, we put n = 2 and define D as the interior of B E,{ t 1 Remaining general cases l ≥ 4. In the case l N l−1 (N l−1 ) = {0}, we put n = 3 and define D as the interior of the set obtained in the following way.• We choose B E,{a 0,j ′ } 3 j ′ =1 ,{r 0,j ′ } 3 j ′ =1satisfying a 0,1 = t1+t l 2 and r 0,1 = (t l −t1) 2 4 .• We choose the previous set as a sufficiently large set. We can choose a family {B E,{a j,j ′ } 3j ′ =1 ,{r j,j ′ } 3 j ′ =1} of n-dimensional ellipsoids in R n satisfying
This seems to be very difficult. At present, we must restrict (our classes of Reeb) graphs to very simple or explicit ones as we do in[7,10,11,12]. For example, related to our comments in Remark 3, let us admit the hypersurfaces S j to be more general ones regarded as some connected components of the real algebraic sets defined by some (single) real polynomials of degrees at most 2. What do we have then?
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Email address: naokikitazawa.formath@gmail. FAX (OfficeTEL (OfficeSumiyoshi-ku Osaka 558-8585Osaka Central Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585, TEL (Office): +81-6-6605-3103, FAX (Office): +81-6-6605-3104, Email address: [email protected] Webpage: https://naokikitazawa.github.io/NaokiKitazawa.html
| [] |
[
"A subsolar oxygen abundance or a radiative region deep in Jupiter revealed by thermochemical modelling",
"A subsolar oxygen abundance or a radiative region deep in Jupiter revealed by thermochemical modelling"
] | [
"T Cavalié \nLaboratoire d'Astrophysique de Bordeaux\nUniv. Bordeaux\nCNRS\nallée Geoffroy Saint-Hilaire\nB18N, 33615, 0000-0002-0649-1192PessacFrance (ORCID\n\nLESIA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\nUPMC Univ\nParis 06\n\nUniv. Paris Diderot\nSorbonne Paris Cité\nMeudonFrance\n",
"J Lunine \nCornell University\nIthacaNYUSA\n",
"O Mousis \nInstitut Origines\nAix Marseille Université\nCNRS\nCNES, LAM\nMarseilleFrance\n"
] | [
"Laboratoire d'Astrophysique de Bordeaux\nUniv. Bordeaux\nCNRS\nallée Geoffroy Saint-Hilaire\nB18N, 33615, 0000-0002-0649-1192PessacFrance (ORCID",
"LESIA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\nUPMC Univ\nParis 06",
"Univ. Paris Diderot\nSorbonne Paris Cité\nMeudonFrance",
"Cornell University\nIthacaNYUSA",
"Institut Origines\nAix Marseille Université\nCNRS\nCNES, LAM\nMarseilleFrance"
] | [] | Jupiter's deep abundances help to constrain the formation history of the planet and the environment of the protoplanetary nebula. Juno recently measured Jupiter's deep oxygen abundance near the equator to be 2.2 +3.9 −2.1 times the protosolar value (2σ uncertainties). Even if the nominal value is supersolar, subsolar abundances cannot be ruled out. Here we use a stateof-the-art one-dimensional thermochemical and diffusion model with updated chemistry to constrain the deep oxygen abundance with upper tropospheric CO observations. We find a value of 0.3 +0.5 −0.2 times the protosolar value. This result suggests that Jupiter could have a carbon-rich envelope that accreted in a region where the protosolar nebula was depleted in water. However, our model can also reproduce a solar/supersolar water abundance if vertical mixing is reduced in a radiative layer where the deep oxygen abundance is obtained. More precise measurements of the deep water abundance are needed to discriminate between these two scenarios and understand Jupiter's internal structure and evolution. | 10.1038/s41550-023-01928-8 | [
"https://export.arxiv.org/pdf/2305.13949v1.pdf"
] | 257,967,643 | 2305.13949 | 8d6bc6775da885cec1f15379c54ba0ca8f0b4254 |
A subsolar oxygen abundance or a radiative region deep in Jupiter revealed by thermochemical modelling
T Cavalié
Laboratoire d'Astrophysique de Bordeaux
Univ. Bordeaux
CNRS
allée Geoffroy Saint-Hilaire
B18N, 33615, 0000-0002-0649-1192PessacFrance (ORCID
LESIA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
UPMC Univ
Paris 06
Univ. Paris Diderot
Sorbonne Paris Cité
MeudonFrance
J Lunine
Cornell University
IthacaNYUSA
O Mousis
Institut Origines
Aix Marseille Université
CNRS
CNES, LAM
MarseilleFrance
A subsolar oxygen abundance or a radiative region deep in Jupiter revealed by thermochemical modelling
10.1038/s41550-023-01928-8received: 20 May 2022 accepted: 23 February 2023
Jupiter's deep abundances help to constrain the formation history of the planet and the environment of the protoplanetary nebula. Juno recently measured Jupiter's deep oxygen abundance near the equator to be 2.2 +3.9 −2.1 times the protosolar value (2σ uncertainties). Even if the nominal value is supersolar, subsolar abundances cannot be ruled out. Here we use a stateof-the-art one-dimensional thermochemical and diffusion model with updated chemistry to constrain the deep oxygen abundance with upper tropospheric CO observations. We find a value of 0.3 +0.5 −0.2 times the protosolar value. This result suggests that Jupiter could have a carbon-rich envelope that accreted in a region where the protosolar nebula was depleted in water. However, our model can also reproduce a solar/supersolar water abundance if vertical mixing is reduced in a radiative layer where the deep oxygen abundance is obtained. More precise measurements of the deep water abundance are needed to discriminate between these two scenarios and understand Jupiter's internal structure and evolution.
Introduction
Giant planets are the true architects of planetary systems given their relatively short formation timescales compared with terrestrial planets, their gravity and migration. Their deep composition holds part of the key to understanding the formation of giant planets in planetary systems, along with other measurements such as their gravity and magnetic field. It can also help to constrain the processes that led to the condensation or trapping of the primordial ices in the protosolar nebula (Helled & Lunine 2014;Bar-Nun et al. 1988).
The Galileo probe plunged in the atmosphere of Jupiter in 1995, reaching the 22 bar level, and measured its elemental and isotopic composition. The main result is the relatively uniform enrichment in volatiles by a factor of two to four with respect to the protosolar value (Wong et al. 2004). The only element that did not follow this trend, and for which there was no internal process to explain its depletion, is oxygen. The accepted hypothesis is that Galileo entered a dry area where water, the main carrier of oxygen at depths, was still not uniformly mixed at 22 bar. The oxygen measurement from Galileo has thus been mostly accepted as a lower limit ever since.
One of the reasons the Juno mission was designed was to fill this gap left by Galileo regarding the deep oxygen abundance in Jupiter. The idea was to observe the microwave spectrum of Jupiter at various emission angles to retrieve the global water abundance (Janssen et al. 2005). The main difficulty then resides in the dependency of the spectrum on the temperature profile and on absorbers other than water, such as ammonia. While Juno confirmed previous observations that ammonia was depleted below its condensation level (de Pater 1986), it surprisingly showed that this depletion was latitude dependent and that it extended as deep as ∼50 bar (Li et al. 2017), except near the equator. Li et al. (2020) used the microwave spectra seen by Juno at this latitude to retrieve the vertical water profile of Jupiter at its equator and derive its deep oxygen abundance. They found that the oxygen was nominally enriched by a factor of 2.7 +2.4 −1.7 times the protosolar value but the 1σ lower bound was weakly determined, and they state that subsolar values are possible. Helled et al. (2022) emphasized that these error bars are sufficiently large that the depletion seen by Galileo may not be a local anomaly, but could instead reflect a global depletion. In what follows, we use protosolar abundances determined by Lodders (2021). The Jovian oxygen abundance of Li et al. (2020) then translates into 2.2 +2.0 −1.4 times the protosolar value. Long before Galileo and Juno, the reported (Beer 1975) detection of CO in the troposphere of Jupiter with an abundance higher than thermochemical equilibrium predictions by orders of magnitude triggered the development of numerous models to constrain the deep oxygen abundance by solving the balance between thermochemistry and vertical mixing (for example, Lunine & Stevenson 1987;Fegley & Prinn 1988;Yung et al. 1988). Below the cloud level, in the deep troposphere, CO and H 2 O are in thermochemical equilibrium resulting from the equilibrium reaction CO + 3H 2 = H 2 O + CH 4 . As the temperature decreases with height, the thermochemical equilibrium is shifted in favour of H 2 O. The detection of CO in the upper troposphere then demonstrates that thermochemical equilibrium is quenched by vertical mixing when the mixing timescale becomes shorter than the chemical conversion timescale. Although the initial studies cited above focused on identifying the rate-limiting reaction, more recent work incorporated comprehensive chemical schemes (Visscher et al. 2010;Wang et al. 2016). By probing the abundance of CO in the upper troposphere and modelling thermochemistry and diffusion, one can then reconstruct the vertical profile of the species and tie it back to the deep water (and thus to the deep oxygen) abundance in the planet.
We used a 1D thermochemical and diffusion model (Cavalié et al. 2017) to fit a CO upper tropospheric mole fraction of 0.9 ± 0.3 ppb, as measured by previous studies (Bézard et al. 2002;Bjoraker et al. 2018). Although the chemical scheme used by Cavalié et al. (2017) was validated globally by the combustion industry over a wide range of temperatures and pressures, Wang et al. (2016) showed that the model failed to agree with most competing chemical schemes regarding the quench chemistry of CO in giant planets. Moses (2014) identified the main source of the disagreement as the kinetics of the conversion reaction of methanol into the methyl radical and water measured by Hidaka et al. (1989). This led Venot et al. (2020) to fully update their CH 3 OH submechanism by adopting the work of Burke et al. (2016), which noticeably provided an explicit logarithmic dependence on pressure of some reaction rates, and removing of the controversial reaction of Hidaka et al. (1989) that produced the methyl radical from methanol (and thus from CO). This updated scheme21 then destroys less CO than that of Venot et al. (2012), in agreement with other studies (Wang et al. 2015(Wang et al. , 2016Moses 2014) (Extended Data Fig. 1). As a consequence, a given CO abundance requires a lower deep oxygen abundance with the new scheme (Methods).
Results
With our nominal parameter set (Methods), we found that a deep oxygen abundance of 0.3 times the protosolar value is required in Jupiter's deep troposphere. The corresponding vertical profiles are displayed in Fig. 1. In our simulations, we allowed three parameters to vary to fit the tropospheric CO and estimate the uncertainty of the deep oxygen abundance with respect to the methane mole fraction y top CH 4 at the top of the troposphere and the vertical eddy diffusion coefficient K zz . We allowed these two parameters to vary within their respective uncertainty ranges; that is, y top CH 4 = 0.00204 ± 0.0050 (ref.
3) and K zz = 10 8 cm 2 ·s −1 within a factor of two (Wang et al. 2016;Grassi et al. 2020). When y top CH 4 was allowed to vary within its uncertainty range and K zz was fixed to 10 8 cm 2 ·s −1 , we obtained our nominal fits to the CO mole fraction with a deep oxygen abundance of 0.3 +0.3 −0.2 times the protosolar value (Fig. 2). Conversely, when we varied K zz and fixed y top CH 4 to 0.00204, we obtained a deep oxygen abundance of 0.3 +0.3 −0.2 times the protosolar value (Fig. 3). The deep carbon abundance was 3.2 ± 0.8 times the protosolar value, depending on the adopted value of y top CH 4 . Finally, if we accounted for the uncertainty ranges of K zz and y top CH 4 , then the result for the deep oxygen was 0.3 +0.5 −0.2 times the protosolar value. According to our model, oxygen is subsolar and the C/O ratio is 6 +10 −5 , suggesting that Jupiter could have a surprisingly carbon-rich envelope.
The deep oxygen abundance in giant planets has long been debated as it is one of the key elements pertaining to the formation of solids and trapping of more volatile species in the protosolar nebula, which are later released to the growing envelopes of the giant planets (Owen et al. 1999;Gautier et al. 2001). Remote sensing observations and in situ measurements provide values ranging from 0.25 to 4.2 times the protosolar value. The Galileo measurement, when translated into an O/H abundance using the protosolar composition of Lodders (2021), is 0.37 ± 0.12 (Wong et al. 2004). This value has often been considered a lower limit because the water abundance was still increasing in the measurements when the signal from the Galileo probe was lost, and the probe entered a dry region of Jupiter's atmosphere (a 5 µm hotspot). Bjoraker et al. (2018) found that their Great Red Spot spectrum around 5 µm was best fitted by fixing the water cloud base at 5 ± 1 bar, which translates nominally into a near-protosolar deep oxygen abundance, but the 1σ range encompasses 0.3 to 3 times the protosolar value. Finally, a deep water abundance of 2.2 +2.0 −1.4 times the protosolar value was obtained (Li et al. 2020) with the Juno Microwave Spectrometer (MWR) in the sole equatorial region where vertical mixing and meteorology seem to maintain a well-mixed atmosphere throughout the probed gas column. Whether this value is representative of the whole planet remains to be verified, especially given the unexpected results for the meridional distribution of ammonia and its depletion at pressures lower than 30 bar (Li et al. 2017) and the role water seems to play in this depletion (Guillot et al. 2020b). The fact that Li et al. (2017) found about half the deep ammonia measured by Galileo could indicate that this oxygen measurement is also not representative of the global value. In any case, their oxygen abundance retrieval presents an error bar in which the lower end is more weakly determined than the higher end. The lower 2σ limit lies at 0.1 times the protosolar value. . The dashed lines correspond to alternative Jupiter abundance and T profiles obtained with a more sluggish mixing (K zz is reduced to 2.5 × 10 6 cm 2 ·s −1 ) to match the nominal Juno O/H value (2.2 times the protosolar value). This lower K zz is then indicative of a radiative region located around the quench level of CO (that is, at pressure p ∼ 0.4-0.5 kbar and T ∼ 1000 K). The dash-dotted lines correspond to a final model in which a deep radiative layer with reduced K zz is inserted to match solar oxygen and produce 0.9 ppb CO in Jupiter's upper troposphere: K zz is set to 1 cm 2 ·s −1 between T = 1400 K and 2200 K and transitions linearly with T towards our nominal value of 10 8 cm 2 ·s −1 at 970 K to ensure that PH 3 and GeH 4 are quenched at ∼800 K. This is a non-unique solution.
Discussion
We nominally found a subsolar deep oxygen abundance in Jupiter. Our results, compatible with the range obtained from similar modelling in Visscher et al. (2010), indicate marginal agreement with the Juno MWR analysis (Li et al. 2020) as discussed above. Cloud models often require one times solar oxygen or higher (Iñurrigarro et al. 2022), but can also accommodate subsolar oxygen (Hueso & Sánchez-Lavega 2001). More problematic are the results from lightning data. While overall modelling of lightning frequency (Aglyamov et al. 2021) permits a subsolar value, the Galileo detection of a potentially deeper lightning flash would imply a water enrichment exceeding solar values (Dyudina et al. 2002). Decreasing K zz from the nominal value of 10 8 cm 2 ·s −1 to 2.5 × 10 6 cm 2 ·s −1 would raise the deep oxygen abundance from 0.3 to 2.2 times solar ( Fig. 1 and Extended Data Fig. 2). However, such a low level of vertical mixing must be confined to the altitudes below which temperatures reach 1000-1100 K, where CO quenching starts, as demonstrated in Fig. 1; at 800 K it must be at or close to our nominal value to fit the data on other disequilibrium species such as PH 3 and GeH 4 (Wang et al. 2016;Grassi et al. 2020), as described in Methods. Our revised chemical network, taken from Venot et al. (2020), probably still suffers from some uncertainties in the reaction rates. However, any change in the rates to produce a solar or supersolar water abundance would require the deep oxygen abundance derived in the ice giants ) to increase as well, raising the problem of the consistency between the deep oxygen abundance and deep D/H ratio when compared with that found in Oort cloud comets (Ali-Dib et al. 2014). Our results therefore require either (1) a carbon-rich envelope in Jupiter or (2) a deep layer of reduced vertical mixing.
Option (1) was already proposed by Mousis et al. (2012), assuming that the Galileo O determination corresponds to the bulk abundance. A high C/O ratio in Jupiter's envelope such as that of 6 +10 −5 found here resulting from a relatively low oxygen abundance was also proposed by previous studies (Lodders 2004;Mousis et al. 2019). In one model (Mousis et al. 2019), the radial drift of pebbles through the amorphous-to-crystalline-ice transition front in the protoplanetary disk releases carbon-rich supervolatiles into the gas while stranding water in the ice. Other scenarios include the agglomeration of building blocks condensed in the vicinity of the condensation lines of C-rich volatiles by the growing Jupiter. The C/O ratio in icy solids formed in those regions is expected to rise steeply, as found by Mousis et al. (2021a) to explain the water-poor composition of Comet C/2016 R2 (PanSTARRS). Another study (Mousis et al. 2021b) suggested that a wide range of protosolar nebula compositions can match Jupiter's metallicity, including several types of icy phase (clathrates and pure condensates). It should be noted that current Jupiter formation and structure models predict a low-metallicity envelope (Helled et al. 2022;Schneider & Bitsch 2021). Option (2) implies a dramatic decrease in vertical mixing at temperatures higher than 1000-1100 K, corresponding to a pressure level of roughly 0.6 kbar. Various mechanisms may lead to stable regions throughout Jupiter's deep atmosphere, and one mechanism that corresponds roughly to the p-T region here is a radiative zone extending from 1200-2000 K (Guillot et al. 1994). Such a zone is the result of low opacity, obtained only in the case of a depletion in the alkali metals (Guillot et al. 2004), and Juno MWR data hint at such a depletion (Bhattacharya et al. 2021). It is therefore plausible that the vertical mixing, represented by K zz in our model, is very low in the region just below the CO quench level, begins to increase at that level and reaches its full convective value by 900 K (300 bar) where PH 3 and GeH 4 quenching determine our nominal value for K zz . A model in which K zz was set to the very low value of 1 cm 2 ·s −1 (it could in principle be as low as the molecular diffusivity) at temperatures higher than 1400 K and to 10 8 cm 2 ·s −1 at temperatures lower than 970 K and then interpolated logarithmically between the two levels produced satisfactory results with solar oxygen (Fig. 1 and Extended Data Fig. 2).
Either of the two possibilities -depleted oxygen or a deep radiative zone -would be important for understanding the nature of Jupiter's interior below its visible atmosphere. Further analysis of Juno MWR data during the extended mission, particularly at the longest wavelength channel, will help to distinguish between them. The results for Jupiter also provide important context for the elemental composition of giant planets in our Solar System, and beyond. In situ probes, despite their inherent limitation of a single entry-point, would provide invaluable compositional data for Saturn and the ice giants, especially regarding noble gases, as presented other works (Mousis et al. , 2018. Cavalié et al. (2020)
Methods
Thermochemical model
We used the 1D thermochemical and diffusion model initially developed in Venot et al. (2012) for warm exoplanets, and adapted in Cavalié et al. (2014Cavalié et al. ( , 2017 to study the deep oxygen abundance in Uranus and Neptune. The model solves the continuity equation as a function of time at each altitude, for 111 carbon, oxygen and nitrogen species through 1912 reactions.
Our model required boundary conditions regarding the composition of the upper troposphere. We took the upper tropospheric mole fraction of He and CH 4 from von Zahn et al. (1998) and Wong et al. (2004), respectively y top He = 0.1359 ± 0.0027 and y top CH 4 = 0.00204 ± 0.0050, both resulting from the Galileo probe measurements. For CO, we adopted an upper tropospheric mole fraction of 0.9 ± 0.3 ppb, following measurements from Bézard et al. (2002) and Bjoraker et al. (2018).
To constrain the deep oxygen abundance of Jupiter from upper tropospheric CO observations, we also needed to make assumptions on the vertical mixing and on the temperature profile. Following Wang et al. (2015) and recent Juno results of Grassi et al. (2020) on tropospheric abundances of disequilibrium species, we nominally adopted a vertical eddy mixing coefficient of 10 8 cm 2 ·s −1 with a factor of two uncertainty. Even though Juno observations of NH 3 (Li et al. 2017) and models to explain downward ammonia transport (Guillot et al. 2020b,a) show that chemical transport does not obey a pure diffusion equation between 0.1 and tens of bars, we assumed that this is the case at several hundred bars in the more homogeneously mixed deeper troposphere where the CO thermochemistry is quenched. Our temperature profile follows the Galileo profile (Seiff et al. 1998) within 1 K down to 22 bar and we extrapolated temperatures using a wet adiabat from the diffusion-dominated upper levels down to the deep levels where thermochemical equilibrium prevails. The deep oxygen abundance measured with Juno for Jupiter (Li et al. 2020) is not high enough to produce a radiative layer, resulting from a mean molecular weight gradient at the water condensation level in which the temperature would sharply increase, as opposed to the case of the ice giants (Cavalié et al. 2017). We stopped our temperature extrapolations at 1700 K, much deeper than the levels where thermochemistry is quenched by vertical mixing. We ensured that our results reached steady state with an integration time of 10 10 s.
We fixed the deep nitrogen abundance to ∼4 times the protosolar value to reproduce the Galileo measurement (Wong et al. 2004). The abundance of N 2 was then ∼10 −5 in the upper troposphere. Our model results regarding oxygen were, however, insensitive to this value, because the nitrogen and oxygen chemistries were mostly uncoupled. We adopted the protosolar abundances reported in Lodders (2021) throughout this Article. We used them to express our model results and to convert previous results to a common scale.
Chemical scheme
The chemical network in Venot et al. (2012) is a C/H/O/N mechanism initially validated for the combustion industry to help to understand the combustion of fuels in car engines and thus limit their environmental impact. It is based on a C 0 -C 2 mechanism to which a nitrogen reaction base was added. It comprises 105 species and 1926 reactions. Although it was validated against experiments for pressures ranging from 0.01 bar to several hundred bars and for temperatures ranging from 300 to 2500 K, Wang et al. (2016) showed that the conversion of H 2 O into CO was less efficient in Jupiter with the network shown in Venot et al. (2012) compared with several others (see their fig. 17), resulting in CO abundance 10 times lower than in simulations involving competing networks. A similar issue with CH 4 ?CO chemistry had been found in applications to hot Jupiters by Moses (2014), even though Venot et al. (2020) found even more compelling differences in cooler planets. Moses (2014) further narrowed down the main difference in the networks to the kinetics of methanol through the H + CH 3 OH = CH 3 + H 2 O reaction. The chemical rate of this reaction had been set in Venot et al. (2012) to that estimated by Hidaka et al. (1989), but was found to be over-estimated by Visscher et al. (2010) in their work on Jupiter's thermochemistry. This led Venot et al. (2020) to fully revise the CH 3 OH sub-network of their chemical scheme. They adopted experimental data (Burke et al. 2016) that are remarkable in several aspects. First, the reaction of Hidaka et al. (1989) is no longer explicitly present in the network. This does not prevent CH 3 OH from being destroyed and producing CH 3 and H 2 O, but this is achieved through other destruction pathways. Second, the kinetic rates of several reactions of this network (more specifically those involving methoxide and the methyl radical) have an explicit logarithmic dependence with pressure defined for up to five pressure decades, which thus increase the accuracy and robustness of the kinetics over this wide range of pressure conditions. This new scheme was validated ) over a wide range of temperature and pressure conditions and showed improved agreement with experimental data. We have produced a CO profile for Jupiter in the same conditions as in fig. 17 of Wang et al. (2016). It is shown in Extended Data Fig. 1 and it fully agrees with the profiles presented in fig. 17 of Wang et al. (2016). When applied to the ice giants this scheme resulted in the production of the observed CO in ice giants with substantially lower oxygen enrichments.
As stated by Moses (2014), "the exact mechanism involved with CH 4 -CO quenching in reducing environments has not been strictly identified". This is caused by the high nonlinearity and high coupling between the various chemical reactions of the scheme. It is thus not possible, as found in initial studies that assumed a rate-limiting reaction in the CO destruction mechanism (Fegley & Prinn 1988;Yung et al. 1988;Bézard et al. 2002), to easily identify a single reaction and quantify uncertainties on the results from the uncertainty on the rate of this reaction. A methodology of uncertainty propagation and global sensitivity analysis has been developed (Dobrijevic et al. 2010), but it required running several hundred simulations following a Monte Carlo scheme. Applying this methodology is beyond the scope of this study.
Deep radiative region in Jupiter?
It has been shown (Wang et al. 2016) that PH 3 and GeH 4 are quenched at ∼700-800 K (p ≈ 0.1 kbar; see their figs. 6 and 11) and the abundances observed with Juno JIRAM (Grassi et al. 2020) in the upper troposphere imply that K zz ≈ 10 7 -10 9 cm 2 ·s −1 from PH 3 and ∼10 8 cm 2 ·s −1 from GeH 4 at this level. For uniform K zz > 10 7 cm 2 ·s −1 our model predicts a subsolar oxygen to fit the observed abundance of CO (Extended Data Table 1).
The first hint that Jupiter's troposphere could harbour a radiative region in the vicinity of the layers where CO is quenched (T ≈ 1000-1100 K; Fig. 1) was then obtained when the deep oxygen abundance was raised to the Juno MWR nominal value of 2.2 times protosolar. This required us to decrease the vertical mixing K zz from its nominal value of 10 8 cm 2 ·s −1 to 2.5 × 10 6 cm 2 ·s −1 (Fig. 1). Fitting the whole range of 1σ uncertainties of the Juno measurement led to the K zz value reported in Extended Data Table 1. We thus found that convection needs to be less efficient in the CO quench region, with K zz lowered by a factor 10 to 100, to obtain solar-to-supersolar oxygen.
A decrease in the Rosseland opacity due to hydrogen and helium opacity between 1200 and 4000 K in Jupiter can result in a radiative region, as initially pointed out by a previous study (Guillot et al. 1994). It was subsequently confirmed (Guillot et al. 2004) that this region may exist between p ≈ 1.5 kbar (T ≈ 1400 K) and p ≈ 8.0 kbar (T ≈ 2200 K) provided that the Jovian atmosphere is also depleted in alkali metals. This depletion seems to be confirmed by recent Juno MWR observations (Bhattacharya et al. 2021). If such a radiative region exists, vertical heat transport and chemical mixing would strongly be inhibited, and our assumption of a vertically uniform K zz would not hold in this region. A previous work (Cavalié et al. 2017) investigated the effect of an insulation layer produced in the ice giants by the rapid change in mean molecular weight gradient where water condenses. Despite the presence of this insulation layer in the altitudes where CO is quenched and vertical transport prevails, they found very limited impact on their results, mostly because the radiative layer was very thin. The radiative layer in Jupiter, which is of a different origin than that in the ice giants, may extend from 1.5 to 8 kbar (Guillot et al. 2004). Even if the radiative layer itself has a limited impact on the CO (15) with our chemical scheme, i.e., that of (21) with revised methanol chemistry kinetics. The profile is obtained for K zz = 10 9 cm 2 ·s −1 and seven times solar oxygen. It is in full agreement with those obtained with other chemical schemes and shown in Figure 17 of (15), which are indicated by the grey area.
O/H (× the protosolar value) Required K zz in the CO quench region 0.3 1×10 8 cm 2 ·s −1 0.8 2.5×10 7 cm 2 ·s −1 2.2 2.5×10 6 cm 2 ·s −1 4.2 4×10 5 cm 2 ·s −1 Table 1: Relationship between Jupiter's deep O/H and the K zz required in the quench region of CO to fit the observed upper tropospheric CO mole fraction. This essentially shows that the higher the deep oxygen abundance, the more inhibited the mixing to produce the right abundance of CO.
profile, because it is located in the region where thermochemical equilibrium between CO and H 2 O prevails over vertical transport, we needed to assess the effect of such a layer and how it connects to upper layers in our simulations.
We found that our model could reconcile solar oxygen with the observed tropospheric CO by adding the presence of a deep radiative layer in which K zz could be as low as the molecular diffusivity. We set K zz to 1 cm 2 ·s −1 for layers with T > 1400 K and interpolated K zz between this value at 1400 K and our nominal value of 10 8 cm 2 ·s −1 at 970 K, such that PH 3 and GeH 4 are quenched in the ∼800 K region as expected from models and observations. This ensures that K zz is low enough where CO is quenched. The resulting CO profile is shown in Fig. 1.
The K zz profiles used in this work that correspond to the results presented in Fig. 1 10 1 10 2 10 3 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9
Temperature [K] K zz [cm 2 .s -1 ] Figure 5: K zz profiles used in this work. The black profile is our nominal model (where K zz =10 8 cm 2 ·s −1 , constant with altitude) which results in an oxygen abundance of 0.3 times the protosolar value. The blue profile (K zz =2.5 × 10 6 cm 2 ·s −1 , constant with altitude) results constrains oxygen to 2.2 times the protosolar value, i.e., the Juno MWR nominal measurement of (7). An intermediate constant value of 2.5 × 10 7 cm 2 ·s −1 (purple line) will produce the observed CO with nearly solar oxygen. The red profile (variable with altitude) indicates the presence of a stable radiative layer at depth with a transition region such that K zz reaches our nominal value at the levels where PH 3 and GeH 4 are quenched.
Figure 1 :
1Abundances and temperature profiles for Jupiter. Vertical temperature (T), CO, H 2 O and CH 4 profiles from our nominal model with K zz = 10 8 cm 2 ·s −1 and y top CH 4 = 0.00204 in which oxygen is subsolar (O/H = 0.3 times the protosolar value)
Figure 2 :
2K zz and oxygen dependence of Jupiter's upper tropospheric CO mole fraction. The CO mole fraction (colour scale) as a function of tropospheric mixing and the deep oxygen abundance relative to solar. The solid line shows the computations that result in 0.9 ppb CO. The grey dashed lines represent the range that results in 0.6 to 1.2 ppb CO (full uncertainty range). The range of possible K zz values, constrained from laboratory experiments and matching observed values of PH 3 and GeH 4 , is shown by the dashed blue lines. The nominal value of our model is shown by the black dot.
Figure 3 :
3have shown how the use of thermochemical simulations can increase the science return of in situ probe measurements. Carbon and oxygen dependence of Jupiter's upper tropospheric CO mole fraction. The CO mole fraction (colour scale) as a function of the carbon abundance (represented by y top CH 4 ) and oxygen water abundance relative to protosolar according to the thermochemical model results when assuming a constant K zz of 10 8 cm 2 ·s −1 . The layout is similar to figFig2.
Figure 4 :
4CO vertical profile in Jupiter computed in the same conditions as in
are shown in Extended Data Fig. 2.
Acknowledgements T.C. acknowledges funding from CNES and the Programme National de Planétologie (PNP) of CNRS/INSU. J.L. acknowledges support from the Juno mission through a subcontract from the Southwest Research Institute.
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| [] |
[
"Support Localization and the Fisher Metric for off-the-grid Sparse Regularization",
"Support Localization and the Fisher Metric for off-the-grid Sparse Regularization"
] | [
"Clarice Poon \nCentre for Mathematical Sciences\nUniversity of Cambridge Wilberforce Rd\nCambridgeUnited Kingdom\n",
"Nicolas Keriven \nDépartement de Mathématiques et Applications\nÉcole Normale Supérieure\n45 rue d'UlmParisFrance\n",
"Gabriel Peyré \nDépartement de Mathématiques et Applications\nÉcole Normale Supérieure\n45 rue d'UlmParisFrance\n"
] | [
"Centre for Mathematical Sciences\nUniversity of Cambridge Wilberforce Rd\nCambridgeUnited Kingdom",
"Département de Mathématiques et Applications\nÉcole Normale Supérieure\n45 rue d'UlmParisFrance",
"Département de Mathématiques et Applications\nÉcole Normale Supérieure\n45 rue d'UlmParisFrance"
] | [] | Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the offthe-grid version of ℓ 1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.S. Dasgupta andA. Gupta. An Elementary Proof of a Theorem of Johnson and Lindenstrauss. Random Structures and Algorithms, 22(1):60-65, 2003. S. Dasgupta and L. J. Schulman. A Two-Round Variant of EM for Gaussian Mixtures. Uncertainty in Artificial Intelligence, pages 152-159, 2000. Y. De Castro and F. Gamboa. Exact reconstruction using Beurling minimal extrapolation. Journal of Mathematical Analysis and applications, 395(1):336-354, 2012. Q. Denoyelle, V. Duval, and G. Peyré. Support recovery for sparse super-resolution of positive measures. | null | [
"https://arxiv.org/pdf/1810.03340v1.pdf"
] | 52,939,700 | 1810.03340 | e8f6fe8af78003fab57b81b04cd6c6858fc46563 |
Support Localization and the Fisher Metric for off-the-grid Sparse Regularization
8 Oct 2018
Clarice Poon
Centre for Mathematical Sciences
University of Cambridge Wilberforce Rd
CambridgeUnited Kingdom
Nicolas Keriven
Département de Mathématiques et Applications
École Normale Supérieure
45 rue d'UlmParisFrance
Gabriel Peyré
Département de Mathématiques et Applications
École Normale Supérieure
45 rue d'UlmParisFrance
Support Localization and the Fisher Metric for off-the-grid Sparse Regularization
8 Oct 2018
Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the offthe-grid version of ℓ 1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.S. Dasgupta andA. Gupta. An Elementary Proof of a Theorem of Johnson and Lindenstrauss. Random Structures and Algorithms, 22(1):60-65, 2003. S. Dasgupta and L. J. Schulman. A Two-Round Variant of EM for Gaussian Mixtures. Uncertainty in Artificial Intelligence, pages 152-159, 2000. Y. De Castro and F. Gamboa. Exact reconstruction using Beurling minimal extrapolation. Journal of Mathematical Analysis and applications, 395(1):336-354, 2012. Q. Denoyelle, V. Duval, and G. Peyré. Support recovery for sparse super-resolution of positive measures.
Introduction
Sparse Regularization
In this work, we consider the general problem of estimating an unknown Radon measure µ 0 ∈ M(X ) defined over some metric space X (for instance X = R d for a possibly large d) from a few number m of randomized linear observations y ∈ C m , Let Φ : M(X ) → C m be defined by
Φµ def. = 1 √ m X ϕ ω k (x)dµ(x) m k=1 , (1.1)
where (ω 1 , . . . , ω m ) are identically and independently distributed according to some probability distribution Λ(ω) on ω ∈ Ω, and for ω ∈ Ω, ϕ ω : X → C is a continuous function, denoted ϕ ω ∈ C (X ). We further assume that ϕ ω (x) is normalized, that is
E ω [|ϕ ω (x)| 2 ] = 1, ∀x ∈ X .
(1.
2)
The observations are y = Φµ 0 + w, where w ∈ C m accounts for noise or modelling errors. Some representative examples of this setting include:
Off-the-grid compressed sensing: off-the-grid compressed sensing, initially introduced in the special case of 1-D Fourier measurements on X = T = R/Z by (Tang et al., 2013), corresponds exactly to measurements of the form (1.1). This is a "continuous" analogous of the celebrated compressed sensing line of works (Candès et al., 2006;Donoho, 2006). Regression using a continuous dictionary: given a set of m training samples (ω k , y k ) m k=1 , one wants to predicts the values y k ∈ R from the features ω k ∈ Ω using a continuous dictionary of functions ω → ϕ ω (x) (here x ∈ X parameterizes the dictionary), as y k ≈ X ϕ ω k (x)dµ(x). A typical example, studied for instance by Bach (2017) is the case of neural networks with a single hidden layer made of an infinite number of neurons, where Ω = X = R p and one uses ridge functions of the form ϕ ω (x) = ψ( x, ω ), for instance using the ReLu non-linearity ψ(u) = max (u, 0). Sketching mixtures: the goal is estimate a (hopefully sparse) mixture of density probability distributions on some domain T of the form ξ(t) = i a i ξ xi (t) where the (ξ x ) x∈X is a family of template densities, and a i 0, i a i = 1. Introducing the measure µ 0 = i a i δ xi , this mixture model is conveniently re-written as ξ(t) = X ξ x (t)dµ 0 (x). The most studied example is the mixture of Gaussians, using (in 1-D for simplicity, T = R) as ξ x (t) ∝ σ −1 e − (t−τ ) 2 2σ 2 where the parameter space is the mean and standard deviation x = (τ, σ) ∈ X = R × R + . In a typical machine learning scenario, one does not have direct access to ξ but rather to n i.i.d. samples (t 1 , . . . , t n ) ∈ T n drawn from ξ. Instead of recording this (possibly huge, specially when T is high dimensional) set of data, following Gribonval et al. (2017), one computes "online" a small set y ∈ C m of m sketches against sketching functions θ ω (t), that is, for k = 1, . . . , m,
y k def. = 1 n n j=1 θ ω k (t j ) ≈ T θ ω k (t)ξ(t)dt.
These sketches exactly have the form (1.1) when defining the functions ϕ ω (x) def.
= T θ ω (t)ξ x (t)dt. A popular set of sketching functions, over T = R d are Fourier atoms θ ω (t) def.
= e i ω, t , for which ϕ · (x) is the characteristic functions of ξ x , which can generally be computed in closed form.
BLASSO. In all these applications, and many more, one is actually interested in recovering a discrete and ssparse measure µ 0 of the form µ 0 = s i=1 a i δ xi where (x i , a i ) ∈ X ×C. An increasingly popular method to estimate such a sparse measure corresponds to solving a infinite-dimensional analogous of the Lasso regression problem min µ∈M(X ) 1 2
Φµ − y 2 2 + λ|µ|(X ).
(P λ (y))
Following De Castro and Gamboa (2012), we call this method the BLASSO (for Beurling-Lasso). Here |µ|(X ) is the so-called total variation of the measure µ, and is defined as
|µ|(X ) def.
= sup {Re f, µ ; f ∈ C (X ), f ∞ 1} .
Note that on unbounded X , one needs to impose that f vanishes at infinity. If X = {x i } i is a finite space, then this corresponds to the classical finitedimensional Lasso problem (Tibshirani, 1996), because |µ|(X ) = a 1 def.
= i |a i | where a i = µ({x i }).
Similarly, if X is possibly infinite but µ = i a i δ xi , one also has that |µ|(X ) = a 1 .
Previous Works. The BLASSO problem (P λ (y)) was initially proposed by De Castro and Gamboa (2012), see also Bredies and Pikkarainen (2013). The first sharp analysis of the solution of this problem is provided by Candès and Fernandez-Granda (2014) in the case of Fourier measurement on T d . They show that if the spikes are separated enough, then µ 0 is the unique solution of (P λ (y)) when w = 0 and λ → 0. Robustness to noise under this separation condition is addressed in (Candès and Fernandez-Granda, 2013;Fernandez-Granda, 2013;Azais et al., 2015). A refined stability results is detailed by Duval and Peyré (2015) which shows that conditions based on minimum separation imply support stability, which means that when w and w /λ are small enough, then the solution of (P λ (y)) has the same number of Diracs as µ 0 , and that both the amplitudes and positions of the spikes converges smoothly as w → 0. These initial works have been extended by Tang et al. (2013) to the case of randomized compressive measurements of the form (1.1), when using Fourier sketching functions ϕ ω . In all these results, the separation condition are given for the Euclidean cases, which is an ad-hoc choice which does not take into account the geometry of the problem, and gives vastly sub-optimal theories for spatially varying operators (such as datadependent kernels in supervised learning, Gaussian mixture estimation and Laplace transform in imaging, see Section 1.2).
While this is not the topic of the present paper, note that for positive spikes, the separation condition is in some cases not needed, see for instance (Schiebinger et al., 2015;Denoyelle et al., 2017). It is important to note that efficient algorithms have been developed to solve (P λ (y)), among which SDP relaxations for Fourier measurements (Candès and Fernandez-Granda, 2013) and Frank-Wolfe (also known as conditional gradient) schemes (Bredies and Pikkarainen, 2013;Boyd et al., 2017). Note also that while we focus here on variational convex approaches, alternative methods exist, in particular greedy algorithms (Gribonval et al., 2017) and (for Fourier measurements) Prony-type approaches (Schmidt, 1986;Roy and Kailath, 1989). To the best of our knowledge, their theoretical analysis in the presence of noise is more involved, see however (Liao and Fannjiang, 2016) for an analysis of robustness to noise when a minimum separation holds.
The Fisher information metric
The empirial covariance operator is defined aŝ K(x, x ′ ) def.
= 1 m i ϕ ωi (x)ϕ ωi (x ′ ) and the deterministic limit as m → +∞ is denoted K with
K(x, x ′ ) def. = Ω ϕ ω (x)ϕ ω (x ′ )dΛ(ω).
(1.3)
Note that many covariance kernels can be written under the form (1.3). By Bochner's theorem, this includes all translation-invariant kernels, for which possible features are ϕ ω (x) = e iω ⊤ x . The associated metric tensor is
H x def. = ∇ x ∇ x ′ K(x, x) ∈ C d×d .
(1.4) Throughout, we assume that H x is positive definite for all x ∈ X . Then, H naturally induces a distance between points in our parameter space X . Given a piecewise smooth curve γ : [0, 1] → X , the length ℓ H [γ] of γ is defined by ℓ H [γ]
def.
= 1 0 H γ(t) γ ′ (t), γ ′ (t) dt. Given two points x, x ′ ∈ X , the distance from x to x ′ , induced by H is d H (x, x ′ ) def. = inf γ∈F ℓ H [γ]
where F is the set of all piecewise smooth paths γ : [0, 1] → X with γ(0) = x and γ(1) = x ′ .
The metric H is closely linked to the Fisher information matrix (Fisher, 1925) associated with Φ: since (1.2) holds, f (x, ω) def.
= |ϕ ω (x)| 2 can be interpreted as a probability density function for the random variable ω conditional on parameter x, and the metric H x is equal (up to rescaling) to its Fisher information matrix, since
∇ (log f (x, ω)) ∇ (log f (x, ω)) ⊤ f (x, ω)dΛ(ω) = 4 E ω [Re ∇ϕ ω (x)∇ϕ ω (x) ⊤ ] = 4H x .
The distance d H is called the "Fisher-Rao" geodesic distance (Rao, 1945) and is used extensively in information geometry for estimation and learning problems on parametric families of distributions (Amari and Nagaoka, 2007). The Fisher-Rao is the unique Riemannian metric on a statistical manifold (Cencov, 2000) and it is invariant to reparameterization, which matches the invariance of the BLASSO problem (P λ (y)) to reparameterization of the space X .
Although d H has been used in conjunction with kernel methods (see for instance Burges (1999)), to the best of our knowledge, it is the first time this metric is put forward to analyze the performance of off-the-grid sparse recovery problems.
Examples
We detail some popular learning and imaging examples.
The Fejér kernel One of the first seminal result of super-resolution with sparse regularization was given by Candès and Fernandez-Granda (2014) for this kernel, which corresponds to discrete Fourier measurements on the torus. We give a multi-dimensional generalization of this result here. Let f c ∈ N, X ∈
T d , Ω = ω ∈ Z d ; ω ∞ f c . Let ϕ ω (x) def.
= e i2πω ⊤ x and Λ(ω) ∝ d j=1 g(ω j ) where g(j) = 1 fc min(j+fc,fc) k=max(j−fc,−fc) (1−|k/f c |)(1−|(j − k)/f c |). Note that this corresponds to sampling discrete Fourier frequencies. Then, the associated kernel is the Fejér
kernel K(x, x ′ ) = d i=1 κ(x i − x ′ i ), where κ(x) def. = sinc 4 fc/2+1 (x) where sinc s (x)
def.
= s −1 sin(πsx)/ sin(πx), which has a constant metric tensor H x = C fc Id and d H (x, x ′ ) = C fc x − x ′ 2 is a scaled Euclidean metric (quotiented by the action of translation modulo 1 on T d ), where C fc = −κ ′′ (0) = π 2 fc(fc+4)
3
.
The Gaussian kernel Let Σ ∈ R d×d be a positive semidefinite matrix, X ⊆ R d and Ω = R d . Let ϕ ω (x) = e iω ⊤ x and Λ(ω) = N (0, Σ −1 ), the centered Gaussian distribution with covariance Σ −1 . This can be interpreted as sampling continuous Fourier frequencies. Then, the associated kernel is K(
x, x ′ ) = e − 1 2 x−x ′ 2 Σ −1 where x Σ = √ x ⊤ Σx, with constant metric H x = Σ −1 , and d H (x, x ′ ) = x − x ′ Σ −1 .
In Section 3, we also detail how to exploit this kernel for Gaussian Mixture Model (GMM) estimation with the BLASSO.
The Laplace transform Letᾱ = (α j ) ∈ R d + , X ⊆ (0, +∞) d and Ω = R d + . A (sampled) Laplace transform is defined by setting ϕ ω (x) = d i=1 2(xi+αi) αi e − x, ω and Λ(ω) = d j=1 (2α j )e − 2ᾱ, ω . Then, K(x, x ′ ) = d i=1 κ(x i + α i , x ′ i + α i ) where κ(a, b) = 2 √ ab a+b , with metric H x as the diagonal matrix with diag- onal (2(x i + α i )) −2 d i=1 and distance d H (x, x ′ ) = i log xi+αi x ′ i +αi 2
. We remark that this kernel, associated to the Laplace transform (which should not be confused with the translation-invariant Laplace kernel exp(− x − x ′ )) appears in some microscopy imaging technique, see for instance Boulanger et al. (2014). Unlike the previous examples, it is not translationinvariant, and therefore the metric H x is not constant.
Our results show that the corresponding Fisher metric is the natural way to impose the separation condition in super-resolution.
Contributions.
Our main contribution is Theorem 1, which states that if the sought after spikes positions X 0 are sufficiently separated with respect to the Fisher distance d H , then the solution to (P λ (y)) is support stable (that is, the solution of the BLASSO is formed of exactly s Diracs) provided that the number of random noisy measurements m is, up to log factors and under the assumption of random signs of the amplitudes a 0 , linear in s, and the noise level w is less than 1/s. In the case of translation invariant kernels, this generalizes existing results to a large class of multi-dimensional kernels, and also provides for the first time a quantitative bounds on the impact of the noise and sub-sampling on the spikes positions and amplitudes errors. For nontranslation kernels, this provides for the first time a meaningful support recovery guarantee, a typical example being the Laplace kernel (see Section 1.2).
Key concepts
Notation for derivatives. Given f ∈ C ∞ (X ), by interpreting the r th derivative as a multilinear map:
∇ r f : (C d ) r → C, so given Q def. = {q ℓ } r ℓ=1 ∈ (C d ) r , ∇ r f [Q] = i1,··· ,ir ∂ i1 · · · ∂ ir f (x)q 1,i1 · · · q r,ir .
and we define the r th normalized derivative of f as
D r [f ] (x)[Q] def. = ∇ r f (x)[{H − 1 2 x q i } r i=1 ] with norm D r [f ] (x) def. = sup ∀ℓ, q ℓ 1 |D r [f ] (x)[Q]|. For i, j ∈ {0, 1, 2}, let K (ij) (x, x ′ ) be a "bi"-multilinear map, defined for Q ∈ (C d ) i and V ∈ (C d ) j as [Q]K (ij) (x, x ′ )[V ] def. = E[D i [ϕ ω ] (x)[Q]D j [ϕ ω ] (x ′ )[V ]] and K (ij) (x, x ′ ) def. = sup Q,V [Q]K (ij) (x, x ′ )[V ] where the supremum is defined over all Q def. = {q ℓ } i ℓ=1 , V def. = {v ℓ } j ℓ=1 with q ℓ 1, v ℓ 1. Note that D 2 [f ] (x) and K (02) (x, x ′ )
can also be interpreted as a matrix in C d×d , and we have the normalization K (02) (x, x) = −Id for all x.
Admissible kernel and separation
In previous studies on the recovery properties of (P λ (y)) (Candès and Fernandez-Granda, 2014;Bhaskar et al., 2013;Bendory et al., 2016;Duval and Peyré, 2015;Fernandez-Granda, 2016), recovery bounds are attained in the context of K being admissible and a separation condition on the underlying positions {x j } j . Namely, given X = {x j } j , that min i =j d H (x i , x j ) is sufficiently large with respect to the decay properties of K. For example, in the case where Φ corresponds to Fourier sampling on a grid, up to frequency f c , this separation condition is min j =ℓ x j − x ℓ 2 1/f c . In fact, if sign(a j ) can take arbitrary values in {+1, −1}, this separation condition is a necessary to ensure exact recovery for the BLASSO (Tang, 2015).
Following the aforementioned works, we introduce the notion of an admissible kernel.
Definition 1. A kernel K will be said admissible with respect to K def. = {r near , ∆, ε i , B ij , s max }, where 0 < r near < ∆/4 is a neighborhood size, ε 0 ∈ (0, 1), ε 2 ∈ (0, r −2
near ) are respectively a distance to 1 and a curvature, ∆ > 0 is a minimal separation, B ij > 0 for i, j = 0, . . . , 2 are some constants and s max ∈ N * is a maximal sparsity level, if
1. Uniform bounds: For (i, j) ∈ {(0, 0), (1, 0)}, sup x,x ′ ∈X K (ij) (x, x ′ ) B ij ; for (i, j) ∈ {(0, 2), (1, 1), (1, 2)} and all x, x ′ such that d H (x, x ′ ) r near or d H (x, x ′ ) > ∆/4, K (ij) (x, x ′ ) B ij ; and finally, sup x∈X K (22) (x, x) B 22 . 2. Neighborhood of each point: For all x ∈ X , K(x, x) = 1 and for all x, x ′ ∈ X with d H (x, x ′ ) r near , Re K (02) (x, x ′ ) −ε 2 Id and Im K (02) (x, x ′ ) cε 2 , where c def. = 1 2 2−ε2r 2 near ε2r 2 near and for d H (x, x ′ ) r near , |K(x, x ′ )| 1 − ε 0 . 3. Separation: For d H (x, x ′ ) ∆/4, for all i, j ∈ {0, . . . , 2} with i + j 3, K (ij) (x, x ′ ) h smax , where h def. = min i∈{0,2} εi 32B1i+32 , 5ε2 16B12+24 . Additionally, there exists C H 0 such that for d H (x, x 0 ) r near : Id − H − 1 2 x0 H 1 2 x C H d H (x, x 0 ). We also denote d H (X, X 0 ) = i d H (x i , x 0,i ) 2 and B def. = i+j 3 B ij and ε def. = min{ε 0 , ε 2 }.
Intuitively, these three conditions express the following facts: 1) the kernel and its derivatives are uniformly bounded, 2) near x = x ′ , the kernel has negative curvature, and otherwise it is strictly less than 1, and 3) for x and x ′ sufficiently separated, the kernel and all its derivatives have a small value.
Almost bounded random features
Ideally, we would like our features and its derivatives to be uniformly bounded for all ω. However this may not be the case: think of e iω ⊤ x where the support of the distribution Λ is not bounded. Hence our results will be dependent on the probability that the derivatives are greater than some value T decays sufficiently quickly as T increases. In the following, for r ∈ {0, 1, 2, 3},
L r (ω) def. = sup x∈X D r [ϕ ω ] (x)
, and let F r be such that P ω (L r (ω) > t) F r (t).
Key assumptions
Our main result will be valid under the following assumptions.
I. On the domain and limit kernel Let X be a compact domain with radius R X def.
= sup x,x ′ ∈X d H (x, x ′ ). Assume the kernel is admissible wrt K def. = {r near , ∆, ε i , B ij , s max }.
II. Assumption on the underlying signal For s s max , let a 0 ∈ C s and let X 0 def.
= (x 0,j ) s j=0 be such that d H (x 0,i , x 0,j ) ∆ for i = j.
The underlying measure is assumed to be µ 0 = s j=1 a 0,j δ x0,j .
III. Assumption on the sampling complexity
For ρ > 0, suppose that m ∈ N and {L i } 3 i=0 ∈ R 4 + are chosen such that 3 j=0 F j (L j ) ρ m , and 3 max j=0 {L 2 j 3 i=0 F i (L i ) + 2 ∞ Lj tF j (t)dt} ε m ,
(2.1) and either one of the following hold:
m C · s · log N d /ρ log (s/ρ) , (2.2) or m C · s 3/2 · log N d /ρ , (2.3) where C def. = ε −2 (L 2 2 B 11 +L 2 1 B 22 +(B 0 +B 2 )L 2 01 ), N def.
= L 3 dR X (r near ε) −1 and L r = max r i=1L i . Remark 1. Our main theorem presents support stability guarantees under the sampling complexity rate (2.2) if sign(a 0 ) = (a 0,i / |a 0,i |) s i=1 forms a Steinhaus sequence, that is, iid uniformly distributed on the complex unit circle. This assumption has been used before in compressed sensing (Candès and Romberg, 2007;Tang et al., 2013) to achieve this optimal complexity (see also Foucart and Rauhut (2013), Chap. 14). As noted in previous works, this random signs assumption is likely to be a proof artefact, however achieving optimal complexity without it may require more involved arguments (Candes and Plan, 2011). When the signs are arbitrary, we prove our results under (2.3). Although this s 3/2 scaling is still sub-optimal in s, we remark it improves upon the previous theoretical rate of s 2 (up to log factors) (Li and Chi, 2017).
Remark 2.
The assumption on the choice ofL r ensures that with high probability, D r [ϕ ω ] (x) is uniformly bounded up to r = 3. Note also that, generally, the {L r } depend on m, through (2.1). However, in all our examples: either a) sup x∈X D r [ϕ ω ] (x) are already uniformly bounded, in which caseL i can be chosen independently of ρ and m (for instance this is the case of the Fejér kernel); or b) the F r (t) are exponentially decaying, in which case we can show that L r = O(log(m/ρ) p ) for some p > 0, which only incurs additional logarithmic terms on the bounds (2.2) and (2.3). This is the case of the Gaussian or Laplace transform kernel.
Main result
Our main theorem below states quantitative exact support recovery bounds under a minimum separation condition according to d H .
Theorem 1. Let ρ > 0, suppose that K is admissible, and that a 0 , X 0 , m andL i satisfy the assumptions of Section 2.3.
Let
D λ0,c0 def. = {(λ, w) ∈ R + × C m ; λ λ 0 , w c 0 λ} where c 0 ∼ min ε0 L0 , ε2 L2 and λ 0 ∼ D/s with D def. = a min r near √ s, ε √ s L 2 2 a , ε CH(B+L 2 2 ) (3.1)
where a = min{|a 0,i | , |a 0,i | −1 }. Suppose that either sign(a 0 ) is a Steinhaus sequence and m satisfies (2.2) or sign(a 0 ) is an arbitrary sign sequence and m satisfies (2.3). Then, with probability at least 1 − ρ,
(i) for all v def.
= (λ, w) ∈ D λ0,c0 , (P λ (y)) has a unique solution which consists of exactly s spikes. Moreover, up to a permutation of indices, the solution can be written as
s i=1 a v i δ x v i , and sign(a v i ) = sign(a 0,i ) for all i = 1, . . . , s (ii) The mapping v ∈ D λ0,c0 → (a v , X v ) is C 1 and we have the error bound a v − a 0 + d H (X v , X 0 ) √ s(λ+ w ) mini|a0,i| (3.2)
We detail below the values relating to the sampling complexity corresponding to each of the examples detailed in Section 1.2.1. The corresponding proofs can be found in Section F of the appendix.
λ = O(s −1 d −2 ), w = O(s −1 d −3 ). Note that our choice of ∆ imposes that x i − x j 2 √ ds 1/4
max /f c whereas the previous result of Candès and Fernandez-Granda (2014) requires x i − x j ∞ C d /f c with no dependency in s max , however, their proof would imply that the constant C d grows exponentially in d. Since we are interested in having a general theory in arbitrary dimension, we have opted to present a polynomial dependency on s max .
Continuous Gaussian Fourier sampling In the appendix we prove that the kernel is admissible with
∆ = O √ log s max , r near = 1/ √ 2, ε 0 = 1 − e − 1 4 , ε 2 = e − 1 4 /2, B ij = O(1) for i + j 3, B 22 = O(d) and L r = d + log dm ρ 2 r 2
(as mentioned before, the dependence in m only incurs additional logarithmic factors in (2.2) and (2.3)). Hence, up to log factors, the sample complexity and noise level for the application of Thm. 1 is the same as for the Fejér kernel.
Laplace sampling The associated kernel is admissible with ∆ = O (d + log(ds max )), r near = 0.2, ε 0 = 0.005, ε 2 = 1.52, B ij = O(1) for i + j 3 and
B 22 = O(d). DefineR X = 1 + RX mini αi d (
where we recall that R X is the radius of X ). Assuming for simplicity that all α j are distinct, we can setL r =
R X (R X + α ∞ ) r √ d + max i 1 αi log dβimRX ραi r
Hence, choosing α i ∼ d, we have thatR X = (1) and up to log factors, (2.2) is O(sd 7 ) and (2.3) is O(s 3/2 d 7 ), and support stability is guaranteed when λ = O(s −1 d −3 ) and w = O(s −1 d −5 ). Note that despite the stronger dependency on d, for practical applications (microscopy), one is typically only interested in the low dimensional setting of d = 2, 3.
Gaussian mixture learning Consider n datapoints z 1 , . . . , z n ∈ R d drawn iid from a mixture of Gaussians i a 0,i N (x 0,i , Σ) with means x 0,i ∈ X ⊂ R d and known covariance Σ, where X is bounded. Consider the following procedure:
draw ω j iid from N (0, Σ −1 /d) (the 1/d normalization is necessary to avoid an exponential dependency in d later on) compute the generalized moments y = 1 √ m n i=1 (e i ωj ,xi ) m j=1 solve the BLASSO with features ϕ ω (x) = e i ω,x e − 1 2 ω 2 Σ , to obtain a distributionμ Then, as described in the introduction, we can in-terpret y as noisy Fourier measurements of µ 0 = i a 0,i δ x0,i in the space of means X , where the "noise" w corresponds to using the empirical average over the z i instead of a true integration. It is easily bounded with probability 1 − ρ by w O log(1/ρ) n , by a simple application of Hoeffding's inequality (Gribonval et al., 2017).
The associated kernel is the Gaussian kernel with covariance (2 + d)Σ and hence, our result states that, if
x i − x j Σ −1 √ d log s,λ 0 = O mini|a0,i| √ sd 2 a0 2
, then, with probability 1 − ρ on both samples z j and frequencies ω j , the distributioñ µ is formed of exactly s Diracs, and their positions and weights converge to the means and weights of the GMM. Let us give a few remarks on this result.
On model selection. Besides convexity (with respect to the distribution of means) of the BLASSO, which is not the case of classical likelihood-or momentsbased methods for learning GMM, the most striking feature of our approach is probably the support stability: with a sample complexity that is polynomial in s and d, the BLASSO yields exactly the right number of components for the GMM. Despite the huge literature on model selection for GMM, to our knowledge, this is one of the only result which is non-asymptotic in sample complexity, as opposed to many approaches (Roeder and Wasserman, 1997;Huang et al., 2013) which guarantee that the selected number of components approaches the correct one when the number of samples grows to infinity.
On separation condition. Our separation condition of √ d log s is, up to the logarithmic term, similar to the √ d found in the seminal work by Dasgupta (1999). This was later improved by different methods (Dasgupta and Schulman, 2000;Vempala and Wang, 2004), until the most recent results on the topic (Moitra and Valianty, 2010) show that it is possible to learn a GMM with no separation condition, provided the sample complexity is exponential in s, which is a necessary condition (Moitra and Valianty, 2010). As mentioned in the introduction, similar results exist for the BLASSO: Denoyelle et al. (2017) showed that in one dimension, one can identify s positive spikes with no separation, provided the noise level is exponentially small with s. Hence learning GMM with the BLASSO and no separation condition may be feasible, which we leave for future work, however we note that the multi-dimensional case is still largely an open problem (Poon and Peyré, 2017).
On known covariance. An important path for future work is to handle arbitrary covariance. When the components all share the same mean and have diagonal covariance, the Fisher metric is related, up to a change of variables, to the Laplace transform kernel case treated earlier. When both means and covariance vary, in one dimension, the Fisher metric is related to the Poincaré half-plane metric (Costa et al., 2015). In the general case, it does not have a closed-form expression. We leave the treatment of these cases for future work.
Sketch of proof
Background on dual certificates
Our approach to establishing that the solutions to (P λ (y)) are support stable is via the study of the associated dual solutions in accordance to the framework introduced in Duval and Peyré (2015). We first recall some of their key ideas. In order to study the support stability properties of (P λ (y)) in the small noise regime, we consider the limit problem as λ → 0 and w → 0, that is min µ∈M(X ) |µ|(X ) subject to Φµ = y.
(P 0 (y))
The dual of (P λ (y)) and (P 0 (y)) are
min p y/λ − p 2 2 ; Φ * p ∞ 1 (D λ (y)) max p { y, p ; Φ * p ∞ 1} . (D 0 (y))
Any solution µ λ of (P λ (y)) to related to the (unique) solution p λ of (D λ (y)) by −p λ = 1 λ (Φµ λ − y) and writing η λ def.
= Φ * p λ , η λ , µ λ = |µ λ | (X ). Note that Supp(µ λ ) ⊆ {x ∈ X ; |Φ * p λ (x)| = 1}, so η λ "certifies" the support of µ λ and is often referred to as a dual certificate. Furthermore, by defining the minimal norm certificate η 0 as η 0 def.
= Φ * p 0 where p 0 = argmin { p 2 ;
p is a solution to (D 0 (y))} (4.1) one can show that p λ converges as λ → 0 to p 0 and hence η λ converges to η 0 def. = Φ * p 0 in L ∞ . When λ and w are sufficiently small, solutions to (P λ (y)) are support stable provided that η 0 (called the minimal norm certificate) is nondegenerate, that is η 0 (x i ) = sign(a i ) for i = 1, . . . , s and ∇ 2 |η 0 | 2 (x i ) is negative definite. This is proven to be an almost sharp condition for support stability, since Duval and Peyré (2017) provided explicit examples where |η 0 (x)| = 1 for some x ∈ {x i } i implies that (P λ (y)) recovers more than s spikes under arbitrarily small noise.
Pre-certificates In practice, the minimal norm certificate is hard to compute and analyse due to the nonlinear ℓ ∞ constraint in (4.1). So, one often introduces a proxy which can be computed in closed form by solving an linear system associated to the following least squares problem: η X def.
= Φ * p where p X def. = argmin{ p 2 ; (Φ * p)(x i ) = sign(a i ), ∇(Φ * p)(x i ) = 0}. (4.2) Note that if η X satisfies η X ∞ 1, then η X = η 0 . Computation of η X For x ∈ X , let ϕ(x) def. = 1 √ m (ϕ ω k (x)) m k=1 . For X = {x i } s i=1 we define Γ X : C s(d+1) → C m as Γ X ([α, β]) def. = s i=1 α i ϕ(x i ) + ∇ϕ(x i ) ⊤ β i where ∇ϕ ∈ C m×d . Then, the minimizer of (4.2) is p X = Γ * , † X sign(a) 0 sd . Furthermore, when Γ X is full rank, we can writeη X (x) def. = iα iK (x i , x) + β i , ∇ 1K (x i , x) , whereα i ∈ C,β i ∈ C d are such that α β = (Γ * X Γ X ) −1 sign(a) 0 sd
, and the hat notation refers to the fact that we are using sub-sampled measurements. The limit precertificate is defined as
η X (x) def. = i α i K(x i , x) + β i , ∇ 1 K(x i , x) , where α β = (E[Γ * X Γ X ]) −1 sign(a) 0 sd .
The key to establishing our recovery results is to show thatη X is nondegenerate. In this paper, we will actually prove a stronger notion of nondegeneracy:
Definition 2. Let a ∈ C s , X = {x i } s i=1 ∈ X s for some s ∈ N, and ε 0 , ε 2 , r > 0. We say that η ∈ C 1 (X ) is (ε 0 , ε 2 )-nondegenerate with respect to a, X and r if for all i, η(x i ) = sign(a i ), ∇η(x i ) = 0 and
∀ x ∈ X far , |η(x)| 1 − ε 0 ∀ x ∈ X near j , |η(x)| 1 − ε 2 d H (x, x j ) 2 where X near j def. = {x ∈ X ; d H (x i , x) r} and X far def. = X \ s j=1 X near j .
Our proof proceeds in three steps:
1. Show that under admissibility of the kernel and sufficient separation, the limit precertificate η X0 is nondegenerate (see Theorem 2). 2. Show that this non-degeneracy transfers toη X when m is large enough and X is close to X 0 . This is the purpose of Section 4.3. 3. As discussed, nondegeneracy ofη X0 automatically guarantees support stability when (λ, w) ∈ D λ0,c0 for λ 0 and c 0 sufficiently small. To conclude we simply need to quantify λ 0 and c 0 . This is the purpose of Section 4.4. In particular, given (λ, w), we construct a candidate solution by means of (a quantitative version of) the Implicit Function Theorem, and show that it is indeed a true solution using the previous results.
Non-degeneracy of the limit certificate
Our first result shows that the "limit precertificate" η X0 is nondegenerate:
Theorem 2. Assume the kernel is admissible wrt K (see Definition 1). Then, for s s max , for all a = (a j ) s j=1 ∈ C s and X = {x j } s j=1 ∈ X s such that d H (x i , x j ) ∆, the function η X0 is ( ε0 2 , ε2
2 )nondegenerate with respect to a, X and r near .
The proof of this result can be found in Appendix B and is a generalization of the arguments of Candès and Fernandez-Granda (2014) (see also Bendory et al. (2016)). We remark that unlike previous works which focus on translation invariant kernels, the Fisher metric provides a natural way to understand the required separation between the points in X and thus open up the possibility of analysing more complex problems such as Laplace transform inversion.
The randomized setting
For the remainder of this paper, we consider solutions of (P λ (y)) given y = Φµ a0,X0 + w for some fixed a 0 ∈ C s and X 0 ∈ X s . The following result shows thatη X is nondegenerate for all X close to X 0 :
Theorem 3. Let ρ > 0. Under the assumptions of Section 2.3, and assuming that either m satisfies (2.2) and sign(a 0 ) is a Steinhaus sequence, or m satisfies (2.3) and sign(a 0 ) is an arbitrary sign sequence, with probability at least 1 − ρ: for all X ∈ X s such that
d H (X, X 0 ) min r near , εr CH √ s max(B,L12Lr) , (4.3) Γ X is full rank andη X is (ε 0 /8, ε 2 /8)-nondegenerate with respect to a 0 , X and r near .
The proof of this result is given in Appendix D. We simply make a remark on the proof here: We first prove thatη X0 is nondegenerate by bounding variations between η X0 andη X0 . The proof of this fact is a generalization of the arguments in Tang et al. (2013) to the multidimensional and general operator case. We then exploit the fact the ϕ is smooth and hence, Γ * X Γ X satisfies certain Lipschitz properties with respect to X, to bound the local variation betweenη X andη X0 .
Quantitative support recovery
This final section concludes the proof of Theorem 1 by quantifying the regions for λ and w for which support stability is guaranteed.
Solution of the noisy BLASSO. Let
Φ X : C s → C m be defined by Φ X a = s i=1 a i ϕ(x i ). Recall that µ a,X = i a i δ xi is a solution to the BLASSO with y = Φµ a0,X0 + w if and only ifη λ = Φ * p λ , with p λ = 1 λ (y − Φ X a), satisfies η λ ∞ 1 andη(x j ) = sign(a j ).
In that case, p λ is the unique solution to the dual of the BLASSO. Moreover, if |η λ (x)| < 1 for x = x i and Φ X is full rank (which follows by Theorem D.2), then µ a,X is also the unique solution of the primal.
Construction
of a solution Following Denoyelle et al. (2017), we define the function
f : C s × X s × R + × C m by f (u, v) def. = Γ * X (Φ X a − Φ X0 a 0 − w) + λ sign(a 0 ) 0 sd where u = (a, X) and v = (λ, w).
Observe that having f (u, v) = 0 ensures the existence ofη λ defined as above that satisfiesη λ (x i ) = sign(a 0,i ) and ∇η λ (x i ) = 0. We will use it to construct a non-degenerate solution to D λ (y) for small λ and w . Now, f is continu-
ously differentiable, with explicit forms of ∂ v f (u, v)
and ∂ u f (u, v) given in (E.1) and (E.2) in the appendix, and in particular,
letting u 0 = (a 0 , X 0 ), ∂ u f (u 0 , 0) = Γ * X0 Γ X0 J a , where J a is the diagonal ma- trix with 1 a ⊗ 1 d ∈ C s(d+1) along its diagonal and Γ X0 is full rank (with probability at least 1 − ρ) by Theo- rem D.2. So, ∂ u f (u 0 , 0) is invertible and f (u 0 , 0) = 0.
Hence, by the Implicit Function Theorem, there exists a neighbourhood V of 0 in C×C m , a neighbourhood U of u 0 in C s × X s and a Fréchet differentiable function
g : V → U such that for all (u, v) ∈ U × V , f (u, v) = 0 if and only if u = g(v)
. So, to establish support stability for (P λ (y)), we simply need to estimate the size of the neighbourhood V on which g is well defined, and given (λ, w) ∈ V , for (a, Z) = g((λ, w)), to check that the associated certificateη λ,w def.
= Φ * p λ,w with p λ,w def. = 1 λ (Φ X a − Φ X0 a 0 − w) is nondegenerate. Indeed, one can prove (see Theorem E.1) that with probability at least 1 − ρ, V contains the ball B r (0) with radius r ∼ 1 √ s min min{rnear,(CHB) −1 } mini|a0,i| , 1 L01L12(1+ a0 )
and given any v ∈ B r (0), (a, X) = g(v) indeed satisfy the error bound (3.2).
Checking that the candidate solution is a true solution It remains to check that g(λ, w) defines a valid certificate and is non-degenerate (and hence, i a i δ xi is the unique solution to (P λ (y))) provided that λ, w satisfy (3.1).
Given (λ, w) ∈ V , let (a, X) = g((λ, w)). Defineη λ,w def. = 1 λ Φ * (Φ X a − Φ X0 a 0 − w) and
following Denoyelle et al. (2017), one can show that
η λ,w =η X + ϕ(·) ⊤ Π X w λ + 1 λ ϕ(·) ⊤ Π X Φ X0 a 0
where Π X is the orthogonal projection onto Im(Γ X ) ⊥ .
Note that since we have the error bound (3.2), our choice of λ and w ensures that (4.3) holds and hence, Theorem D.2 implies thatη X is nondegenerate with probablity at least 1 − ρ. To conclude, it is sufficient to show that the two remaining terms are sufficiently small, so thatη λ,w remains non-degenerate.
UnderĒ, D r [ϕ ω ] (·) L r , and for any z ∈ C m , D r ϕ ⊤ z · L r z . Therefore, since Π X is a projection, we have D r ϕ(·) ⊤ Π X w λ ε r when w /λ ε r /L r . Finally, since Φ X0 a 0 = s j=1 ϕ(x 0,j ),
by Taylor expansion of ϕ(x 0,j ) around x j and applying Π X (see Lemma E.1 for this computation), we have A Notations.
1 λ Π X Γ X0 a 0 0 sd L 2 λ a 0 ∞ d H (X, X 0 ) 2 . Since g satisfies (3.2) our choice of λ 0 = O(s −1 ) ensures that we can upper bound this byL 2 a 0 ∞ s(λ+ w 2 /λ) min|a0,i| 2 ε and consequently, 1 λ D r ϕ(·) ⊤ Π X Φ X0 a 0 ε r . C. Fernandez-Granda.
In this section, we recall and introduce some notation which will be used throughout the appendix.
Block norms. By default, · is the Euclidean norm for vector and spectral norm for matrices. For a vector
x = [x 1 , . . . , x s ] ∈ C sd formed of s blocks x i ∈ C d , 1 i s, we define the block norm x block def. = sup 1 i s x i 2 For a vector q = [q 1 , . . . , q s , Q 1 , . . . , Q s ] ∈ C s(d+1) decomposed such that q i ∈ C and Q i ∈ C d , we define q * ,∞ def. = s max i=1 {|q i | , Q i }.
Kernel The empirical kernel is defined aŝ
K(x, x ′ ) = 1 m m k=1 ϕ ω k (x)ϕ ω k (x ′ ) and the limit kernel is K(x, x) def. = E ω [ϕ ω (x)ϕ ω (x ′ )].
The metric tensor associated to this kernel is
H x def. = E ω [∇ϕ ω (x)∇ϕ ω (x) ⊤ ] Given an event E, we write K E (x, x ′ ) def. = E ω [K(x, x ′ )|E] to denote the conditional expectation on E.
Derivatives Given f ∈ C ∞ (X ), by interpreting the r th derivative as a multilinear map:
∇ r f : (C d ) r → C, so given Q def. = {q ℓ } r ℓ=1 ∈ (C d ) r , ∇ r f [Q] = i1,··· ,ir ∂ i1 · · · ∂ ir f (x)q 1,i1 · · · q r,ir .
and we define the r th normalized derivative of f as
D r [f ] (x)[Q] def. = ∇ r f (x)[{H − 1 2 x q i } r i=1 ] with norm D r [f ] (x) def. = sup ∀ℓ, q ℓ 1 |D r [f ] (x)[Q]|.
We will sometimes make use the the multiarray interpre-
tation: D 0 [f ] = f , D 1 [f ] (x) = H − 1 2 x ∇f (x) ∈ C d , D 2 [f ] (x) = H − 1 2 x ∇ 2 f (x)H − 1 2 x ∈ C d×d .
For a bivariate function K : X × X → C, ∂ 1,i (resp. ∂ 2,i ) designates the derivative with respect to the i th coordinate of the first variable (resp. second variable), and similarly ∇ i and ∇ 2 i denote the gradient and Hessian on the i th coordinate respectively.
For i, j ∈ {0, 1, 2}, let K (ij) (x, x ′ ) be a "bi"-multilinear map, defined for Q ∈ (C d ) i and V ∈ (C d ) j as [Q]K (ij) (x, x ′ )[V ] def. = E[D i [ϕ ω ] (x)[Q]D j [ϕ ω ] (x ′ )[V ]] and K (ij) (x, x ′ ) def. = sup Q,V [Q]K (ij) (x, x ′ )[V ] where the supremum is defined over all Q def. = {q ℓ } i ℓ=1 , V def. = {v ℓ } j ℓ=1 with q ℓ 1, v ℓ 1. When i+j 2, an equivalent definition is K (ij) (x, x ′ ) = E[D i [ϕ ω ] (x)D j [ϕ ω ] (x ′ ) ⊤ ]
, and we note that K (00) = K, and we have normalized so that Re K (11) (x, x) = −Re K (02) (x, x) . Finally, we will make use of the still equivalent definition:
[q]K (12) (x, x ′ ) = E[q ⊤ D 1 [ϕ ω ] (x)D 2 [ϕ ω ] (x ′ ) ⊤ ] ∈ C d×d . Kernel constants For for i, j ∈ {(0, 0), (0, 1)}, define B ij def. = sup x,x ′ ∈X K (ij) (x, x ′ ) , for (i, j) ∈ {(0, 2), (1, 2)}, B ij def. = sup K (ij) (x, x ′ ) ; d H (x, x ′ ) r near or d H (x, x ′ ) > ∆/2 . and define for i = 1, 2 B ii def. = sup x∈X K (ii) (x, x) . For convenience, we define B i def. = B 0i + B 1i + 1, B def. = i,j∈{0,1,2} i+j 3 B ij + 1. (A.1)
Matrices and vectors We will make use of the following vectors and matrices throughout: Given X def.
= {x j } s j=1 ∈ X s and a ∈ C s which are always clear from context, define the vector γ X (ω) ∈ C s(d+1) as
γ X (ω) def. = ϕ ω (x i ) s i=1 , D 1 [ϕ ω ] (x i ) ⊤ s i=1 ⊤ , (A.2)
and
Υ X def. = E ω [γ(ω)γ(ω) * ] ∈ C s(d+1)×s(d+1) f X (x) def. = E ω [γ(ω)ϕ ω (x)] ∈ C s(d+1) α def. = Υ −1 X u s , u s = sign(a) 0 sd .
Note that the diagonal of Υ has only 1's. For ω 1 , . . . , ω m , we denote their empirical versions as:
Υ X def. = 1 m m k=1 γ(ω k )γ(ω k ) * , f X (x) def. = 1 m m k=1 γ(ω k )ϕ ω k (x),α def. =Υ −1 X u s .
which will serve us to construct our certificate, using the properties of their respective limit version.
We remark that G
−1/2 X Γ * X Γ X G −1/2 X =Υ X ,
where Γ X is defined in the main paper and
G X = Id s 0 H x1 . . . 0 H xs
The vanishing derivative pre-certificateη X isα ⊤f X (·) and the limit pre-certificate is η X def.
= α ⊤ f X (·). When the set of points X is clear from context, we will drop the subscript X and write instead γ, Υ, f , η, and so on.
Metric induced distances Given
X = (x j ) s j=1 ∈ X s and X ′ = (x ′ j ) s j=1 ∈ X s , denote d H (X, X ′ ) def. = j d H (x j , x ′ j ) 2 .
Observe also that G X is positive definite for all X and induces a metric on R s × X s so that given a, a ′ ∈ R s and X, X ′ ∈ X s ,
d G ((a, X), (a ′ , X ′ )) = a − a ′ 2 2 + d H (X, X ′ ) 2 .
Stochastic gradient bounds For r ∈ N,
L r (ω) = sup x∈X D r [ϕ ω ] (x) , and L ij (ω) def. = L i (ω) 2 + L j (ω) 2 . For i = 0, 1, 2, 3, let F i be such that P ω (L j (ω) > t) F i (t),
Throughout, for (L j ) 3 j=0 ∈ R 4 + , the eventĒ is defined as
E def. = m k=1 E ω k where E ω def. = {L j (ω) L j , ∀j = 0, 1, 2, 3}. (A.3)
B Proof of Theorem 2
In this section, we consider the (limit) vanishing derivative pre-certificate
η(x) = u ⊤ Υ −1 X f X (x).
Note that
D 2 [η] (x) = s i=1 α 1,i K (02) (x i , x) + [α 2,i ]K (12) (x i , x) where we have decomposed α = [α 1,1 , . . . , α 1,s , α 2,1 , . . . , α 2,s ] ∈ C s(d+1) where α 2,i ∈ C d .
We aim to prove that η is nondegenerate if K is an admissible kernel. Our first lemma shows that nondegeneracy of η within each small neighbourhood of x i can be established by controlling the real and imaginary parts of D 2 [η] in each small region:
Lemma B.1. Let ε > 0. Let a 0 = 0, x 0 ∈ X and let σ ∈ C be such that |σ| = 1. Suppose that η ∈ C 2 (X ; C) is such that η(x 0 ) = σ, ∇η(x 0 ) = 0 and Re (σD 2 [η] (x 0 )) ≺ −εId. Then, ∇ 2 |η| 2 (x 0 ) ≺ −2εId. If in addition, we have c, r > 0 with εr < 1 and c 2 (1 − εr 2 )/(εr 2 ) such that for all x such that d H (x, x 0 ) r, Re (σD 2 [η] (x)) ≺ −εId and Im (σD 2 [η] (x)) cε, then, |η(x)| 2 1 − ε 2 d H (x, x 0 ) 2 for all x such that d H (x, x 0 ) r.
Proof. The first claim follows immediately from the computation: by writing η = η r (x) + iη i (x) where η i and η r are real valued functions,
1 2 D 2 |η| 2 = Re D 1 [η]D 1 [η] ⊤ + D 2 [η] η ,
and evaluation at x 0 gives the required result.
Let γ : [0, 1] → X be a piecewise smooth path such that
γ(0) = x 0 , γ(1) = x. η(x) = η(x 0 ) + 1 0 (1 − t) ∇ 2 η(γ(t))γ ′ (t), γ ′ (t) dt = η(x 0 ) + 1 0 (1 − t) D 2 [η] (γ(t))H 1 2 γ(t) γ ′ (t), H 1 2 γ(t) γ ′ (t) dt. So, Re sign(a 0 )η(x) = 1 + inf γ Re sign(a 0 ) 1 0 (1 − t) D 2 [η] (γ(t))H 1 2 γ(t) γ ′ (t), H 1 2 γ(t) γ ′ (t) dt 1 − εd H (x, x ′ ) 2
if we minimise over all paths from x to x 0 . Similarly,
Im sign(a 0 )η(x) cεd H (x, x 0 ) 2 Therefore, |η(x)| 2 1 − εd H (x, x 0 ) 2 2 + cεd H (x, x 0 ) 2 2 1 − 2εd H (x, x 0 ) 2 + ε 2 d H (x, x 0 ) 4 + c 2 ε 2 d H (x, x 0 ) 4 = 1 − εd H (x, x 0 ) 2 − εd H (x, x 0 ) 2 1 − εd H (x, x 0 ) 2 1 + c 2 1 − εd H (x, x 0 ) 2 .
Proof of Theorem 2. In order to show that η is (ε 0 /2, ε 2 /2)-nondegenerate, it is enough to show that
∀x ∈ X far , |η(x)| 1 − ε 0 /2 (B.1) ∀x ∈ X near , Re sign(a j )D 2 [η] (x) ≺ − ε 2 2 Id and Im sign(a j )D 2 [η] (x) p 4 ε 2 (B.2)
where p = 1−ε2r 2 near /2 ε2r 2 near /2 . We first prove that the matrix Υ is invertible. To this end, we write
Υ = Υ 0 Υ ⊤ 1 Υ 1 Υ 2 (B.3) where Υ 0 def. = (K(x i , x j )) s i,j=1 ∈ C s×s , Υ 1 def. = (K (10) (x i , x j )) s i,j=1 ∈ C sd×s , and Υ 2 def. = (K (11) (x i , x j )) s i,j=1 ∈ C sd×sd . By definition of K (ij)
, Υ (and also Υ 0 and Υ 2 ) has only 1's on its diagonal.
To prove the invertibility of Υ, we use the Schur complement of Υ, and in particular it suffices to prove that Υ 2 and the Schur complement Υ S def.
= Υ 0 − Υ 1 Υ −1 2 Υ ⊤ 1 are both invertible. To show that Υ 2 is invertible, we define A ij = K (11) (x i , x j ).
So Υ 2 has the form:
Υ 2 = Id A 12 . . . A 1s A 21 Id . . . . . . . . . . . . . . . . . . A s1 . . . . . . Id
and by Lemma G.6, we have
Id − Υ 2 block max i j A ij 1/4.
Since Id − Υ 2 block < 1, Υ 2 is invertible, and we have Υ −1 2 block 1 1− I−Υ2 block 4 3 . Next, again with Lemma G.6, we can bound
I − Υ 0 ∞ = max i j =i |K(x i , x j )| ε 0 16 Υ 1 ∞→block max i j K (10) (x i , x j ) h since K (10) (x, x) = 0 Υ ⊤ 1 block→∞ max i j K (10) (x j , x i ) h
Hence, we have
I − Υ S ∞ I − Υ 0 ∞ + Υ ⊤ 1 block→∞ Υ −1 2 block Υ 1 ∞→block ε 0 16 + 4 3 h 2 ε 0 8 (B.4)
since h ε0 32 . Therefore the Schur complement of Υ is invertible and so is Υ.
Expression of η. By definition, η = satisfies η(x i ) = sign(a i ) and ∇η(x i ) = 0.
We divide:
α = Υ −1 u s = α 1 α 2 where α 1 ∈ C s and α 2 ∈ C sd , and we denote α 2,i ∈ C d blocks such that α 2 = [α 2,1 , . . . , α 2,s ].
The Schur's complement of Υ allows us to express α 1 and α 2 as
α 1 = Υ −1 S sign(a), α 2 = −Υ −1 2 Υ 1 Υ −1 S sign(a) (B.5)
and therefore we can bound
α 1 ∞ 1 1 − ε 0 /8 (B.6) α 2 block 8 3 h 4h (B.7)
Moreover, we have
α 1 − sign(a) ∞ I − Υ −1 S ∞ Υ −1 S ∞ I − Υ S ∞ 1 4 (B.8)
Non-degeneracy. We can now prove that η is non-degenerate.
Let x be such that d H (x i , x) r near . We need to prove that for all x such that d H (x, x i ) r, Re sign(a i )D 2 [η] (x) ≺ − ε 2 2 Id and Im sign(a i )D 2 [η] (x) ε 2 2 2 − εr 2 near ε 2 r 2 near .
Then, since r near ∆/2 and the
x i 's are ∆-separated, for all j = i we have d H (x, x j ) ∆/2. Then, we have sign(a i )D 2 [η] (x) = sign(a i ) α 1,i K (02) (x i , x) + j =i α 1,j K (02) (x j , x) + [α 2,i ]K (12) (x i , x) + j =i [α 2,j ]K (12) (x j , x) Re sign(a i )D 2 [η] (x) (1 − α 1 − sign(a) ∞ )Re K (02) (x i , x) + α 1 ∞ j =i K (02) (x j , x) Id + K (12) (x i , x) + j =i K (12) (x j , x) α 2 block Id − 3 4 ε 2 + 1 1 − ε 0 /8 ε 2 16 + 4h(B 12 + 1) Id ε 2 − 3 4 + 1 4 Id − ε 2 2 Id .
Taking the imaginary part, we have
Im sign(a i )D 2 [η] (x) (1 + α 1 − sign(a) ) Im K (02) (x i , x) + α 1 ∞ j =i K (02) (x j , x) + K (12) (x i , x) + j =i K (12) (x j , x) α 2 block 5cε 2 4 + 1 (1 − ε 0 /8) h + 4h(B 12 + 1) 5cε 2 4 + h (4B 12 + 6) ε 2 2 2 − εr 2 near ε 2 r 2 near . So, by Lemma B.1, for each i = 1, . . . , s, |η(x)| 1 − ε 2 /2d H (x, x i ) for all x ∈ X such that d H (x, x i ) r near .
Next, for any x such that d H (x, x i ) r near for all x i 's, we can say that there exists (at most) one index i such that d H (x, x i ) r near and for all j = i we have d H (x, x j ) ∆/2. We have
|η(x)| = α 1,i K(x i , x) + j =i α 1,j K(x j , x) + K (10) (x i , x) ⊤ α 2,i + j =i K (10) (x j , x) ⊤ α 2,j α 1 ∞ |K(x i , x)| + j =i |K(x j , x)| + α 2 block K (10) (x i , x) + j =i K (10) (x j , x) 1 − ε 0 + ε 0 /16 1 − ε 0 /8 + 4h(B 10 + 1) 1 − ε 0 2 .
Remark B.1. Assuming that the derivatives of the kernel decay like a function f ( x − x ′ ) when, there is always a separation ∆ ∝ f −1 (1/(Cs max ))) such that the kernel is admissible. Ex: when f = x −p , we have ∆ ∝ s 1/p max (eg Cauchy). When f = e −x p , we have ∆ ∝ log 1/p (s max ) (eg Gaussian).
C Preliminaries
In this section, we present some preliminary results which will be used for proving our main results. We assume that K is admissible, and given a set of points X ∈ X s , let X near
j def. = {x ∈ X ; d H (x, x j ) r near }, X near def. = s j=1 X near j
and X far def. = X \ X near .
C.1 On the determistic kernel
For an admissible kernel, we have the following additional bounds that will be handy.
Lemma C.1. Assume K is an admissible kernel, let X ∈ X s be ∆-separated points. Then we have the following:
(i) We have seen that Υ is invertible. Additionally it satisfies
Id − Υ 1 2 and Id − Υ * ,∞ 1 2 . (C.1)
(ii) For any vector q ∈ C s(d+1) and any x ∈ X far , we have
f (x) B 0 and q ⊤ f (x) B 0 q * ,∞ (C.2)
(iii) For any vector q ∈ C s(d+1) and any x ∈ X near we have the bound:
D 2 q ⊤ f (.) (x) q B 2 and D 2 q ⊤ f (.) (x) q * ,∞ B 2 (C.3) Proof. We bound the spectral norm of Id − Υ. Define y ∈ C s(d+1) decomposed as y = [y 1 , . . . , y s , Y 1 , . . . , Y s ] where Y i ∈ R d , such that y 1. We have (Id − Υ)y 2 = s i=1 j =i K(x i , x j )y j + s j=1 K (10) (x i , x j ) ⊤ Y j 2 + j y j K (10) (x i , x j ) + j =i K (11) (x i , x j )Y j 2 s i=1 j =i |K(x i , x j )| |y j | + s j=1 K (10) (x i , x j ) Y j 2 + j |y j | K (10) (x i , x j ) + j =i K (11) (x i , x j ) Y j 2 max dH(x,x ′ ) ∆ |K(x, x ′ )| , K (10) (x, x ′ ) , K (11) (x, x ′ ) 2 i 2 j |y j | + Y j 2 4s 2 max dH(x,x ′ ) ∆ |K(x, x ′ )| , K (10) (x, x ′ ) , K (11) (x, x ′ ) 2
by Cauchy-Schwartz inequality and since K (10) (x, x) = 0 for all x ∈ X . Since by hypothesis we have
max dH(x,x ′ ) ∆ |K(x, x ′ )| , K (10) (x, x ′ ) , K (11) (x, x ′ ) 1 4s max , we obtain Id − Υ 1 2 (C.4)
and we deduce (i). A near identical argument also yields Υ − Id * ,∞ 1 4 . For (ii), let x ∈ X far , then we have
f (x) s i=1 |K(x i , x)| 2 + K (10) (x i , x) 2 1 2 B 2 00 + (s − 1)ε 2 0 (16s max ) 2 + B 2 10 + (s − 1) s 2 max 1 2 B 0
for which, similar to the proof above, we have used the fact that x is ∆/2-separated from at least s − 1 points x i . Similarly, for any vector q = [q 1 , . . . , q s , Q 1 , . . . , Q s ] ∈ C s(d+1) and any x ∈ X far , we have
q ⊤ f (x) s i=1 |q i | |K(x i , x)| + Q i K (10) (x i , x) q * ,∞ B 00 + (s − 1)ε 0 32s max ) + B 10 + (s − 1)ε 0 32s max B 0 q * ,∞ .
For any x ∈ X near we have the bound:
D 2 q ⊤ f (x) = s i=1 q i K (02) (x i , x) + [Q i ]K (12) (x i , x) q s i=1 K (02) (x i , x) 2 + K (12) (x i , x) 2 1 2 q B 2 and D 2 q ⊤ f (x) = s i=1 q i K (02) (x i , x) + [Q i ]K (12) (x i , x) q * ,∞ s i=1 K (02) (x i , x) + K (12) (x i , x) q * ,∞ B 2
C.2 Lipschitz bounds
Lemma C.2 (Local Lipschitz constant of ϕ ω and higher order derivatives).
Suppose that D j [ϕ ω ] (x) L j for all x ∈ X . For all x, x ′ with d H (x, x ′ ) r near , we have (i) |ϕ ω (x) − ϕ ω (x ′ )| L 0 d H (x, x ′ ), (ii) D 1 [ϕ ω ] (x) − D 1 [ϕ ω ] (x ′ ) L 1 d H (x, x ′ ), (iii) D 2 [ϕ ω ] (x) − D 2 [ϕ ω ] (x ′ ) L 2 d H (x, x ′ ), where L 0 def. =L 1 , L 1 def.
=L 1 C H +L 2 (1 + C H r near ) and L 2 def.
=L 2 C H + C 2 H r near + 1 +L 3 (1 + C H r near ) 2 . As a consequence, for all X = (x j ) and
X ′ = (x ′ j ) such that d H (x j , x ′ j ) r near , we have sup q =1 D r q ⊤ (f X −f X ′ ) (y) L r L 2 0 + L 2 1 d H (X, X ′ ). Proof. Let x, x ′ ∈ X with d H (x, x ′ ) r near . Recall that H 1 2 x ′ H − 1 2 x − Id C H d H (x, x ′ ), and so, H 1 2 x ′ H − 1 2 x 1 + C H r near .
Let p : [0, 1] → X be a piecewise smooth path such that p(0) = x ′ , p(1) = x. Then, by Taylor's theorem,
ϕ ω (x) − ϕ ω (x ′ ) = 1 t=0 H − 1 2 p(t) ∇ϕ ω (p(t)), H 1 2 p(t) p ′ (t) dt L 1 1 0 H 1 2 p(t) p ′ (t) dt (C.5) so taking the minimum over all paths p yields |ϕ ω (x) − ϕ ω (x ′ )| L 1 d H (x, x ′ ).
Given q ∈ R d , by Taylor's theorem,
D 1 [ϕ ω ] (x)[q] = ∇ϕ(x)[H − 1 2 x q] = ∇ϕ(x ′ )[H − 1 2 x q] + ∇ 2 ϕ ω (p(t))[H − 1 2 x q, p ′ (t)]dt = D 1 [ϕ ω ] (x ′ )[q] + D 1 [ϕ ω ] (x ′ )[(H 1 2 x ′ H − 1 2 x − Id)q] + D 2 [ϕ ω ] (p(t))[H 1 2 p(t) H − 1 2 x q, H 1 2 p(t) p ′ (t)]dt (C.6) Therefore, D 1 [ϕ ω ] (x) − D 1 [ϕ ω ] (x ′ ) L 1 C H d H (x, x ′ ) +L 2 (1 + C H r near )d H (x, x ′ ).
Finally, for all q 1 , q 2 ∈ R d , by Taylor's theorem
D 2 [ϕ ω ] (x)[q 1 , q 2 ] − D 2 [ϕ ω ] (x ′ )[q 1 , q 2 ] = ∇ 2 ϕ ω (x)[H − 1 2 x q 1 , H − 1 2 x q 2 ] − ∇ 2 ϕ ω (x ′ )[H − 1 2 x ′ q 1 , H − 1 2 x ′ q 2 ] = D 2 [ϕ ω ] (x ′ )[H 1 2 x ′ H − 1 2 x q 1 , (H 1 2 x ′ H − 1 2 x − Id)q 2 ] + D 2 [ϕ ω ] (x ′ )[(H 1 2 x ′ H − 1 2 x − Id)q 1 , q 2 ] + D 3 [ϕ ω ] (p(t))[H 1 2 p(t) H − 1 2 x q 1 , H 1 2 p(t) H − 1 2 x q 2 , H 1 2 p(t) p ′ (t)]dt. (C.7) Therefore, D 2 [ϕ ω ] (x) − D 2 [ϕ ω ] (x ′ ) L 2 ((1 + C H r near )C H + 1) +L 3 (1 + C H r near ) 2 d H (x, x ′ ).
By applying these Lipschitz bounds, we obtain
sup q =1 D r q ⊤ (f X −f X ′ ) (y) 2 s j=1 K (0r) (x j , y) −K (0r) (x ′ j , y) 2 + s j=1 K (1r) (x j , y) −K (1r) (x ′ j , y) 2 s j=1 L 2 0L 2 r d H (x j , x ′ j ) 2 + s j=1 L 2 1L 2 r d H (x j , x ′ j ) 2 = L 2 0 + L 2 1 L 2 r d H (X, X ′ ) 2 Lemma C.3 (Local Lipschitz constant ofK (ij) ). Let x 1 , x 0 ∈ X . Let i, j ∈ {0, 1, 2} with i + j 3. Define A ij = sup x K (ij) (x, x 0 )
where x ranges over d H (x, x 1 ) r near . Then, for all x such that d H (x, x 1 ) r near ,
K (0j) (x, x 0 ) −K (0j) (x 1 , x 0 ) A 1j d H (x, x 1 ) K (1j) (x, x 0 ) −K (1j) (x 1 , x 0 ) (C H A 1j + (1 + C H r near )A 2j ) d H (x, x 1 )
The same results hold if we replaceK by K.
Proof. The Lipschitz bounds onK ij follow by combining
[q 1 , . . . , q i ](K (ij) (x, x 0 ) −K (ij) (x 1 , x 0 ))[v 1 , . . . , v j ] =ÊRe (D i [ϕ ω ] (x) − D i [ϕ ω ] (x 1 ))[q 1 , . . . , q i ]D j [ϕ j ] (x 0 )[v 1 , . . . , v j ]
whereÊ indicates either empirical expectation or true expectation with (C.5), (C.6) and (C.7).
C.3 Probability bounds
In the proof of our main results, we will often assume that eventĒ (see (A.3)) holds since our assumptions in Section 2.3 imply that P(Ē c ) ρ/m. The following lemma shows that our assumptions also imply that
E ω [L i (ω) 2 1 E c ω ] ε m
. and this is a condition which our proofs will often rely upon.
Lemma C.4. The following holds.
P(E c ω ) i F i (L i ) and E ω [L j (ω) 2 1 E c ω ] 2 ∞ Lj tF j (t)dt +L 2 j i F i (L i ) Proof. Let E ω,j be the event that L r (ω) L r , so E ω = ∩ 3 j=0 E ω,j . By the union bound, P(E c ω ) j P(E c ω,j ) i F i (L i ). For the second claim, observe that E c ω = ∪ i E c ω,i so that E[L j (ω) 2 1 E c ω ] i E[L j (ω) 2 1 E c ω,i ] and we have E[L j (ω) 2 1 E c ω,i ] = ∞ 0 P(L j (ω) 2 1 E c ω,i t)dt = ∞ 0 P (L j (ω) 2 t) ∩ (L i (ω) L i ) dt L 2 j F i (L i ) + ∞ L 2 j F j ( √ t)dt =L 2 j F i (L i ) + 2 ∞ Lj tF j (t)dt where we have bounded P (L j (ω) 2 t) ∩ (L i (ω) L i ) by respectively P(L i (ω) L i ) F i (L i )
in the first term and by P(L j (ω) 2 t) F j ( √ t) in the second term.
C.3.1 Concentration inequalities
The following result is an adaption of the Matrix Bernstein inequality for dealing with conditional probabilities.
Lemma C.5 (Adapted unbounded Matrix Bernstein). Let A j ∈ R d1×d2 be a family of iid matrices for j = 1, . . . , m.
Let Z = 1 m m j=1 A j and letZ = E[Z]. Let t ∈ (0, 4 E[A 1 ] ]. Let events E j be independent events such that E j ⊆ { A j L} and let E = ∩ j E j . Suppose that we have P(E c j ) t t + 4 E[A 1 ] and E[ A j 1 E c j ] t 4
Then a first consequence is that we have
E E [Z] = E Ej [A j ] for all j and E[Z] − E E [Z] t 2 . Finally, assuming that σ 2 def. = max j { E Ej [A j A * j ] , E Ej [A * j A j ] } < ∞ we have P E ( Z − E[Z] t) (d 1 + d 2 ) exp − mt 2 /4 σ 2 + Lt/3 . Proof. We first bound E[Z] − E E [Z] . First observe that E[Z] = E E1 [A 1 ] and E E Z = E E1 [A 1 ] since A j are iid. Moreover, E[A 1 ] = E[A 1 1 E1 ] + E[A 1 1 E c 1 ] = E[A 1 |E 1 ]P(E 1 ) + E[A 1 1 E c 1 ]. Hence, E[A 1 ] − E E1 [A 1 ] = (P (E 1 ) − 1)E E1 [A 1 ] + E[A 1 1 E c 1 ] P(E c 1 ) E[A 1 ] + P (E c 1 ) E[A 1 ] − E E1 [A 1 ] + E[ A 1 1 E c 1 ]. Therefore, E[A 1 ] − E E1 [A 1 ] P (E c 1 ) E[A 1 ] + E[ A 1 1 E c 1 ] 1 − P(E c 1 ) t 2
For the second statement,
P E ( Z − E[Z] t) P E ( Z − E E [Z] t − E[Z] − E E [Z] ) P E ( Z − E E [Z] t/2).
To conclude, we apply Bernstein's inequality (Lemma G.
2) to Y j = A j −E[A j |E] = Y j = A j −E[A j |E j ] conditional to E. Observe that 0 E E [Y j Y ⊤ j ] E E [A j A ⊤ j ] − E E [A j ]E E [A j ] ⊤ ] E E [A j A ⊤ j ], which yields E E [Y j Y ⊤ j ] E[A j A ⊤ j ] and similarly, E E [Y ⊤ j Y j ] E E [A ⊤ j A j ] . So by Bernstein's inequality P E ( Z − E E [Z] t/2) 2(d 1 + d 2 ) exp − mt 2 /4 σ 2 + Lt/3 . Corollary C.1. Let x, x ′ ∈ X . If P(E c ω ) t t + 4 K (ij) (x, x ′ ) and E[L ij (ω)1 E c ω ] t 4 then K (ij) E (x, x ′ ) − K (ij) (x, x ′ ) t/2.
Proposition C.1. Let t > 0 and assume that
P(E c ω ) t t + 6 and E[L 01 (ω) 2 1 E c ω ] t 4s then Υ − ΥĒ t/2 and PĒ( Υ −Υ t) 4(d + 1)s exp − mt 2 /4 sL 2 01 (3 + t/3)
Consequently,
PĒ( Υ −1 −Υ −1 t) 4(d + 1)s exp − mt 2 16sL 2 01 (3 + 2t)
.
Proof. We apply Lemma C.5 to A j = γ(ω j )γ(ω j ) * with the following observations:
• for each ω, γ(ω)γ(ω) * γ(ω) 2 s max x∈X { D 1 [ϕ ω ] (x) 2 + |ϕ ω (x)| 2 },
so under eventĒ, A j sL 2 01 .
• By Lemma C.1, E[A j ] = Υ 3/2,
• We may set σ 2 =L 01 (3/2 + t/2) since
0 EĒ[A 1 A * 1 ] = EĒ[A * 1 A 1 ] = EĒ[ γ(ω j ) 2 γ(ω j )γ(ω j ) * ] L 01 ( E[A j ] + t/2)Id. The last claim is because Υ −Υ t implies that Υ 3/2 + t, Υ −1 Υ 1− Υ−Υ Υ −1 3 2−4t and Υ −1 −Υ −1 Υ −1 Υ −Υ Υ −1 3t
1−2t and writingt = 3t 1−2t is equivalent to t =t/(3 + 2t).
Bounds onf X applied to a fixed vector Proposition C.2. Let t ∈ (0, 1), r ∈ {0, 2}, q ∈ C s(d+1) and y ∈ X r , where X 0 def. = X and X 2 def.
= X near . If
P(E c ω ) t t + 4B r and E[L 01 (ω)L r (ω)1 E c ω ] t 4 √ s then PĒ D r (f X0 − f X0 ) ⊤ q (y) t q 2d exp −mt 2 /4 2L 2 r +L rL01 t/(3 √ s) whered = 1 if r = 0 andd = d if r = 2.
As a consequence, since √ 2s q * ,∞ q 2 , we have
P E D r (f X0 −f X0 ) ⊤ q (y) t q * ,∞ 2d exp −mt 2 16s(L 2 r + 8L rL01 t/(3 √ 2)) provided that P(E c ω ) t t + 4 √ 2sB r and E[L 01 (ω)L r (ω)1 E c ω ] t 4 √
2s .
Proof. Without loss of generality, assume that q = 1. First note that
D r (f X0 − f X0 ) ⊤ q (y) = 1 m m k=1 q ⊤ γ(ω k )D r [ϕ ω k ] (y) − E[q ⊤ γ(ω k )D r [ϕ ω k ] (y)].
We first consider the case of r = 0. We apply Lemma C.5 to A k def.
= q ⊤ γ(ω k )ϕ ω k (y) ∈ C: Note that |A k | √ sL 01 (ω k )L 0 (ω k ) and |E[A k ]| B 0 . • Under event E ω k , A k L 2L01 √ s def. = L. • EĒ |A k | 2 = EĒ[ γ(ω k )γ(ω k ) * q, q |ϕ ω k (y)| 2 ] L 2 0 ΥĒ (3/2 + t/2)L 2 0 2L 2 0 def. = σ 2 .
For the case r = 2, we apply Lemma C.5 with A k def.
= q ⊤ γ(ω k )D 2 [ϕ ω k ] (y) ∈ C d×d . Then, A k √ sL 01 (ω k )L 2 (ω k ), E[A k ] B 2 , under event E ω k , A k L 2L01 √ s def. = L and EĒ[A k A * k ] = EĒ[A * k A k ] = EĒ[D 2 [ϕ ω k ] (y) * D 2 [ϕ ω k ] (y) q ⊤ γ(ω k ) 2 ] L 2 2 EĒ[ q ⊤ γ(ω k ) 2 ] 2L 2 2 def. = σ 2 .
Lemma C.6. Assume that
P(E c ω ) t t + 6 √ 2s and E[L 01 (ω) 2 1Ēc] t 4 √ 2s 3/2
Let q ∈ C s(d+1) . Then, for all t 2
√ 2sL01L1 m + 8s 2L2 01L 2 1 m 2 + 144sL 2 1 m , we have for each x i ∈ X, P E D 1 q ⊤ (f X −f X ) (x i ) 2 > 2t q * ,∞ 28 exp − mt 2 /(4s) 2L 2 1 + √ 2tL 1L01 /3 . Proof. For each x i ∈ X, D 1 (EĒ[q ⊤f X ] − q ⊤ f X ) (x i ) Υ − ΥĒ q t √ 2s q ,
by Proposition C.1. For convenience, we drop the subscript X from f X . Fix i ∈ {1, . . . , s}. Observe that
P E D 1 q ⊤ (f −f ) (x i ) 2 > 2t q * ,∞ P E D 1 q ⊤ (f −f ) (x i ) 2 > 2t √ 2s q 2 P E D 1 q ⊤ (EĒ[f ] −f ) (x i ) 2 > t √ 2s q 2
The claim of this lemma follows by applying Lemma G.3: Let
Y k = D 1 [ϕ ω k ] (x i )γ(ω k ) ⊤ q − EĒD 1 [ϕ ω k ] (x i )γ(ω) ⊤ q ∈ C d , and observe that D 1 q ⊤ (f − EĒ[f ]) (x i ) = 1 m k Y k .
Without loss of generality, assume that q 2 = 1. We apply Lemma G.3. Observe that conditional on event E,
• Y k 2 2 q 2 γ(ω k ) 2 D 1 [ϕ ω k ] (x i ) 2 2 √ sL 01L1 . • E E Y k 2 E E [ γ(ω k ) ⊤ q 2 D 1 [ϕ ω k ] (x i )D 1 [ϕ ω k ] (x i ) ⊤ ] L 2 1 Υ E . So, σ 2 mL 2 1 Υ E mL 2 1 (t + Υ ) mL 2
1 (t/2 + 3/2) 2mL 2 1 (here we are talking about the σ 2 in Lemma G.3).
Therefore, for all
t 2 √ 2sL 01L1 m + 8s 2L2 01L 2 1 m 2 + 144sL 2 1 m P 1 m m k=1 Y k 2 t √ 2s 28 exp − mt 2 /(4s) 2L 2 1 + √ 2tL 1L01 /3
Proposition C.3 (Block norm bound onΥ applied to a fixed vector). Suppose that
P(E c ω ) t t + 6 √ s(B 0 + 1) and E[L 01 (ω) 2 1Ēc] t 4s 3/2 (1 + 4B 0 )
Then, for all
t 4 √ 2sL 01L1 m + 32s 2L2 01L 2 1 m 2 + 576sL 2 1 m we have P E (Υ −Υ)q * ,∞ t q * ,∞ 32s exp − mt 2 s 32L 2 1 + 34tL 1L01 . (C.8) Proof. Let S 0 def. = {1, .
. . , s} and S j def.
= {s + (j − 1)d + 1, . . . , s + jd} for j = 1, . . . , s. Observe that by the union bound
P E (Υ −Υ)q * ,∞ t q * ,∞ P E ((Υ −Υ)q) S0 ∞ t q * ,∞ + s j=1 P E ((Υ −Υ)q) Sj 2 t q * ,∞ s j=1 P E ((Υ −Υ)q) j t q * ,∞ + s j=1 P E ((Υ −Υ)q) Sj 2 t q * ,∞ . (C.9)
To bound the first sum, observe that (
(Υ −Υ)q) j = (f (x j ) −f (x j )) ⊤ q and ((Υ −Υ)q) Sj = D 1 q ⊤ (f −f ) (x j ).
So, the first sum can be bounded by applying Proposition C.2. The second sum can be bounded by applying Lemma C.6.
Norm bounds forf We will repeatedly make use of the following result onf X . This result is due to concentration bounds on the kernelK which are derived subsequently.
Proposition C.4 (Bound onf X ). Let X ∈ X s . Let ρ > 0. Assume that for all (i, j) ∈ {(0, 0), (1, 0), (0, 2), (1, 2)},
P(E c ω ) t t + 4 √ s max{B 0 , B 2 } , E[L i (ω)L j (ω)1 E c ω ] t 4 √
s Support Localization and the Fisher Metric for off-the-grid Sparse Regularization Then, given any y ∈ X ,
PĒ f X (y) − f X (y) t 4sd exp − mt 2 /8 3sL 2 01 .
(C.10) and given any y ∈ X near , writingf X = (f j ) p j=1 and f X = (f j ) p j=1 with p = s(d + 1), we have
PĒ sup q =1 p j=1 D 2 f j − f j (y)q 2 > t s(3d + d 2 ) exp − mt 2 /8 s(L 2 2 B 11 +L 2 1 B 22 +L 01L2 ) . (C.11) Proof. Let i, j ∈ N 0 with i + j 2. Let [s] def.
= {1, . . . , s} and I def.
= {(0, 0), (1, 0)}, By Lemma C.7 and the union bound,
PĒ ∃(i, j) ∈ I, ∃ℓ ∈ [s], K (ij) (x ℓ , y) − K (ij) (x ℓ , y) t √ s 4sd exp − mt 2 /4 3sL 2 01 . (C.12)
So, (C.10) follows because
f X (y) − f X (y) s i=1 K (x i , y) − K(x i , y) 2 + K(10) (x i , y) − K (10) (x i , y) 2 √ 2t.
By Lemma C.7, Lemma C.9 and the union bound, letting I 2 def.
= {(0, 2), (1, 2)}, we have PĒ ∃(i, j) ∈ I 2 , ∃ℓ ∈ [s], K (ij) (x ℓ , y) − K (ij) (x ℓ , y) t √ s 2sd exp − mt 2 /4 2s(L 2 2 +L 0L2 ) + s(d + d 2 ) exp − mt 2 /4 s(L 2 2 B 11 +L 2 1 B 22 +L 1L2 )
.
(C.13) and (C.11) follows since given q ∈ C d , q = 1, we have
p j=1 D 2 f j − f j (y)q 2 s j=1 K (02) (x j , y) − K (02) (x j , y) 2 + K (12) (x j , y) − K (12) (x j , y) 2 2t 2 Lemma C.7 (Concentration on kernel). Let t > 0, x, x ′ ∈ X . Let i, j ∈ N 0 with i + j 2. Assume P(E c ω ) t t + 4 K (ij) (x, x ′ ) , E[L i (ω)L j (ω)1 E c ω ] t 4 then PĒ K (ij) (x, x ′ ) − K (ij) (x, x ′ ) t 2d exp − mt 2 L 2 p (b ij + 1) +L iLj t/3 where p = max (i, j) and b ij = 1 if min (i, j) = 0 and b ij def. = K (11) (x, x ′ ) otherwise.
Proof. It is an immediate application of Lemma C.5 with
A k = Re D i [ϕ ω k ] (x)D j [ϕ ω k ] (x ′ ) ⊤ for k = 1, . . . , m. Note that A k ∈ (R d ) i+j if (i, j) ∈ {(0, 0), (0, 1), (1, 0)} and A k ∈ R d×d if max(i, j) = 2. noting that under E, A k L iLj . Next, we need to bound EĒ[A k A * k ] and EĒ[A * k A k ] .
We present only the argument for (i, j) = (0, 2), since all the other cases are similar:
0 EĒA k A * k EĒ[ ϕ ω k (x ′ ) 2 D 2 [ϕ ω k ] (x)D 2 [ϕ ω ] (x) * ] L 2 2 EĒ ϕ ω k (x ′ ) 2 Id =L 2 2 |KĒ(x ′ , x ′ )| Id (1 + t/2)L 2 2 Id so EĒA k A * k (1 + t/2)L 2 2 . Similarly, EĒA * k A k (1 + t/2)L 2 2 and EĒA * k A k , EĒA k A * k L 2 p (B qq + t/2)
where p = max (i, j) and q = min (i, j).
Applying a grid on X near , we get a uniform version.
Lemma C.8. Let i, j ∈ N 0 with i + j 2, and assume that
P(E c ω ) t t + 16B ij , E[L i (ω)L j (ω)1 E c ω ] t 16 . Then PĒ ∃ x, x ′ ∈ X near , K (ij) (x, x ′ ) − K (ij) (x, x ′ ) t 2ds 2 exp − mt 2 /16 L 2 p (B qq + 1) +L iLj t/12 + 2d log 4(L iLj +L i L j ) t .
where p = max (i, j) and q = min (i, j) and L i , L j are as in Lemma C.2
Proof. We define a δ-covering of X near for the metric d H with δ = min r near , t 4(LiLj +LiLj ) of size s rnear δ d .
Let this covering be denoted by X grid .
By the union bound and Lemma C.7,
PĒ ∃x, x ′ ∈ X grid s.t. K (ij) (x, x ′ ) − K (ij) (x, x ′ ) t/4 2ds 2 r near δ 2d exp − mt 2 /16 L 2 p (B qq + 1) +L iLj t/12
where p = max (i, j) and q = min (i, j). This gives the required upper bound: Given any x, x ′ ∈ X , let
x grid , x ′ grid ∈ X grid be such that d H (x, x grid ), d H (x ′ , x ′ grid ) δ.
Then, under eventĒ, by Lemma C.2,
K (ij) (x, x ′ ) −K (ij) (x grid , x ′ grid ) (L iLj +L i L j )δ t/4.
By Jensen's inequality and since
K (ij) E (x, x ′ ) − K (ij) (x, x ′ ) t/4 for all x, x ′ , we have K (ij) (x, x ′ ) − K (ij) (x grid , x ′ grid ) t/2.
We now derive analogous results for the kernel differentiated 3 times.
Lemma C.9 (Concentration on order 3 kernel). Let x, x ′ ∈ X near . Assume that
P(E c ω ) t t + 4 max{B 12 , B 22 } , E[(L 1 (ω)L 2 (ω) + L 2 2 (ω))1 E c ω ] t 4
For j = 1, . . . , m, let a i = (D 1 ϕ ωj (x)) i ∈ C, D def.
= D 2 [ϕ ω ] (x ′ ) ∈ C d×d and A j def. = a 1 D a 2 D · · · a d D ⊤ ∈ C d 2 ×d (C.14) Let Z def. = 1 m m j=1 (A j − E[A j ])
. Then, given
g(x ′ ) def. = (g i (x ′ )) d i=1 def. = m k=1 D 1 [ϕ ω k ] (x)ϕ ω (x ′ ) − E[D 1 [ϕ ω k ] (x)ϕ ω (x ′ )] =K (10) (x, x ′ ) − K (10) (x, x ′ ), (i) sup q∈C d , q 1 d i=1 D 2 [g i ] (x ′ )q 2 = Z 2 , (ii) sup q∈C d , q 1 D 2 q ⊤ g (x ′ ) = K (12) (x, x ′ ) − K (12) (x, x ′ ) Z . and PĒ ( Z t) (d + d 2 ) exp − mt 2 /4 B +L 1L2 t/3 whereB def.
= max{L 2 2 (B 11 + t/2),L 2 1 (B 22 + t/2)}.
Proof. The claim (i) is simply by definition, since
Zq = (D 2 [g i ] (x ′ )q) d i=1 ∈ C d 2 .
For (ii), the first equality is simply be definition, and for the inequality, observe that
sup q∈C d , q 1 D 2 q ⊤ g (x ′ ) = sup q∈C d , q 1 sup p∈C d , p 1 d i=1 q i D 2 [g i ] (x ′ )p sup q∈C d , q 1 sup p∈C d , p 1 q d i=1 D 2 [g i ] (x ′ )p 2 Z .
Finally, the probability bound follows by applying Lemma C.5: First note that underĒ,
A j L 1L2 . It remains to bound EĒ[A * j A j ] and EĒ[A j A * j ] : sup q 1 EĒ A * j A j q, q = sup q 1 E E d i=1 (D 1 ϕ ωj (x)) i 2 D 2 [ϕ ω ] (x ′ )q 2 sup q k 1L 2 1 EĒ D 2 [ϕ ω ] (x ′ )[q 1 , q 2 ]D 2 [ϕ ω ] (x ′ )[q 3 , q 4 ] L 2 1 K (22) E (x, x) L 2 1 (B 22 + t/2). Given p i ∈ C d for i = 1, . . . , d such that i p i 2 1, write P = p 1 p 2 · · · p d ∈ C d×d andp = p ⊤ 1 p ⊤ 2 · · · p ⊤ d ⊤ ∈ C d 2 .
Then,
E E A j A * jp ,p = E E d i=1 (D 1 ϕ ωj (x)) i D 2 ϕ ωj (x ′ )p i 2 = E E D 2 ϕ ωj (x ′ )P D 1 ϕ ωj (x) 2 L 2 2 E E i k p i,k (D 1 ϕ ωj (x)) k 2 =L 2 2 i K (11) E (x, x)p i , p i L 2 2 K (11) E (x, x) 2 i p i 2 L 2 2 (B 11 + t/2).
Lemma C.10 (Uniform concentration on order 3 kernel). Assume
P(E c ω ) t t + 16 max{B 12 , B 22 } , E[L 1 (ω)L 2 (ω)1 E c ω ] t 16 then PĒ ∃x, x ′ ∈ X near , K (12) (x, x ′ ) − K (12) (x, x ′ ) t s 2 (d + d 2 ) exp − mt 2 /16 B +L 1L2 t/6 + 2d log 8(L 1L2 +L 2 L 2 ) t whereB def.
= max{L 2 2 (B 11 + t/2),L 2 1 (B 22 + t/2)}, L 1 , L 2 are as in Lemma C.2.
Proof. Let X grid be a δ-covering of X near for the metric d H with δ = min r near ,
PĒ ∃x, x ′ ∈ X grid , K (ij) (x, x ′ ) − K (ij) (x, x ′ ) t/2 s 2 (d + d 2 ) 8(L 1L2 +L 2 2 ) t 2d exp − mt 2 /16 L 2 2 (B 11 + t/4) +L 1L2 t/6 def. = ρ.
Moreover, under eventĒ, given any x, x ′ ∈ X near , there exists grid points x grid , x ′ grid such that
d H (x, x grid ), d H (x ′ , x ′ grid ) δ and K (12) (x, x ′ ) − K (12) (x, x ′ ) K (12) (x grid , x ′ grid ) − K (12) (x grid , x ′ grid ) + K (12) (x, x ′ ) −K (12) (x grid , x ′ grid ) + K (12) (x, x ′ ) − K (12) (x grid , x ′ grid ) ,
and by Lemma C.2, under eventĒ,
K (12) (x, x ′ ) −K (12) (x grid , x ′ grid ) (L 1L2 + L 2L2 )δ t/8.
and by Jensen's inequality and since K (12) (x, y) − K
E (x, y) t/8,
K (12) (x, y) − K (12) (x grid , y) 3t/8.
Therefore, conditional onĒ, K (12) (x, y) − K (12) (x, y) < t with probability at least 1 − ρ.
D Proof of Theorem 3
In all the rest of the proofs we fix X 0 ∈ X s to be ∆-separated points, a 0 ∈ C s , and let u = (sign(a 0 ), 0 sd ). We denote X near i = {x ∈ X ; d H (x, x 0,i ) r near } and X near = ∪ i X near i and X far = X \X near .
Since K is an admissible kernel, from (B.2) and (B.1) in the proof of Theorem 2 η X0 satisfies (i) for all y ∈ X far , |η X0 (y)| 1 − 1 2 ε 0 , (ii) for all y ∈ X near (i), −Re (sign(a i )D 2 [η X0 ] (y)) 1 2 ε 2 Id and Im (sign(a i )D 2 [η X0 ] (y)) ( p 2 ) 1 2 ε 2 . p def.
= (1 − ε 2 r 2 near /2)/(ε 2 r 2 near /2) 1, since ε 2 r 2 near 1 by assumption of K being admissible. We aim to show that, for X close to X 0 ,η X is
nondegenerate by showing that D r [η X ] − D r [η X0 ]
cε r for some positive constant c sufficiently small.
D.1 Nondegeneracy ofη X0
We first establish the nondegeneracy ofη X0 , our proof can be seen as a generalisation of the techniques in Tang et al. (2013) to the multidimensional setting with general sampling operators:
Theorem D.1. Let ρ > 0 and assume that the assumptions in Section 2.3 hold. Assume also that either (a) or (b) holds:
(a) sign(a 0 ) is a Steinhaus sequence and
m C · s · log N d ρ log s ρ (b) sign(a 0 )
is an arbitrary sequence from the complex unit circle, and
m C · s 3/2 · log N d ρ
where C, N are defined in the main paper. Then with probability at least 1 − ρ, the following hold: For all y ∈ X far , |η X0 (y)| 1 − 7 16 ε 0 , and for all y ∈ X near (i), −Re (sign(a i )D 2 [η X0 ] (y)) 7 16 ε 2 Id and Im (sign(a i )D 2 [η X0 ] (y)) ( p 2 + p 8 ) 1 2 ε 2 and hence,η X0 is ( 7 16 ε 0 , 7 16 ε 2 )-nondegenerate.
Proof. Note that 8 7 p 2 + p 8 = 5 8 p < 1 − 7ε 2 r 2 near /16 7ε 2 r 2 near /16 soη X0 is ( 7 16 ε 0 , 7 16 ε 2 )-nondegenerate by Lemma B.1 Let c def.
= 1/32. Observe that by assumption and Lemma C.4, P(Ē) ρ/2. Therefore, it is sufficient to prove that conditional onĒ, with probability at least 1 − δ with δ def.
= ρ/2,η X0 is nondegenerate.
We will repeatedly use the fact that our assumptions (by Lemma C.4) also imply that
P(E c ω ) ε m , E[L i (ω)L j (ω)1 E c ω ] ε m
for all (i, j) ∈ {(0, 0), (1, 0), (0, 2), (1, 2)},
Step I: Proving nondegeneracy on a finite grid.
Let X far grid ⊂ X far and X far grid ⊂ X near , be finite point sets. Let
Q r (y) def. = D r [η X0 ] (y) − D r [η X0 ] (y) , r = 0, 2.
We first prove that conditional onĒ, with probability at least 1 − δ where δ def.
= ρ/2, that Q 0 (y) cε 0 for all y ∈ X far grid and Q 2 (y) cε 2 for all y ∈ X far grid . Let us first recall some facts which were proven in the previous section: Let a, t ∈ (0, 1) and write f = (f j )
s(d+1) j=1 andf = (f j ) s(d+1) j=1 . Let q 0 def. = Υ −1 u, so q 0 2 √ s. Let F be the event that (a) Υ −1 −Υ −1 t, (b) ∀y ∈ X far grid , f X0 (y) − f X0 (y) aε 0 , (c) ∀y ∈ X near grid , sup q∈C d , q =1 p j=1 D 2 f j −f j (y)q 2 aε 2 ,
Let G be the event that
(d) ∀y ∈ X far grid , (f X0 (y) − f X0 (y)) ⊤ q 0 2aε 0 (e) ∀y ∈ X near grid , D 2 (f X0 − f X0 ) ⊤ q 0 (y) 2aε 2 then provided that P(E c ω ) u u + max{4 √ sB ij , 6} , E[L i (ω)L j (ω)1 E c ω ] u 4s (D.1) where u = min{aε i , t}, we have PĒ(F c ) 4(d + 1)s exp − mt 2 16sL 2 01 (3 + 2t) + 4sd X far grid exp − m(aε 0 ) 2 /8 s(L 2 01 (B 11 + 1) +L 2 01 ) + s(3d + d 2 ) X near grid exp − m(aε 2 ) 2 /8 s(L 2 2 B 11 +L 2 1 B 22 ) +L 01L2 ) PĒ(G c ) 2 X far grid exp − ma 2 ε 2 0 s(8L 2 0 + 4 3L 0L01 aε 0 ) + 2d X near grid exp − ma 2 ε 2 2 s(8L 2 2 + 4 3L 2L01 aε 2 ) , (D.2)
where for PĒ(F c ), the first term on the right is due to Proposition C.1, the second and third are due to Proposition C.4 while the bound on PĒ(G c ) is due to Proposition C.2 (noting that, since this probability bound over the ω j is valid for all fixed u, and the ω j and the signs are independent, it is valid with the same probability over both ω j and u).
Observe that
D j [η X0 ] (y) − D j [η X0 ] (y) = D j (α X0 − α X0 ) ⊤f X0 (y) + D j α ⊤ X0 (f X0 − f X0 ) (y) D j u ⊤ (Υ −1 − Υ −1 )f X0 + Υ −1 (f X0 − f X0 ) (y) (D.3)
Step I (a): Random signs
We first bound (D.3) in the case where u is a Steinhaus sequence.
Let β 1 (y) def. = (Υ −1 − Υ −1 )f X0 (y) and β 2 (y) def. = Υ −1 (f X0 (y) − f X0 (y))
. Then, event F implies that β 1 (y) t(B 0 + aε 0 ) for all y ∈ X far grid , and event G implies that u ⊤ β 2 (y) 2aε 0 . So, PĒ ∃y ∈ X far grid , u ⊤ (β 1 + β 2 )(y) > cε 0 P F ∩Ē ∃y ∈ X far grid , u ⊤ β 1 (y) > c 2 ε 0 PĒ(F ) + PĒ (F c )
+ P G∩Ē ∃y ∈ X far grid , u ⊤ β 2 (y) > c 2 ε 0 PĒ(G) + PĒ (G c ) P F ∩Ē ∃y ∈ X far grid , u ⊤ β 1 > c 2 ε 0 + PĒ (F c ) + PĒ (G c ) 4 X far grid e − (c/4) 2 ε 2 0 8t 2 (B 0 +aε 0 ) 2 + PĒ(F c ) + PĒ (G c ) . (D.4)
where we set a = c/4 for the second inequality and the last inequality follows from Lemma G.4 and because u consists if random signs.
Now consider Q 2 (y) = D 2 u ⊤ β (y). Under event G, D 2 u ⊤ β 2 (y) c 2 ε 2 . Writing M = (Υ −1 − Υ −1 ), we have D 2 u ⊤ β 1 (y) = D 2 u ⊤ Mf X0 (y) = p ℓ=1 u ℓ p j=1 M ℓj D 2 [f j ] (y) . (D.5)
We aim to bound (D.5) by applying the Matrix Hoeffding's inequality (Corollary G.1): let
Y ℓ def. = Re p j=1 M ℓj D 2 [f j ] (y) ∈ R d×d which is a symmetric matrix. Note that p ℓ=1 Y 2 ℓ = sup q∈R d , q =1 p ℓ=1 Y 2 ℓ q, q = sup q∈R d , q =1 d ℓ=1 Y ℓ q 2 sup q∈R d , q =1 p j=1 M ℓ,j (D 2 [f j ] (y)q) 2 .
Then, for a vector q of unit norm, let V j,n def.
= (D 2 [f j ] (y)q) n for j = 1, . . . , p and n = 1, . . . , d, then
p ℓ=1 p j=1 M ℓ,j (D 2 [f j ] (y)q) 2 = p ℓ=1 d n=1 p j=1 M ℓ,j V j,n 2 = d n=1 M V ·,n 2 M 2 d n=1 V ·,n 2 = M 2 d n=1 p j=1 |V j,n | 2 = M 2 p j=1 D 2 [f j ] (y)q 2 .
Under event F , we have M 2 p j=1 D 2 [f j ] (y)q 2 t 2 (B 2 + aε 2 ) 2 . Then,
P F ∩Ē D 2 u ⊤ Re Mf X0 (y) cε 2 √ 2 2d exp − (c/2) 2 ε 2 2 4t 2 (B 2 + aε 2 ) 2 .
By repeating this argument for the imaginary part, we obtain
P F ∩Ē D 2 u ⊤ Im Mf X0 (y) cε 2 √ 2 2d exp − (c/2) 2 ε 2 2 4t 2 (B 2 + aε 2 ) 2 . So, PĒ ∃y ∈ X near grid , D 2 u ⊤ β(y) > cε 2 P F ∩Ē ∃y ∈ X near grid , D 2 u ⊤ Re Mf X0 (y) c 2 ε 2 + PĒ(F c ) + PĒ(G c ) 4d X near grid exp − (c/2) 2 ε 2 2 4t 2 (B 2 + aε 2 ) 2 + PĒ(F c ) + PĒ(G c ).
(D.6) Therefore, 1 − P Q 0 (y 0 ) cε 0 and Q 2 (y 2 ) cε 2 , ∀y 0 ∈ X far grid , ∀y 2 ∈ X near grid 4 X far grid exp − (c/2) 2 ε 2 0 32t 2 (B 0 + aε 0 ) 2 + 4d X near grid exp − (c/2) 2 ε 2 2 16t 2 (B 2 + aε 2 ) 2 + 2PĒ(F c ) + 2PĒ(G c ).
The first 2 terms are each bounded by δ/7 by setting t such that
1 t 2 = 2 13 log 112Nd δ B + 1 c 2 ε 2 whereB def. = max{B 0 , B 2 }, ε def.
= min{ε 0 , ε 2 } andN = max X near grid , X far grid . The first term of (D.2) is bounded by δ/7 if m 1 t 2 log 28(d + 1)s δ 64sL 2 01 = sL 2 01 2 19 B + 1 c 2 ε 2 log 112Nd δ log 28(d + 1)s δ and the last 4 terms of (D.2) are each bounded by δ/7 provided that m log 28(s + d)dN δ 16s(L 2 2 B 11 +L 2 1 B 22 +L 01L2 ) c 2 ε 2 So, to summarise, recalling that δ = ρ/2,η X0 is nondegenerate on X near grid and X far grid with probability at least 1 − δ (conditional onĒ) provided that
m log sdN ρ log sd ρ s(L 2 2 B 11 +L 2 1 B 22 +BL 2 01 +L 01L2 ) ε 2 and P(E c ω ) ε B 3/2 √ s log(N d/ρ) and , E[L i (ω)L j (ω)1 E c ω ] ε 4s √ B log(N d/ρ)
Step I (b): Deterministic signs Assume now that u consists of arbitrary signs. We will show that (D.3) can be bounded by cε when m is chosen as in condition (b) of this theorem. Let F ′ be the event that
(a') Υ −Υ t s 1/4 and Υ −1 −Υ −1 t s 1/4 (b') ∀y ∈ X far grid , (f X0 (y) − f X0 (y)) aε0 s 1/4 (c') ∀y ∈ X near grid , sup q =1 D 2 (f X0 − f X0 ) ⊤ q (y) aε2 s 1/4 (f) (Υ −Υ)Υ −1 u * ,∞ aε Υ −1 u * ,∞ 2aε.
Then, provided that
P(E c ω ) u u + 6s(B 0 + B 2 )
and E[L 01 (ω) 2 1Ēc] u 4Bs 3/2 , with u = min{aε i , t} as before, we have PĒ((F ′ ) c ) 4(d + 1)s exp − mt 2 16s 3/2L2 01 (3 + 2t) + 4sd X far grid exp − m(aε 0 ) 2 /8 s 3/2 (L 2 01 (B 11 + 1) +L 2 01 ) + s(3d + d 2 ) X near grid exp − m(aε 2 ) 2 /8 s 3/2 (L 2 2 B 11 +L 2 1 B 22 +L 01L2 )
+ 32s exp − m4a 2 ε 2 s 32L 2 1 + 68aεL 1L01 .
where the first bound is from Proposition C.1, the second and third are from Proposition C.4 and the final bound is due to Proposition C.3.
To bound (D.3), we first observe that if event G holds, then just as observed previously, D r u ⊤ β 2 (y) 2aε r . To bound u ⊤ β 1 (y) , observe that
u ⊤ β 1 (y) = u ⊤ (Υ −1 −Υ −1 )(f X0 − f X0 ) + u ⊤ (Υ −1 −Υ −1 )f X0 = u ⊤ (Υ −1 −Υ −1 )(f X0 − f X0 ) + u ⊤ Υ −1 (Υ − Υ)Υ −1 f X0 = u ⊤ (Υ −1 −Υ −1 )(f X0 − f X0 ) + u ⊤ Υ −1 (Υ − Υ)(Υ −1 − Υ −1 )f X0 + u ⊤ Υ −1 (Υ − Υ)Υ −1 f X0 Under event F ′ , • u ⊤ (Υ −1 −Υ −1 )(f X0 − f X0 ) √ s Υ −1 −Υ −1 f X0 − f X0 taε • u ⊤ Υ −1 (Υ − Υ)(Υ −1 − Υ −1 )f X0 √ s · 2 · Υ − Υ Υ −1 − Υ −1 B 0 2t 2 B 0 • Υ −1 (Υ − Υ)Υ −1 u * ,∞ Υ −1 * ,∞ (Υ − Υ)Υ −1 u * ,∞ 4aε.
Finally, given any vector q such that q * ,∞ 4aε, we have q ⊤ f X0 4aεB 0 . Therefore,
u ⊤ β 1 (y) ta + 2t 2 + 4aεB 0 ,
and in a similar manner, we can show that the same upper bound holds for D 2 u ⊤ β 1 (y) .
Therefore,
D r u ⊤ β (y) cε r (D.7)
if both F ′ and G hold, so conditional onĒ, (D.7) holds with probability at least 1 − δ provided that m s 3/2 · (L 2 2 B 11 +L 2 1 B 22 +BL 2 01 +L 01L2 ) ε 2 · log N ds ρ and
P(E c ω ) ε B 3/2 s log(N d/ρ) and , E[L i (ω)L j (ω)1 E c ω ] ε s 3/2 √ B log(N d/ρ)
Step II: Extending to the entire space To prove thatη X0 is nondegenerate on the entire space X , we first show thatη X0 is locally Lipschitz (and hence determine how fine our grids X near grid , X far grid need to be): for
x, x ′ ∈ X with d H (x, x ′ ) r near , D r [η X0 ] (x) − D r [η X0 ] (x ′ ) = 1 m m k=1 D r Re (Υ −1 X u) ⊤ γ(ω k )ϕ ω k (x) (D.8) − D r Re (Υ −1 X u) ⊤ γ(ω k )ϕ ω k (x ′ ) = 1 m m j=1 Re (Υ −1 X u) ⊤ γ(ω k ) · (D r [ϕ ω k ] (x) − D r [ϕ ω k ] (x ′ )) Υ −1 X u √ sL 01 D r [ϕ ω k ] (x) − D r [ϕ ω k ] (x ′ ) (D.9) 4sL 01 d H (x, x ′ )L r cε r . (D.10)
where we have applied Lemma C.2 to obtain the last line.
Choosing X far grid to be a δ 0 def. = cε0 4L0L01s -covering of X near (of size at most O(R X /δ 0 )), X far grid to be a δ 2 def.
= cε2 4L2L01scovering of X far (of size at most O(R X /δ 2 )). Then for any x ∈ X near and x ′ ∈ X near grid such that d H (x,
x ′ ) δ 0 , |η X0 (x)| |η X0 (x ′ )| + |η X0 (x) −η X0 (x ′ )| 1 − ε 0 + 2cε 0 .
and given any x ∈ X far , let x ′ ∈ X far grid be such that d H (x, x ′ ) δ 2 , so
Re sign(a i )D 2 [η X0 ] (x) Re sign(a i )D 2 [η X0 ] (x ′ ) + D 2 [η X ] (x) − D 2 [η X ] (x ′ ) Id (−ε 2 + 2cε 2 )Id, and Im sign(a i )D 2 [η X0 ] (x) Im sign(a i )D 2 [η X0 ] (x ′ ) + cε 2 (c 2 + c)ε 2 .
D.2 Nondegeneracy transfer toη X .
We are now ready to prove Theorem 3, which we restate below for clarity.
Theorem D.2. Under the assumptions of Theorem D.1, the following holds with probability at least 1 − ρ: for all X such that
d H (X, X 0 ) min r near , ε r (C H B √ s) −1 , ε r (C HL12Lr √ s) −1 , (D.11) we have (i) for all y ∈ X far , |η X (y)| 1 − 13 32 ε 0 (ii) for all y ∈ X near (i), −Re sign(a i )D 2 [η X ] (y) 13ε2 32 Id and Im sign(a i )D 2 [η X ] (y) ( p 2 + 3p 16 ) 1 2 ε 2 .
Hence,η X is ( 13 32 ε 0 , 13 32 ε 2 )-nondegenerate.
The proof essentially exploits the fact thatΥ X ,f X are locally Lipschitz in X with respect to the metric d H , and consequently nondegeneracy ofη X0 implies nondegeneracy ofη X whenever d H (X, X 0 ) is sufficiently small.
D.2.1 Proof of Theorem D.2
We begin with a lemma which shows thatΥ X is locally Lipschitz in X.
Lemma D.1 (Lipschitz bound ofΥ X ). Let X 0 ∈ X s be ∆-separated points. Assume that for all i + j 3
P(E c ω ) 1 1 + 16 √ sB ij , E[L i (ω)L j (ω)1 E c ω ] 1 16
√ s for all i, j = 0, ..., 2. Let ρ > 0 and m s(L 2 2 B 11 +L 2 1 B 22 +L 01L2 ) log
sd ρ + d log sC H 3 max i=0L i
Then, conditional on eventĒ, with probability at least 1 − ρ, the following hold:
• (i) for all X such that d H (x i , x 0,i ) r near , we have Υ X −Υ X0 C H Bd H (X, X 0 ) . • (ii) for all X such that d H (X, X 0 ) min r near , 1 CHB , we have Id −Υ X 3 4 and G − 1 2 X Γ * X 1.
Proof. By Lemma C.8 and Lemma C.10, with probability at least 1 − ρ conditonal onĒ, for all (i, j) ∈ {(0, 0), (0, 1), (1, 1), (1, 2)} and all x, y ∈ X near ,
K (ij) (x, y) K (ij) (x, y) + 1 √ s , note that this also holds forK (ji) (x, y) sinceK (ij) (x, y) =K (ij) (y, x).
In particular, for all x,
x ′ such that d H (x, x ′ ) ∆/4, we have K (ij) (x, x ′ ) 2 √ s . Take any X such that d H (x i , x 0,i )
r near , we have that both x i , x 0,i are at least ∆/4-separated from x j and x 0,j . Therefore, for k, ℓ ∈ {0, 1}, using Lemma C.3:
K (kℓ) (x i , x j ) −K (kℓ) (x i,0 , x j,0 ) C H √ s d H (x i , x 0,i ) 2 + d H (x j , x 0,j ) 2 K (kℓ) (x i , x i ) −K (kℓ) (x i,0 , x i,0 ) C H (B k+1,ℓ + B k,ℓ+1 ) d H (x i , x 0,i ) (D.12)
and therefore by Lemma G.6:
Υ X −Υ X0 2 s i,j=1 1 k,ℓ=0 K (kℓ) (x i , x j ) −K (kℓ) (x 0,i , x 0,j ) 2 2 s i,j=1 1 k,ℓ=0 K (kℓ) (x i , x j ) −K (kℓ) (x 0,i , x j ) 2 + K (ℓk) (x j , x 0,i ) −K (ℓk) (x 0,j , x 0,i ) 2 C 2 H k,l∈{0,1,2} k+ℓ 3 B kℓ 2 i d H (x i , x 0,i ) 2 + 1 s j =i d H (x j , x 0,j ) 2
which yields the desired result.
For the second statement, using Proposition C.1, PĒ( Υ X0 − Υ X0 > 1 8 ) ρ, so conditional onĒ, we have with probability 1 − ρ, Υ X −Υ X0 1 8 and the claim follows since Id − Υ X0 1 2 (due to Lemma C.1) implies that Id −Υ X 3 4 and Υ X 7/4 and G − 1 2 X Γ * X = Υ X √ 7/2.
Proof of Theorem D.2. Sinceη X0 is nondegenerate with probability at least 1 − ρ, the conclusion follows if we prove that for all x ∈ X far and all y ∈ X near ,
D 2 [η X −η X0 ] (x) ε 0 /32 and D 2 [η X −η X0 ] (y) pε 2 /32 (D.13)
with probability at least 1 − ρ. We first writê
η X (y) −η X0 (y) =α ⊤ X (f X −f X0 ) + (α X −α X0 ) ⊤f X0 (y).
Conditional onĒ, with probability at least 1 − ρ/2, we have by Lemma D.1 (note that our assumptions imply the assumptions of Lemma D.1),
Υ X − Υ X0 C H Bd H (X, X 0 ) and Υ −1 X 4. So, D r (α X −α X0 ) ⊤f X0 (y) √ s Υ −1 X − Υ −1 X0 8 √ s Υ X −Υ X0 √ sC H Bd H (X, X 0 ).
By Lemma C.2, ifĒ occurs, then
D r α ⊤ X (f X −f X0 ) (y) C r α X d H (X, X 0 ) C r Υ −1 X √ sd H (X, X 0 ) 4C r √ sd H (X, X 0 ),
where C r (1 + C H )L rL12 . Finally, since P(Ē c ) ρ/2, we have with probability at least 1 − ρ, for all y ∈ X , (D.13) holds provided that (D.11) holds. Combining with the nondegeneracy ofη X0 , the conclusion follows with probability 1 − 2ρ.
E Supplementary results to the proof Theorem 1
Recall that in the proof of Theorem 1, we defined the function f :
C s × X s × R + × C m by f (u, v) def. = Γ * X (Φ X a − Φ X0 a 0 − w) + λ sign(a 0 ) 0 sd
where u = (a, X) and v = (λ, w). This function f is differentiable with
∂ v f (u, v) = sign(a 0 ) 0 sd , −Γ * X ∈ C s(d+1)×m , (E.1)
and
∂ u f (u, v) is Γ * X Γ X J a + 0 1×s A 11 0 · · · 0 0 1×s 0 A 12 · · · 0 . . . . . . . . . . . . . . . 0 1×s 0 0 · · · A 1s 0 d×s A 21 0 · · · 0 0 d×s 0 A 22 · · · 0 . . . . . . . . . . . . . . . 0 d×s 0 0 · · · A 2s (E.2) where A 1j def. = ∇ x ϕ(x j ), z ⊤ , A 2j def. = ∇ 2 x ϕ(x j ), z , z def. = (Φ X a − Φ X0 a 0 − w) and J a ∈ R s(d+1)×s(d+1) is a the diagonal matrix: J a = Id s×s 0 a 1 Id d×d . . . 0 a s Id d×d .
Letting u 0 = (a 0 , X 0 ) and v 0 = (0, 0), ∂ u f (u 0 , v 0 ) = Γ * X0 Γ X0 J a is invertible and f (u 0 , v 0 ) = 0. Hence, by the Implicit Function Theorem, there exists a neighbourhood V of v 0 in C × C m , a neighbourhood U of u 0 in C s × X s and a Fréchet differentiable function g : V → U such that for all (u, v) ∈ U ×V , f (u, v) = 0 if and only if u = g(v). To conclude, we simply need to bound the size of the region on which g is well defined, and to bound the error between g(v) and g(0). Let us first remark that our assumptions imply that P(Ē c ) ρ/2 and P(E c ω )
1 1 + 16 √ sB ij , E[L i (ω)L j (ω)1 E c ω ] 1 16 √ s , (E.3)
for all i, j = 0, ..., 2. Therefore, it is sufficient to prove the existence of g conditional on eventĒ:
Theorem E.1. Assume that for all i + j 3 P(E c ω ) 1 1 + 16 √ sB ij , E[L i (ω)L j (ω)1 E c ω ] 1 16
√ s for all i, j = 0, ..., 2. Let ρ > 0 and suppose that m s(L 2 2 B 11 +L 2 1 B 22 +L 01L2 ) log
sd ρ + d log (sC H L 3 )
where L r def.
= max i r L r . Then, conditional on eventĒ, with probability at least 1 − ρ: there exists a C 1 function g such that, for all v = (λ, w) such that v r with r satisfying
r = O 1 √ s min min{rnear,(CHB) −1 } mini|a0,i| , 1 L01L12(1+ a0 ) , (E.4) we have f (g(v)
, v) = 0 and g(0) = u 0 . Furthermore, given (λ, w) in this ball, (a, X) def.
= g((λ, w)) satisfies
a − a 0 + d H (X, X 0 ) √ s(λ + w ) min i |a 0,i | . (E.5)
We begin with some preliminary results before presenting the proof of this theorem in Section E.2.
E.1 Preliminary results
Theorem E.2 (Quantitative implicit function theorem, adapted from Denoyelle et al. (2017)). Let F : H × Y → C n be a differentiable mapping where H is a Hilbert space, Y ⊆ C s × R sd , n = s(d + 1), · be a norm on H. For each y ∈ Y, suppose that there exists a positive definite matrix G y , and let d G be the associated metric. Assume that F (x 0 , y 0 ) = 0, and that for x ∈ B · (x 0 , r 1 ), y ∈ B dG (y 0 , r 2 ), ∂ y F (x, y) is invertible and we have
G − 1 2 y ∂ x F (x, y) D 1 and G 1 2 y ∂ y F (x, y) −1 G 1 2 x D 2 .
Then, defining R = min r2 D1D2 , r 1 , there exists a unique Fréchet differentiable mapping g : B · (x 0 , R) → B dG (y 0 , r 2 ) such that g(x 0 ) = y 0 and for all x ∈ B · (x 0 , R), F (x, g(x)) = 0, and furthermore
dg(x) = −(∂ y F (x, g(x))) −1 ∂ x F (x, g(x)) and consequently G 1 2 g(x) dg(x) D 1 D 2 . Proof. Let V * = ∪ V ∈V V , where V is the collection of all open sets V ∈ R m such that 1. x 0 ∈ V ,
2. V is star-shaped with respect to x 0 , 3. V ⊂ B · (x 0 , r 1 ), 4. there exists a C 1 function g : V → B dG (y 0 , r 2 ) such that g(x 0 ) = y 0 and F (x, g(x)) = 0 for all x ∈ V .
Observe that V is non-empty by the (classical) Implicit Function Theorem. Moreover, V is stable by union: indeed, all conditions expect the last one are easy to check. Now, let V,Ṽ ∈ V and g,g be corresponding functions. The set V = {x ∈ V ∩Ṽ , g(x) =g(x)} is non-empty (it contains x 0 ), and closed in V ∩Ṽ . Moreover, it is open: for any x ∈ V , by our assumptions ∂ y F (x, g(x)) is invertible and the Implicit Function theorem applies at (x, g(x)), and by the uniqueness of the mapping resulting from it we obtain an open set around x in which g andg coincide. Hence V is both closed and open in V ∩Ṽ , and by the connectedness of it V = V ∩Ṽ . Therefore, there exists a function g ′ defined on V ∪Ṽ that satisfies condition 4. above (it is defined as g on V andg onṼ , which is well-posed for their intersection), and V is indeed stable by union.
Hence V * ∈ V, let g * be its corresponding function. It is unique by the arguments above, satisfies F (x, g * (x)) = 0 and
G 1 2 g * (x) dg * (x) = −G 1 2 g * (x) (∂ y F (x, g * (x))) −1 ∂ x F (x, g * (x)) = −(G − 1 2 g * (x) ∂ y F (x, g * (x))G − 1 2 g * (x) ) −1 G − 1 2 g * (x) ∂ x F (x, g * (x))
for all x ∈ V * . Note that by our assumptions G 1 2
g * (x) dg * (x) D 1 D 2 .
We finish the proof by showing that V * contains a ball of radius r 2 /(D 1 D 2 ). Let x ∈ R m with x = 1, R x = sup{R, x 0 + Rx ∈ V * }, and x * = x 0 + R x x ∈ ∂V * . Clearly 0 < R x r 1 since V * is open, assume R x < r 1 . Our goal is to show that in that case R x r1 D1D2 . Since dg * is bounded, g * is uniformly continuous on V * and it can be extended on ∂V * , and by continuity F (x * , g * (x * )) = 0. By contradiction, if g * (x * ) ∈ B dG (y 0 , r 2 ), by our assumptions we can apply the Implicit Function Theorem at (x * , g * (x * )), and therefore extend g * on an open set V that is not included in V * such that V ∪ V * ∈ V, which contradicts the maximality of V * . Hence d G (g * (x * ), y 0 ) = r 2 . Let γ : [0, 1] → Y be defined by γ(t) def.
= g * (x * + t(x 0 − x * )), so γ ′ (t) = dg * (γ(t))(x 0 − x * ). Then,
r 2 = d G (g * (x * ), g * (x 0 )) 1 0 G g * (γ(t)) γ ′ (t), γ ′ (t) dt = 1 0 G 1 2 g * (γ(t)) dg * (γ(t))(x 0 − x * ) 2 dt D 1 D 2 R x .
Lemma E.1. Asssume that eventĒ occurs. Then, for all X such that d H (x i , x 0,i ) r near ,
Π X Γ X0 a L 2 a 1 max i d H (x i , x 0,i ) 2 L 2 a ∞ d H (X, X 0 ) 2 Proof. Recall that Im(Γ X ) = {ϕ(x i ), J ϕ (x i )} i , and Π X is a projector on Im(Γ X ) ⊥ . Also note that for d H (x i , x 0,i ) r near , we have H − 1 2 x0,i H 1 2
xi 1, and therefore underĒ:
H − 1 2 x0,i ∇ 2 ϕ ωj (x i )H − 1 2 x0,i D 2 ϕ ωj (x i ) L 2
Let γ i : [0, 1] → X be any piecewise smooth curve such that γ i (1) = x 0,i and γ i (0) = x i . Then, by Taylor expanding ϕ(γ i (t)) about t = 0, we obtain
ϕ(x 0,i ) = ϕ(x i ) + ∇ϕ(x i ), γ ′ i (0) + 1 0 1 2 ∇ 2 ϕ(γ i (t))γ ′ i (t), γ ′ i (t) dt.
Therefore,
Π X Γ X0 a = Π X s i=1 a i ϕ(x 0,i ) = Π X s i=1 a i 2 1 0 ∇ 2 ϕ(γ i (t))γ ′ i (t), γ ′ i (t) dt
Taking the norm implies
Π X Γ X0 a s i=1 |a i | 2 1 0L 2 H γi(t) γ ′ i (t) 2 dt
and taking the infimum over all paths γ i yields
Π X Γ X0 a L 2 i |a i | d H (x i , x 0,i ) 2 .
E.2 Proof of Theorem E.1
Our goal is to apply Theorem E.2. Let u = (a, X), u 0 = (a 0 , X 0 ), v = (λ, w) and v 0 = (0, 0). We must control
G − 1 2 X ∂ v f (u, v) and G 1 2 X ∂ u f (u, v) −1 G 1 2
X for (u, v) sufficiently close to (u 0 , v 0 ). Using Lemma D.1, conditional on eventĒ, with probability 1 − ρ we have (u, v) and bound the norm of its inverse.
G − 1 2 X ∂ v f (u, v) u + G − 1 2 X Γ X √ s To control G 1 2 X ∂ u f (u, v) −1 G 1 2 X , first observe that G −1/2 X ∂ u f (u, v)G −1/2 X = G −1/2 X Γ * X Γ X G −1/2 X + M (u, v) J a where M (u, v) def. = 0 1×s 1 a1 H − 1 2 x1 ∇[ ϕ, z ](x 1 ) ⊤ · · · 0 . . . . . . . . . . . . 0 1×s 0 · · · 1 as H − 1 2 xs ∇[ ϕ, z ](x s ) ⊤ 0 d×s 1 a1 H − 1 2 x1 ∇ 2 [ ϕ, z ](x 1 )H − 1 2 x0,1 · · · 0 . . . . . . . . . . . . 0 d×s 0 · · · 1 as H − 1 2 xs ∇ 2 [ ϕ, z ](x s )H − 1 2 x0,s , (E.6) where z = (Φ X a − Φ X0 a 0 − w). Now, let us study the invertibility of G − 1 2 X Γ * X Γ X G − 1 2 X + M
Lemma E.2 (Bound on M (u, v)). Let u = (a, X), v = (λ, w) and let M (u, v) be as defined in (E.6). Assume thatĒ occurs and given ε > 0, let c ε def. = ε mini|a0,i| 2L12
. Then, for all X ∈ X s , a ∈ R s and w ∈ C m such that
a − a 0 c ε 3L 0 , w c ε /3 and d H (X, X 0 ) min r near , c ε 3L 1 a 0 , we have M (u, v) ε and M (u, v) * ,∞ ε
Proof. First note that for r ∈ N 0 ,
D r ϕ ⊤ z (x i ) 1 √ m m j=1 z j D r ϕ ωj (x i ) L r z
Now, forq = [q 1 , . . . , q s , Q 1 , . . . , Q s ] ∈ C s(d+1) , where q i ∈ C and Q i ∈ C d , and q = 1, we have
M (u, v)q 2 = s i=1 1 a i H − 1 2 xi ∇[ϕ ⊤ z](x i ) ⊤ Q i 2 + 1 a i H − 1 2 xi ∇ 2 [ϕ ⊤ z](x i )H − 1 2 xi Q i 2 4 min i |a 0,i | 2 q 2 max i H − 1 2 xi ∇[ϕ ⊤ z](x i ) 2 + H − 1 2 xi ∇ 2 [ϕ ⊤ z](x i )H − 1 2 xi 2 = 4 min i |a 0,i | 2 max i D 1 ϕ ⊤ z (x i ) 2 + D 2 ϕ ⊤ z (x i ) 2 4 min i |a 0,i | 2 (L 2 1 +L 2 2 ) z 2
where we have used the fact that min i |a i | min i |a 0,i | /2. If q * ,∞ = 1, then
M (u, v)q * ,∞ = max i { H − 1 2 xi ∇[ϕ ⊤ z](x i ) ⊤ Q i , H − 1 2 xi ∇[ϕ ⊤ z](x i )H − 1 2 xi Q i 2 } max i { H − 1 2 xi ∇[ϕ ⊤ z](x i ) , H − 1 2 xi ∇[ϕ ⊤ z](x i )H −= (ϕ ω k (x)) m k=1 , we have z = i (a i ϕ(x i ) − a 0,i ϕ(x 0,i )) − w L 0 a − a 0 + a 0 max k i |ϕ ω k (x i ) − ϕ ω k (x 0,i )| 2 + w L 0 a − a 0 + a 0 L 1 d H (X, X 0 ) + w
where the last inequality follows from Lemma C.2.
The bound on M (u, v) from Lemma E.2 allows us to conclude that under eventĒ, taking
c def. = min i |a 0,i | 16L 12 (E.7)
for all X ∈ X s , a ∈ R s and w ∈ C m such that
a − a 0 c 3L 0 , w c/3 and d H (X, X 0 ) min r near , c 3L 1 a 0 , we have M (u, v) 1 8 . Combining this with Lemma D.1 gives Id − (G − 1 2 X Γ * X Γ X G − 1 2 X + M (u, v)) Id − G − 1 2 X Γ * X Γ X G − 1 2 X + M (u, v) < 7 8
and therefore it is invertible and
(G − 1 2 X Γ * X Γ X G − 1 2 X + M (u, v)) −1 1 1 − Id − (G − 1 2 X Γ * X Γ X G − 1 2 X + M (u, v)) = O (1) .
In this case, ∂ u f (u, v) is invertible, and we have
(G − 1 2 X ∂ u f (u, v)G − 1 2 X ) −1 = J −1 a (G − 1 2 X Γ * X Γ X G − 1 2 X + M (u, v)) −1 1 min i |a 0,i | since a − a 0 min i |a 0,i | by assumption.
Therefore we can apply Theorem E.2 with (recalling the definition of c in (E.7))
r 1 = c, D 1 = O √ s , r 2 = O min r near , c L1 a0 , c L0 , 1 CHB , D 2 = O 1 mini|a0,i| with B = i+j 3 B ij , we obtain that g(v) is defined for v ∈ V def. = B · 2 (0, r) with r def. = min r2 D1D2 , r 1 = r2 D1D2 = O min rnear √ s mini|a0,i| , 1 √ sL1L12 a0 , 1 √ sL12L0 , 1 √ s mini|a0,i|CHB such that g is C 1 , f (g(v)
, v) = 0, g(v 0 ) = u 0 , where we recall that u 0 = (a 0 , X 0 ) and v 0 = (0, 0).
Finally, from Theorem E.2 we also have that
G X dg(v) D 1 D 2 √ s min i |a 0,i | and by defining γ(t) = g(v 0 + t(v − v 0 )) for t ∈ [0, 1],
we have the following error bound between u = g(v) and
u 0 = g(v 0 ): d G (u, u 0 ) = a − a 0 2 2 + d H (X, X 0 ) 2 1 0 G γ(t) γ ′ (t), γ ′ (t) dt = 1 0 G γ(t) dg(tv)v, dg(tv)v dt √ s min i |a 0,i | v .
F Examples
F.1 Fejér kernel
Let f ∈ N and X ∈ T d the d-dimensional torus. We consider the Fejér kernel
K(x, x ′ ) = d i=1 κ(x i − x ′ i ), where κ(x) def. = sin f 2 +1 πx f 2 +1 sin(πx) 4 , with constant metric tensor H x = C f Id and d H (x, x ′ ) = C − 1 2 f x − x ′ 2 . where C f def. = −κ ′′ (0) = π 2 3 f (f + 4) ∼ f 2 . Note that K (ij) = C −(i+j)/2 f ∇ i 1 ∇ j 2 K andΩ = ω ∈ Z d ; ω ∞ f , ϕ ω (x) def. = e i2πω ⊤ x , and Λ(ω) = d j=1 g(ω j ) where g(j) = 1 f min(j+f,f ) k=max(j−f,−f ) (1 − |k/f |)(1 − |(j − k)/f |).
Note that this corresponds to sampling discrete Fourier frequencies. In this case, the derivatives of the random features are uniformly bounded with ∇ j ϕ ω (x) = ω j = O(C j/2 f d j/2 ). So, we can setL i = O(d i/2 ). The remainder of this section is dedicated to proving this theorem. The uniform bounds on B ij are due to Lemma F.4 (uniform bounds), and the bound on ∆ and h are due to Lemma F.3. From Lemma F.1, we see that by setting
r near def. = 1 8 √ 2 , for all d H (x, x ′ ) r near , K (20) (x, x ′ ) ≺ −ε 2 Id with ε 2 = (1 − 6r 2
near )(1 − r 2 near /(2 − r 2 near ) − r 2 near ) 0.941. Finally, from Lemma F.2, we have that for for all d H (x, x ′ ) r near , |K| 1 − 1/(8 3 · 2), so we can set ε 0 def. = 0.00097.
Before proving these lemmas, we first summarise in Section F.1.3 some key properties of the univariate Fejér kernel κ when f 128 which were derived in Candès and Fernandez-Granda (2014).
For notational convenience, write t i def.
= x i − x ′ i , κ i def. = κ(t i ), κ ′ i def.
= κ ′ (t i ), and so on. Let
K i def. = d k=1 k =i κ k , K ij def. = d k=1 k =i,j κ k and K ijℓ def. = d k=1 k =i,j,ℓ κ k .
With this, we have:
∂ 1,i K(x, x ′ ) = κ ′ i K i ∂ 1,i ∂ 2,i K(x, x ′ ) = − κ ′′ i K i , and ∀i = j, ∂ 1,i ∂ 2,j K(x, x ′ ) = −κ ′ i κ ′ j K ij .
Where convenient, we sometimes write
K(t) = K(x − x ′ ) def. = K(x, x ′ ).
F.1.3 Properties of κ
From (Candès and Fernandez-Granda, 2014, Equations (2.20)-(2.24) and (2.29)), for all t ∈ [−1/2, 1/2] and ℓ = 0, 1, 2, 3:
1 − C f 2 t 2 κ(t) 1 − C f 2 t 2 + 8 1 + 2/f 1 + 2/(2 + f ) 2 C 2 f t 4 1 − C f 2 t 2 + 8C 2 f t 4 |κ ′ (t)| C f t, |κ ′′ (t)| C f , |κ ′′′ (t)| 3 1 + 2/f 1 + 2/(2 + f ) 2 C 2 f t 12C 2 f t κ ′′ −C f + 3 2 1 + 2/f 1 + 2/(2 + f ) 2 C 2 f t 2 −C f + 6C 2 f t 2 .
(F.1)
By (Candès and Fernandez-Granda, 2014, Lemma 2.6),
κ (ℓ) (t) π ℓ H ℓ (t) (f +2) 4−ℓ t 4 , t ∈ [ 1 2f , √ 2 π ] π ℓ H ∞ ℓ (f +2) 4−ℓ t 4 , t ∈ [ √ 2 π , 1 2 ),
where H ∞ = α 4 (t)β ℓ (t), with
α(t) def. = 2 π(1 − π 2 t 2 6 ) ,β(t) def. = α(t) f t = 2 f tπ(1 − π 2 t 2 /6) and β 0 (t) def. = 1, β 1 (t) def. = 2 + 2β(t), β 2 def.
= 4 + 7β(t) + 6β(t) 2 and β 3 (t) def.
= 8 + 24β + 30β(t) 2 + 15β(t) 3 . Let us first remark thatβ is decreasing on I def.
= [ 1 2f , √ 2 π ]
, so β (t) β (1/(2f )) ≈ 1.2733, and a(t) a( √ 2/π) = 3 π on I. Therefore, on I, H 0 (t) 3 π , H 1 (t) 3.79, H 2 (t) 18.83 and H 3 (t) 98.26, and we can conclude that on
[ 1 2f , 1 2 ), we have κ (ℓ) (t) π ℓH ∞ ℓ (f + 2) 4−ℓ t 4 whereH ∞ 0 = 1,H ∞ 1 def. = 4,H ∞ 2 def. = 19,H ∞ 3 def. = 99. Combining with (F.1), we have κ (ℓ) ∞ κ ∞ ℓ where κ ∞ 0 def. = 1, κ ∞ 2 def. = C f , κ ∞ 1 def. = C f max 2π 4 ( 1 2 + 1 f ) 3 f C f , C f 2f = O( C f ) κ ∞ 3 def. = (C f ) 3/2 max 99π 3 ( 1 2 + 1 f ) 2f C f 4 , 6 C f f = O((C f ) 3/2 ).
Finally, given p ∈ (0, 1),
(f + 2) 4 t 4 (1 + p(f + 2) 2 t 2 ) 2 , ∀ t 1 (1 − p)(f + 2) .
Choosing p = 1 2 and using (f + 2) 2 = ( 3 π 2 C f + 4)
3 π 2 C f , we have
κ (ℓ) (t) κ ∞ ℓ (1 + 3 2π 2 C f t 2 ) 2 , ∀ t 2 2π 2 3C f , (F.2) F.1.4 Bounds in neighbourhood of x ′ = x Lemma F.1. Suppose that C f t 2 2 c with c > 0 such that ε def. = (1 − 6c) 1 − c 2 − c − c > 0 Then,K 02 (t) −εId.
Proof. We need to show that λ min (−K (02) (t)) b. Let q ∈ R d , and note that
− ∇ 2 2 Kq, q = − i q i κ ′′ i K i − κ ′ i j =i q j κ ′ j K ij q i = − i q 2 i κ ′′ i K i − i q i κ i j =i q j κ j K ij q 2 − max i {κ ′′ i K i } − j κ ′ j 2 . (F.3)
We first consider κ ′′ i K i :
κ ′′ i −C f + 6C 2 f t 2 i , K i j =i 1 − C f 2 t 2 i 1 − C f 2 t 2 2 − C f 2 t 2 2 3 − C f 2 t 2 2 5 − · · · 1 − C f t 2 2 2(1 − C f 2 t 2 2 )
. and hence,
κ ′′ i K i −C f + 6C 2 f t 2 2 1 − C f t 2 2 2(1 − C f 2 t 2 2 ) For the second term, j κ ′ j 2 C 2 f t 2 2 .
Therefore,
λ min (−K (02) (t)) 1 − 6C f t 2 2 1 − C f t 2 2 2(1 − C f 2 t 2 2 ) − C f t 2 2
Lemma F.2. Assume that 1 8 √ C f t 2 Then,
K(t) 1 − C f 4 t 2 2 + 16C 2 f t 4 2 .
and otherwise, we have |κ
′ i K i | |κ ′ (t i )| j =i |κ(b j )| κ ∞ 1 (1+ 3 4π 2Ā 2 d √ smax) 2 , In a similar manner, writing V def. = 1 + 3 4π 2Ā 2 d √ s max −2 , we can deduce that |κ ′ i K i | κ max 1 V, |κ ′′ i K i | κ max 2 V, κ ′ i κ ′ j K ij 2 (κ max 1 ) 2 V |κ ′′′ i K i | 3 κ max 3 V, κ ′′ i κ ′ j K ij 3 κ max 2 κ max 1 V, κ ′ i κ ′ j κ ′ ℓ K ijℓ (κ max 1 ) 3 V.
Therefore,
K (10) = 1 C f ∇ 1 K 1 C f d j=1 κ ′ j K j 2 κ ∞ 1 C f V √ d 1 A 4 d 3/2 s max .
Using Gershgorin theorem, we have
∇ 2 2 K(x, x ′ ) max 1 i d {|κ ′′ i K i | + |κ ′ i | j =i κ ′ j |K ij |}
and hence,
K (02) = 1 C f ∇ 2 2 K 1 C f d max i=1 {|κ ′′ i K i | + |κ ′ i | j =i κ ′ j K ij } 1 C f V κ max 2 + (κ max 1 ) 2 (d − 1) max{κ ∞ 2 , (κ ∞ 1 ) 2 } C f V d 1 A 4 ds max .
Note also that K (11) = K (02) . Finally, since
∂ 1,i ∇ 2 2 K(x, x ′ ) max |κ ′′′ i K i | + |κ ′′ i | j =i κ ′ j |K ij | , max j =i { κ ′′ j κ ′ i K ij + κ ′ j κ ′′ i K ij + |κ ′ i | κ ′ j l =i,j |κ ′ l | |K ijℓ |} ,
we have
K (12) = 1 C 3/2 f ∇ 1 ∇ 2 2 K 1 C 3/2 f √ dV max κ max 3 + κ max 2 κ max 1 (d − 1), 2κ max 2 κ ∞ 1 + (d − 1)(κ ∞ 1 ) 3 d 3/2 max{κ ∞ 3 , κ ∞ 1 κ ∞ 2 , (κ ∞ 1 ) 3 } 1 C 3/2 f V 1 A 4 d 1/2 s max F.1.6 Uniform bounds Lemma F.4. If r near ∼ 1/ C f , then B 0 = O(1), B 01 = O( √ d), B 02 = B 12 = B 11 = O(1) and B 22 = O(d).
Proof. We have |K| 1, and ∇K
2 i |κ i | 2 |K i | 2 d(κ ∞ 1 ) 2 C f d, so B 01 = O( √ d).
From (F.3), for all q = 1,
∇ 2 2 K(t)q, q max i |κ ′′ i | q 2 2 + q 2 2 i |κ i | 2 C f + C 2 f t 2 = O(C f ), for t 1/ C f . So, since r near 2/ C f , K 02 (t) 2 def.
= B 02 . The norm bound for K 11 is the same.
K (12) = sup q = p =1 1 C 3/2 f k k =i ∂ 1,i ∂ 2 2,k Kp i q 2 k + ∂ 1,i ∂ 2,i ∂ 2,k Kp i q i q k + i k j ∂ 1,i ∂ 2,j ∂ 2,k p i p j p k + i j =i ∂ 1,i ∂ 2,i ∂ 2,j Kp i q i q j + i ∂ 1,i ∂ 2 2,j Kp i q 2 i = sup q = p =1 1 C 3/2 f k k =i κ ′ i κ ′′ k K ik p i q 2 k + κ ′′ i κ ′ k K ik p i q i q k + i k j κ ′ i κ ′ k κ ′ j K ijk p i p j p k + i j =i κ ′′ i κ ′ j K ij p i q i q j + i κ ′ i κ ′′ j K ij p i q 2 i 1 C 3/2 f 3 κ ′′ ∞ i |κ ′ k | 2 + i |κ ′ k | 2 3/2 + κ ′ ∞ κ ′′ ∞ 1 C 3/2 f 3C 2 f t + C 3 f t 3 + O(C 3/2 f ) = O(1)
for t 1/C Then, we have
K (10) (x, x ′ ) = K (01) (x, x ′ ) = d H (x, x ′ )κ(d H (x, x ′ )) K (02) (x, x ′ ) = K (11) (x, x ′ ) (d H (x, x ′ ) 2 + 1)κ(d H (x, x ′ )) K (02) (x, x ′ ) (d H (x, x ′ ) 2 − 1)κ(d H (x, x ′ ))Id and for q ∈ R d with q = 1, since i (Σ 1 2 ∇ϕ ω ) i q i = ∇ϕ ⊤ ω (Σ 1 2 q) = i ∂ i ϕ ω (q ⊤ Σ 1 2 e i )
we can write
K (12) (x, x ′ )q = d i=1 (q ⊤ Σ 1 2 e i )Σ 1 2 ∂ 1,i ∇ 2K Σ (t)Σ 1 2
Thus we examine each term in ∂ 1,i ∇ 2K Σ . We have
i (q ⊤ Σ 1 2 e i )Σ 1 2 Σ −1 tf ⊤ i Σ 1 2 = Σ − 1 2 t i q ⊤ Σ 1 2 e i e ⊤ i Σ − 1 2 = Σ − 1 2 tq ⊤ and similarly i (q ⊤ Σ 1 2 e i )Σ 1 2 f i t ⊤ Σ −1 Σ 1 2 = qt ⊤ Σ 1 2 . Then i (q ⊤ Σ 1 2 e i )(t ⊤ Σ −1 e i )Σ 1 2 Σ −1 Σ 1 2 = t ⊤ Σ −1 ( i e i e ⊤ i )Σ 1 2 q = (t ⊤ Σ 1 2 q)Id and similarly i i (q ⊤ Σ 1 2 e i )(t ⊤ Σ −1 e i )Σ 1 2 Σ −1 tt ⊤ Σ −1 Σ 1 2 = (t ⊤ Σ 1 2 q)Σ − 1 2 tt ⊤ Σ − 1 2 .
Hence at the end of the day
K (12) (x, x ′ ) (3d H (x, x ′ ) + d H (x, x ′ ) 3 )κ(d H (x, x ′ ))
and this bound is automatically valid for K (21) as well.
Finally, note that
K (22) (x, x) = sup p 1 Σ 1/2 ∇ 2 ∇ 2 · Σ 1/2 K (2,0) (x, x)p , p
where ∇ 2 · is the divergence operator on the 2nd variable, and one can show that K (22) (x, x) = (d + 1).
We are then going to use the fact that for any q 1 the function f (r) = r q e − 1 2 r 2 defined on R + is increasing on [0,
√ q] and decreasing after, and its maximum value is f ( √ q) = q e q/2 . Furthermore, it is easy to see that we have f (r) = r q e −r 2 /2 2q 2 q 2 e −r 2 /4 and therefore f (r) ε if r 2 log 1 ε + q 2 log 2q e . We define r near = 1/ √ 2 and ∆ = C 1 log(s max ) + C 2 for some C 1 and C 2 . and note that K (11) = K (02) , so for all i + j 3 B ij = O (1).
2.
Near 0 For d H (x, x ′ ) r near , we have
K (02) − e − 1 4 2 Id L j tF j (t)dt = 2 d 2 L j t exp − t 2 j 4 dt = 2 d 2 L 2 j j (j/2)t j−1 exp − t 4 dt = 2 d 2 (j/2) L 2 j j t j−1 exp − t 8 exp − t 8 dt 2 d 2 (j/2) 8(j − 1) e j−1 L 2 j j exp − t 8 dt = 2
F.2.2 Gaussian mixture model learning
We apply the mixture model framework with the base distribution:
P θ = N (θ, Σ)
The random features on the data space are ϕ ′ ω (x) = Ce iω ⊤ x with Gaussian distribution ω ∼ Λ = N (0, A) for some constant C and matrix A. Then, the features on the parameter space are ϕ ω (θ) = E x∼P θ ϕ ′ ω (x) = Ce iω ⊤ θ e − 1 2 ω 2 Σ (that is, the characteristic function of Gaussians). Then, it is possible to show (Gribonval et al., 2017) that the kernel is
K(θ, θ ′ ) = C 2 A −1 1 2 |2Σ + A −1 | 1 2 e − 1 2 θ−θ ′ 2 (2Σ+A −1 ) −1
Hence we choose A = cΣ −1 , C = (1 + 2c)
F.3 The Laplace transform kernel
Let α ∈ R d + and let X ⊂ R d + be a compact domain. Define for x ∈ X and ω ∈ R d + ,
ϕ ω (x) def. = exp(− x, ω ) d i=1 (x i + α i ) α i and Λ(ω) def. = exp(− 2α, ω ) d i=1 (2α i ),
The associated kernel is K(
x, x ′ ) = d i=1 κ(x i + α i , x ′ i + α i ) where κ is the 1D Laplace kernel κ(u, v) def. = 2 √ uv (u + v)
.
A direct computation shows that H x ∈ R d×d is the diagonal matrix with (h xi+αi ) d i=1 where h x def.
= ∂ x ∂ x ′ κ(x, x) = (2x) −2 . Note that d κ (s, t) = max{s,t} min{s,t} (2x + 2α) −1 dx = log t + α s + α (F.5) and so, d H (x,
x ′ ) = d i=1 log xi+αi x ′ i +αi 2 .
We have the following results concerning the boundedness of D j [ϕ ω ] and the admissiblity of K:
Theorem F.2 (Stochastic gradient bounds). Assume that the α i 's are all distinct. Then,L 0 (ω) L 0 def.
= 1 + RX mini αi d and for j = 1, 2, 3,
P(L j (ω) t) F j (t) def. = d i=1 β i exp −α i 1 2(R X + α ∞ ) t L 0 1/j − √ d
and we have that i F j (L j ) δ andL 2 j i F i (L i ) + 2 ∞ Lj tF j (t)dt δ provided that
L j ∝L 0 (R X + α ∞ ) j √ d + max i 1 α i log dβ iL0 (R X + α ∞ ) δα i j .
where β i = j =i αj αj −αi . Note that α i ∼ d implies thatL 0 ∼ (1 + R X /d) d ∼ e RX . Theorem F.3 (Admissiblity of K). The Laplace transform kernel K is admissible with r near = 0.2, C H = 1.25, The first result Theorem F.2 is proved in Section F.3.1 and the second result, Theorem F.4 is a direct consequence of Theorem F.4 and Lemma F.5 in Section F.3.2.
F.3.1 Stochastic gradient bounds
Proof of Theorem F.2. Let V def.
= (1 − 2(x i + α i )ω i ) d i=1 ∈ R d . Then, V = i (1 − 2(x i + α i )ω i ) 2 i 1 + 4(x i + α i ) 2 ω 2 i d + 4(R X + α ∞ ) 2 w 2 √ d + 2(R X + α ∞ ) w
We have the following bounds:
|ϕ ω (x)| d i=1 1 + x i α i 1 + R X min i α i d def. =L 0 , D 1 [ϕ ω ] (x) = ϕ ω (x)V =⇒ D 1 [ϕ ω ] (x) L 0 V D 2 [ϕ ω ] (x) = ϕ ω (x)(V V ⊤ − 2Id) =⇒ D 2 [ϕ ω ] (x) L 0 min{ V 2 , 2}.
and given u, q ∈ R d , D 3 [ϕ ω ] (x)[q, q, u] = ϕ ω (x) u, V q, V 2 − 2 q 2 − 4 u, q q, V + 8
i q 2 i u i , so D 3 [ϕ ω ] (x) |ϕ ω (x)| V 3 + 10 + 4 V L 0 5( V 3 + 3),
And therefore, in general,
D j [ϕ ω ] (x) L j (ω) def. =R j+1 X √ d + ω j D j [ϕ ω ] (x) L j (ω) def. =L 0 √ d + 2(R X + α ∞ ) w j
Assuming for simplicity that all α j are distinct, we have Akkouchi:
P( w t) P( ω 1 t) = d i=1 β i e −αit
where β i = j =i αj αj −αi , using the fact that ω 1 is a sum of independent exponential random variable.
Hence, for all 1 j 3 and t d j 2 we have P(L j (ω) t) P w 1 2(R X + α ∞ )
t L 0 1/j − √ d F j (t) def. = d i=1 β i exp −α i 1 2(R X + α ∞ ) t L 0 1/j − √ d δ and F j (L j ) δ ifL j L 0 2 j (R X + α ∞ ) j √ d + max i 1 α i log dβ i δ j
Next, in a similar manner to the Gaussian case, we compute
L j tF j (t)dt = d i=1 β i L j t exp −α i 1 2(R X + α ∞ ) t L 0 1/j − √ d dt =L 2 0 j d i=1 e αi √ d β i (Lj/L0) 1/j exp −α i u 2(R X + α ∞ ) u 2j−1 du (2j − 1)4(R X + α ∞ ) eα i 2j−1L 2 0 j d i=1 e αi √ d β i (Lj/L0) 1/j exp −α i u 4(R X + α ∞ ) du 4(R X + α ∞ ) α i 2j 2j − 1 e 2j−1L 2 0 j d i=1 e αi √ d β i exp −α i (L j /L 0 ) 1/j 4(R X + α ∞ ) δ
if for all i = 1, . . . , d,
4(R X + α ∞ ) α i 2j log 4(2j − 1)(R X + α ∞ ) eα i + log(L 2 0 j) + α i √ d + log dβ i δ L j L 0 1/j that is,L j L 0 2 j (R X + α ∞ ) j √ d + max i 1 α i log dβ i δ j .
It remains to boundL j F ℓ (L ℓ ) with ℓ, j ∈ {0, 1, 2, 3}: LetL ℓ L 0 M ℓ for some M to be determined. Then,
L j F ℓ (L ℓ ) L 0 M j d i=1 β i exp −α i 2(R X + α ∞ ) M + α i √ d =L 0 d i=1 β i M j exp −α i 4(R X + α ∞ ) M exp −α i 4(R X + α ∞ ) M e αi √ d L 0 e −j d i=1 4j(R X + α ∞ ) α i j β i exp −α i 4(R X + α ∞ ) M e αi √ d L 0 e −3 d i=1 12(R X + α ∞ ) α i 3 β i exp −α i 4(R X + α ∞ ) M e αi √ d δ if for each i = 1, . . . , d M 4(R X + α ∞ ) √ d + max i 1 α i log L 0 dβ i δe 3 12(R X + α ∞ ) α i 3 .
Therefore, similar to the Gaussian case, the conclusion follows forL 0 = 1 + RX mini αi d , and for j = 1, 2, 3,
L j ∝L 0 (R X + α ∞ ) j √ d + max i 1 α i log dβ iL0 (R X + α ∞ ) δα i j .
F.3.2 Admissiblity of the kernel
Metric variation We have the following lemma on the variation of the Fisher metric:
Lemma F.5. Suppose that d H (x, x ′ ) c, then Id − H 1/2
x H x ′ (1 + ce c )d H (x, x ′ ) . Proof. Note that |1 − |(x i + α i )/(x ′ i + α i )|| max{e dκ(xi,x ′ i ) − 1, 1 − e −dκ(xi,x ′ i ) } d κ (x i , x ′ i )(1 + ce c ) for all d κ (x i , x ′ i ) c. Therefore, Id − H x H x ′ 2 = i |1 − |(x i + α i )/(x ′ i + α i )|| 2 (1 + ce c )d H (x, x ′ ) provided that d H (x, x ′ ) c.
Admissiblity of the kernel The following theorem provides bounds for K and its normalised derivatives.
Theorem F.4. 1. |K(x, x ′ )| min{2 d e − 1 2 dH(x,x ′ ) , 8 8+dH(x,x ′ ) 2 }. 2. K (10) (x, x ′ ) min{2 √ d |K| , √ 2}.
3. K (11) min{9d |K| , 8}
4. K (20) min{10d |K| , 8} and λ min (−K (20) ) 2 − 12d H (x, x ′ ) 2 K.
5. K (12) min{66 |K| d 3/2 , 16 √ d + 49} and K (12) (x, x ′ ) 34 if d H (x, x ′ ) 1.
K (22)
16d + 9.
In particular, for d H (x, x ′ ) 2d log(2) + 2 log 52d 3/2 smax h , we have K (ij) (x, x ′ ) h smax .
To prove this result, we first present some bounds for the univariate Laplace kernel in Section F.3.3 before applying these bounds in Section F.3.4.
F.3.3 1D Laplace kernel
In the following κ (ij) (x,
x ′ ) def. = h −i/2 x h −j/2 x ′ ∂ i x ∂ j x ′ κ(x, x ′ ). Lemma F.6. We have (i) κ(x, x ′ ) = sech dκ(x,x ′ ) 2 2e − 1 2 dκ(x,x ′ ) ,
(ii) κ (10) (x, x ′ ) = 2 tanh dκ(x,x ′ ) 2 κ(x, x ′ ) , and κ (10) 2 |κ|.
(iii) κ (11) 4 |κ| 3 + 4 |κ| (iv) κ (20) 6 |κ| and −κ (20) 2κ(x, x ′ ) 1 − 2 tanh dκ(x,x ′ ) 2 .
(v) κ (12) 49 |κ|.
(vi) κ (22) (x, x) = 9 for all x.
def.
= E[|x i | 2 ] σ 2 for all i's.
Then for all t > 0 we have
X 1 n n i=1 x i t 4 exp − nt 2 /4 σ 2 + M t/(3 √ 2) . (G.1)
Lemma G.2 (Matrix Bernstein (Tropp (2015), Theorem 6.1.1)). Let Y 1 , ..., Y m ∈ C d1,d2 be complex random matrices with EY j = 0, Y j L, v(Y j ) := max( EY j Y * j , EY * j Y j ) M for each index 1 j m. Introduce the random matrix
Z = 1 m j Y j .ThenM x 2 = s i=1 s j=1 a ij x j + Q ⊤ ij X j 2 + s j=1 R ij x j + A ij X j 2 s i=1 x s j=1 a 2 ij + Q ij 2 2 + x s j=1 R ij 2 + A ij 2 2 i,j a 2 ij + Q ij 2 + R ij 2 + A ij 2 Now, if x Block = 1, we have M x Block = max i s j=1 a ij x j + Q ⊤ ij X j , s j=1 R ij x j + A ij X j max i s j=1 |a ij | + Q ij , s j=1 R ij x j + A ij X j
Discrete
Fourier sampling The Fejer kernel of order f c 128 is admissible with ∆ = O( √ d 4 √ s max ), r near = 1/(8 √ 2), ε 0 = 0.00097, ε 2 = 0.941, B 01 = O(d), B 11 = B 02 = B 12 = O(1) and B 22 = O(d). Moreover,L r = O(d r/2 ). Hence, up to logarithmic terms, Thm. 1 is applicable with m = O(sd 3 ) when the random signs assumption holds, and m = O(s 3 2 d 3 ) in the general case, with guaranteed support stability when
and the number of measurements and sample complexity satisfy, up to logarithmic terms, m = O s 3 2 d 3 , n = O s 2 d 6 / min i |a 0,i | 2 and
.
By Lemma C.9 and the union bound,
F.1. 2
2Admissibility of the kernel Theorem F.1. Suppose that f 128. Then, K is an admissible kernel with r near = 1/(8 √ 2), ε 2 = 0.941, ε 0 = 0.00097, h = O(d −1/2 ) and ∆ = O(d , B 00 = B 11 = B 20 = O(1), B 01 = O(d 1/2 ) and B 22 = O(d).
Hence this quantity is bounded byδ ifL j ∝ d + log 1 δ j 2 . Then we haveL 2 j F i (L i )
d 4 .
4, and we come back to the previous case K(θ, θ ′ ) = e Henceε i = O (1), B ij = O (1), d H (θ, θ ′ ) = θ − θ ′Admissible features. Unlike the previous case, the features are directly bounded and Lipschitz. We have |ϕ ω (
For all i + j 3, B ij = O(1), B 22 = O(d), ∆ = O(d + log d 3/2 s max ) and h = O(1).
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since the metric is constant, we can set C H F.1.1 Discrete Fourier sampling A random feature expansion associated with the Fejér kernel is obtained by choosingdef.
= 0.
1 .
1Global Bounds. From what preceeds, we haveK (10)
1
√
e
,
K (02)
2
e
+ 1,
K (12)
3
√
e
+
3
e
3
2
Lemma G.3 (Vector Bernstein for complex vectors Minsker(2017)). Let Y 1 , . . . , Y M ∈ C d be a sequence of 1. Let M ∈ R sd×sd be a matrix formed by blocks :A 11 . . . A1s . . . . . . . . . A s1 . . . A ss Then we have M block = sup Now, let P ∈ R sd×s be a rectangular matrix formed by stacking vectors Q ij ∈ R d : a 11 . . . a 1s Q ⊤ 11 . . . Q ⊤ 1s . . . . . . . . . . . . . . . . . . a s1 . . . a ss Q ⊤ s1 . . . Q ⊤ ss R 11 . . . R 1s A 11 . . . A 1s . . . R s1 . . . R ss A s1 . . . A ss|a ij | + Q ij ,3. Taking x = [x 1 , . . . , x s , X 1 , . . . , X s ] ∈ R s(d+1) with x = 1, we haveP ( Z
t) 2(d 1 + d 2 )e − mt 2 /2
M +Lt/3
(G.2)
M =
x block =1
M x block
max
1 i s
s
j=1
A ij
(G.6)
M =
Q 11 . . . Q 1s
. . .
. . .
. . .
Q s1 . . . Q ss
Then,
M ∞→block
max
1 i s
s
j=1
Q ij 2 ,
M ⊤
block→∞
max
1 i s
s
j=1
Q ji 2
(G.7)
2. Consider A ∈ R s(d+1)×s(d+1) decomposed as
M =
. . .
. . .
. . .
. . .
. . .
Then,
M
i,j
a 2
ij + Q ij
2 + R ij
2 + A ij
2 ,
M Block max
i
{
j
. Then we must choose the L j such that L j tF j (t)dt is bounded by some δ. Taking L j d j/2 in any case, we have
AcknowledgementsWe would like to thank Ben Adcock for a helpful conversation regarding the stochastic gradient bounds. This work was partly funded by the CFM-ENS chair "Modèles et Sciences des données" and the European Research Council, NORIA project.and all t such that t 2 c,Proof. First note thatdef.and note that g(u) ∈ (0, C f 2 ) for u ∈ (0, 1/(8 C f ). So, writing t = (t i ) d i=1 and g j def.= g(t j ), we haveNote that − j =k =ℓ t 2 j t 2 k t 2 ℓ · g j g k g ℓ + j =k =ℓ =n t 2 j t 2 k t 2 ℓ t 2 n · g j g k g ℓ g n − j =k =ℓ t 2 j t 2 k t 2 ℓ · g j g k g ℓ + j =k =ℓ t 2 j t 2 k t 2 ℓ · g j g k g ℓ n t 2 n g n − j =k =ℓ t 2 j t 2 k t 2 ℓ · g j g k g ℓ 1 −by assumption. So,Finally, observe that the functionis positive and increasing on the interval [0, 1, we want to make use of the form (F.2). We can do this for each t j such that |t j |, by our assumptions onĀ. Therefore, we may assume that we have some d p 1 such thatSo, by applying the fact that |κ| 1, κ ∞ 0 = 1 and (F.2), we haveF.2 The Gaussian kernelWe consider the Gaussian kerneland for u ∈ R, κ(u) = exp − 1 2 u 2 . Denote by {e i } the canonical basis of R d , and by f i = Σ −1 e i the i th row of Σ −1 . We have the following:, we have CK = 1, C H = 0 (that is, the metric tensor of the kernel is constant, and d H is defined as the corresponding normalized norm). Hence we can define the F j such that, for all t, and F j (L j ) is smaller than some δ ifL j ∝ d + log 1 δ j Proof. We first state the partial derivatives of κ:so κ (11) 4 |κ| 3 + 4 |κ|. Support Localization and the Fisher Metric for off-the-grid Sparse Regularization (iv). We first prove that(iii) K (11) |K| g 2 2 + 5(iv) K (20) |K| g 2 2 + 6 and λ min K (20) K 2 − 3 g 2 2 .(v) K (12) |K| g 3 2 + 16 g 2 + 49The result would then follow because• sech(x) 2e −x and sech(x) (1 + x 2 /2) −1 .• |tanh(x)| min{x, 1}, so g min{d H (x, x ′ ), 2 √ d},For example, K (12)In the following, we write κ = j =i κ j . Moreover, we will make use of the inequalities for κ (ij) derived in Lemma F.6.iii p 2 i q i |K| g 3 2 + 16 g 2 + 49 .(vi)since κ (10) (x, x) = κ (01) (x, x) = 0, and κ (11) (x, x) = 4 from the proof of (iii) in Lemma F.6.G ToolsG.1 Probability tools Lemma G.1 (Bernstein's inequality (Sridharan(2002), Thm. 6)). Let x 1 , . . . , x n ∈ C be i.i.d. bounded random variables such that Ex i = 0, |x i | M and V ar(x i )We were only able to find a reference for this result in the case where u is a Rademacher sequence, however, by the contraction principle (see(Ledoux and Talagrand, 2013, Theorem 4.4)), a similar statement is true for Steinhaus sequences (we write only for the case of real symmetric matrices because this is all we require in this paper, but of course, the same argument extends to complex self-adjoint matrices):Corollary G.1. Let the components of u ∈ C k i.i.d. from a symmetric distribution on the complex unit circle or 0 and let B 1 , . . . , B M ∈ R d×d be symmetric matrices. Set σ 2 def. = M ℓ=1 B 2 ℓ . Then, for t > 0,Proof. By the union bound,By the contraction principle (Ledoux and Talagrand, 2013, Theorem 4.4),where ξ is a Rademacher sequence, and the same argument applies to the case of Im (u ℓ ). Therefore by Lemma G.5, we have P M ℓ=1 u ℓ B ℓ t 4d exp − t 2 4σ 2 .G.2 Linear algebra toolsThe following simple lemma will be handy.Lemma G.6. For 1 i, j s, take any scalars a ij ∈ R, vectors Q ij , R ij ∈ R d and square matrices A ij ∈ R d×d .
On the convolution of exponential distributions. M Akkouchi, M. Akkouchi. On the convolution of exponential distribu- tions.
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| [] |
[
"HFB calculations with a microscopic pairing interaction",
"HFB calculations with a microscopic pairing interaction"
] | [
"T Duguet ",
"P Bonche \nService de Physique Théorique\nCEA Saclay\n91191Gif sur Yvette CedexFrance\n",
"\nPhysics Division\nArgonne National Laboratory\n9700 South Cass Avenue60439ArgonneILUSA\n"
] | [
"Service de Physique Théorique\nCEA Saclay\n91191Gif sur Yvette CedexFrance",
"Physics Division\nArgonne National Laboratory\n9700 South Cass Avenue60439ArgonneILUSA"
] | [] | Hartree-Fock-Bogolyubov (HFB) calculations making use of a recently proposed microscopic effective pairing interaction are presented. The interaction was shown to reproduce the pairing properties provided by the realistic AV 18 force very accurately in infinite matter. Although finite-ranged and non-local, it makes 3D HFB calculations in coordinate space tractable. As a first application, basic pairing properties of calcium isotopes in their ground-state are studied. By comparing the results with those obtained using a standard Density-Dependent Delta Interaction, the crucial isovector character of the microscopic interaction is highlighted. 2 Even if some significant differences remains, one can argue that the Gogny force behaves almost like a bare force in the 1 S 0 and 1 D 2 channels, especially when the D1S [13] parameterization is used[14,15]. | 10.1063/1.1905323 | [
"https://export.arxiv.org/pdf/nucl-th/0411023v1.pdf"
] | 16,542,169 | nucl-th/0411023 | 27e6b450cb6159dd948920dce88063b892981605 |
HFB calculations with a microscopic pairing interaction
5 Nov 2004
T Duguet
P Bonche
Service de Physique Théorique
CEA Saclay
91191Gif sur Yvette CedexFrance
Physics Division
Argonne National Laboratory
9700 South Cass Avenue60439ArgonneILUSA
HFB calculations with a microscopic pairing interaction
5 Nov 2004
Hartree-Fock-Bogolyubov (HFB) calculations making use of a recently proposed microscopic effective pairing interaction are presented. The interaction was shown to reproduce the pairing properties provided by the realistic AV 18 force very accurately in infinite matter. Although finite-ranged and non-local, it makes 3D HFB calculations in coordinate space tractable. As a first application, basic pairing properties of calcium isotopes in their ground-state are studied. By comparing the results with those obtained using a standard Density-Dependent Delta Interaction, the crucial isovector character of the microscopic interaction is highlighted. 2 Even if some significant differences remains, one can argue that the Gogny force behaves almost like a bare force in the 1 S 0 and 1 D 2 channels, especially when the D1S [13] parameterization is used[14,15].
INTRODUCTION
The structure of the nucleus and the properties of extended nuclear systems strongly depend on their possible superfluid nature. In finite nuclei, pairing constitutes the main part of the residual interaction and has a strong influence on most of low-energy properties of the system [1]. In extended systems such as neutron stars, pairing is a decisive ingredient of dynamical and thermal evolutions [2,3]. Despite its major role, the present knowledge of the pairing force and the nature of pairing correlations in nuclei, that is, the way Cooper pairs are formed in the nuclear medium out of the strong nucleon-nucleon (NN) force, is quite poor. Properties such as the range of the effective pairing interaction, its link to the bare force, its possible surface character in finite nuclei and its density dependence (in particular isovector) still have to be clarified [4,5,6,7].
Regarding self-consistent mean-field calculations of finite nuclei, only phenomenological forces such as the Gogny force [8] or (Density-Dependent) Delta Interactions ((DD)DI) [5,6,9,10] have been used in the pairing channel so far. One exception exists however [11], where the Bardeen-Cooper-Schrieffer (BCS) gap equation was solved in the Hartree-Fock (HF) basis of 120 Sn using realistic AV 18 bare force [12]. It required the treatment of single-particle states up to 800 MeV! Although successful in describing low-energy nuclear structure over the known mass table [1], the Gogny force and DDDI lack a clear link to the bare nucleon-nucleon (NN) interaction 2 . This feature strongly limits the reliability of their analytical structure such as their possible density dependence. Also, their direct fit to nuclear data through mean-field calculations makes probable the re-normalization of beyond-mean-field effects. This is a significant limitation if one wants to go explicitly beyond that level of approximation. Finally, their fits performed onto very limited sets of nuclei around stability make their extrapolated use toward the drip-lines unsafe. For instance, while such phenomenological forces all provide similar and reasonable pairing properties around stability, the predicted location of the two-neutron drip-line can differ by up to ten/twenty mass units depending on the force used [16].
To improve on that situation, a microscopic effective interaction explicitly linked to the bare NN force, and equivalent to it at the mean-field level, was proposed recently to treat pairing correlations in the 1 S 0 channel [15]. BCS pairing properties provided in infinite matter by AV 18 were reproduced very accurately. These properties dealt not only with the gap at the Fermi surface as a function of density, but also with the momentum dependence of the gap at fixed density [15].
In the present paper, we discuss the first results of 3D HFB calculations using that force and we focus on the isovector and low-density properties of the interaction. In section 2, we briefly outline the formalism and the characteristics of the new pairing force. Results obtained along the calcium isotopic chain are discussed in section 3. They are compared to those obtained using a standard surface peaked DDDI and a zero-range approximation to the new microscopic force. Conclusions are given in section 4.
FORMALISM
To treat pairing, one needs to specify the many-body technique used and the appropriate interaction to insert into the calculation at the chosen level of approximation. The latter depends both on the situation and on the system. In the present case, we concentrate on a (self-consistent) mean-field description of finite nuclei using the HFB method. Eventually, calculations beyond the mean-field are to be performed. Typically, correlations associated with symmetry restorations (Projected Mean Field Method) and large amplitude motion (Generator Coordinate Method) are considered [17,18,19]. Thus, one has to identify the appropriate vertices to be used coherently at each level of approximation.
While variational calculations are of no direct help in that respect, perturbative methods using Green's function or Goldstone formalisms provide guides to do so. In particular, such many-body theories show unambiguously that the interaction to be used in the particle-particle (p-p) channel at lowest order in irreducible vertices is the bare NN force 3 . At the next order, the irreducible pairing vertex involves the so-called polarization diagrams. This situation is in contrast to the particle-hole (p-h) channel where one may need to regularize the repulsive core of the bare interaction from the outset through the definition of an in-medium two-body vertex like the G-matrix [20]. This stresses the fact that the effective forces may differ in the two channels at a given level of approximation.
The approximate ground-state energy E HFB of a nucleus is a functional of the one-body density matrix ρ q ji = Φ | c † i c j | Φ and pairing tensor κ q jl = Φ | c l c j | Φ , where | Φ is the HFB state 4 . For a general presentation of the HFB formalism, we refer to Ref. [17]. Also, the two-basis method used to solve the HFB problem iteratively is discussed in Ref. [21]. Finally, a detailed presentation of the method applied to the new interaction will be soon available [22].
The pairing field ∆ q i j reads as:
∆ q i j = ∂ E HFB ∂ κ q * i j = 1 2 ∑ kl V 1 S 0 i jkl κ q kl ,(1)
where, as already explained, V 1 S 0 is the bare NN force. In Ref. [15], the effective vertex was then introduced by recasting the previous gap equation written in the canonical basis 5 into a fully equivalent expression [15]:
∆ q i ≡ ∆ q ii = − 1 2 ∑ m V 1 S 0 iīmm ∆ q m 2 E q m = − 1 2 ∑ m 2 ρ q m D 1 S 2q 0 (0) iīmm ∆ q m 2 E q m ,(2)
where D 1 S 2q 0 (0) is an off-shell in-medium two-body matrix summing p-p and h-h ladders in the superfluid phase. In Eq. 2, the expression of the pairing tensor in its canonical form κ q
mm = ∆ q m /2E q m was used, where E q m = E q m = (h q m − µ q ) 2 + ∆ q 2 m , while h q m
is the diagonal matrix element of the HF field in the canonical basis and µ q the chemical potential. Identifying the two sides of Eq. 2, the effective pairing vertex is naturally defined through its antisymmetrized matrix elements in the canonical basis of the Bogolyubov transformation as:
V 1 S 2q 0 e f f iīmm = 2 ρ q m D 1 S 2q 0 (0) iīmm .(3)
3 This constitutes our motivation for the mean-field level. With the G-matrix [20] in the p-h channel, such a mean-field theory precisely aims at treating the nucleus as a system of independent pairs, including the correlations associated with the existence of Cooper bound-states in the medium. 4 The quantities are defined in an arbitrary basis and the isospin quantum number q (1/2 for neutrons and −1/2 for protons) is specified to make clear that isospin mixing is considered neither in the p-h channel nor in the p-p channel. 5 The canonical basis corresponds to the single-particle basis diagonalizing the one-body density matrix ρ q ji = ρ q i δ ji = ρ q i and putting the pairing tensor in its canonical form κ q kl = κ q kk δ lk . ρ q m plays in the canonical basis the role of the usual BCS occupation number. Which states (i,ī) are paired is a by-product of the Bogolyubov transformation solution of the problem. States are not paired a priori as in the BCS approximation.
In Ref. [15], D 1 S 2q 0 (0) was calculated explicitly in infinite matter starting from a separable form of the bare NN interaction in the 1 S 0 channel. It was studied in detail and shown to take a closed form in coordinate space:
D 1 S 2q 0 (0) ( r 1 , r 2 , r 3 , r 4 ) = λ 1 S 0 1 − P σ 2 d r f [ρ q ( r )] e − ∑ 4 i=1 | r− r i | 2 /2α 2 (2π) 6 α 12 ,(4)
where P σ is the spin-exchange operator 6 incorporates the density dependence of the T z = 2q component of the effective interaction [15]. The density dependence stems from the re-summation of p-p and h-h ladders in the medium. The effective vertex is thus finite ranged, non local, total-momentum dependent and density dependent 7 . However, the computing cost of corresponding 3D HFB calculations is, through the two-basis method [21,23], of the same order as for a zero-range interaction. By re-summing the effect of pairs scattered at high-energy into the effective vertex, the latter is soft even if the bare force has a hard core. Also, a smooth cut-off 2 ρ q m emerges naturally in the gap equation through its recast. This cut-off further limits the necessity to use large basis sets as in Ref. [11] and makes zero-range approximations of the effective vertex meaningful. The pairing problem is regularized in a similar way to what was proposed in Refs. [7,24,25] using re-normalization techniques. It is worth noting that the derived cut-off differs from all ad-hoc ones used in connection with usual DDDI. Thus, a zero-range (ZR) approximation providing identical gaps at the Fermi surface in infinite matter was defined in Ref. [15]. The coefficients entering the functional f [ρ q ( r )] differ from the ones used in the finite range case. Performing such an approximation, the roles of the range and of the density dependence of the interaction could be disentangled [15]. In particular, the surface-enhanced character of phenomenologically optimized DDDI [5,6] was demonstrated and shown to be, to a large extent, a way of re-normalizing the range of the interaction. Also it was shown that usual DDDI miss the low-density behavior of the effective pairing force.
Both the finite-range microscopic force and its zero-range approximation depend on the density ρ q ( r ) of the interacting particles rather than on the total matter density. It is to be contrasted with usual DDDI which often take the form:
D 1 S 2q 0 ( r 1 , r 2 , r 3 , r 4 ) = λ 1 S 0 1 − P σ 2 1 − ρ 0 ( r 1 ) ρ c δ ( r 1 − r 3 ) δ ( r 2 − r 4 ) δ ( r 1 − r 2 ) ,(5)
where ρ 0 ( r) denotes the total matter density (the local scalar-isoscalar part of the one-body density-matrix) while ρ c is equal to (one-half) the saturation density for the surface (half-surface) type pairing force. The role of the finite range, the isovector density-dependence and the low density behavior of the pairing force, as well as the regularization scheme used together with contact approximations has to be addressed in detail. While we briefly focus on the isovector properties of the interaction in the present communication, we refer to Ref. [22] for an extensive study of all other issues.
RESULTS
We performed 3D HFB calculations of Ca ground-states from proton to neutron drip-lines [21]. Good particle-numbers were approximately restored before variation through the Lipkin-Nogami (LN) procedure [26]. Self-consistent blocking and time reversal symmetry breaking were included in the calculations of odd isotopes. The Sly4 Skyrme force [27] was used in the p-h channel. Each calculation was repeated three times using the microscopic finite range force defined through Eq. 4, its zero range approximation and a standard surface peaked DDDI as given by Eq. 5 [10]. The latter was adjusted together with a phenomenological cut-off defining an active window (±5MeV ) around the Fermi energy.
By keeping the force in the p-h channel fixed, we probe the vertex used in the pairing channel, including the selfconsistent coupling between the two channels. Of course, properties of the force in the p-h channel have an impact on the results. In that respect, it is worth noting that the considered DDDI was adjusted on properties of (non-exotic) nuclei using the SLy4 parameterization in the p-h channel [10], and thus, is consistent with the isoscalar effective mass m * /m = 0.7 predicted by the latter. The microscopic pairing forces were adjusted once for all without any reference to finite nuclei with the property of not depending explicitly on the effective mass appearing in the p-h channel [15]. However, according to our definition of the mean-field, they should be used at the HFB level together with a p-h vertex providing an effective mass consistent with a G-matrix supplemented by a three-body force [28]. In Fig 1, calculated one and two-neutron separation energies are compared to experiment [29]. Overall, the agreement is (at least) of the same quality as the one obtained from other calculations of the same type [7,30]. There are interesting differences which are beyond the scope of the present discussion. We rather concentrate on the effect of the pairing force used in the region of unknown exotic nuclei, because it is where its choice is of crucial importance. As an example, its influence on the position of the two-neutron drip-line is clearly seen on the right panel of Fig. 1. While the usual DDDI predicts it to be located at N = 44, the stability against two-neutron emission extends up to N = 50 when the microscopic forces are used. Note that a difference of two mass units in the predicted position of the drip-line for such light semi-magic nuclei can translate into a difference of ten mass units for lead isotopes [16].
Average neutron and proton pairing gaps calculated with the three pairing forces are plotted in Fig 2 along the Ca isotopic chain 8 . We see non-trivial differences between the predictions of the phenomenological DDDI and of the microscopic forces. We refer to Ref. [22] for a detailed discussion. Here we simply stress the different isovector trend of those predictions. While the magnitude of the neutron gaps are of the same order for nuclei around the stability line, the phenomenological DDDI provides much too strong pairing in neutron rich nuclei as anticipated in Ref. [15]. The overshoot of simple DI by usual DDDI near the drip-line was also identified [16] and often thought to be a result of the surface-peaked character of the latter 9 . In fact, in both cases, the primary cause of the overshoot is the improper isovector character of usual surface-enhanced DDDI associated with their dependence on the total (isoscalar) density 10 . Indeed, such a DDDI, when adjusted on nuclei having very similar neutron and proton densities, will provide stronger (weaker) pairing in a region of neutron rich matter than a force independent of the density or depending on the neutron (proton) density. This situation is also illustrated in the right panel of Fig. 2. While the LN prescription is responsible for the non-zero value of the proton gap at Z = 20, the latter should not evolve much with neutron number. However, < ∆ p (A) > presents an artificial slope when using the DDDI whose origin goes precisely along the line of the previous argument. When repeating the calculation without LN, the proton gap even switches on artificially for N ≤ 24 in the 8 Values appear for odd and even particle numbers. For odd particle numbers, no blocking was considered here as such HFB states constitute the proper reference on top of which the blocking as to be eventually performed to describe the final odd state [31]. The average gaps are not compared with experiment because we do not want to discuss here how they are related to odd-even mass differences [32]. 9 The way each interaction couples to the continuum also plays a role. Usually, pairing correlations are diminished by a strong coupling to the continuum. The way the continuum is treated numerically also influences the results near the drip-line [33]. 10 It is proved here not to be related to the range of the interaction. The gaps obtained with the finite range force do not differ significantly from those obtained with its zero-range approximation. The results obtained with the three pairing forces used in the present paper are compared to the results derived from the realistic AV18 NN force [12]. To make the theoretical comparison clear, all gaps are calculated with free kinetic energies as single-particle energies.
case of the DDDI. In order to confirm that interpretation, the BCS neutron gap at the Fermi surface in symmetric nuclear matter and in pure neutron matter is plotted in Fig. 3 as a function of the neutron Fermi momentum k n F . Because we use free kinetic energies as single-particle energies, ∆ n (k n F ) obtained from AV18 or from the density-dependent microscopic forces are the same in symmetric matter and in pure neutron matter. On the other end, while the DDDI adjusted on stable nuclei reproduces rather well the gap in symmetric matter, it strongly overshoots it in neutron rich matter. This is due to its dependence on the total density and cannot be related here to any surface effect. Furthemore, beyond mean-field effects in the bulk one could eventually incorporate will always lead to a decrease of ∆ n (k n F ) in infinite neutron matter [34]. As a last point, we discuss the low-density behavior of the pairing force. It was shown in Ref. [15] that the intensity of the effective microscopic force strongly rises at low density due to the very large scattering length of the NN interaction in the 1 S 0 channel. The influence of this strong attraction can be seen in the right panel of Fig. 1 where nuclei are stabilized beyond the sub-shell N = 40 and the two-neutron drip-line pushed back by ten mass units. Indeed, the gaps obtained from the microscopic force resist their otherwise decreasing trend near the drip-line because of the increasing importance of low densities. Such an effect is not seen for the (still too strong) gaps calculated from the DDDI. Strong low-density dependence of DDDI were simulated phenomenologically in Ref. [6] and shown to bring about pathologies. This highlights the fact that such a density dependence should be used in connection with the corresponding microscopically derived cut-off.
CONCLUSIONS
We presented results of the first (3D) HFB calculations performed in finite nuclei using a recently proposed microscopic effective pairing interaction. The isoscalar and isovector density-dependences derived ab-initio provide the pairing force with a strong predictive power when extrapolated toward the drip-lines. We concentrated here on that aspect by studying basic pairing properties of calcium isotopes in their ground-state. By comparing the results with those obtained from a standard Density-Dependent Delta Interaction, the crucial isovector character of the microscopic interaction was highlighted.
In the near future, more systematic HFB calculations will be presented to identify the role of the bare NN force in building pairing in finite nuclei. Through comparisons with experiment, an indirect measure of missing correlations in the p-p channel will be realized. Local theories of pairing will be challenged by probing the importance of the finite range of the force, especially when describing low-energy excited states. Later, the use of the microscopic force will be extended to beyond mean-field calculations.
, while α = √ 0.52 f m and λ 1 S 0 = −840 MeV. f m 3 denote the range and the intensity of the force, respectively. No further adjustment is to be made in finite nuclei. The functional f [ρ q ( r )]
FIGURE 1 .
1Left panel: one-neutron separation energy S N (A) = E(N − 1, Z) − E(N, Z) in Ca isotopes. Right Panel: two-neutron separation energy S 2N (A) = E(N − 2, Z) − E(N, Z) for the same nuclei. Experimental [29] (diamonds) and theoretical values for three different pairing forces are displayed.
FIGURE 2 .FIGURE 3 .
23Neutron (left panel) and proton (right panel ) average pairing gaps < ∆ q (A) >= ∑ n ∆ Neutron pairing gap at the (neutron) Fermi surface as a function of the neutron density. Left panel: symmetric nuclear matter. Right panel: pure neutron matter.
The force acting only in the relative S-wave, the projection on the spin singlet corresponds to a simultaneous projection on the isospin triplet. The T = 1 neutron-proton pairing (T z = 0 component) is not considered here.7 Although the effective vertex breaks total-momentum and angular-momentum conservation of the interacting pair, as well as isospin symmetry, it does so in such a way that the energy functional itself remains invariant under rotation in isospin and real spaces, as well as under translation[22].
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| [] |
[
"HOCHSCHILD COHOMOLOGY, MONOID OBJECTS AND MONOIDAL CATEGORIES",
"HOCHSCHILD COHOMOLOGY, MONOID OBJECTS AND MONOIDAL CATEGORIES"
] | [
"Magnus Hellstrøm-Finnsen "
] | [] | [] | This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in[1]. This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided. | 10.1007/s41980-020-00443-0 | [
"https://arxiv.org/pdf/2201.09401v1.pdf"
] | 225,278,418 | 2201.09401 | b20756e930dce1666b0ee5dd8b393576051c17a8 |
HOCHSCHILD COHOMOLOGY, MONOID OBJECTS AND MONOIDAL CATEGORIES
23 Jan 2022
Magnus Hellstrøm-Finnsen
HOCHSCHILD COHOMOLOGY, MONOID OBJECTS AND MONOIDAL CATEGORIES
23 Jan 2022
This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in[1]. This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided.
Introduction
Hochschild cohomology was initially studied by G. Hochschild in [2] and [3], and provides a cohomology theory for associative algebras. In [4], M. Gerstenhaber discovered that the cohomology ring has a rich structure, which later has been called a Gerstenhaber algebra. The rich structure provides interesting implications, not only restricted to mathematics, but also to physics and related fields.
At Isfahan School & Conference on Representations of Algebras (ISCRA), April 2019, I reported from [1]. This article gives a description of Hochschild cohomology in terms of monoid objects ("ring like objects") in Ab-enriched monoidal categories. Monoidal categories were independently discovered by J. Bénabou and S. Maclane in the beginning of the 1960's (see [5], [6] and [7]), and they provided an axiomatic system to describe the categories with tensor product, like modules over a ring R with tensor product over R i.e. ⊗ R , or abelian groups with the tensor product over the integers, i.e. ⊗ Z , etc.
In this paper, we will first improve the construction of the Hochschild cochain complex given in [1], by taking a more intuitive (and perhaps less combinatorial) approach to this complex. Thereafter, we approach Hochschild cohomology by a cosimplicial object in the category of abelian groups. We will discover that these formulations are equivalent.
Monoidal categories, monoid objects and module objects
First, we recall the definition of a monoidal category. Definition 1.1. A category K is said to be a monoidal category if it is equipped with a bifunctor ⊗ : K × K → K , called the tensor product or monoidal product, and an object 1 ∈ K , called the tensor unit or monoidal unit, together with three natural isomorphisms:
• The associator, α : (? ⊗?) ⊗? ⇀? ⊗(? ⊗?), which has components:
α X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗(Y ⊗ Z),
for all objects X, Y and Z in K . • The left unitor, λ : 1 ⊗? ⇀?, which has components: λ X : 1 ⊗ X → X, for every object X in K . • The right unitor, ρ :? ⊗ 1 ⇀?, which has components:
ρ X : X ⊗ 1 → X,
for every object X in K .
Such that the pentagon diagram:
(W ⊗ X) ⊗(Y ⊗ Z) ((W ⊗ X) ⊗ Y ) ⊗ Z W ⊗(X ⊗(Y ⊗ Z)), (W ⊗(X ⊗ Y )) ⊗ Z W ⊗((X ⊗ Y ) ⊗ Z) α W ⊗ X ,Y ,Z α W ,X ,Y ⊗ Z αW,X,Y ⊗ 1Z αW,X ⊗ Y,Z 1W ⊗ αX,Y,Z
where W , X, Y and Z are arbitrary objects in K , and the triangle diagram:
(X ⊗ 1) ⊗ Y X ⊗(1 ⊗ Y ) X ⊗ Y, αX,I,Y ρX ⊗ 1Y 1X ⊗ λY
where X and Y are arbitrary objects in K , commute. This category is denoted (K , ⊗, 1, α, λ, ρ).
Remark 1.2. We recall some facts about natural coherence in monoidal categories from [8,Section VII.2]. This was originally proposed by G. M. Kelly and S. MacLane (see [7] and [9]). Natural coherence asserts that every formal diagram involving instances (of compositions) of the natural isomorphisms (α, λ and ρ, possibly tensored with suitable identities) commutes. MacLane's coherence theorem in [7] can equivalently be stated as: Each monoidal category is monoidally equivalent to a strict one. A monoidal category is strict whenever the natural isomorphisms α, λ and ρ are identities. For a strict monoidal category, the construction of the Hochschild cochain complex (in Definition 2.5) is immediate. The originality of the present paper is not to rely on this theorem. How a monoidal category can be turned into a strict one is elaborated upon in [10].
The first formulation of natural coherence above (in Remark 1.2) will be important in the formulation of the Hochschild cochain complex in Section 2. In particular, it is important in the proof that this construction actually is a cochain complex (see [1,Theorem 3.2]).
The next objective is to establish the notion of "ring like objects" in a monoidal category, that will be utilized in this paper. Furthermore, we will also establish an appropriate notion of arrows between these objects. Definition 1.3. Let (K , ⊗, 1, α, λ, ρ) be a monoidal category. A monoid object is an object M in K equipped with a multiplication rule µ M : M ⊗ M → M and a multiplicative unit η M : 1 → M . These morphisms satisfy the following relations:
• The associative relation: the multiplication rule is associative in the sense that the following diagram commutes:
(M ⊗ M ) ⊗ M M ⊗(M ⊗ M ) M ⊗ M M ⊗ M M. αM,M,M µM ⊗ 1R µM 1M ⊗ µM µM
• The unitarity relation: the multiplication rule admits a left unit and a right unit in the sense that the following diagram commutes:
M 1 ⊗ M M ⊗ M M ⊗ 1 M M. λ −1 M ηM ⊗ 1M µM 1M ρ −1 M 1M ⊗ ηM 1M
We denote a monoid object as a triple (M, µ M , η M ), and often the subscripts are skipped when the monoid object is not changed.
The next objective is to define an appropriate notion of arrows between monoid objects in a monoidal category K , and observe that monoid objects and arrows of monoid objects form a subcategory of K . Over a "ring-like" object M in K , there are notions of left, right and bi "module-like" objects in K . The next objective is to define these. Definition 1.5. Let (K , ⊗, 1, α, λ, ρ) be a monoidal category and let (M, µ, η) be a monoid object in K . A right module object over M is an object X in K equipped with a right action on X from M , which is a morphism ω : X ⊗ M → X in K , that makes the following two diagrams commute:
(X ⊗ M ) ⊗ M X ⊗(M ⊗ M ) X ⊗ M X ⊗ M X αX,M,M ω ⊗ 1M ω 1X ⊗ µ ω and (N ⊗ X) ⊗ M N ⊗(X ⊗ M ) X ⊗ M N ⊗ X X. αN,X,M ν ⊗ 1M ω 1N ⊗ ω ν
We often denote bimodule objects by a triple:
( N X M , ν N X , ω M X ) = (X, ν, ω).
When N = M , we simply say that (X, ν, ω) is an M -bimodule object.
In the classical case of S-R-bimodules over two rings R and S, a morphism of S-R-bimodules is simply a morphism S-R-modules. Next, we observe that a similar result holds for bimodule objects in a monoidal category, i.e. a morphism of left and right module objects respects the bimodule object structure.
Proposition 1.8. Let (K , ⊗, 1, α, λ, ρ) be a monoidal category, let (M, µ M , η M ) and (N, µ N , η N ) be two monoid objects in K , and let ( N X M , ν N X , ω M X ) = (X, ν X , ω X ) and ( N Y M , ν N Y , ω M Y ) = (Y, ν Y , ω Y ) be N -M -bimodule objects. A morphism of N -M -module objects f : (X, ν X , ω X ) → (Y, ν Y , ω Y ) (i.e.
f respects the left action and the right action) does also respect the bimodule object structure.
Proof. Consider the following diagram:
(N ⊗ X) ⊗ M (N ⊗ Y ) ⊗ M N ⊗(X ⊗ M ) N ⊗(Y ⊗ M ) X ⊗ M Y ⊗ M N ⊗ X N ⊗ Y X Y X Y. α N ,X ,M νX ⊗ 1M ωX 1N ⊗ ωX α N ,Y ,M νY ⊗ 1M ωY 1N ⊗ ωY νY (1N ⊗ f ) ⊗ 1M f f νX 1N ⊗(f ⊗ 1M )
The top square commutes since α is a natural transformation. The back part of the diagram (the back "hexagon") commutes since f is a morphism of N -M module objects. The front part/hexagon commutes by the same reason. The left side of the diagram commutes since X is an N -M -bimodule. Similarly, the right part commutes since Y is an N -M -bimodule. The bottom square commutes by composition with identities. Hence, the diagram commutes, which proves the proposition.
A consequence of Proposition 1.8 is that no further assumptions on the morphisms of bimodule objects are needed, than those already given by the module objects. The definition of the category of bimodule objects follows next.
Tuples of a monoid object, the Hochschild cochain complex and Hochschild cohomology
In this section our monoidal category K will be Ab-enriched, which we define next.
Definition 2.1. An arbitrary category K is said to be Ab-enriched if the hom-sets are abelian groups and the composition is bilinear over the integers.
This means that K is enriched over the category of abelian groups. However, it is not assumed that K is additive, since we do not assume that our category has finite biproducts.
We recall the basic definitions of a cochain complex and cohomology.
Definition 2.2. Let K be an Ab-enriched category (or an other category with a zero object). A (Z-graded) cochain complex in K is a sequence of objects and arrows
(C • , d • ) : . . . d −2 −−→ C −1 d −1 −−→ C 0 d 0 − → C 1 d 1 − → · · · ,
such that two adjacent arrows compose to zero,
d k • d k−1 = 0 for all k ∈ Z. A morphism of chain complexes f : (A • , d • A ) → (B • , d • B )
is a degree wise collection of morphisms in K , f k : A k → B k for all k ∈ Z, such that all squares in the following diagram commute:
· · · A k−1 A k A k+1 · · · · · · B k−1 B k B k+1 · · · . d k−1 A d k A d k−1 B d k B f k−1 f k f k+1
The category of cochain complexes over K is denoted by coCh(K ).
Note, with this definition, a chain complex is a "cochain complex" where the Z-grading is reversed, or equivalently, a cochain complex in K op . Definition 2.3. Let K be an abelian category. The k-th cohomology group of the cochain complex (C
• , d • ) is defined to be ker(d k )/ im(d k−1 ).
The next objective is to define the Hochschild cochain complex for a monoid object M in an Ab-enriched monoidal category with values in a bimodule object X over M . As stated in the Introduction, we will use a slightly different method, than that used in the construction of this cochain complex given in [1]. In particular, we will deal with the associators slightly differently, and perhaps more intuitively, in order to get a less complicated description of the differentials. The method of construction of the cochain complex in the present paper is more motivated by the basic ideas behind Hochschild cohomology, while the operations used in [1] were more "tensor combinatorially" motivated. Hence, we will now discuss these ideas carefully to find a more intuitive formulation, but first we agree on some conventions. Convention 2.4. Let M be a monoid object in a monoidal category K . We denote the k-tuple tensor product by
M ⊗ k = (· · · ((M ⊗ M ) ⊗ M ) · · · ) ⊗ M,
where M occurs k times and the parenthesis are arranged to the left side, i.e. all the left parenthesis are grouped together.
Recall the basic idea of the formulation of the Hochschild cochain complex given in [1, Definition 2.1]. The objects in the cochain complex here are given by
C k = 0
for k < 0
Hom K (1, X) for k = 0 Hom K (M ⊗ k , X) for k ≥ 1,
for a monoidal category K , a monoid object M in K and a bimodule object X over M .
The differential, d k , provides a morphism d k (f ) : M ⊗(k+1) → X, for every morphism f : M ⊗ k → X in K . The differential is given by an alternating sum. In this sum, we will later distinguish between what is referred to as inner and outer summands.
For the inner summands, we will apply the multiplication rule µ on a pair of adjacent objects in the tuple M ⊗(k+1) , before we apply f on the "remaining" M ⊗ k . Therefore, we need a procedure to isolate a pair of objects from the tuple M ⊗ k and a procedure to "rearrange" it back (when we have one isolated object). There are two outer summands. For the first one, we will isolate and apply f on the last k occurrences of M in the tuple M ⊗(k+1) , then we apply the left action from the remaining monoid object M on the bimodule object X. The second outer summand is dual to the first, we apply f to the first k occurrences of M in the tuple M ⊗(k+1) , and then we apply the right action on X from M . For the first of the outer summands, we need a procedure to isolate a single object at the beginning of a tuple. Such prosedure is not needed for the second outer summand, since a singe object is already isolated on the right in M ⊗(k+1) (see Convention 2.4). As mentioned, a rather combinatorial, but nevertheless a general procedure to isolate a tuple with i objects in a tuple with k objects in position j by the operation α i,j k is described in [1, 2.4]. In this paper, we describe a different procedure to isolate an adjacent pair of objects, which will be useful for the inner summands. First, we observe the following
M ⊗ k = M ⊗(k−1) ⊗ M = (M ⊗(k−2) ⊗ M ) ⊗ M = ((M ⊗(k−3) ⊗ M ) ⊗ M ) ⊗ M = · · · = = (· · · (M ⊗ 2 ⊗ M ) ⊗ M · · · ) ⊗ M,
and we can apply µ to the isolated pair in the front of this tuple by applying
M ⊗ k = (· · · (M ⊗ 2 ⊗ M ) ⊗ M · · · ) ⊗ M (···(µ ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−− → (· · · (M ⊗ M ) ⊗ M · · · ) ⊗ M = M ⊗(k−1) .
To isolate a pair of adjacent objects at other places in the tuple, than the pair already isolated in the front, we use the associative relation as follows:
M ⊗ k (···(α M ⊗ i ,M,M ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M −−−−−−−−−−−−−−−−−−−−−−−→ (· · · (M ⊗ i ⊗ M ⊗ 2 ) ⊗ M · · · ) ⊗ M.
We apply µ to the isolated pair on the right side of the expression above
(· · · (M ⊗ i ⊗ M ⊗ 2 ) ⊗ M · · · ) ⊗ M (···(1 M ⊗ i ⊗ µ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−− → (· · · (M ⊗ i ⊗ M ) ⊗ M · · · ) ⊗ M = M ⊗(k−1) .
This procedure (in contrast to that in [1]) has the advantage of that we get M ⊗(k−1) directly on the right hand side, and there is no need to "rearrange" the parentheses back again after applying µ. For the first outer summand, we need to isolate one object in the beginning of a tuple. We do this by a composition of associators (tensored with the unit) of the form
M ⊗ k (···(αM,M,M ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M −−−−−−−−−−−−−−−−−−−−−→ (· · · (M ⊗ M ⊗ 2 ) ⊗ M · · · ) ⊗ M (···(α M,M ⊗ 2 ,M ) ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−−−−−−− → (· · · (M ⊗ M ⊗ 3 ) ⊗ M · · · ) ⊗ M (···(α M,M ⊗ 3 ,M ) ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−−−−−−− → · · · α M,M ⊗(k−2) ,M − −−−−−−−−− → M ⊗ M ⊗(k−1) .
We denote this composition by
α 1 M ⊗ k := α M,M ⊗(k−2) ,M • · · · • (· · · (α M,M ⊗ 2 ,M ) ⊗ 1 M ) ⊗ 1 M · · · ) ⊗ 1 M • (· · · (α M,M,M ⊗ 1 M ) ⊗ 1 M · · · ) ⊗ 1 M : M ⊗ k → M ⊗ M ⊗(k−1) .
It should be noted that any procedure to get a specific reconfiguration of a tuple is equivalent, since natural coherence in monoidal categories (Remark 1.2) provides that any diagram constructed to determine the relationship between such procedures commutes. Now, we can define the differential in the Hochschild cochain complex.
Definition 2.5. Let (K , ⊗, 1, α, λ, ρ) be an Ab-enriched monoidal category, let (M, µ, η) be a monoid object in K and let (X, ν, ω) be a bimodule object over M . We define the Hochschild cochain complex (C • , d • ) to have objects C k (M ; X) as given above, i.e.
C k (M ; X) = 0 for k < 0 Hom K (1, X) for k = 0
Hom K (M ⊗ k , X) for k > 0, and the differentials,
d k : C k (M ; X) → C k+1 (M ; X),
are defined as:
• For k < 0: d k = 0.
• For k = 0: Let f ∈ C 0 (M ; X) = Hom K (1, X). The differential, d 0 :
Hom K (1, X) → Hom K (M, X)
, evaluated on f is defined to be the sum of the compositions of:
M λ −1 M −−→ 1 ⊗ M 1M ⊗ f −−−−→ X ⊗ M ω − → X − M ρ −1 M −−→ M ⊗ 1 f ⊗ 1M −−−−→ M ⊗ X ν − → X.
• For k > 0: We distinguish between the inner and outer summands, as discussed above. Denote the summands by χ i . Let f ∈ C k (M ; X) = Hom K (M ⊗ k , X). The differential, d k :
Hom K (M ⊗ k , X) → Hom K (M ⊗(k+1) , X)
, evaluated on f is defined to be the alternating sum, k i=0 (−1) i χ i , where the χ i 's are defined as follows: For i = 0, we get the first outer summand χ 0 , which is defined to be the composition of
M ⊗(k+1) α 1 M ⊗(k+1) −−−−−−→ M ⊗ M ⊗ k 1M ⊗ f −−−−→ M ⊗ X ν − → X.
For i = k, we get the other outer summand χ k , which is defined to be the composition of
M ⊗(k+1) = M ⊗ k ⊗ M f ⊗ 1M −−−−→ X ⊗ M ω − → X.
For 0 < i < k, we get the inner summands χ i , which are defined to be the compositions of the form
M ⊗(k+1) (···(α M ⊗ i ,M,M ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M −−−−−−−−−−−−−−−−−−−−−−−→ (· · · (M ⊗ i ⊗ M ⊗ 2 ) ⊗ M · · · ) ⊗ M (···(1 M ⊗ i ⊗ µ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−− → M ⊗ k f − → X.
The formulation of the Hochschild cochain complex is a coherently equivalent description to that in [1, Definition 3.1]. Hence, [1,Theorem 3.2] confirms that (C • (M ; X), d • ) is, in fact, a cochain complex, and its cohomology groups are well-defined.
Definition 2.6. The Hochschild cohomology groups are defined to be the cohomology of the cochain complex (C • (M ; X), d • ) given in Definition 2.5, i.e.
HH i (M ; X) = ker(d i )/ im(d i−1 ).
Some of the classical results for the lower dimensional Hochschild cohomology groups are also proved in [1].
Here, HH 0 (M ; M ) = HH 0 (M ) is used to define some notion of "quasi centre" of M . Similarly, we can define the quasi centre, Z(X), for a bimodule object X to be:
Z(X) := HH 0 (M ; X) ∼ = ker d 0 .
With this formulation of Hochschild cohomology, it is proved that the Hochschild cohomology ring
Cosimplicial description of the Hochschild cochain complex
The objective for this section is to formulate the Hochschild cochain complex of a monoid object M in a monoidal category K with values in a bimodule X in terms of a cosimplicial object in Ab. We argue that this formulation is equivalent to that which was given in the previous section (Section 2). In [11,Section 9.1], Hochschild cohomology for a k-algebra is introduced as a cosimplicial object in the vector space over k (k is a field). with the ordering 0 ≤ 1 ≤ · · · ≤ k. This category is called the simplex category or the ∆-category. Let K be an arbitrary category. A cosimplicial object A in K is a functor A : ∆ → K . We write A([n]) = A n .
We should first note that A, in the definition above, is often referred to as a K -valued cosimplicial object. It should also be noted that a K -valued simplicial object B is a functor B : ∆ op → K .
The category ∆ can be described several ways. Since we try to explore category theoretical properties of the Hochschild cochain complex, we should perhaps describe the ∆-category categorically.
Remark 3.2. The ∆-category can equivalently be defined as the category, where the objects are free categories on linear single arrowed and directed graphs/quivers and the morphisms are functors.
We recall the face and degeneracy maps in ∆ (see [11,Section 8.1]).
Definition 3.3. For each [k] ∈ ∆, let ǫ i : [k − 1] → [k]
denote the (i-th) face map, that is the unique injective map that misses i:
ǫ i (j) = j if j < i j + 1 if j ≥ i.
Dually, for each [k] ∈ ∆, let ζ i : [k + 1] → [k] denote the (i-th) degeneracy map, that is the unique surjective map that sends two elements to i:
ζ i (j) = j if j ≤ i j − 1 if j > i.
Face and degeneracy maps satisfies certain identities (see [11,Exercise 8.1.1]). Let A be a cosimplicial object in an arbitrary category K , then the face maps and degeneracy maps generalise to coface operations and codegeneracy operations on A, respectively. We recall [11,Corollary 8.1.4], which states how we can describe cosimplicial objects.
Proposition 3.4. To describe a cosimplicial object A in an arbitrary category K , it is sufficient and necessary to give a sequence of objects A 0 , A 1 , . . . together with coface operations
δ i : A k−1 → A k
and codegeneracy operations
σ i : A k+1 → A k ,
such that the following identities are satisfied:
δ j δ i = δ i δ j−1 if i < j, σ j σ i = σ i σ j+1 if i ≤ j, σ j δ i = δ i σ j−1 if i < j 1 if i = j or i = j + 1 σ i−1 δ j if i > j + 1.
Definition 3.5. We will refer to the identities given in Proposition 3.4 as the cosimplicial identities.
The objective now is to construct a cosimplicial object in Ab, which we later will reformulate to the Hochschild cochain complex (in Theorem 3.8).
Definition 3.6. Let (K , ⊗, 1, α, λ, ρ) be an Ab-enriched monoidal category, (M, µ, η) a monoid object in K and (X, ν, ω) a bimodule object over M . The object A in Ab is constructed in the following way. To each
[k] ∈ ∆ we assign:
[k] → A 0 = Hom K (1, X) for k = 0 A k = Hom K (M ⊗ k , X) for k > 0.
The coface operations are given by: For k = 0, let f ∈ A 0 = Hom K (1, X). Then δ i : A 0 → A 1 , for i ∈ {0, 1}, evaluated on f , is defined as the composition of:
δ 0 (f ) = M λ −1 M −−→ 1 ⊗ M f ⊗ 1M −−−−→ X ⊗ M ω − → X δ 1 (f ) = M ρ −1 M −−→ M ⊗ 1 1M ⊗ f −−−−→ M ⊗ X ν − → X.
For k > 0, let f ∈ A k = Hom K (M ⊗ k , X). Then δ i : A k → A k+1 , for i ∈ {0, 1, . . . , k}, evaluated on f , is defined as the composition of:
δ 0 (f ) = M ⊗(k+1) α 1 M ⊗(k+1) −−−−−−→ M ⊗ M ⊗ k 1M ⊗ f −−−−→ M ⊗ X ν − → X δ i (f ) = M ⊗(k+1) (···(α M ⊗ i ,M,M ⊗ 1M ) ⊗ 1M ··· ) ⊗ 1M −−−−−−−−−−−−−−−−−−−−−−−→ (· · · (M ⊗ i ⊗ M ⊗ 2 ) ⊗ M · · · ) ⊗ M (···(1 M ⊗ i ⊗ µ) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−− → M ⊗ k f − → X for 0 < i < k δ k (f ) = M ⊗(k+1) = M ⊗ k ⊗ M f ⊗ 1M −−−−→ X ⊗ M ω − → X.
The codegeneracy operations are given by: For k = 0, let f ∈ A 1 = Hom K (M, X). Then σ 0 : A 1 → A 0 , evaluated on f , is defined as the composition of:
σ 0 (f ) = 1 η − → M f − → X.
For k > 0, let f ∈ A k+1 = Hom K (M ⊗(k+1) , X). Then σ i : A k+1 → A k , for i ∈ {0, 1, . . . , k}, evaluated on f , is defined as the composition of: We refer to this construction, (A, δ, σ), as the Hochschild cosimplicial object.
In the definition of the codegeneracy maps for 0 < i < k (in the definition of the Hochschild cosimplicial object above), we should observe that the inverse of the procedure to isolate a pair 1 is now used to rearrange the parenthesis back to M ⊗(k+1) .
Theorem 3.7. The Hochschild cosimplicial object is, in fact, a cosimplicial object in Ab.
Proof. This is a straightforward verification of the identities given in Proposition 3.4. We will only show δ 1 δ 0 = δ 0 δ 0 for Hom K (1, X) − → Hom K (M, X) − → Hom K (M ⊗ 2 , X).
non-negative k and 0 otherwise, i.e.
d k = 0 if k < 0 δ 0 − δ 1 if k = 0 k+1 i=0 (−1) i δ i if k > 0.
Moreover, this cochain complex equals the Hochschild cochain complex defined in Definition 2.5.
Definition 1 . 4 .
14Let K be a monoidal category, and let (M, µ M , η M ) and (N, µ n , η N ) be two monoid objects inK . A morphism of monoid objects f : (M, µ M , η M ) → (N, µ N , η N )is a morphism in K such that multiplication is preserved, in the sense that the following diagram commutes: and the units are preserved, in the sense that the following diagram commutes: and morphisms of monoid objects form a category, denote this category by MonObj(K ).
Definition 1 . 9 .
19Let (K , ⊗, 1, α, λ, ρ) be a monoidal category and let (M, µ M , η M ) be a monoid object in K . The full subcategory of leftModObj(M ) and rightModObj(M ) generated by bimodule objects over M is called the category of bimodule objects over M , and it is denoted by biModObj(M ).
HH i (M ) := HH i (M ; M )) is graded commutative with the cup product (see [1, Theorem 5.5]).
Definition 3. 1 .
1Let ∆ denote category of non-empty finite ordinals and order preserving maps. The non-empty finite ordinals are denoted by [k] = {0, 1, . . . , k}
MM
⊗ k (···((λ −1 M ⊗ 1M ) ⊗ 1M )··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−−− → (· · · ((1 ⊗ M ) ⊗ M ) ⊗ M · · · ) ⊗ M (···(((η ⊗ 1M ) ⊗ 1M ) ⊗ 1M )··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−−−−−− → M ⊗(k+1) ⊗ k (···(1 M ⊗ i ⊗ λ −1 M ) ⊗ 1M ··· ) ⊗ 1M −−−−−−−−−−−−−−−−−−−−→ (· · · (M ⊗ i ⊗(1 ⊗ M )) ⊗ M · · · ) ⊗ M (···(1 M ⊗ i ⊗(η ⊗ 1M )) ⊗ 1M ··· ) ⊗ 1M − −−−−−−−−−−−−−−−−−−−−−− → (· · · (M ⊗ i ⊗(M ⊗ M )) ⊗ M · · · ) ⊗ M
This procedure to isolate a pair of objects in a tuple was described in Section 2.
Acknowledgements. I would like to thank the organisers of ISCRA, Professor Javad Asadollahi and his crew for a wonderful convention in Isfahan, Iran, April 2019. It was a true pleasure and I am grateful to be given the opportunity to speak at the conference. I thank Professor Aslak Bakke Buan and my former employer the Department of Mathematical Sciences at NTNU for providing funding in this occasion. I am further thankful for the invitation to submit this work to the Bulletin of the Iranian Mathematical Society. I thank the two anonymous reviewers for valuable comments and suggestions, and Professor Majid Soleimani-damaneh, Editor in Chief, Bulletin of the Iranian Mathematical Society, for the correspondence. Finally, I thank Eirik Hellstrøm Finnsen for his time and competence concerning this article.
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S Mac Lane, Natural associativity and commutativity. 49S. Mac Lane, "Natural associativity and commutativity," The Rice University Studies, vol. 49, no. 4, pp. 28-46, 1963.
Categories for the working mathematician. S Mac Lane, Graduate Texts in Mathematics. 5Springer-Verlagsecond edition ed.S. Mac Lane, Categories for the working mathematician, vol. 5 of Graduate Texts in Mathematics. New York: Springer-Verlag, second edition ed., 1998.
On MacLane's conditions for coherence of natural associativities, commutativities, etc. G M Kelly, J. Algebra. 1G. M. Kelly, "On MacLane's conditions for coherence of natural associativities, commutativities, etc," J. Algebra, vol. 1, pp. 397-402, 1964.
Turning monoidal categories into strict ones. P Schauenburg, New York J. Math. 7P. Schauenburg, "Turning monoidal categories into strict ones," New York J. Math., vol. 7, pp. 257-265, 2001.
An introduction to homological algebra. C A Weibel, of Cambridge Studies in Advanced Mathematics. CambridgeCambridge University Press38C. A. Weibel, An introduction to homological algebra, vol. 38 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1994.
NO-1757 Halden, Norway. Email address: mhellstroemfinnsen@gmail. Magnus Hellstrøm-Finnsen, Postboks. 700Avdeling for ingeniørfag, Høgskolen i ØstfoldMagnus Hellstrøm-Finnsen, Avdeling for ingeniørfag, Høgskolen i Østfold, Postboks 700, NO-1757 Halden, Norway. Email address: [email protected]
| [] |
[
"STRONG RATIONAL CONNECTEDNESS OF TORIC VARIETIES",
"STRONG RATIONAL CONNECTEDNESS OF TORIC VARIETIES"
] | [
"Yifei Chen ",
"Vyacheslav Shokurov "
] | [] | [] | In this paper, we prove that: For any given finitely many distinct points P1, . . . , Pr and a closed subvariety S of codimension ≥ 2 in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there exists a rational curve f : P 1 → X passing through P1, . . . , Pr, disjoint from S \ {P1, . . . , Pr} (see Main Theorem). As a corollary, we prove that the smooth loci of complete toric varieties are strongly rationally connected. | 10.4310/mrl.2011.v18.n6.a12 | [
"https://arxiv.org/pdf/0905.1430v1.pdf"
] | 15,535,389 | 0905.1430 | 624e061585e69139ded69806727b28b3d255d5dd |
STRONG RATIONAL CONNECTEDNESS OF TORIC VARIETIES
9 May 2009
Yifei Chen
Vyacheslav Shokurov
STRONG RATIONAL CONNECTEDNESS OF TORIC VARIETIES
9 May 2009
In this paper, we prove that: For any given finitely many distinct points P1, . . . , Pr and a closed subvariety S of codimension ≥ 2 in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there exists a rational curve f : P 1 → X passing through P1, . . . , Pr, disjoint from S \ {P1, . . . , Pr} (see Main Theorem). As a corollary, we prove that the smooth loci of complete toric varieties are strongly rationally connected.
Introduction
The concept of rationally connected varieties is independently invented by Kollár-Miyaoka-Mori ( [KMM92b]) and Campana ([Ca92]). This kind of variety has interesting arithmetic and geometric properties.
A class of proper rationally connected varieties comes from the smooth Fano varieties ( [Ca92], [KMM92a] or [Kol96]). Shokurov ([Sh00]), Zhang ([Zh06]), Hacon and McKernan ([HM07]) proved that FT (Fano type) varieties are rationally connected.
An interesting question is whether the smooth locus of a rationally connected variety is rationally connected. In general the answer of the question is NO. However, for the FT (or log del Pezzo) surface case, Keel and McKernan gave an affirmative answer, that is, if (S, ∆) is a log del Pezzo surface, then its smooth locus S sm is rationally connected ( [KM99]), but this does not imply the strong rational connectedness.
The concept of strongly rationally connected varieties (see Definition 5) was first introduced by Hassett and Tschinkel ( [HT08]). A proper and smooth separably rationally connected variety X over an algebraically closed field is strongly rationally connected ( [KMM92b] 2.1, or [Kol96] IV.3.9). Xu ([Xu08]) announced that the smooth loci of log del Pezzo surfaces are not only rationally connected but also strongly rationally connected, which confirms a conjecture of Hassett and Tschinkel ([HT08], Conjecture 20). It is expected that the smooth locus of an FT variety is strongly rationally connected (cf. Example 2 and Main Theorem).
Throughout the paper, we are working over an uncountably algebraically closed field of characteristic 0. It is interesting that whether Main Theorem holds for any algebraically closed (or perfect) field.
Main Theorem. Let X be a complete toric variety. Let P 1 , . . . , P r be finitely many distinct points in X (P i possibly singular). Then there is a geometrically free rational curve f : P 1 → X over P i , 1 ≤ i ≤ r (see Definition 7). Moreover, f is free over P i if all points P i are smooth.
Main Theorem can be rephrased as follows: Let X be a complete toric variety. For any given distinct points P 1 , . . . , P r ∈ X (possibly singular) and any given codimension ≥ 2 subvariety S ⊆ X, there is a rational curve f : P 1 → X passing through P 1 , . . . , P r , disjoint from S \ {P 1 , . . . , P r }.
If all points P i are smooth, then we get the following corollary.
Corollary 1. The smooth locus of a complete toric variety is strongly rationally connected.
Preliminaries
When we say that x is a point of a variety X, we mean that x is a closed point in X.
A rational curve is a nonconstant morphism f : P 1 → X. A normal projective variety X is called FT (Fano Type) if there exists an effective Q-divisor D, such that (X, D) is klt and −(K X + D) is ample. See [PSh09] Lemma-Definition 2.6 for other equivalent definitions.
Let N ∼ = Z n be a lattice of rank n. A toric variety X(∆) is associated to a fan ∆, a finite collection of convex cones σ ⊂ N R := N ⊗ Z R (see [Fu93] or [Oda88]).
Example 2. Projective toric varieties are FT. Let K be the canonical divisor of the projective toric variety X(∆), T be the torus of X, and Σ = X \ T = D i be the complement of T in X. Then K is linearly equivalent to −Σ. Since X is projective, there is an ample invariant divisor L. Suppose that L = d i D i . Let the polytope L = {m ∈ M | m, e i +d i ≥ 0, ∀e i ∈ ∆(1)}, where M is the dual lattice of N , and ∆(1) is the set consisting of 1-dimensional cones in ∆. Let u be an element in the interior of L . Let χ u be the corresponding rational function of u ∈ M (see [Fu93] section 1.3), and div χ u be the divisor of χ u . Then D =div χ u + L is effective and ample and has support Σ. That is, D = d ′ i D i and all d ′ i > 0. Let ǫ be a positive rational number, such that all coefficients of prime divisors in ǫD are strictly less than 1. Then Σ − ǫD is effective. It is easy to check that (X, Σ − ǫD) is klt, and −(K + Σ − ǫD) ∼ ǫD is ample. Hence X is FT.
Definition 3. An isogeny of toric varieties is a finite surjective toric morphism. Toric varieties X and Y are said to be isogenous if there exists an isogeny X → Y . The isogeny class of a toric variety X is a set consisting of all toric varieties Y such that X and Y are isogenous.
Theorem 4. Let f : X → Y be a finite surjective toric morphism. Then there exists a finite surjective toric morphism g : Y → X.
Proof. Let f : X → Y be a finite surjective toric morphism of toric varieties and ϕ : (N ′ , ∆ ′ ) → (N, ∆) be the corresponding map of lattices and fans. Then we can identify N ′ as a sublattice of N and ∆ ′ = ∆.
There is an positive integer r such that rN is a sublattice of N ′ . Let g be the corresponding toric morphism of (rN, ∆) → (N ′ , ∆). Since (rN, ∆) and (N, ∆) induce an isomorphic toric variety, we get g : Y → X is a finite surjective toric morphism.
The properties of isogeny: 1) Isogeny is an equivalence relation. 2) If a toric variety Y is in the isogeny class of X and µ : X → Y is the isogeny, then there is a one-to-one correspondence between the set of orbits {O X i } of X and the set of orbits
{O Y i = µ(O X i )} of Y . Hence dim O X i = dim O Y
i for all i, and the number of orbits is independent of the choice of toric varieties in an isogeny class of X.
A variety X over a characteristic 0 field is rationally connected, if any two general points x 1 , x 2 ∈ X can be connected by a rational curve of X of a bounded family.
Definition 5. ([HT08] Definition 14.) A smooth rationally connected variety Y is strongly rationally connected if any of the following conditions hold:
(1) for each point y ∈ Y , there exists a rational curve f : P 1 → Y joining y and a generic point in Y ;
(2) for each point y ∈ Y , there exists a free rational curve containing y;
(3) for any finite collection of points y 1 , . . . , y m ∈ Y , there exists a very free rational curve containing the y j as smooth points;
(4) for any finite collection of jets
Spec k[ǫ]/ ǫ N +1 ⊂ Y, i = 1, . . . , m
supported at distinct points y 1 , . . . , y m , there exists a very free rational curve smooth at y 1 , . . . , y m and containing the prescribed jets.
Definition 6. Let X be a complete normal variety, B be a set of finitely many closed points in P 1 , and g : B → X be a morphism. A rational curve f : P 1 → X is called weakly free over g if there exist an irreducible family of rational curves T and an evaluation morphism ev:
P 1 × T → X such that 1) f = f t 0 = ev| P 1 ×t 0 for some t 0 ∈ T ,
2) for any t ∈ T , f t = ev| P 1 ×t is a rational curve and f t | B = g, 3) the evaluation morphism ev:
P 1 × T → X by ev(x, t) = f t (x) is domi- nant.
We say that a rational curve f ′ : P 1 → X is a general deformation of f , or f ′ is a sufficiently general weakly free rational curve, if there is an open dense subset U of T , such that f ′ = f t and t ∈ U ⊆ T . We say that a weakly free rational curve g : P 1 → X is a general deformation of f , if there is an irreducible family T ′ , such that T ∩ T ′ contains an open dense subset in T , g = g t ′ for some t ′ ∈ T ′ and g is weakly free in its own family.
Definition 7. Let X be a complete normal variety, B be a set of finitely many closed points in P 1 , and g : B → X be a morphism. A rational curve f : P 1 → X is called geometrically free over g if there exist an irreducible family of rational curves T and an evaluation morphism ev:
P 1 × T → X such that 1) f = f t 0 = ev| P 1 ×t 0 for some t 0 ∈ T , 2) for any t ∈ T , f t = ev| P 1 ×t is a rational curve and f t | B = g, 3) for any codimension 2 subvariety Z in X, f t (P 1 ) ∩ Z ⊆ g(B) for general t ∈ T (general meaning t belongs to a dense open subset in T , depending on Z).
If X is smooth over an uncountable field of characteristic 0, then weak freeness over g is equivalent to usual freeness over g if |B| ≤ 2.
Remark. In our application, we usually assume g is one-to-one.
Let P i = g(Q i ) where B = {Q i }.
Without confusion, we say f is geometrically free over {P i } (resp. weakly free over {P i }) instead of saying that f is geometrically free over g (resp. weakly free over g).
Weak freeness and geometric freeness are generalizations of usual freeness (see [Kol96] II.3.1 Definition) if the curve passes through singularities. To consider weakly free rational curves or geometrically free rational curves, we think of them as general members in a certain family. In particular, we can suppose that the morphism ev is flat.
Example 8. Let X be a projective cone over a conic. Let T be the family of all lines through the vertex O. Then l ∈ T is not free. However l is weakly free and geometrically free over O by construction.
We need a resolution as follows.
Theorem 9. Let X be a toric variety. Let Σ be the invariant locus of X. Let P 1 , . . . , P r ∈ X be r points. Let f : P 1 → X be a sufficiently general weakly free rational curve over P 1 , . . . , P r . Then there exists a resolution π :X → X, such that
1) π −1 (Σ ∪ {P i }) is a divisor with simple normal crossing; 2) π −1 (P j ) ⊆ π −1 (Σ ∪ {P i }) is a divisor for each point P j ; 3) π :X → X is an isomorphism over X \ (Sing X ∪ {P i }); 4) sufficiently generalf (P 1 ) intersects π −1 (Σ ∪ {P i }) over each P j only in divisorial points of π −1 (Σ ∪ {P i })
, wheref : P 1 →X is the proper birational transformation of a general deformation of f and is a (weakly) free rational curve.
More generally, let f j : P 1 → X, 1 ≤ j ≤ m be finitely many sufficiently general weakly free rational curve over a subset of {P i }, where {P i } is a set of finitely many distinct points in X. Then there exists a resolution π :X → X such that
1') π −1 (Σ ∪ {P i }) is a divisor with simple normal crossing; 2') π −1 (P i ) ⊆ π −1 (Σ ∪ {P i }) is a divisor for each point P i ; 3') π :X → X is an isomorphism over X \ (Sing X ∪ {P i }); 4') For each j, sufficiently generalf j (P 1 ) intersects π −1 (Σ ∪ {P i }) over each P i only in divisorial points of π −1 (Σ ∪ {P i }), wheref j : P 1 →X is the proper birational transformation of a general deformation of f j and is a (weakly) free rational curve.
Proof. When the ground field is of characteristic 0, 1)-3) follow from usual facts in the resolution theory, e.g. see [KM98] Theorem 0.2. However, in the toric or toroidal case, the same result holds for any field. More precisely, if all P i are invariant, we can use a toric resolution. If some P i are not invariant, they can be converted into toroidal invariant points P i after a toroidalization.
We say thatf (P 1 ) intersects π −1 (Σ ∪ {P i }) over each P i in a divisorial point x if x belongs to only one prime divisor of π −1 (Σ ∪ {P i }) for some i and the prime divisor is over P i . To fulfill 4), we need extra resolution over intersections of the divisorial components of π −1 (Σ ∪ {P i }) through which generalf is passing over P i . Termination of such resolution follows from an estimation by the multiplicities of intersection for f (P 1 ) with Σ. The last resolution is independent of the choice of a general rational curve by Lemma 12 below. However it depends on the choice of intersections of divisorial components. For more details, see the proof of Lemma 4.3.4 in [Ch09].
For the general statement, we can get 1')-3') in a similar manner above. To fulfill 4'), we just need extra resolutions over each point P i .
We discuss some examples of rational curves on projective spaces and quotient projective spaces.
Example 10. For any given subvariety S of codimension ≥ 2 in P n , any points P 1 , . . . , P r ∈ P n , and any integer d ≥ r, there exists a rational curve C of degree d, such that each P i ∈ C and C ∩ S = ∅.
Indeed, we can construct a tree T with r branches, such that each P i is a smooth points on a unique branch and disjoint from S. The tree can be smoothed into a rational curve C passing through P 1 , . . . , P r , disjoint from S. The rational curve C has degree r. For d ≥ r, we can attach d − r rational curves to the tree T , and smooth it.
Applying Example 10, we get Example 11. Let π : P n → X be a finite morphism, S be a codimension ≥ 2 subvariety in X, and {P i } m i=1 be a set of m points outside S. Then there exists a rational curve C, such that each P i ∈ C and C ∩ S = ∅.
In particular, the same result holds if X is a quotient space P n /G, where G is a finite group, for example, if X is a weighted projective space. It is well known that if X is a complete Q-factorial toric variety with Picard number one, then there exist a weighted projective space Y and a finite toric morphism π : Y → X. So the same result holds for rational curves on a complete Q-factorial toric variety with Picard number one. It is a very special case of our Main Theorem.
Proof of Main Theorem
In this section we prove Main Theorem. Let us first prove Main Lemma, which is a special weak case of Main Theorem.
Main Lemma. Let X be a complete toric variety. Let P, Q ∈ X be two distinct points (P, Q possibly singular). Let S ⊆ X be a closed subvariety of codimension ≥ 2. Then there exists a weakly free rational curve on X over P, Q, disjoint from S \ {P, Q}.
To prove Main Lemma, we need some preliminaries.
Lemma 12. Let f be a weakly free rational curve on X, and F 1 , . . . , F s ⊆ X be s proper irreducible subvarieties in X. Then there exist s ′ , 0 ≤ s ′ ≤ s, subvarieties among {F j } (after renumbering we assume they are F 1 , . . . , F s ′ ) such that a general deformation of f intersects F 1 , . . . , F s ′ , and is disjoint from F s ′ +1 , . . . , F s .
The proof of this Lemma is a standard exercise in incidence relations. See [Ch09] Lemma 4.3.2 for a detailed proof.
Lemma 13. Let X be a complete toric variety. Let P, Q ∈ X be two points (possibly singular), and S be a closed subvariety of codimension ≥ 2. Let F 1 , . . . , F s be all the irreducible components of Sing X. Let f : P 1 → X be a sufficiently general weakly free rational curve over P, Q. Suppose f (P 1 ) intersects F 1 \ {P, Q}, . . . , F s ′ \ {P, Q}, and is disjoint from F s ′ +1 \ {P, Q}, . . . , F s \{P, Q}. Then there exists a weakly free rational curve f ′ over {P, Q}, which is a general deformation of f , such that
f ′ (P 1 ) is disjoint from ((S \ Sing X) ∪ F s ′ +1 ∪ . . . ∪ F s ) \ {P, Q}. Moreover, for any fixed closed subvariety Z of X, if f (P 1 )∩(Z \{P, Q}) = ∅, then f ′ (P 1 )∩(Z \{P, Q}) = ∅.
Proof. Applying Theorem 9 to the toric variety X and two points {P, Q}, we get a resolution π :X → X satisfying 1)-3) in the theorem and a weakly free rational curvef : P 1 →X satisfying 4) in the theorem. A general deformationf ′ off is weakly free, sof ′ is free by [Kol96] II.3.11 (Here we need the assumption that the ground field is uncountable and of characteristic 0.) Moreover, we can assume thatf ′ is disjoint from (S \ Sing X) \ π −1 {P, Q} by [Kol96] II.3.7.
On the other hand, let Σ be the invariant locus of X. Notice that Sing X ⊆ Σ. Then by Theorem 9,f (P 1 ) intersects π −1 (Σ ∪ {P, Q}) divisorially over P, Q, andf (P 1 ) is disjoint from the closure of π −1 (F s ′ +1 \ {P, Q}), . . . , π −1 (F s \ {P, Q}). So the general deformationf ′ off intersects open subsets of divisors π −1 (P ) and π −1 (Q), disjoint from the closure of ((S \ Sing X) \ π −1 {P, Q}) ∪ π −1 (F s ′ +1 \ {P, Q}) ∪ · · · ∪ π −1 (F s \ {P, Q}). We apply Lemma 14 by replacing f ′ byf ′ , dominant morphism µ by π :X → X, {P i } by {P, Q}, and S by (S\ Sing X) ∪ F s ′ +1 ∪ · · · ∪ F s . Then we get the weakly free rational curve f ′ = πf ′ : P 1 → X is a general deformation of f (see Definition 6), passing through points P, Q and disjoint from
((S \ Sing X) ∪ F s ′ +1 ∪ · · · ∪ F s ) \ {P, Q}.
Moreover, we can assume that f ′ is a weakly free rational curve over P, Q, by a base change of the family to which f ′ belongs (For details, see the proof of Lemma 4.3.1 in [Ch09]).
The last statement can be proved similarly.
Lemma 14. Let X, X ′ be two complete varieties with dim X > 0. Let µ : X ′ → X be a dominant morphism. Then the image of a weakly free rational curve on X ′ is weakly free on X in the following sense: Let P 1 , P 2 , . . . , P r ∈ µ(X) be r distinct points, and S ⊆ X be a closed subvariety. Let S ′ = µ −1 S, and P ′ 1 , P ′ 2 , . . . , P ′ r ∈ X ′ be points such that µ(P ′ i ) = P i for i = 1, . . . , r. If f ′ : P 1 → X ′ is a weakly free rational curve over P ′ 1 , P ′ 2 , . . . , P ′ r , disjoint from S ′ \ {P ′ 1 , P ′ 2 , . . . , P ′ r }, then f = µ • f ′′ is a weakly free rational curve on X over P 1 , P 2 , . . . , P r , disjoint from S \ {P 1 , P 2 , . . . , P r }, where f ′′ is a general deformation of f ′ .
Proof. Since f ′ is weakly free, ev:
P 1 × T ′ → X ′ is dominant, where T ′ is the family associated to f ′ . Since µ : X ′ → X is dominant, ev: P 1 × T ′ → X ′ → X is dominant. Hence for general deformation f ′′ ∈ T ′ of f ′ , f = µ • f ′′ is a weakly free rational curve on X.
Lemma 15. Let X be a Q-factorial toric variety, and O be a singular orbit of X. Then there exists an isogeny µ :
Y → X, such that µ −1 (O) is smooth.
Proof. Let (N, ∆) be the lattice and fan associated to X. Let N ′ be the sublattice generated by the primitive elements of the simplicial cone σ such that O is contained in the affine open subset σ corresponding to. Let Y be the toric variety corresponding to (N ′ , ∆) and µ be the natural finite dominant morphism corresponding to (N ′ , ∆) → (N, ∆). By construction of µ, µ −1 (O) is smooth.
Proof of Main Lemma.
Step 1. After Q-factorization q : X ′ → X, we can assume that X is a complete Q-factorial toric variety ([Fj03] Corollary 3.6). Indeed, a weakly free rational curve on X ′ gives a weakly free rational curve on X by Lemma 14.
Step 2. A weakly free rational curve can be moved from any smooth variety of codimension ≥ 2 in the sense of Lemma 13. So we can reduce the proof of Main Lemma to the case S = I(X), where I(X) denotes the union of orbits of X of codimension ≥ 2. Since X is a toric variety, Sing X ⊆ I(X).
Indeed, for any subvariety S ⊆ X of codimension ≥ 2, suppose there is a sufficiently general weakly free rational curve f : P 1 → X over P, Q ∈ X, disjoint from I(X) \ {P, Q}. Apply Lemma 13 to the subvariety S, and the weakly free rational curve f . Since Sing X ⊆ I(X), s ′ = 0 in Lemma 13, that is, f (P 1 ) is disjoint from F 1 \ {P, Q}, . . . , F s \ {P, Q}. Then there exists a weakly free rational curve f ′ , which is a general deformation of f ,
such that f ′ (P 1 ) is disjoint from ((S \ Sing X) ∪ F 1 ∪ . . . ∪ F s ) \ {P, Q} = ((S \ Sing X) ∪ Sing X) \ {P, Q} = S \ {P, Q}.
Step 3. Suppose that I(X) consists ofs distinct orbits O 1 , . . . , Os. Let f : P 1 → X be a sufficiently general weakly free rational curve over P, Q. By Lemma 12, we can assume that f (P 1 ) intersects with Notice that s ′ depends on the points P, Q and the variety X. However, since s ′ is bounded bys, ands is independent of choice of X in an isogeny class, there exists ans such that for any toric variety Y in the isogeny class of X, and two distinct points P ′ , Q ′ ∈ Y , there exists a weakly free rational curve f ′ s :
P 1 → Y over P ′ , Q ′ , such that f ′ s (P 1 ) intersects with at most O Y 1 \ {P ′ , Q ′ }, . . . , O Ȳ s \ {P ′ , Q ′ }, and is disjoint from O Ȳ s+1 \ {P, Q}, . . . , O Ỹ s \ {P, Q}, where O Y i are orbits of Y of codimension ≥ 2. Furthermore, we can assume that dim O Y 1 ≥ dim O Y 2 ≥ · · · ≥ dim O Y s ′ ≥ dim O Y s ′ +1 ≥ · · · ≥ dim O Ỹ
s . This order is good for us, because ∪ j≥s O Y j is closed for any s. We fix a complete toric variety X, two points P, Q and a weakly free rational curve fs over P, Q. By Lemma 14 and 15, we can suppose that the orbit Os is smooth. Indeed, by Lemma 15, there is an isogeny µ : Y → X such that O Ȳ s = µ −1 (Os) is smooth. Let P ′ , Q ′ ∈ Y such that µ(P ′ ) = P, µ(Q ′ ) = Q. Then existence of a weakly free rational curve f ′ : P 1 → Y over P ′ , Q ′ , disjoint from O Ȳ s ∪ · · · ∪ O Ỹ s , implies existence of a weakly free rational curve f : P 1 → X over P, Q, disjoint from Os ∪ · · · ∪ Os, by Lemma
14 with X ′ = Y, {P i } = {P, Q} and S = O Ȳ s ∪ O Ȳ s+1 ∪ · · · ∪ O Ỹ s .
Step 4. Now, we prove that there is a weakly free rational curve fs −1 over P, Q such that fs −1 (P 1 ) intersects at most O 1 \{P, Q}, . . . , Os −1 \{P, Q}, and is disjoint from Os \ {P, Q}, . . . , Os \ {P, Q}. Indeed, we have the following two cases: 1) If fs(P 1 ) is disjoint from Os \ {P, Q}, then let fs −1 = fs.
2) If fs(P 1 ) intersects Os \ {P, Q}, we apply Lemma 13 with Z = Os +1 ∪ · · · ∪ Os and S = Os ∪ Z. Notice that S and Z are closed subvarieties of X of codimension ≥ 2, and Os is smooth. In particular, S\ Sing X ⊇ Os. By assumption, fs(P 1 ) ∩ (Z \ {P, Q}) = ∅. Therefore, by the Lemma, there exists a weakly free rational curve fs −1 on X, which is a general deformation of fs, such that fs −1 (P 1 ) intersects at most O 1 \ {P, Q}, . . . , Os −1 \ {P, Q}, and is disjoint from (Os ∪ Z) \ {P, Q} = (Os \ {P, Q}) ∪ (Os +1 \ {P, Q}) ∪ · · · ∪ (Os \ {P, Q}).
Step 5. By induction ons, there is a weakly free rational curve f 0 over P, Q, disjoint from I(X) \ {P, Q}.
Proof of Main Theorem.
Step 1. First, let us consider S = Sing X.
There is a free rational curve f 0 : C 0 ∼ = P 1 → X disjoint from {P i } ∪ S. Indeed, we can apply Main Lemma to the subvariety {P i } ∪ S and any two smooth points P, Q ∈ {P i } ∪ S in X. Since f 0 (P 1 ) is in the smooth locus of X, f 0 is free and disjoint from {P i } ∪ S.
We construct a comb of smooth rational curves C and a morphism f : C → X ′ as follows.
I. Assume that P 1 , . . . , P r ′ are smooth points for some r ′ , 1 ≤ r ′ ≤ r, and P r ′ +1 , . . . , P r are singular points of X. Choosing points t 1 , . . . , t r ∈ C 0 , such that P ′ i = f 0 (t i ) ∈ X are distinct. For each j, applying the Main Lemma to S = Sing X ∪ {P i } and points P = P j , Q = P ′ j , there is a weakly free rational curve f j : C j ∼ = P 1 → X over P j , P ′ j for each 1 ≤ j ≤ r, disjoint from S \ {P j , P ′ j }.
Applying the general statement of Theorem 9 to weakly free rational curves f 0 , f 1 , . . . , f r and the set {P i } = {P i } i≥r ′ +1 , we get a resolution π : X ′ → X.
For each 1 ≤ i ≤ r ′ , since P i and P ′ i are smooth points, f i (P 1 ) is contained in the smooth locus of X. Therefore f i is free for each 1 ≤ i ≤ r ′ by [Kol96] II.3.11. We identify the curve f i : C i ∼ = P 1 → X birationally with a free rational curve f i : C i ∼ = P 1 → X ′ . We also identify P i ∈ X with P i ∈ X ′ for 1 ≤ i ≤ r ′ , and
P ′ i ∈ X with P ′ i ∈ X ′ for 1 ≤ i ≤ r. More precisely, f i (0 i ) = P i , where 0 i ∈ C i , 1 ≤ i ≤ r ′ , and f i (∞ i ) = P ′ i where ∞ i ∈ C i , 1 ≤ i ≤ r.
For each r ′ + 1 ≤ j ≤ r, P j is singular. Let f ′ j : C j ∼ = P 1 → X ′ be the proper birational transformation of a sufficiently general deformation of f j . Since π : X ′ → X is a resolution in Theorem 9, f ′ j (C j ) intersects π −1 P j divisorially over P j for r ′ + 1 ≤ j ≤ r, and is disjoint from the closure of π −1 (S \{P i }). Let Q j be a point in f ′ j (C j )∩π −1 P j over P j for r ′ +1 ≤ j ≤ r. We can suppose that f i is very free for 1 ≤ i ≤ r ′ and f ′ j is very free for r ′ + 1 ≤ j ≤ r by [KMM92a] 1.1. or [Kol96] II.3.11.
By construction of f i , 1 ≤ i ≤ r ′ and f ′ j , r ′ + 1 ≤ j ≤ r, f i (C i ) and f ′ j (C j ) are disjoint from the closure of π −1 (S \ {P 1 , . . . , P r }) = π −1 (S \ {P r ′ +1 , . . . , P r }).
II. Gluing ∪ r i=0 C i , we get a comb of smooth rational curves C = r i=0 C i and a morphism f : C → X ′ . Indeed, we identify points ∞ i ∈ C i with t i ∈ C 0 for each 1 ≤ i ≤ r. Then we have a comb of smooth rational curves C = r i=0 C i and a morphism f : C → X ′ because f 0 (t i ) = f i (∞ i ) = P ′ i . Notice that f (C) is disjoint from the closure of π −1 (S \ {P 1 , . . . , P r }).
In the end, f : C → X ′ can be smoothed into a rational curve f ′ : P 1 → X ′ such that f ′ is free over P i , 1 ≤ i ≤ r ′ and Q j , r ′ + 1 ≤ j ≤ r, and is disjoint from the closure of π −1 (S \ {P 1 , . . . , P r }) (We can generalize the proof of [Kol96] II.7.6 for comb to get f ′ is a free rational curve over {P 1 , . . . , P r ′ , Q r ′ +1 , . . . , Q r }, not only with {P 1 , . . . , P r ′ , Q r ′ +1 , . . . , Q r } fixed, as stated in [Kol96] II.7.6. Or we can attach additional rational curves to enlarge of the family of f ′ , such that f ′ is a free rational curve over {P 1 , . . . , P r ′ , Q r ′ +1 , . . . , Q r } after a base change).
Step 2. Now we consider any closed subvariety S of codimension ≥ 2. By Step 1, there is a free rational curve f ′ : P 1 → X ′ over P 1 , . . . , P r ′ , Q r ′ +1 , . . . , Q r , disjoint from the closure of π −1 (Sing X \{P 1 , . . . , P r }), where π : X ′ → X is the resolution in Step 1. On the other hand, π −1 ((S \ Sing X)\{P 1 , . . . , P r }) is a codimension ≥ 2 subvariety on X ′ by Theorem 9 3'). So a general deformation f ′′ of f ′ is free over P 1 , . . . , P r ′ , Q r ′ +1 , . . . , Q r , disjoint from π −1 ((S \ Sing X) \ {P 1 , . . . , P r }) by [Kol96] II.3.7. Since f ′ is disjoint from the closure of π −1 (Sing X \ {P 1 , . . . , P r }), f ′′ is disjoint from π −1 (Sing X \ {P 1 , . . . , P r }). Hence f ′′ is disjoint from π −1 (Sing X \ {P 1 , . . . , P r }) ∪ π −1 ((S \ Sing X) \ {P 1 , . . . , P r }) = π −1 (S \ {P 1 , . . . , P r }). Therefore, πf ′′ is a general deformation of πf ′ over P 1 , . . . , P r , disjoint from S \ {P 1 , . . . , P r }, and thus πf ′ is a geometrically free rational curve over P 1 , . . . , P r on X.
O 1 \
1{P, Q}, . . . , O s ′ \ {P, Q}, and is disjoint from O s ′ +1 \ {P, Q}, . . . , Os \ {P, Q} for some s ′ .
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| [] |
[
"Ground state of composite bosons in low-dimensional graphs",
"Ground state of composite bosons in low-dimensional graphs"
] | [
"Cecilia Cormick \nCONICET\nInstituto de Física Enrique Gaviola\nUniversidad Nacional de Córdoba\nCiudad UniversitariaX5016LAECórdobaArgentina\n",
"Leonardo Ermann \nDepartamento de Física Teórica\nGAIDI\nComisión Nacional de Energía Atómica\nBuenos AiresArgentina\n\nCONICET\nGodoy Cruz2290 (C1425FQB) CABAArgentina\n\nECyT-UNSAM\nCampus Miguelete, 25 de Mayo y Francia1650Buenos AiresArgentina\n"
] | [
"CONICET\nInstituto de Física Enrique Gaviola\nUniversidad Nacional de Córdoba\nCiudad UniversitariaX5016LAECórdobaArgentina",
"Departamento de Física Teórica\nGAIDI\nComisión Nacional de Energía Atómica\nBuenos AiresArgentina",
"CONICET\nGodoy Cruz2290 (C1425FQB) CABAArgentina",
"ECyT-UNSAM\nCampus Miguelete, 25 de Mayo y Francia1650Buenos AiresArgentina"
] | [] | We consider a system of composite bosons given by strongly bound fermion pairs tunneling through sites that form a low-dimensional network. It has been shown that the ground state of this system can have condensate-like properties in the very dilute regime for two-dimensional lattices but displays fermionization for one-dimensional lattices. Studying graphs with fractal dimensions, we explore intermediate situations between these two cases and observe a correlation between increasing dimension and increasing condensate-like character. However, this is only the case for graphs for which the average path length grows with power smaller than 1 in the number of sites, and which have an unbounded circuit rank. We thus conjecture that these two conditions are relevant for condensation of composite bosons in arbitrary networks, and should be considered jointly with the well-established criterion of high entanglement between constituents. | 10.1103/physreva.107.043324 | [
"https://export.arxiv.org/pdf/2304.14834v1.pdf"
] | 258,395,483 | 2304.14834 | cf9055072ffdbbd877cd8a395c8fd22777aa0aea |
Ground state of composite bosons in low-dimensional graphs
Cecilia Cormick
CONICET
Instituto de Física Enrique Gaviola
Universidad Nacional de Córdoba
Ciudad UniversitariaX5016LAECórdobaArgentina
Leonardo Ermann
Departamento de Física Teórica
GAIDI
Comisión Nacional de Energía Atómica
Buenos AiresArgentina
CONICET
Godoy Cruz2290 (C1425FQB) CABAArgentina
ECyT-UNSAM
Campus Miguelete, 25 de Mayo y Francia1650Buenos AiresArgentina
Ground state of composite bosons in low-dimensional graphs
(Dated: April 28, 2023)
We consider a system of composite bosons given by strongly bound fermion pairs tunneling through sites that form a low-dimensional network. It has been shown that the ground state of this system can have condensate-like properties in the very dilute regime for two-dimensional lattices but displays fermionization for one-dimensional lattices. Studying graphs with fractal dimensions, we explore intermediate situations between these two cases and observe a correlation between increasing dimension and increasing condensate-like character. However, this is only the case for graphs for which the average path length grows with power smaller than 1 in the number of sites, and which have an unbounded circuit rank. We thus conjecture that these two conditions are relevant for condensation of composite bosons in arbitrary networks, and should be considered jointly with the well-established criterion of high entanglement between constituents.
I. INTRODUCTION
Several recent developments have opened the doors to the exploration of systems with very complex geometries, approaching the freedom to build arbitrary networks for many-body physics. This is possible due to the engineering capabilities achieved with optical tweezer arrays [1,2] and most notably with circuit quantum electrodynamics [3,4]. Other examples in this direction are given by the implementation of frustrated lattices and quasicrystalline structures in optical traps [5,6], photonic quasicrystals [7] and polaritonic systems [8], as well as artificial fractal lattices for electrons [9] and tree-like couplings of atom arrays inside optical cavities [10]. Implementations in the context of Josephson-junction arrays have also inspired the study of Bose-Hubbard models on various graphs [11][12][13].
It is well known that the dimensionality of a physical system can have tremendous impact on its properties. One remarkable illustration of this phenomenon is given by the fact that hard-core bosons exhibit fermionization in one-dimensional (1D) systems [14,15], thus questioning intuitive expectations of what bosonic behaviour means. Another striking example is the existence of anyonic particles only in two dimensions [16]. In more general scenarios, other graph properties besides dimension have been shown to play a relevant role as well, for instance for the observation of Bose-Einstein condensates in low-dimensional graphs [11], propagation of solitons in networks [17], or frustration in spin systems [18].
The number of spatial dimensions has been recently studied in connection with the validity of the so-called "coboson ansatz" for the ground state of a system of N composite bosons [19] . This ansatz is a canonicalensemble analogue of the BCS wavefunction and can in many cases be used as a compact approximation of the ground state of a many-particle system [20]. We note that the ansatz describes approximate condensation to the ground state of a single composite particle, as opposed to a more general kind of condensation to an arbitrary state as studied in [21].
Loosely speaking, the coboson ansatz is expected to be suitable for dilute systems with short-range interactions and high entanglement between the constituents of each composite boson [20,22]. It has been noted that dimensionality also plays an important role for the validity of the ansatz, and that 1D systems fulfilling the previous conditions do not generally obey the ansatz [23,24]. This is not surprising in the light of the already mentioned fermionized behaviour. However, fermion pairs in a ladder geometry, i.e. an n × m lattice with n → ∞ while m is kept fixed, behave as in a 1D system although fermionization is not applicable [23]. We note that this dimension-dependent change in behaviour, fermionizing in 1D but condensing in 2D for low enough densities, was first studied for standard hard-core bosons, see for instance [25].
We now extend the previous analysis of the relation between lattice dimensionality and validity of the coboson ansatz to models with dimensions between 1 and 2. To this aim, we consider fractal geometries, and compare the performance of the ansatz for fractals with different Hausdorff dimensions [26]. The lack of translational invariance makes this study numerically costly, so that the systems studied have moderate sizes. Nevertheless, several conclusions can be drawn from our results. Indeed, the graph dimension does seem to play an important role for the validity of the ansatz. However, a new aspect that appears in our analysis is the observation that fermion pairs do not exhibit condensation in tree-like graphs. Besides, we find that the ansatz is generally a much better approximation to the ground state in systems with closed boundaries, while the number of neighbours has little impact on the results.
Because of the computational cost, we focus on systems of a few fermion pairs, i.e. two or three hard-core bosons or equivalently four or six paired fermions. The structure of coboson theory is such that the coboson ansatz for an arbitrary number N of pairs leads, for low densities, to expectation values dominated by the results of one and two pairs, accompanied by known N -dependent prefactors [20]. Then, if the ansatz is shown to give wrong results for two composite bosons, it cannot provide a good approximation of the ground state of N pairs either.
This article is organized as follows: in Sec. II we introduce the model we study for different networks. Section III discusses basic concepts regarding the coboson ansatz that are essential to understand our work, whereas Sec. IV introduces relevant graph properties. In Sec. V B we present our numerical results for various graphs with dimensions between 1 and 2. Finally, in Sec. VI we summarize our conclusions.
II. THE MODEL
The system we consider consists of a graph of M sites with two species of fermions that can hop along them. Fermions of different species experience a very strong attraction, so that if the numbers of particles of both species are equal then the low energy manifold has all particles in pairs. More precisely, the Hamiltonian takes the form:
H = −U 0 M j=1 a † j a j b † j b j + J 2 <i,j> (a † i a j + b † i b j + H.c.) (1)
where a j (a † j ) destroys (creates) a particle of type a in site j, b j (b † j ) does the same for a particle of type b, and < i, j > indicates neighbouring sites in the graph. For simplicity, we do not consider spatial variations of local energies or tunneling strength.
We are interested in the limit when the interaction energy is much stronger than the hopping, i.e. U 0 J, and apply perturbation theory to find the ground state of the system. As will be shown in the following, the restriction to the limit when particles always tunnel in pairs makes our system an instance of the hard-core Bose-Hubbard model, which is equivalent to a Heisenberg model [27][28][29][30]. Heisenberg models on fractal lattices have been studied for instance in [31][32][33]. An important point to keep in mind is that for a coboson system with N composite particles only a particular subspace of the equivalent spin system with a fixed total spin projection S z is relevant [23].
This limit of very strongly bound pairs is the one studied in [23,34], and it is particularly relevant for our purposes since it is the situation where the coboson description should be most appropriate. The Hilbert space of N fermions of each kind divides into a ground manifold composed by the states where all fermions are paired (i.e. occupying the same site as one of the other species), and many excited manifolds with unpaired particles. The effective Hamiltonian within the ground manifold can be found with perturbation theory. We note that the steps involved are the same as in [23,34], but now generalized to arbitrary graph geometries.
To zero order in the hopping the energy of the ground manifold is −N U 0 . When second-order terms in the hopping are introduced, one obtains an overall shift of the energy of this subspace, Hamiltonian terms describing correlated tunneling of pairs, and an additional term coming from the fact that hopping of a particle into a given site is forbidden if there is already an identical fermion there. This leads to the form:
H (N ) g −N U 0 + J eff + H t + H d .(2)
Here, J eff is the effective tunneling strength for a pair,
J eff = J 2 U 0 ,(3)
and H t is the tunneling contribution,
H t = − J eff 2 <i,j> (T i,j + T j,i )(4)
with T i,j the operators that correspond to hopping of a pair from site i to j. The interactions between nearestneighbours are contained in a contribution H d that will be diagonal in our basis and is of the form:
H d = J eff <i,j> N i N j .(5)
In this expression, N j is the number of pairs in site j, with double occupations being forbidden because of the fermionic character of the constituents. Our strongly bound fermion pairs are thus the same as hard-core bosons with a specific relation between tunneling strength and nearest-neighbour repulsion. The hopping term tends to delocalize the cobosons, whereas the repulsive interaction together with the hard-core character compete with the hopping. Therefore one expects that the ground state will have delocalized pairs but which are unlikely to be found next to each other. We note that previous work [35] has considered the inclusion of longer-ranged attractive interactions that can lead to the formation of larger aggregates. Such generalizations imply a richer variety of behaviours and are beyond the scope of the present study.
From Eq. (2) we can write down the matrix form of the Hamiltonian for any particular subspace with fixed N . For brevity, in the following we ignore the overall energy −N (U 0 + J eff ). For N = 1 the basis of our restricted Hilbert space is composed by states of the form |k where k labels the location of the bound pair in the graph, and the Hamiltonian contains only tunneling:
H t k,l = − J eff 2 A kl .(6)
Here A kl is the corresponding element of the adjacency matrix, which is 1 when sites k and l are neighbours and vanishes otherwise. For N = 2 the basis of the Hilbert space is composed by the states |k, l where k, l label the locations of each of the pairs in the graph, and using Pauli exclusion principle we can take k < l. In this basis, the diagonal part is:
H d kl,kl = J eff A kl(7)
while the tunneling part of the Hamiltonian is of the form:
H t kl,mn = − J eff 2 (δ ln A km + δ lm A kn + δ kn A lm + δ km A ln ) .
(8) For N = 3 the basis is of the form |k, l, m with k < l < m. The diagonal part of the Hamiltonian is:
H d klm,klm = J eff (A kl + A km + A lm )(9)
whereas the tunneling term can be written as:
H t ijk,lmn = − J eff 2 (A il δ jm δ kn + all permutations) (10)
where by "all permutations" we mean all permutations of indices i, j, k and l, m, n separately.
III. COBOSON ANSATZ FOR THE GROUND STATE
In this work we focus on identical composite bosons, each made of two distinguishable fermions. This section provides a brief overview of the coboson ansatz for the ground state of N such pairs. For a more complete introduction to the coboson formalism, we refer the reader to [20]. For a given Hamiltonian corresponding to a single pair, the ground state |ψ defines the coboson creation operator B † , namely the operator which acts on the vacuum state |v creating a single pair in the ground state, |ψ = B † |v .
One can write a normalized state of N composite bosons obtained after acting N times with the coboson creation operator in the form [20,36]
|N = B † N √ N !χ N |v .(11)
Here χ N is a normalization factor which accounts for the Pauli exclusion principle [36,37]. For composite bosons made of two fermions the normalization factor depends on the Schmidt coefficients λ j of |ψ and takes the form [36][37][38],
χ N = N ! p N >p N −1 >...>p1 λ p1 λ p2 ...λ p N .(12)
For the case N = 2, the normalization coefficient is equal to χ 2 = 1 − P , with P the purity of the reduced density matrix of one of the constituent particles of a pair in the ground state |ψ . Thus, a high entanglement in state |ψ is a key aspect behind effectively bosonic features. In general, the behavior of pairs as approximate elementary bosons can be related to the normalization coefficients, and bosonic behavior is recovered when χ N /χ N −1 1 [22,36,38,39].
In particular, for the model we consider the purity P is determined by the distribution of occupation probabilities p 1 (j) of the various sites j in the single-pair ground state, according to:
P = M j=1 p 2 1 (j) .(13)
For a fixed number of sites, the maximum entanglement or minimal purity corresponds to a uniform probability distribution.
The idea that the state |N of Eq. (11) provides a good approximation of the ground state of a system of N cobosons is an important element of coboson theory, and we refer to this in the following as the coboson ansatz. This ansatz provides an enormous simplification of the description of the many-body ground state, in the cases when it is applicable. Nonetheless, we note that the coboson formalism is a powerful machinery also when this simplification is not possible, as illustrated by several examples in the literature, see for instance [24,40,41].
A useful property of state |N is that for dilute systems one can derive simple expressions for the expectation values of many observables of interest in terms of an expansion in the density of particles. Following the steps in [20], one obtains approximations of the form:
O N N O 1 + N (N − 1) 2 ( O 2 − 2 O 1 )(14)
up to corrections of third order in particle density. Here, O N denotes the mean value of operator O evaluated over state |N . The derivation in [20] takes O to be the system Hamiltonian, but the same steps can be carried out for any observable which can be written as a onebody operator either in terms of cobosons or in terms of elementary fermions. Problems with full translational invariance, like the ring and torus studied in [23], lead to very simple forms for the state |N . The ground state of a single pair in such a lattice is the same as in the fully connected graph, i.e.:
|ψ = 1 √ M M j=1 a † j b † j |v .(15)
The structure of this state greatly simplifies many calculations, because all Schmidt coefficients of this state are equal to 1/M . For states of the form of Eq. (15) the coboson ansatz has full symmetry between all sites for arbitrary N .
In general, the coboson ansatz is expected to be a satisfactory approximation when the system is very dilute, interactions are sufficiently short-ranged, and the single-pair ground state |ψ exhibits high entanglement between the pair constituents. Previous work, however, has shown that the validity of the ansatz may also depend on the dimensionality [23]. This is to be expected since the coboson ansatz does not properly capture the strong spatial correlations among pairs that build up for one-dimensional or quasi one-dimensional systems.
IV. GRAPH DEFINITIONS AND RELEVANT PROPERTIES
In the light of the previous results, in this work we consider more general graphs with both integer and noninteger dimensions, to better study the interplay between dimensionality and applicability of the coboson ansatz. Before presenting our findings, in the following we introduce the fractal graphs that we will consider, as well as some concepts of graph theory that will be important for our analysis.
In the next Section we numerically examine the ground state of two and three fermion pairs in different graphs with dimensions between 1 and 2, considering the Hausdorff dimension for the case of fractals [26]. In contrast with regular lattices, which have integer dimensions, the Hausdorff dimension can take non-integer values to describe the geometry of fractal graphs. It is related to the box-counting dimension, associated with the growth of the number N (r) of balls of radius at most r required to cover a graph region completely, in the limit when r goes to zero.
We note that fractal dimension is only well defined in the limit of infinitely many points in the graph, whereas our systems are always of finite size. Our interest, however, is precisely the behavior of the ground state as the infinite-size limit is approached and the system becomes more dilute. We regard fractal dimension as a good measure for our purposes since the increase in system size in our study is achieved by considering successive levels of the iterative construction of the fractals.
We take the following instances of graphs: 1) onedimensional lattices given by simple chains (which we label 1D); 2) two-dimensional square lattices (2D); 3) fractal graphs given by the Sierpinski gasket, also called Sierpinski triangle, with dimension log 2 (3)
1.58 and which we label S; 4) Vicsek fractals, with dimensions d = log(ν + 1)/ log (3) 1.26 for ν = 3 and 1.47 for ν = 4 [42,43], labelled V3 and V4 respectively. Both the Sierpinski and the Vicsek fractals are built iteratively, so that the actual fractal structure corresponds to the limit of infinitely many iterations. The step-wise construction of these fractals is illustrated in Fig. 1. In the case of the Vicsek fractals, the parameter ν is associated with the number of copies of the graph that are appended to it to build the next iteration, and the first level is a star graph with ν+1 points.
The dimension of the graphs we consider is related to the growth of the average path length between graph nodes as the size of the graph is increased. Here, the path length is taken to be the number of edges along the shortest path between nodes; this quantity is then averaged over all pairs of nodes. In Fig. 2 we show the growth of the average path length for the system sizes we explore. For all graphs except the Vicsek fractals we consider both open and closed boundary conditions; for the Sierpinski gaskets the modified boundary conditions are obtained adding extra edges linking the outer vertices of the triangle. As can be seen in the plots, the numerical fits of the growth of the average path length indicate a power that coincides with the inverse of the graph dimension and that is independent of the boundary conditions. Apart from the dimensionality, the quality of the coboson ansatz may also be affected by other graph properties. Two concepts from graph theory that will be relevant in the following are those of circuit rank and betweenness. The circuit rank of a connected graph is equal to the minimum number of edges that must be removed to turn the graph into a tree [44], and so in comparison with the total number of edges, it gives an idea of how close the graph is to being a tree. For instance, the circuit rank is equal to zero for a tree, whereas it grows quadratically with the number of nodes for a fully connected graph and linearly for a regular lattice of dimension larger than 1.
The betweenness centrality or betweenness g of each node in a graph is a measure of its centrality in the graph [45]. It is related to the fraction of shortest paths among graph nodes that pass through the particular vertex considered. Among the graph families we study, the distribution of betweenness of the different nodes varies significantly. For the graphs with closed boundary conditions, all sites have the same value of g; nontrivially, this is also valid for the Sierpinski gasket. Among the graphs with open boundary conditions, regular lattices have nodes with larger betweenness at the center, but the variation is smooth. In contrast, for Vicsek fractals the central nodes have values of g that are much larger than those of the rest.
Actually, betweenness centrality is related to another aspect that is relevant to our study, namely the purity P of the reduced density matrix of each fermion in the single-pair ground state, which depends on the kind of graph considered. As explained in the previous section, bosonic behaviour of the coboson creation operator is directly associated with P . To take this into account, we characterized the effective size S of our system as S = 1/P for each given lattice. The reason to consider S as an effective size is that a value of S smaller than M implies that a pair in the ground state is not spread over all sites equally, and we wish to correct for this effect when we study the validity of the coboson ansatz.
It turns out that, although there is no one-to-one correspondence, the single-pair ground state tends to localize at the nodes with larger betweenness, as illustrated in Fig. 3. We note that the 2D case is not shown in Fig. 3 but looks similar to the 1D chain. The very uneven occupation probability found for the Vicsek fractals can be better appreciated observing the dashed horizontal lines that indicate the average over all sites.
Thus, graphs with very unbalanced betweenness values also have larger differences between the number of sites M and the effective size S. The relation between M and S for various graphs is displayed in Fig. 4. Indeed, the lowest values of S for a given M are found for Vicsek fractals. The limit value S = M is shown with a dashed line and is found for the one-dimensional ring and the torus (i.e. the square lattice with periodic boundary conditions). Closed boundary conditions always lead to larger values of S for fixed M , thus implying larger effective sizes without increasing the numerical cost. However, for large systems the relative difference in effective size becomes rather small, as boundary effects become less important.
V. FIDELITY BETWEEN THE GROUND STATE AND THE COBOSON ANSATZ FOR VARIOUS GEOMETRIES
A. Preliminary considerations
The goal of this study is to test the validity of the ansatz in different graphs with dimensions between 1 and 2. For this purpose, we first find the ground state of a single composite particle in the lattice, of the form:
|φ (1) GS = M j=1 c j |j .(16)
where |j labels the location of the pair on the lattice. Because the two fermions forming a pair are always at the same site, the Schmidt decomposition is trivial, with Schmidt coefficients given by λ j = c 2 j , since for our model c j ∈ R.
From state (16), we can build the coboson ansatz for an arbitrary number of pairs. Our aim is to assess the quality of the coboson ansatz for the ground state in different graphs, and we do so by computing the fidelity between the ground state |φ (N ) GS obtained numerically and the coboson ansatz |N according to:
F N = N |φ (N ) GS 2 .(17)
If the coboson ansatz is applicable, the fidelity F N is expected to approach 1 in the infinitely dilute limit, i.e. when the number of sites M tends to infinity. We note that F N depends on both N and M and also on the kind of graph considered; this dependence will be left implicit to keep the notation simple. One could consider, instead of calculating the fidelity, a comparison of the ground-state energies obtained from numerical calculation and from the coboson ansatz. A clear disagreement between these values would already imply a failure of the ansatz. However, as has been shown in [23], it is possible that the energies agree to a good approximation while at the same time other properties such as spatial correlations are not correctly reproduced.
In the following we will focus on the cases N = 2, 3. For two composite particles we get:
|N = 2 = 2 χ 2 j<k c j c k |j, k .(18)
If the ground state for two pairs has coefficients given by
|φ (2) GS = j<k α jk |j, k(19)
then the fidelity takes the form:
F = 2 χ 2 j<k α jk c j c k 2(20)
where we are using that all coefficients are real. Along the same lines, for the case of three pairs, N = 3, the expression for the fidelity takes the form:
F 3 = 6 χ 3 j<k<l α jkl c j c k c l 2(21)
with α jkl the coefficients of the ground state for N = 3 in the basis |j, k, l . A few cases can be readily analyzed: the fully connected graph, due to its symmetry, has the coboson ansatz as exact ground state, and thus F N = 1 for any number of sites as long as it can acommodate N pairs. Another example that is solvable because of its high symmetry is the star graph. We note, however, that the creation operator of the single-pair ground state on the star cannot exhibit proper bosonic behaviour, because χ N /χ N −1 does not tend to 1 as the number of sites goes to infinity.
The 1D chain with closed boundary conditions has been studied in [23]. The single-pair ground state has the fully symmetric form of Eq. (15), and the fidelity for N = 2 with M → ∞ is equal to 8/π 2 0.81, as has been shown using the mapping to the Heisenberg model. A 1D chain with open boundaries has a non-uniform single pair ground state, and the asymptotic fidelity for N = 2 is 2 17 /(π 4 45 2 ) 0.66 as can be seen using fermionization [15]. Thus, not surprisingly, 1D systems do not exhibit condensation in the dilute limit and the coboson ansatz does not approach the ideal fidelity of 1.
Previous work [23] has also numerically studied 2D square lattices with periodic boundary conditions, observing a fidelity which increases monotonically with system size and suggesting an asymptotic value of F 2 = 1. On the contrary, the fidelity for ladder systems, i.e. n×m lattices with n → ∞ and m fixed, has a behaviour that resembles that of 1D chains. It seems then that the effective dimension of the lattice for our purposes is best captured by the power of the growth of the average path length between sites: ladder models, indeed, have average path lengths that asymptotically grow linearly in the total number of sites, just as 1D systems. It is natural to conjecture that an average path length growth with power 1 is associated with the failure of the coboson ansatz.
B. Numerical results
In the following we analyze the behavior of the fidelity between the coboson ansatz and the numerical ground state for increasing sizes of graphs with different geometries. We note that, as observed in [23], the convergence of the fidelity with increasing system size is rather slow. The study of large systems in [23] was largely simplified by exploiting translational invariance, which is no longer possible in the present work.
In Fig. 5 we show the fidelity for two pairs, F 2 , as a function of S for the different networks considered. We note that apart from the cases plotted and already described, we performed calculations with open square, triangular and hexagonal 2D lattices, leading to very similar results as the open square lattice. Calculations with the Hanoi graph, or dual Sierpinski gasket, also lead to very similar results as the Sierpinski gasket. Just as the various geometries for two-dimensional lattices, the Hanoi graph and the Sierpinski gasket have the same dimension but different number of neighbours. From these comparisons, we conclude that the graph degree does not play a relevant role for our purposes. Several features of Fig. 5 are interesting: in general, the plots always show a higher fidelity for closed boundary conditions (empty symbols), even after correcting for the effective size using S instead of M . Leaving aside the case of the Vicsek fractals, a higher dimension is associated with a higher fidelity, and graphs with dimension higher than 1 seem to have fidelities that improve as the graph size is increased. Of course, this does not imply that the fidelity approaches the ideal value of 1 as the number of sites approaches infinity; it may well be that convergence is observed at much larger values than the ones numerically accessible.
The case of the Vicsek fractals is strikingly different from the other graphs studied; indeed, in Vicsek fractals the coboson ansatz actually gets worse as the system size is increased, and the corresponding fidelities lie clearly below the case of 1D chains. We associate this behaviour with the tree-like character of the Vicsek fractals (see Fig. 1). We thus conjecture that the validity of the coboson ansatz for dilute systems in general graphs is to be expected when not only the average path length grows with a power lower than 1, but also the graph should have an unbounded circuit rank.
As explained in Sec. III, a failure of the coboson ansatz for N = 2 generally leads to wrong predictions for higher numbers of particles. However, as a further check, we also study the fidelity for increasing graph size for systems with N = 3. In this case, we restrict to smaller graphs to keep the full Hilbert space tractable. The results are shown in Fig. 6, and indeed display a similar behaviour as the previous figure: the fidelity increases with graph size for the 2D lattice and the Sierpinski gasket, whereas it exhibits convergence to a value clearly below 1 for the 1D chain and it decreases for the Vicsek fractals. There is actually an intuitive explanation why treelike graphs can be expected to behave very differently from other graphs. Indeed, the presence of a first pair on one site blocks that site for a second pair; thus one can think of that second pair as moving in a graph which is modified by the removal of the blocked site. In the case of 2D graphs, it is clear that this effect is minor; on the contrary, for tree-like graphs removing a single site generally splits the graph in two or more isolated parts.
Along this line, one can then think that the key feature determining whether the ground state can have condensate-like properties is given by the vertexconnectivity, i.e. the minimum number of nodes that have to be removed to make the graph disconnected. This, however, does not seem to be the case, since Sierpinski triangles have low connectivity and their behaviour is more similar to 2D graphs than to Vicsek fractals. There is also a reason why the connectivity is not necessarily relevant: it is not only important that removing a node will modify the properties of the graph; it also matters how likely it is to find a pair at a particular given site.
This point can be better examined considering once more the probability distribution p 1 of occupation of the various nodes for the single-pair ground state as displayed in Fig. 3. For the case of the Vicsek fractals, we notice a few sites have an occupation probability that is way larger than the average. These sites are precisely those whose removal has the largest impact in the properties of the remaining graph, in particular the central node. This is not the case in the Sierpinski gasket, or in 1D and 2D lattices, for which many nodes have comparably high occupation. We conjecture that this strong relation between centrality and occupation probability makes treelike graphs particularly unfavorable for condensation.
VI. CONCLUSIONS
We have studied a system of a few (N = 2, 3) strongly bound fermion pairs in graphs of dimension between 1 and 2. In order to characterize the behaviour of the composite particles at zero temperature, we have examined the fidelity between the ground state found numerically and the so-called "coboson ansatz". The latter is a condensate-like state but taking into account the fermionic character of the constituents of each pair. Although we restricted to systems with 2 or 3 pairs, the structure of coboson theory is such that at low densities many expectation values are dominated by the results for few pairs, and so our conclusions are relevant also for the many-body scenario [20].
Our results indicate a poor performance of the coboson ansatz for 1D chains and Vicsek fractals. From our findings we conjecture that condensation of fermion pairs in the very dilute limit can be expected when: i) the average path length grows with a power strictly lower than 1, and ii) the graph has unbounded circuit rank and is thus "very different from a tree". These conditions should be considered together with the already known criteria associated with the entanglement between the constituents of a single pair in its ground state and the short-ranged interactions.
Among the graphs fulfilling conditions i) and ii) above, we find a strong correlation between the dimension and the quality of the coboson ansatz: the larger the dimension, the better the description provided by the ansatz. Boundary conditions also play an important role: lattices with closed boundary conditions consistently display larger fidelities than their open-boundary counterparts. On the other hand, the comparison of different kinds of 2D lattices, and of the Sierpinski gasket with its dual, suggests that the number of neighbours does not affect the results significantly.
We note that the same conclusions apply to standard hard-core bosons, since we observed little difference when the nearest-neighbour interactions resulting from Pauli exclusion were removed. Given the slow convergence of the fidelity with system size, it would be desirable to extend our analysis to larger graphs. This will require more sophisticated numerical techniques; direct diagonalization is too costly due to the quadratic scaling of the Hilbert space and the lack of translational invariance. We hope that the present work sparks further interest in the problem of condensation of composite particles in arbitrary geometries.
VII. ACKNOWLEDGMENTS
The authors acknowledge funding from grants PICT 2017-2583, PICT 2018-02331 and PICT 2020-SERIEA-00959 from ANPCyT (Argentina).
FIG. 1 :
1Small-size versions of the fractal graphs we consider. a) and b) First and second levels of the Vicsek fractal with ν = 3 respectively. c) and d) First and second levels of the Vicsek fractal with ν = 4. e) Sierpinski gasket with 15 sites. FIG. 2: Growth of the average path length between sites (in logarithmic scale) as a function of the total number M of sites in the graph, for: a) 1D chains, 2D square lattices, and b) Sierpinski gaskets and Vicsek fractals with ν = 3, 4. For regular lattices and Sierpinski gaskets we consider both open and closed boundary conditions, the latter labelled by "CBC". In each case the inverse of the power of the growth of the average path length, α, obtained from the numerical fit and represented with dotted line, coincides with the graph dimension.
FIG. 3 :
3Occupation probability p1 of each site in the singlepair ground state vs. centrality of the node, as quantified by the betweenness centrality g, for graphs with open boundary conditions and numbers of sites M ∼ 1000. The subplots correspond to: a) 1D chain with M = 900, b) Sierpinski gasket with M = 1095, c) Vicsek fractal with ν = 3 and M = 1024, and d) Vicsek fractal with ν = 4 and M = 900. Dashed horizontal lines represent values of uniform probability given by 1/M .
FIG. 4 :
4Effective size quantified by log10(S) as a function of log10(M ) for the graphs studied, with S = 1/P the effective system size and M the total number of sites. The dashed line shows the identity M = S that is obtained for uniformly distributed single-pair ground states. The lowest values of S(M ) are obtained for Vicsek fractals.
FIG. 5 :
5Fidelity between the coboson ansatz and the true ground state for two pairs, quantified by F2 as a function of log 10 (S) for different graphs.
FIG. 6 :
6Fidelity between the coboson ansatz and the true ground state for three pairs, quantified by F3 as a function of log10(S) for different graphs.
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"BRENT: Bidirectional Retrieval Enhanced Norwegian Transformer",
"BRENT: Bidirectional Retrieval Enhanced Norwegian Transformer"
] | [
"Lucas Georges \nLanguage Technology Group\nUniversity of Oslo\n\n",
"Gabriel Charpentier \nLanguage Technology Group\nUniversity of Oslo\n\n",
"Sondre Wold \nLanguage Technology Group\nUniversity of Oslo\n\n",
"David Samuel \nLanguage Technology Group\nUniversity of Oslo\n\n",
"Egil Rønningstad \nLanguage Technology Group\nUniversity of Oslo\n\n"
] | [
"Language Technology Group\nUniversity of Oslo\n",
"Language Technology Group\nUniversity of Oslo\n",
"Language Technology Group\nUniversity of Oslo\n",
"Language Technology Group\nUniversity of Oslo\n",
"Language Technology Group\nUniversity of Oslo\n"
] | [
"Proceedings of the 24th Nordic Conference on Computational Linguistics (NoDaLiDa)"
] | Retrieval-based language models are increasingly employed in questionanswering tasks. These models search in a corpus of documents for relevant information instead of having all factual knowledge stored in its parameters, thereby enhancing efficiency, transparency, and adaptability. We develop the first Norwegian retrieval-based model by adapting the REALM framework and evaluate it on various tasks. After training, we also separate the language model, which we call the reader, from the retriever components, and show that this can be fine-tuned on a range of downstream tasks. Results show that retrieval augmented language modeling improves the reader's performance on extractive question-answering, suggesting that this type of training improves language models' general ability to use context and that this does not happen at the expense of other abilities such as part-of-speech tagging, dependency parsing, named entity recognition, and lemmatization. Code, trained models, and data are made publicly available. 1 | 10.48550/arxiv.2304.09649 | [
"https://www.aclanthology.org/2023.nodalida-1.21.pdf"
] | 258,212,897 | 2304.09649 | 9c84531a1a2f27c16f53c091c91ef5403fb83548 |
BRENT: Bidirectional Retrieval Enhanced Norwegian Transformer
Association for Computational LinguisticsCopyright Association for Computational LinguisticsMay 22-24, 2023 c 2023
Lucas Georges
Language Technology Group
University of Oslo
Gabriel Charpentier
Language Technology Group
University of Oslo
Sondre Wold
Language Technology Group
University of Oslo
David Samuel
Language Technology Group
University of Oslo
Egil Rønningstad
Language Technology Group
University of Oslo
BRENT: Bidirectional Retrieval Enhanced Norwegian Transformer
Proceedings of the 24th Nordic Conference on Computational Linguistics (NoDaLiDa)
the 24th Nordic Conference on Computational Linguistics (NoDaLiDa)Association for Computational LinguisticsMay 22-24, 2023 c 2023
Retrieval-based language models are increasingly employed in questionanswering tasks. These models search in a corpus of documents for relevant information instead of having all factual knowledge stored in its parameters, thereby enhancing efficiency, transparency, and adaptability. We develop the first Norwegian retrieval-based model by adapting the REALM framework and evaluate it on various tasks. After training, we also separate the language model, which we call the reader, from the retriever components, and show that this can be fine-tuned on a range of downstream tasks. Results show that retrieval augmented language modeling improves the reader's performance on extractive question-answering, suggesting that this type of training improves language models' general ability to use context and that this does not happen at the expense of other abilities such as part-of-speech tagging, dependency parsing, named entity recognition, and lemmatization. Code, trained models, and data are made publicly available. 1
Introduction
Retrieval-based language models meet some important shortcomings associated with pre-trained language models (PLMs): they are more dynamic, allowing for updating of knowledge without having to re-train the model from scratch; they are more transparent, allowing backtracking the source of returned statements; and they are more efficient, as retrieval provides a non-parametric memory. The accentuated benefit of these models has been the * The authors contributed equally to this work 1 https://github.com/ltgoslo/brent OpenQA task -where they have established new state-of-the-art results on datasets like NaturalQuestions (Kwiatkowski et al., 2019) and WebQuestions (Berant et al., 2013). There, models first fetch a relevant passage from a data source in order to be able to answer a question -as compared to extractive QA, where a passage with the correct answer is provided explicitly as additional input to the model, also referred to as machine reading comprehension.
In this work, we develop the first Norwegian retrieval-based model, BRENT: Bidirectional Retrieval Enhanced Norwegian Transformer, based on the general approach proposed by Guu et al. (2020). Our model consists of two encoders that respectively learn to embed documents and queries into dense vector representations, and a reader module that learns to utilize the retrieved context for prediction, as shown in Figure 1. These are trained jointly and end-to-end, and we start their training from an already pre-trained Norwegian LM. Compared to previous work, we use a relatively small retrieval corpus consisting of 730k Wikipedia documents.
The learning objective is masked language modeling (MLM), and the top k most relevant documents are retrieved from the retrieval corpus through a maximum inner product search (MIPS).
The size of our retrieval corpus allows us to update the search index synchronously during training and do exact matching, as opposed to the asynchronous updates and approximations done in Guu et al. (2020). Furthermore, we do not consider OpenQA as an evaluation task, but instead, we study how retrieval-augmented language modeling can be used as a continued pre-training step in order to improve context utilization in the reader model. That is, we evaluate the reader as a stand-aloneextracting it from the overall pipeline so that it can be distributed and used as a normal LM.
In order to analyze the effect of this continued pre-training, we also benchmark the reader against other NLP tasks that by intuition should not benefit from this type of training, such as part-ofspeech-tagging, named entity recognition, dependency parsing, and lemmatization. We find that the retrieval-augmented training procedure increases the reader's performance on extractive QA without decreasing performance on these tasks. However, we find that it decreases performance on both targeted and sentence-level sentiment analysis. To summarize, our contributions are:
• We develop and release the first Norwegian retrieval-based language model.
• We study how retrieval improves the reader's ability to use context on extractive QA while still performing on par with comparable baselines on morpho-syntactic tasks.
• We analyze the different components of a retrieval-based system through a series of ablations, addressing problems associated with common design choices.
Related work
The basic setup for most retrieval-based approaches to NLP is that for a query q, be it a question for QA or a premise in natural language inference, the model must retrieve a set of passages relevant to q. Relevant candidates are then typically appended to q before being passed to a classification layer. While earlier work approached this using heuristics and sparse retrieval methods like BM25 (Robertson et al., 2009), recent work has focused on learning this retrieval step. Most of these use an architecture with an encoder and a reader: the encoder learns to represent q and the retrieved passages in a representation space that makes it possible to match documents using the inner product operation, while the reader learns how to utilize the retrieved passage for downstream prediction. Recent work by Jiang et al. (2022) shows how it is also possible to model this interaction using a single transformer model (Vaswani et al., 2017) with a retrieval as attention technique, as compared to having separate encoders and readers. note that it is computationally impractical to learn to make predictions conditioned on a retrieval corpus from scratch and thus proposed to pre-train the encoders with an inverse cloze task (ICT) in order to "prime" the model for retrieval. This is also done in Sachan et al. (2021). We outline more details on this and how we use ICT in the following section.
The most direct application of retrieval is to use a supervision signal such as OpenQA to train the context and passage encoders, such as in Khattab et al. (2021). However, Guu et al. (2020) show how this setup can also be used for language modeling. Using English Wikipedia as the retrieval corpus, they perform MLM conditioned on retrieved passages. Passages are retrieved using MIPS over an index that is asynchronously updated during training. They also use a masking technique that prioritizes named entities in order to incentivize the usage of world knowledge from the retrieved passages. A similar approach to language modeling is also done in Borgeaud et al. (2022), but over a corpus consisting of trillions of tokens. For both works, the LMs are trained for a number of steps with retrieval before being fine-tuned on a downstream task, as is the typical workflow with PLMs. Lewis et al. (2020b) demonstrate how the encoder-reader architecture can be used for language generation as well. They propose both a sequence model where the generation is conditioned on the same set of retrieved documents for the entire sequence and a token model where a different document is used per target token. The retriever is based on Dense Passage Retrieval (DPR) (Karpukhin et al., 2020), which uses the same general approach to retrieval as Guu et al. (2020), where a PLM like BERT (Devlin et al., 2019) is 203 used as the encoder. The reader model is swapped with a generator based on BART (Lewis et al., 2020a).
Method
As in Guu et al. (2020), the architecture of BRENT can be separated into two parts: a retriever and a reader. Our architecture is modified to improve the speed of training, to ensure that the retrieved documents affect the predictions, and to incentivize the retrieval of world knowledge from the retrieval corpus instead of the reader memorizing it. This section puts forth the architecture and these modifications.
Architecture
Retriever The first part of BRENT is the retriever, which consists of two components: the Query Encoder (Enc query ) and the Document Encoder (Enc doc ). Both have their own sets of weights and in our case have a BERT-style architecture and tokenizer. However, these can be initialized with other types of dense representation learners and could potentially also share weights for faster training.
The retriever receives as input the query, q, which is the masked sentence from the pre-training corpus, and passes it to Enc query to get a dense representation. Enc doc encodes all the documents in the retrieval corpus. Once all the documents and the query are encoded, a similarity score is calculated between each document d and the query q:
sim(q, d) = Enc query (q) T Enc doc (d) √ h dim ,
where h dim represents the encoding dimension of the retriever encoders. Since the query and doc vectors are not normalized, the inner product can be very large. In order to stabilize the training, we scale the inner product by dividing it by the square root of the hidden dimension. Once all the similarity scores are calculated, we use softmax to create a probability distribution over all the documents for a given query:
p(d|q) = exp(sim(q, d)) d ′ ∈D exp(sim(q, d ′ ))
.
Finally, we create the inputs to the reader by appending the representations of each d to q, i.e. [q; d]. In other words, if we have a retrieval corpus D with N documents, then a single query generates N inputs to the reader -effectively multiplying by N the batch size passed to the reader. However, it is unfeasible to do this for the whole corpus, therefore we only retrieve the top-k documents based on the similarity scores.
Reader The reader is a single pre-trained language model taking as input the document d appended to query q ([q; d]). During continued pretraining, the reader is optimized for MLM. For each input to the model, we get predictions on what the masked words in the query are -given the context provided by document d. Formally, each input generates the following probability for the correct masked words y:
p(y|d, q) = y i ∈Mq p(y i |d, q),
where y i is the i-th masked word in query q and M q is the set of all masked words in q. However, we want p(y|q). Therefore, for a query q, we need to do k forward passes to get all the p(y|d, q). Finally, to obtain p(y|q) we marginalize:
p(y|q) = d∈D p(y|d, q) p(d|q).
Loss With p(y|q) we can calculate the loss. During the loss function (cross-entropy) derivation, the error backpropagation is spread between the reader and the retriever. For the reader, this is the same as for any transformer-based model trained on the MLM task except that it averages over the batch size, number of retrieved documents, and the number of masked tokens. For the retriever, it is updated based on whether p(y|d, q) was better or worse than p(y|q). Specifically, if p(y|d, q) is higher than p(y|q), then the similarity score between q and d should increase. This can be seen with the following equation:
∇ θ log p(y|q) = d∈D u(d)p(d|q)∇ θ sim(d, q) u(d) = p(y|d, q) p(y|q) − 1 ,
where θ represents the parameters of Enc doc . 2 The same derivation applies to the parameters of Enc query .
Null Document
As pointed out in Guu et al. (2020), not all masked words need world knowledge to be predicted correctly. Therefore, we also add a null document appended to the query q. There are two ways to encode the null document. The first, and most obvious, is to pass the empty string to Enc doc and use the resulting encoding as the null document. However, we use a parameter tensor initialized with all zeros instead. This saves us one forward and backward pass of Enc doc per step, without affecting performance.
Corpus
We use a snapshot of the Norwegian Wikipedia from October 2022 as our corpus, limited to the Bokmål written standard. We pre-process the articles into chunks of token length 128, padding the last chunk of each article so that no chunk contains text from two different sources. After processing, the corpus consists of 735 000 documents, with an average number of words per chunk being about 100 (µ = 102, σ = 25). We use this corpus for sampling sentences to mask for MLM and as a retrieval corpus during continued pre-training.
Search index
As described in our architecture, we use the documents d in the retrieval corpus D to improve the model's predictions. Ideally, we would use all the documents of the retrieval corpus to make the prediction. Then, the model would assign close to zero probabilities to most documents, while simultaneously having access to all documents, and therefore identifying the most relevant ones. However, this is not feasible, as it would require a very high amount of resources (which would keep increasing as we increase our retrieval corpus) and be unreasonably time-consuming. Therefore, we instead only retrieve the top-k documents in terms of similarity score. To be able to efficiently retrieve and find these documents, we use a search index. We build this index using the encoding of the documents produced by Enc doc . Since we update Enc doc at every backward pass, it follows that we should re-index the documents at each backward pass. However, this is too time-consuming and we therefore only reindex each s steps. We want to note here that there has been recent work on how to more efficiently retrieve from such an index (Alon et al., 2022;He et al., 2021). Since we use the same corpus for both MLM training and retrieval, the first retrieved document is often the same document from which the query comes from, as this will naturally have a high similarity score. To avoid directly giving our model the answer with the unmasked token in it, we make sure to remove this document.
Inverse cloze task
We warm up the encoders for both the query and the document with the ICT task from Lee et al. (2019) on 68k Wikipedia article introductions limited to 128 tokens, from a snapshot from October 2020. For each pass, the model must predict the relevant pseudo-document for a pseudo-question from a set of distractors. The question is a random sentence and the document is the text surrounding it, the distractors are sampled from the same batch.
Span Masking
For the MLM task, we combine both salient masking (Guu et al., 2020), where only named entities and dates that require world knowledge are masked, and random masking. We identify entities using an off-the-shelf named entity recognizer and dates with a simple parsing algorithm. 3 We use 15% salient masking, making sure to mask at least one salient span for each sample, and 3.75% random span masking (which is 25% of 15%). By doing this, we encourage the network to learn to retrieve spans requiring world knowledge while ensuring that the model is still able to model linguistic features.
Experiments
We evaluate BRENT on a wide range of Norwegian NLP tasks. We do this both without retrieval using the extracted reader, and with retrieval turned on using the full model. By doing this, we highlight both the improved capacity of the reader to use context and show how retrieval in general affects performance on NLP tasks other than QA. This section describes the specific datasets and models we use during experimentation.
Models
NorBERT2 A baseline Norwegian LM, originating from Kutuzov et al. (2021).
NorBERT2 50k A NorBERT2 model trained for 50k additional steps on Wikipedia using MLM as described in Section 3.6, with a batch size of 1024. We show the performance of this model in order to get a more fair comparison, showcasing the improvements gained from the actual retrievalaugmented pre-training as compared to just doing regular pre-training for 50k more steps on the same corpora.
BRENT The entire model with retrieval turned on during fine-tuning. This is akin to a Norwegian version of REALM (Guu et al., 2020), but with our modifications. When subscripted, this indicates the source of the retrieval corpus, which could be either from Wikipedia or a task-specific dataset.
BRENT reader The reader model extracted after continued pre-training, used without any retrieval during fine-tuning on the downstream tasks.
Datasets
NorQuAD A Norwegian question answering dataset for machine reading comprehension (Ivanova et al., 2023) based on the SQuAD format (Rajpurkar et al., 2016). For a given question, the model must predict the correct span in a provided passage that answers the question. NorQuAD includes three domain splits: one sourced from the Norwegian Wikipedia (N = 2351), one from Norwegian news articles (N = 2398), and one split that combines both of them (N = 4749). We use an 80 − 10 − 10 split on all three domains for training, validation, and testing.
NoReC fine A fine-grained sentiment analysis dataset for Norwegian (Øvrelid et al., 2020). The texts are a subset of the NoReC dataset (Velldal et al., 2018), a multi-domain dataset of full-text professional reviews published in Norwegian online news sources. Each sentence in NoReC fine is annotated for sentiment holders, targets, polar expressions, expression polarities, and polar intensities. A version for targeted sentiment analysis (TSA) is released on GitHub where only the sentiment targets are labeled. 4
NoReC sent A sentence-level sentiment analysis dataset for Norwegian derived from NoReC fine Kutuzov et al., 2021). This dataset is generated by aggregating the entity sentiments in each sentence. The sentences are then labeled as either positive, negative, or neutral. We use the version only containing positive and negative sentiments. Both versions of the dataset (with and without neutral sentiment sentences) are available on GitHub. 5
Morpho-syntactic tasks This group of tasks is based on annotations from the Norwegian Dependency Treebank (Solberg et al., 2014), which were converted to the Universal Dependencies (UD) format by Øvrelid and Hohle (2016) and later enriched with named-entity types by Jørgensen et al. (2020). The resulting dataset is called NorNE and we use its latest version. 6 The source of NorNE is mostly news texts, but also government reports, parliament transcripts, and blogs. We evaluate the models on all available UD tasks for Norwegian Bokmål (UPOS and UFeats tagging, lemmatization, and dependency parsing; Nivre et al., 2016), 7 as well as on named entity recognition (NER). 8
Implementation details
Since running these models is resource intensive, we do not do a hyperparameter search. Instead, we base our hyperparameters on previous research where available. The following paragraphs outline the details of our experiments.
Search Index
We use the FlatIndexIP from the FAISS (Johnson et al., 2019) library to construct our index. This allows us to get the most relevant documents rather than an approximation of the best documents. We can do this since our corpus of documents is relatively small. We retrieve the top-7 documents and append the null document, in essence retrieving 8 documents in total. We reindex every 100 steps.
ICT We use NorBERT2 as the initialization for the ICT warmup. For this, we use a learning rate of 1 * 10 −4 and batch size of 128 for 10 epochs with early stopping on a single NVIDIA A100 GPU. After the warmup, these weights are then used as the starting point for Enc query and Enc doc in the 5 https://github.com/ltgoslo/norec_ sentence/ 6 https://github.com/ltgoslo/norne 7 We use the official evaluation script from CoNLL 2018 shared task (Zeman et al., 2018, https://universaldependencies.org/ conll18/evaluation.html). 8 We employ the evaluation method from SemEval 2013 task 9.1 (Segura-Bedmar et al., 2013), re-implementated in https://github.com/davidsbatista/ NER-Evaluation. Table 2: Results on the morpho-syntactic tasks: accuracy of UPOS and UFeats tagging, the accuracy of lemmatization, the labeled attachment scores of dependency parsing, F1 scores of named entity recognition, where the evaluation requires an exact match on both span and label. Results are reported as the mean and standard deviation over five random seeds.
retriever, while the reader uses NorBERT2 without any warmup.
Pre-training We then train BRENT for 50k steps with a batch size of 1024 divided over 128 AMD MI250X GPUs, 9 a learning rate of 2 * 10 −5 , using the AdamW optimizer, and a Cosine scheduler with a warmup, on the chunked Wikipedia corpus. A full description of the model and the hyperparameters can be found in Appendix A.2.
Fine-tuning We run all experiments using five different seeds and report the average result and standard deviation. For the fine-tuning of the retrieval-enhanced models, we test both with and without re-indexing, i.e., fine-tuning Enc doc . In both cases, we continue to fine-tune Enc query . When fine-tuning, we use a higher learning rate for the retriever as compared to the reader, since we saw experimentally that this obtained better results. When re-indexing, we do it every 100 steps and at the end of each epoch. Hyperparameters for all evaluation tasks can be found in Appendix A.3. We fine-tune all models on a single GPU. Table 3: Results of the binary sentiment analysis task (BSA) on the NoReC sent dataset and targeted sentiment analysis (TSA) on the NoReC fine dataset. Evaluation is on the test set and is based on the best model found during training. Results are reported as the mean and standard deviation over five random seeds. nri stands for no re-indexing. The NoReC subscript represents the training dataset being used as the retrieval corpus.
Results
Extractive QA
NorQuAD dataset. BRENT reader outperforms all other approaches on the three domain splits, especially with respect to the EM metric, which we explain by the salient masking technique. Although BRENT reader was only trained on Wikipedia, the improvement in performance is significant also for questions in the news category. Naturally, NorBERT2 50k also improves a bit compared to NorBERT2 on the Wikipedia split, but not by the same margin, and not at all on the news category. This indicates that BRENT reader actually learns to use context better, that this generalizes beyond 207 0 2000 4000 6000 8000 10000
Step 10 2 10 3 Perplexity NorBERT2 -50k BRENT Figure 2: Training perplexity during the first 10 000 steps. The values are smoothed with an exponential moving average, using α = 0.99.
the style of Wikipedia, and that this could not be achieved by simply training the same underlying LM for 50k additional steps on the same corpus with the same MLM setup.
Sentiment analysis
As for sentiment analysis, Table 3 shows that BRENT reader performs worse compared to the baseline of NorBERT2 on the binary sequence classification task, indicating that the continued pretraining with retrieval does not actually help for this task, but rather impedes performance. This is also the case for the NorBERT2 50k model, albeit with a smaller impediment to performance, suggesting that it might be the continued training on Wikipedia reducing the performance of the models on this task. When retrieval is used, as can be seen in the bottom half of Table 3, the performance is better, but still short of the baseline. For TSA, the reader performs on par with the baselines but turning retrieval on substantially decreases performance. When retrieving from a corpus during finetuning, our model retrieves reviews that are related with respect to inner product similarity, not necessarily sentiment. If classifying a negative review of a TV, our model could end up retrieving another review about some other electronic apparatuswhich might be positive. This is clearly not helpful for the task at hand. In order to teach the retrievers to retrieve based on sentiment, we would need a bigger dataset to fine-tune on. Despite this, manual inspection shows that the retrieved contexts are sometimes very relevant for the query when the retrieval corpus is NoReC. When the retrieval corpus is Wikipedia, however, the contexts are of low relevance. Examples of queries and retrieved contexts for BRENT Wiki and BRENT NoReC on binary sentiment analysis (BSA) can be found in Appendix A.1.1 and Appendix A.1.2.
We also note that not re-computing the search index decreases performance. However, as performance is relatively similar, it might not be worth it as re-indexing is a lot more resource-demanding. With Wikipedia as the retrieval corpus on our computing setup, TSA fine-tuning takes about 7 hours with re-indexing, compared to 2.5 hours without. Table 2 shows the results of the reader compared to the baselines on a series of Norwegian tokenlevel labeling tasks. BRENT reader performs on par with the baseline models, which strengthens our hypothesis that the continued pre-training with retrieval does not impede the model's ability to perform morpho-syntactic tasks while simultaneously increasing performance on extractive QA. This claim is further supported by the fact that the same happens with NorBERT2 50k , which indicates that adding the retrieval is no worse than just continuing to do MLM over the same corpus for additional steps. Figure 2 shows the perplexity values of BRENT and NorBERT2 50k during the first 10k steps of continued pre-training on the Wikipedia corpus. After the initial convergence phase, NorBERT2 50k establishes itself on values around 40, while BRENT sits at around 20. As we do mainly salient masking, perplexity is a proxy for how well the models predict the correct named entities and dates. The difference between the two shows how retrieval is helpful for predicting masked entities.
Morpho-syntactic
Analysis of the pre-training
Ablations
As with other retrieval-augmented LMs, BRENT is a pipeline model -consisting of multiple parts that interact according to a series of design choices that impact the outcome. Due to the computational cost of pre-training, it is not feasible to quantitatively determine the effect of all these choices, resulting in a poor understanding of some aspects of these models. To mitigate this, we study the effect of some of these choices with respect to the overall loss during pre-training with a series of ablations. We do this for a reduced number of steps, but with 208 0 500 1000 1500 2000
Step 3 the same retrieval corpus and with the same GPU setup as described in Section 3.3 and Section 4.3. Figure 3 shows the effect of the ICT warmup task with respect to the loss for 2000 steps. When ICT is turned off, Enc doc and Enc query are initialized with the same weights as the reader. As can be seen from the figure, the loss converges slower when the ICT task is not used, but it is quickly matching the setting when it is used. Guu et al. (2020) claims that without ICT one would encounter a cold-start problem where the retrieved documents will likely be unrelated to the query at the beginning of training, causing a cycle where the encoders do not receive meaningful gradients. We find that this is not the case and that the effect of ICT warmup is minimal.
ICT
The effect of the null document
As mentioned in Section 3.2, we use a parameter tensor initialized with all zeros for representing the null document. This is optimized jointly with the rest of the weights. Figure 3 shows how the model behaves when the null document is removed, which is done by making the probability of the null document zero, as compared to the test model which has it included. Contrary to Guu et al. (2020), we find that the effect of the null document is questionable. It makes sense to have a "sink" to use when no retrieval is necessary, but we do not find the null document to fulfill this need. Figure 4 shows how the number of retrieved documents influences training with respect to loss. For the first 2 000 training steps, k = 16 converges a bit quicker than the k = 8. However, we see that the result is minimal after that point, which is also the conclusion in Guu et al. (2020). Given that it is more computationally expensive to train with a higher k and that the gain of going from 8 to 16 is negligible, we keep k at 8.
Varying the number of documents to retrieve
Conclusion
We develop the first Norwegian retrieval augmented language model, BRENT, based on the REALM method proposed by Guu et al. (2020). The model uses an encoder-reader architecture, and we train it on a relatively small corpus consisting of 735k Wikipedia documents. In addition to the model itself, our contribution has been to demonstrate how the use of continued pre-training with retrieval benefits the context utilization of the reader, which we extract from the pipeline. The reader performs better than comparable baselines on the extractive QA task without losing performance on morpho-syntactic tasks. We also evaluate our full retriever model on sentiment analysis with two different corpora as the retrieval corpus, but here we observe a decrease in performance overall. Contrary to some previous work, our ablation studies find that the effect of having a null document and using ICT as a warmup task is minimal.
209 7 Future work A future direction for our work is to study in greater detail how retrieval influences the language modeling task. In particular, we would like to train a retrieval model from scratch. Another direction, which has also been pointed out in related work, is to experiment with cross-lingual retrieval, especially in the case where the retrieval corpus is from a high-resource language. This would be useful in scenarios where a large knowledge source like English Wikipedia could be used to augment a lower resource language, like Norwegian, which does not have such an extensive source available. Examples of retrieved contexts from the BRENT Wiki model fine-tuned on the BSA task with Wikipedia as the retrieval corpus.
• Query: Men så kommer de skjaerende lydene 'But then the squeaky sounds appear' -Retrieved context: Øynene 'The eyes'.
• Query: Broen går seg vill i sitt eget ønske om a vaert artsy '"The Bridge" is lost in its own wish to be "artsy"' -Retreived context: Sverige 'Sweden'
A.1.2 NoReC examples
Examples of retrieved contexts from the BRENT NoReC model fine-tuned on the BSA task with the N oReC dataset as the retrieval corpus.
• Query: Men så kommer de skjaerende lydene 'But then the squeaky sounds appear'
-Retrieved context: Allerede under første låt får vi slengt alle klisjeene i trynet. 'Already during the first song we are hit in the face with all the clichés'.
• Query: Begeistringen var uvanlig stor og applausen deretter da det 70 minutters lange verket var fullført 'The enthusiasm was unusually great and so was the applause that followed when the 70-minute long piece was over'.
-Retreived context: En helt utrolig konsertopplevelse 'A wonderful concert experience'.
• Query:Å vaere eksperimentell er ikke positivt i seg selv; de mange sjangrene og retningene i musikken gjør helehetsinntrykket rotete og meningsløst 'Being experimental is not positive in and of itself; the many genres and directions makes the music seem messy and meaningless'.
-Retrieved context: Automatisk to-soners klimaanlegg 'Automatic two-zone aircondition'.
A.2 Model
The hyperparameters used for the continued pretraining can be found in Table 4.
A.3 Hyperparameters
A.3.1 NorQuAD
For comparison, we use the same set of hyperparameters as in Ivanova et al. (2023), visible in Table 5.
A.3.2 Sequence labeling
For the task of targeted sentiment analysis, we finetune and report the average test results over five runs, from the epoch providing the best results on the development set. Hyperparameters can be found in Table 6.
A.4 Binary Sentiment Analysis
For the task of binary sentiment analysis, we finetune for three epochs and select the best model based on the development set's f1 score. We average our test results over five runs. Hyperparameters can be found in table Table 7.
A.5 Morpho-syntactic
For morpho-syntactic tasks, we fine-tune for 10 epochs and select the best model based on the average performance on the development split. We average our test results over five runs. Hyperparameters can be found in Table 8.
Figure 1 :
1query Gauss regnet ut den nøyaktige posisjonen hvor en kunne forvente å observere [MASK] igjen Allerede som barn viste Gauss seg som svaert begavet ... ledet Gauss til studiet av dvergplaneters bevegelse Gauss ble professor i astronomi i 1807 og var direktør for observatoriet i. The proposed architecture, based on the REALM method fromGuu et al. (2020).
Figure 3 :
3Loss curves of having no null document or no ICT warmup compared to the tested model. The values of all three runs are smoothed with an exponential moving average, using α = 0.99.
Figure 4 :
4Loss curves of the first 2 000 training steps with varying number of retrieved documents k. The values are smoothed with an exponential moving average, using α = 0.99.
Table 1 :
1NorBERT2 50k 59.14 ±0.55 73.98 ±1.05 64.89 ±1.44 77.22 ±0.57 63.88 ±0.49 77.05 ±0.55 BRENT reader 62.57 ±1.77 76.45 ±1.40 68.10 ±2.87 80.40 ±1.71 66.56 ±1.36 80.01 ±1.16Results on different domains of the NorQuAD dataset. Results are reported as the mean and standard deviation over five random seeds. *Human performance is the mean performance of two annotators as reported inIvanova et al. (2023)) Model
Wiki
News
All
EM
F1
EM
F1
EM
F1
Human*
72.65
88.84
83.61
93.43
78.13
91.14
NorBERT2
57.76 ±1.15 71.89 ±0.89 64.05 ±1.27 76.93 ±1.15 64.64 ±1.40 77.86 ±0.65
Model
UPOS
UFeats
Lemma
LAS
NER
NorBERT2
98.65 ±0.04 97.58 ±0.06 98.18 ±0.03 93.15 ±0.05 88.13 ±0.34
NorBERT2 50k 98.64 ±0.04 97.54 ±0.04 98.12 ±0.06 93.10 ±0.22 88.41 ±0.45
BRENT reader
98.62 ±0.06 97.55 ±0.02 98.09 ±0.04 92.96 ±0.15 87.70 ±0.49
Table 1
1shows the exact match (EM) and token-
level F1 scores of different approaches on the
9 These resources were made available to us
through the EuroHPC JU project:
https://www.
lumi-supercomputer.eu/
Table 4 :
4Hyperparameters for the continued pre-training of both BRENT and NorBERT2. k represents the number of documents retrieved including the null document.Hyperparameter
Value
Batch size
16
Epochs
3
lr
5 * 10 −5
Scheduler
Linear
Optimizer
AdamW
Seeds
[42, 437, 4088, 3092, 9720]
Table 5 :
5Hyperparemeters for fine-tuning on the NorQuAD datasetHyperparameter
Value
Batch size
32
Epochs
8
lr reader
5 * 10 −5
lr retriever
1.5 * 10 −4
k
4
re-indexing frequency 100 steps + end of epoch
Scheduler
Linear
Optimizer
AdamW
Seeds
[101, 202, 303, 404, 505]
Table 6 :
6Hyperparemeters for fine-tuning on the NoREC fine dataset. k represents the number of documents retrieved including the null document. The lr reader is for both the retrieval and non-retrieval models.213
Table 7 :
7Hyperparemeters for fine-tuning on the NoREC sent dataset. k represents the number of documents retrieved including the null document. The lr reader is for both the retrieval and non-retrieval models.Hyperparameter
Value
Batch size
32
Epochs
10
LR reader
1 * 10 −4
LR heads
1 * 10 −3
Scheduler
Cosine
Optimizer
AdamW
Seeds
[1234, 2345, 3456, 4567, 5678]
Table 8 :
8Hyperparemeters for fine-tuning on the morpho-syntactic tasks. k represents the number of documents retrieved including the null document. The learning rate is different for the fine-tuned language model and for the classification heads.
The full derivation can be found in the appendix ofGuu et al. (2020), where z = d, x = q and f represents the function sim
spaCy: https://spacy.io/
https://github.com/ltgoslo/norec_tsa
AcknowledgementsParts of the work documented in this publication have been carried out within the NorwAI Centre for Research-based Innovation, funded by the Research Council of Norway (RCN), with grant number 309834.
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| [
"https://github.com/ltgoslo/brent",
"https://github.com/ltgoslo/norec_",
"https://github.com/ltgoslo/norne",
"https://github.com/davidsbatista/",
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[
"THE PRESCRIBED MEAN CURVATURE MEASURE EQUATION IN NON-PARAMETRIC FORM",
"THE PRESCRIBED MEAN CURVATURE MEASURE EQUATION IN NON-PARAMETRIC FORM"
] | [
"Gian Paolo Leonardi ",
"Giovanni E Comi "
] | [] | [] | We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data, and show existence and several properties of BV solutions under natural assumptions on the prescribed measure. Our approach is direct, as it does not rely on approximate solutions, and requires the combination of various ingredients, including the theory of λ-pairings, a new refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct an example of non-uniqueness showing the sharpness of the continuity assumption for the validity of the maximum principle.2020 Mathematics Subject Classification. Primary: 49Q10. Secondary: 35J93, 49Q20, 46N10. Key words and phrases. prescribed mean curvature, functions of bounded variation, divergence-measure fields, Gauss-Green formulas, convex analysis. This work has been financially supported by GNAMPA -INdAM. The first author is particularly grateful to Lorenzo Brasco and Giorgio Saracco for their support and encouragement during the preparation of the paper. | null | [
"https://export.arxiv.org/pdf/2302.10592v1.pdf"
] | 257,050,809 | 2302.10592 | e0447eb41e953af15a0339294f6f806850fb5228 |
THE PRESCRIBED MEAN CURVATURE MEASURE EQUATION IN NON-PARAMETRIC FORM
21 Feb 2023
Gian Paolo Leonardi
Giovanni E Comi
THE PRESCRIBED MEAN CURVATURE MEASURE EQUATION IN NON-PARAMETRIC FORM
21 Feb 2023
We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data, and show existence and several properties of BV solutions under natural assumptions on the prescribed measure. Our approach is direct, as it does not rely on approximate solutions, and requires the combination of various ingredients, including the theory of λ-pairings, a new refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct an example of non-uniqueness showing the sharpness of the continuity assumption for the validity of the maximum principle.2020 Mathematics Subject Classification. Primary: 49Q10. Secondary: 35J93, 49Q20, 46N10. Key words and phrases. prescribed mean curvature, functions of bounded variation, divergence-measure fields, Gauss-Green formulas, convex analysis. This work has been financially supported by GNAMPA -INdAM. The first author is particularly grateful to Lorenzo Brasco and Giorgio Saracco for their support and encouragement during the preparation of the paper.
Introduction
Let Ω ⊂ R n be a bounded open set, and let µ be a signed Radon measure on Ω with finite total variation (µ ∈ M(Ω) in short). Our aim is to study the Prescribed Mean Curvature Measure equation div ∇u
1 + |∇u| 2 = µ in Ω,(PMCM)
and in particular to determine the appropriate weak formulation, as well as the natural assumptions on the prescribed measure µ leading to existence of solutions in BV (Ω).
1.1. Historical notes. The left-hand side of (PMCM) is the minimal surface operator applied to u. When u ∈ C 2 (Ω), the minimal surface operator evaluates the mean curvature of the graph of u at any point (x, u(x)). In the case µ = H dx with H(x) Lipschitz-continuous on Ω, (PMCM) reduces to the classic prescribed mean curvature equation in non-parametric form, which has been considered by several authors in the past (see, e.g., [14,[16][17][18]23]). The mathematical theory of capillarity, with its long history starting from the seminal works of Young, Laplace, and Gauss, represents one of the main motivations for the study of prescribed mean curvature problems. However, a mathematical model of capillarity in presence of singular or concentrated forces has not yet been fully developed, although it would be of interest not only from a theoretical perspective, but also in view of applications like, e.g., the study of waterwalking devices [20].
A necessary condition for the existence of a classical solution to (PMCM) can be formally derived by integrating the equation on smooth subdomains. Indeed if we assume that u ∈ C 2 (Ω) satisfies (PMCM), then setting for any nonempty A ⊂⊂ Ω with smooth boundary, where ν A , H n−1 , and P (A) denote, respectively, the outer normal to ∂A, the Hausdorff measure of dimension n − 1, and the perimeter of A. Note that the strict inequality is due to the fact that |T (u)(x)| < 1 for all x ∈ Ω. In the seminal paper [18], Giusti showed that, when µ = H dx with H and ∂Ω both Lipschitz, the necessary condition
A H(x) dx < P (A), ∀ A ⊂⊂ Ω smooth,(1.1)
is also sufficient for the existence of solutions. An extension of Giusti's result to a larger class of domains Ω, called weakly-regular, has been recently obtained in [21]. The method of proof goes as follows. One first distinguish between two cases: non-extremal and extremal. More precisely, the non-extremal case is verified when there exists a constant 0 < L < 1 such that (1.1) is replaced by the following stronger condition:
A H(x) dx ≤ L P (A), ∀ A ⊂⊂ Ω smooth. (1.2) Conversely, the extremal case holds when (1.1) holds true, and additionally the inequality is saturated on the whole Ω:
Ω H(x) dx = P (Ω) .
In the first, non-extremal case, one proceeds by exploiting the variational nature of the problem. In this case a solution can be found by minimizing an auxiliary energy functional of the form
J[u] = B 1 + |Du| 2 + Ω u(x) H(x) dx
among BV functions on a ball B ⊃⊃ Ω that coincide with a given function φ ∈ W 1,1 (B) (a weak Dirichlet datum) on B \ Ω. The functional J is clearly lower-semicontinuous with respect to L 1 convergence, as it is the sum of a lower semicontinuous functional -the generalized area of the graph of u -and a continuous integral term. What is crucial here is then the compactness of sequences of BV functions with equibounded energy, a direct consequence of the non-extremality assumption (1.2).
In the second, extremal case, one proceeds by approximating Ω from inside with a sequence of smooth subdomains falling within the non-estremal case, and then by showing that the corresponding sequence of variational solutions admits a locally convergent subsequence, up to vertical translations. Even though this last case, together with its characterisation, is the most subtle due to the lack of compactness, which reflects the possibility of having solutions with infinite BV norm, it is clear that the non-extremal case represents the first, fundamental step of the whole proof.
1.2. Description of the main results. The first goal of the paper is to propose the appropriate weak formulation of (PMCM). To this aim we shall focus on the case of µ being a non-extremal measure 1 with a possibly nontrivial singular part with respect to the Lebesgue measure. The natural questions about existence and properties of solutions to this weak formulation, as well as the variational approximation issue, will reveal themselves as particularly challenging from a technical point of view, as we will explain later on.
In order to single out a natural class of measures, we observe that the condition of being the distributional divergence of a sub-unitary vector field implies some restriction on µ, and in particular that µ ≪ H n−1 (see [6]). Consequently, we are forced to exclude measures that are even partially concentrated on sets of Hausdorff dimension less than n − 1. Other conditions on µ, like the property of belonging to the dual of the space BV (Ω), will be discussed later on.
The first step leading to the weak formulation of (PMCM) relies on an alternative characterisation of the vector field T (u). Following the approach of Scheven and Schmidt [25], one notes that the vector field T (u) is the unique, sub-unitary vector field satisfying the identity T (u) · ∇u = 1 + |∇u| 2 − 1 − |T (u)| 2 in a pointwise sense, which can be shown through an elementary computation (here we can assume u ∈ C 1 for more simplicity). Now one observes that the right-hand side of the identity can be interpreted as a difference of measures (the area of the graph minus the absolutely continuous measure with L ∞ density given by 1 − |T (u)| 2 ), so it would be natural to give a consistent measure-theoretic meaning to the scalar product between T (u) and ∇u. This is achieved through the theory of divergence-measure fields and their pairings with distributional gradients of weakly differentiable functions, initially proposed in the seminal work of Anzellotti [3]. A p-summable divergence-measure field on an open set Ω is a vector field F ∈ L p (Ω; R n ) such that its distributional divergence div F is a finite Radon measure on Ω, and we denote by DM p (Ω) their space. When p ∈ [1, ∞) such fields can be naturally paired with scalar functions u ∈ W 1,p ′ (Ω), where p ′ is the conjugate exponent of p, given that the scalar product F ·∇u is well posed in L 1 (Ω). On the other hand, the case p = ∞ is more complex and interesting to study, since p ′ = 1, so that the case of u ∈ BV (Ω) is included. The theory of pairings between essentially bounded divergence-measure fields and scalar functions with bounded variations was developed by many authors (see for instance [1,6,24]) until some recent contributions due to Scheven-Schmidt [25] and Crasta-De Cicco-Malusa [9,10]. As we will see later on, this framework will provide the solution to our problem.
In particular, let F ∈ DM ∞ (Ω), u ∈ BV (Ω) ∩ L ∞ (Ω), and λ : Ω → [0, 1] be a given Borel function. It turns out that uF ∈ DM ∞ (Ω) (see [6]), and then, following the approach of [10], one can define the λ-pairing distribution (F, Du) λ := div(uF ) − u λ div F , where u λ is defined H n−1 -almost everywhere as the convex combination of u + and u − (the upper and the lower approximate limits of u) by means of the coefficients λ and 1 − λ, respectively. Note that the λ-pairing is a distribution of order zero (i.e., a measure) (see [10,Proposition 4.4]). Moreover the classical pairing introduced by Anzellotti is the one corresponding to λ ≡ 1 2 , and in this case u λ coincides H n−1 -almost everywhere with u * , the precise representative of u. The need of considering a whole class of pairings extending Anzellotti's pairing is readily explained. Consider the natural counterpart of the auxiliary functional J [u] in the case of a prescribed mean curvature measure:
J µ [u] = B 1 + |Du| 2 + Ω u dµ .
The problem with this expression is that u ∈ BV (Ω) cannot be uniquely specified on null sets, which might be an issue when µ has a nontrivial singular part with respect to Lebesgue measure and thus the meaning of the integral of u with respect to µ is unclear. We thus need to choose a representative of u defined almost everywhere with respect to H n−1 . This can be done by a suitable choice of the Borel function λ, which thus enters the variational formulation as an extra variable:
J µ [u, λ] = B 1 + |Du| 2 + Ω u λ dµ .
We can then minimize J µ [u, λ] first with respect to λ, which corresponds to choosing λ = λ µ = the characteristic function of a Borel set satisfying (1− λ)µ = µ + . In other words, if µ = µ + − µ − with µ ± nonnegative measures, then λ µ is such that
Ω u λµ dµ = Ω u − dµ + − Ω u + dµ − .
This gives the natural candidate λ-pairing to be used in the weak formulation of (PMCM). One can show that if u is a weak solution of (PMCM), such that u ∈ W 1,1 (Ω), then the vector field T satisfying (1.3)-(1.5) coincides with ∇u √ 1+|∇u| 2 a.e. on Ω (we will actually prove in Section 8 that the same is true for any weak solution in BV (Ω)). Moreover, if µ = H dx with H Lipschitz (this assumption is made e.g. in [18]) then u ∈ C 2 (Ω) and thus the weak formulation reduces to the classical one. Moreover, owing to (1.3) and (1.5), by the divergence theorem we obtain a first version of the necessary condition for the existence of weak solutions of (PMCM), that is
|µ(A)| ≤ P (A) , ∀ A ⊂⊂ Ω smooth.
However, we stress that the strict inequality cannot be enforced as in (1.1) because in this general case one cannot guarantee that |T | < 1 H n−1 -a.e. on Ω (even when u ∈ W 1,1 (Ω), one cannot exclude that |T | = 1 on some Lebesgue null set). More general versions of the necessary condition, also depending on the (weak) regularity property of Ω, are proved in Lemma 4.1.
Another, natural assumption to be required on the prescribed measure is that it is admissible, i.e., that |µ| belongs to the dual of BV (Ω), see Definition 4.5. This is consistent with the fact that for any vector field T such that div T = µ, the pairing (T, Du) λ is well-defined for any BV function u and any Borel function λ with values in [0, 1]. In general, this assumption might be difficult to check, however a relevant subclass of measures in the dual of BV (Ω) is given by those whose absolutely continuous part admit a density in L q (Ω) for some q > n, and their singular part is of the form µ s = αH n−1 Γ + − βH n−1 Γ − , with Γ ± ⊂ Ω compact, disjoint, and with H n−1 (Γ ± ) finite, and α, β bounded nonnegative Borel functions (see Definition 4.6). Note that we do not require any rectifiability of Γ in the above definition of µ s .
In order to show existence of weak solutions of (PMCM) we apply as usual the direct method of Calculus of Variations, i.e., we prove coercivity (see Section 5.1) and lower semicontinuity (Theorem 5.1) for the functional
J µ [u] = 1 + |Du| 2 (B) + Ω u − dµ + − Ω u + dµ −
restricted to the class of BV functions on B that agree with some prescribed Dirichlet boundary datum φ ∈ W 1,1 (B) on B \ Ω. In particular, for technical reasons we are able to prove the semicontinuity theorem for measures of the form described above, even though we expect that the result holds for the larger class of non-extremal measures µ such that |µ| belongs to the dual of BV (Ω). The main technical difficulty in the proof of lower semicontinuity is due to the fact that while the area term 1 + |Du| 2 (B) is lower semicontinuous w.r.t. L 1 convergence, the other integral terms defining J µ [u] are not. One thus has to exploit the non-extremality assumption on µ to show that any error in the lower semicontinuity of the integral terms is actually compensated by a lower semicontinuity drop of the area term. To do that, we implement a one-sided uniform truncation of a sequence of functions obtained by contradiction, and converging in L 1 sense to a limit function u ∈ BV (Ω). This is done using Lemmas 5.2 and 5.3. Another very useful tool, that is used in the proof of Theorem 5.1, is Theorem A.2, a refined version of Anzellotti-Giaquinta approximation that guarantees not only the standard properties of the sequence of smooth approximations, but also a pointwise H n−1 -a.e. convergence to any given λ-representative u λ of u ∈ BV (Ω) (furthermore, this result seems also interesting in itself, and particularly with respect to the theory of λ-pairings cited before).
Once a minimizerū of the functional J µ exists, we prove in Section 6 that it is a weak solution of (PMCM). This follows from a duality argument showing the existence of a vector field T ∈ DM(Ω) such that the pair (ū, T ) satisfies (1.3), (1.4), and (1.5) with λ = λ µ .
In Section 7 we prove Theorem 7.2, a Gamma-convergence result for a sequence of functionals J µ j , in which µ j is a suitable sequence of C ∞ functions on Ω, such that the measures µ j L n converge to µ in weak- * sense, as j → ∞. The way of constructing µ j starting from µ is slightly delicate as it requires an Anzellotti-Giaquinta type regularization of a vector field F such that div F = µ (see Proposition 7.1), which is reminiscent of a technique used for instance in [11].
Section 8 collects some further properties of weak solutions of (PMCM). More specifically, Theorem 8.1 shows the uniqueness of the vector field T = T (u) associated with a weak solution u, while Theorem 9.1 states a maximum principle valid for continuous weak solutions, which implies the uniqueness of continuous weak solutions with the same Dirichlet boundary conditions (Corollary 9.2).
Then in Section 10 we first characterize radial solutions on annuli with a prescribed radial measure supported on a sphere, and then we apply this preliminary analysis to the more elaborate construction of a one-parameter family of radial solutions, for a suitable, non-negative and radial measure supported on two spheres, all attaining the same Dirichlet boundary datum while providing at the same time an example of non-uniqueness.
We finally collect in the Appendix some standard results concerning the approximation of the area functional in BV , for which we could not find an exhaustive reference.
Preliminaries
Through the rest of the paper, we work in an open set Ω ⊂ R n . We denote by M(Ω) the space of finite Radon measures on Ω, and by M H (Ω) the space of measures in M(Ω) that are absolutely continuous with respect to H n−1 ; that is,
M H (Ω) := {µ ∈ M(Ω) : |µ|(B) = 0 for all Borel sets B ⊂ Ω such that H n−1 (B) = 0}.
Given two measures µ 1 , µ 2 ∈ M(Ω), we say that µ 1 ≤ µ 2 on Ω if µ 1 (B) ≤ µ 2 (B) for all Borel sets B ⊆ Ω. If µ ∈ M(Ω) satisfies µ ≥ 0 on Ω, then we say that µ is nonnegative. Thanks to the Hahn decomposition theorem, we know that for any µ ∈ M(Ω) there exist two nonnegative measures µ + and µ − (the positive and negative part of µ, respectively), which satisfy µ = µ + − µ − and are concentrated on mutually disjoint sets. In particular, we also have |µ| = µ + + µ − .
We say that u ∈ L 1 (Ω) is a function of bounded variation, and we write u ∈ BV (Ω), if its distributional gradient Du is a vector valued Radon measure whose total variation |Du| is a finite measure on Ω. Following the notation of [2, Section 3.6], we say that a function u ∈ L 1 loc (Ω) has
approximate limit at x ∈ Ω if there exists z ∈ R such that lim r→0 − Br(x) |u(y) − z| dy = 0 ,(2.1)
and we denote by u(x) the value, z, of the approximate limit of u at x. For a given measurable set E, we denote by E 1 the set of Lebesgue points of E, that is, points where the approximate limit of the characteristic function χ E equals 1.
As customary, the approximate discontinuity set S u is defined as the set of points where the approximate limit does not exist. In addition, we say that x belongs to J u (the set of approximate jump points of u) if there exists a, b ∈ R, a = b, and ν ∈ S n−1 such that
lim r→0 − B + r (x,ν) |u(y) − a| dy = 0 lim r→0 − B − r (x,ν) |u(y) − b| dy = 0, (2.2) where B ± r (x, ν) := {y ∈ B r (x) : ±(y − x) · ν ≥ 0}. The triplet (a, b, ν)
is uniquely determined by (2.2) up to a permutation of (a, b) and a change of sign of ν, and we denote it by (u + (x), u − (x), ν u (x)). For the approximate traces u ± (x), we adopt the convention of having u + (x) > u − (x). We can actually extend the approximate traces also for x ∈ Ω \ S u , by setting
u + (x) = u − (x) = u(x).
Let k > 0 and T k be the truncation operator, that is, the 1-Lipschitz map defined as
T k (t) = k if t > k, t if |t| ≤ k, −k if t < −k.
We recall the statement of [2, Proposition 3.69 (c)]: if u ∈ L 1 loc (Ω) and x ∈ J u , then x ∈ J T k (u) if and only if T k (u + (x)) = T k (u − (x)), and in this case we have T k (u) ± (x) = T k (u ± (x)). Otherwise, x / ∈ S T k (u) and
T k (u)(x) = T k ( u)(x) = T k (u + (x)) = T k (u − (x)).
Hence, if we extend the approximate traces of T k (u) as above, we get
T k (u) ± (x) = T k (u ± )(x) for all x ∈ Ω \ (S u \ J u ).
In particular, this means that, if x ∈ J u , u − (x) < k and u + (x) > −k, then x ∈ J T k (u) and
T k (u) ± (x) = T k (u ± (x)). Instead, if u − (x) ≥ k, then T k (u) ± (x) = k; and, if u + (x) ≤ −k, then T k (u) ± (x) = −k.
Let now u ∈ BV loc (Ω). Thanks to [2,Corollary 3.80], the precise representative u * (x) is well defined for H n−1 -a.e. x ∈ Ω and
u * (x) = u + (x) + u − (x) 2 ,
hence u * = u on Ω \ S u . In addition, H n−1 (S u \ J u ) = 0, thanks to [2,Theorem 3.78]. For future reference, we recall the easy and well-known estimate
|T k (u) * | ≤ |u * |. (2.3)
Given a function u ∈ BV (Ω) we define 1 + |Du| 2 as the distributional area factor of the graph {(x, t) : u(x) = t} ⊂ Ω × R, as done in [19]. While the expression makes sense for L n -a.e.
x ∈ Ω for a function u ∈ W 1,1 loc (Ω), when u ∈ BV (Ω) we more generally define
U 1 + |Du| 2 := sup Ω η + u div φ dx : (φ, η) ∈ C 1 c (U ; R n × R), |(φ, η)| ≤ 1 ,
for any open set U ⊂ Ω. It is then easy to see that, if we denote the absolutely continuous part of Du by ∇u L n , and the singular part by D s u, we have
1 + |Du| 2 = 1 + |∇u| 2 L n + |D s u|. (2.4)
Given a Borel set E, the perimeter of E in Ω is defined by P (E; Ω) = |Dχ E |(Ω). When Ω = R n we simply write P (E). We refer to [2,22] for the definition and properties of ∂ * E, the reduced boundary of E, for which one has P (E; Ω) = H n−1 (∂ * E ∩ Ω), where by H n−1 we denote the Hausdorff measure of dimension n − 1.
If Ω has Lipschitz boundary, we denote by Tr ∂Ω (u) the trace of u over ∂Ω and recall that the trace operator Tr ∂Ω : BV (Ω) → L 1 (Ω) is linear, continuous, and surjective. In the following theorem we recall the well-known Poincaré-Wirtinger and Poincaré-trace inequalities. (2.5)
In particular, if E ⊂ Ω has finite perimeter in Ω, then
min Tr ∂Ω (χ E ) L 1 (∂Ω) , Tr ∂Ω (χ Ω\E ) L 1 (∂Ω) ≤ C ′ P (E; Ω) , (2.6) where C ′ = C|Ω| 1−n n P (Ω) is scale-invariant.
When Ω is a ball, we will write C P T in place of C ′ .
Proof. For the proof of (2.5) we refer to [27,Theorem 5.11.1]. The proof of (2.6) easily follows from (2.5) and the concavity of the map t → t 1− 1 n , where u = χ E and
G(u) = P (Ω) −1 ∂Ω Tr ∂Ω (u) dH n−1 .
Then, we recall the notions of divergence-measure field and of pairing between an essentially bounded function of bounded variation and an essentially bounded divergence measure field.
Definition 2.2. A vector field F ∈ L p (Ω; R n ) for some 1 ≤ p ≤ ∞ is called a divergence-measure field, denoted as F ∈ DM p (Ω), if div F ∈ M(Ω). A vector field F is a locally divergence-measure field, denoted as F ∈ DM p loc (Ω), if the restriction of F to U is in DM p (U ) for any U ⊂⊂ Ω open.
In the case p = ∞, F will be called a (locally) essentially bounded divergence-measure field.
It has been proved by Chen and Frid [6] (see also [15], for an improved proof), that, if F ∈ DM ∞ (Ω) and u ∈ BV (Ω) ∩ L ∞ (Ω), then the product uF belongs to DM ∞ (Ω); and an analogous result holds also locally. Furthermore, we recall the fact that, if F ∈ DM ∞ loc (Ω), then | div F | ≪ H n−1 , for which we refer to [6] and [26]. Hence, if u * is the precise representative of u, the measure u * div F is well defined, u * being defined H n−1 -a.e. on Ω.
We now introduce a suitable generalization of u * , i.e., the λ-representative of u. We fix a Borel function λ : Ω → [0, 1] and set
u λ = λu + + (1 − λ)u − .
Then, following the approach of [10] we define the notion of λ-pairing of a divergence-measure field and a scalar function of bounded variation. Definition 2.3. Given a vector field F ∈ DM ∞ loc (Ω), a scalar function u ∈ BV loc (Ω) ∩ L ∞ loc (Ω) and a Borel function λ : Ω → [0, 1], we define the λ-pairing between F and Du as the distribution (F, Du) λ given by
(F, Du) λ := div(uF ) − u λ div F. (2.7)
Thanks to [10], the λ-pairing is indeed a Radon measure. Roughly speaking, the notion of λ-pairing extends the classical dot product between F and Du. We also remark that in the case λ ≡ 1 2 one recovers the classical pairing first considered by Anzellotti in [3].
Definition 2.4. We denote by BV (Ω) * the dual of the space BV (Ω); that is, the space of linear functionals T : BV (Ω) → R for which there exists a constant C > 0 such that
|T(u)| ≤ C u BV (Ω) for all u ∈ BV (Ω).
It is well known that there are some elements in the dual of BV whose action can be represented as the integration of a suitable representative of the BV function against a Radon measure (see for instance [24]). This leads us naturally to the following definition.
Given µ ∈ M H (Ω), the linear functional
T µ (u) = Ω u * dµ for all u ∈ BV (Ω) ∩ L ∞ (Ω)
is well defined, although not necessarily continuous.
Definition 2.5. We denote by BV (Ω) * ,M the space of functionals T ∈ BV (Ω) * such that there exists µ ∈ M(Ω) for which T(u) = T µ (u) for all u ∈ BV (Ω).
It is proved in [24,Theorem 7.4] that a necessary condition to have T µ ∈ BV (Ω) * is that µ ∈ M H (Ω), and that a sufficient condition for the continuity of T µ on BV c (Ω) ∩ L ∞ (Ω) is that, additionally, |µ(U )| ≤ C P (U ) for all U ⊂⊂ Ω with smooth boundary and for some C > 0. With a little abuse of notation, we shall write µ ∈ BV (Ω) * ,M in the case T µ ∈ BV (Ω) * .
Given a pair (µ, λ) ∈ M H (Ω) × B b (Ω), it is possible to define
T µ,λ (u) = Ω u λ dµ for all u ∈ BV (Ω) ∩ L ∞ (Ω).
However, unless λ ≡ 1 2 , this mapping is not linear. It is worth remarking that for a relevant class of measures the functional T µ,λ can be extended to a continuous functional defined on the whole space BV (Ω), see Definition 4.5 and Corollary 4.9.
An approximation result
As customary, we say that any function ρ ∈ C ∞ c (B 1 (0)) such that ρ ≥ 0, ρ(−x) = ρ(x) and
B 1 (0) ρ dx = 1 is a standard mollifier. Now, given u ∈ BV (Ω), we have that T k (u) ∈ BV (Ω)
for any k > 0, hence the λ-representative of T k (u) is well-defined, and we can get convergence properties and bounds for T k (u) λ that are uniform in k. However, we notice that we cannot obtain the estimate |T k (u) λ (x)| ≤ |u λ (x)|, unless λ ∈ {0, 1 2 , 1}. As an example, let k > 0, λ ≡ 1 3 and
u(x) = 2k if x ∈ B(0, 1), − 5 4 k if x / ∈ B(0, 1).
Then, for all x ∈ ∂B(0, 1), we have u + (x) = 2k and u − (x) = − 5 4 k, so that
u λ (x) = 1 3 u + (x) + 2 3 u − (x) = 2 3 k 1 − 5 4 = − k 6
for all x ∈ ∂B(0, 1).
On the other hand, notice that
T k (u) λ (x) = 1 3 T k (u + )(x) + 2 3 T k (u − )(x) = k 3 − 2k 3 = − k 3
for all x ∈ ∂B(0, 1), so that we do not have |T k (u) λ | ≤ |u λ | on ∂B(0, 1). Nevertheless, we are able to obtain the following result (for which we also refer to [10, Proposition 3.4]).
|u * (x)| + λ(x) − 1 2 |u + (x)| + |u − (x)| . (3.1) Then |T k (u) λ (x)| ≤ M [u, λ](x) for H n−1 -a.e. x ∈ Ω. (3.2) Moreover, we have T k (u) λ (x) = u λ (x) (3.3) for all x ∈ Ω such that −k ≤ u − (x) ≤ u + (x) ≤ k, which in turn implies T k (u) λ (x) → u λ (x) for H n−1 -a.e. x ∈ Ω, as k → +∞. Proof. Assuming x ∈ Ω is such that −k ≤ u − (x) ≤ u + (x) ≤ k, we have T k (u) ± (x) = u ± (x).
This immediately implies that
T k (u) λ (x) = λ(x)T k (u) + (x) + (1 − λ(x))T k (u) − (x) = λ(x)u + (x) + (1 − λ(x))u − (x) = u λ (x)
and this shows (3.3). Since u + (x) and u − (x) are well-defined and finite for H n−1 -a.e. x ∈ Ω, we notice that (3.3) easily implies T k (u) λ (x) → u λ (x) as k → +∞ for H n−1 -a.e. x ∈ Ω. Now, in order to prove (3.2), we must show two separate inequalities. The first inequality directly follows from the definition of T k :
|T k (u) λ (x)| ≤ λ(x)|T k (u) + (x)| + (1 − λ(x))|T k (u) − (x)| ≤ λ(x)|u + (x)| + (1 − λ(x))|u − (x)| .
The second inequality is obtained by adding and subtracting T k (u) * and by using (2.3), that is, for H n−1 -a.e. x ∈ Ω we have
|T k (u) λ (x)| ≤ |T k (u) * (x)| + |T k (u) λ (x) − T k (u) * (x)| ≤ |u * (x)| + |(λ(x) − 1/2)T k (u) + (x) + (1/2 − λ(x))T k (u) − (x)| ≤ |u * (x)| + |λ(x) − 1/2| |u + (x)| + |u − (x)| .
By combining the two previous estimates, we get (3.2) as wanted.
Remark 3.2. We observe that (3.2) gives a finer upper bound for T λ k (u) than the one given in [10,Proposition 3.4]. In addition, we notice that, if λ ≡ 1 2 , then (3.2) reduces to the standard estimate
|T k (u) * (x)| ≤ |u * (x)| for H n−1 -a.e. x ∈ Ω.
The following theorem represents one of the key tools in the proofs of the main results of Section 5. Its proof is based on a delicate perturbation of the classic construction by Anzellotti and Giaquinta (see Theorem A.2). Theorem 3.3. Let λ : Ω → [0, 1] be a given Borel function. Then for every u ∈ BV (Ω) there exists a sequence {u λ k } k ⊂ C ∞ (Ω) such that we have the following:
(1) u λ k → u in BV (Ω)-strict, (2) u λ k → u λ H n−1 -a.e. on Ω, as k → +∞, (3) lim k→+∞ 1 + |Du λ k | 2 (Ω) = 1 + |Du| 2 (Ω). In addition, if u ∈ BV (Ω) ∩ L ∞ (Ω), then we have |u λ k (x)| ≤ 1 + 1 k u L ∞ (Ω)
for all x ∈ Ω and k ∈ N.
Proof. We first prove the theorem under the assumption u L ∞ (Ω) < +∞. It is enough to construct a sequence {u λ k } k of smooth functions that converge to u in BV (Ω)-strict, to u * = u = u λ H n−1 -a.e. on Ω \ J u , and to u λ locally in measure with respect to µ = H n−1 J u . Then, we conclude by extracting a suitable subsequence that converges µ-almost everywhere to u λ , hence H n−1 -a.e. on Ω.
Let {u ε } ε>0 be the smooth approximation of u given by Theorem A.2. We choose a sequence ε = 1 k , and we set u k := u 1 k for simplicity. Since u k (x) → u * (x) for H n−1 -a.e. x ∈ Ω, the idea is to define u λ k as a suitable perturbation of u k near the jump set J u , and then show that u λ k satisfies the convergence in measure stated above. The proof will be split into some steps.
Step one: local construction and estimates. We fix ε > 0 and a Borel set A ⊂⊂ Ω, then we consider the set
S = S(u, A, ε) = x ∈ A ∩ J u : u + (x) − u − (x) > ε .
Notice that H n−1 (S) < +∞, hence µ(S) < +∞. Up to choosing the parameter m in the proof of Theorem A.2 to be large enough, we can assume that u k = u * ρ k on A, where ρ k (|x|) = k n ρ(k|x|) is a standard mollifier with support in B 1/k , for k sufficiently large. For H n−1 -a.e. x ∈ S, we define the blow-up of u at x as the step function u x,∞ : R n → R defined by
u x,∞ (y) = u + (x) if (y − x) · ν u (x) ≥ 0, u − (x) otherwise. Therefore we have lim r→0 1 r n Br(x) |u(y) − u x,∞ (y)| dy = 0 . For H n−1 -a.e. x ∈ S we define the Borel function τ = τ k (x) ∈ [−1, 1] as the unique implicit solution of u x,∞ * ρ k (x + τ ν u (x)/k) = u λ (x) . (3.4)
Note that τ k (x) can be written as the composition of a continuous function, depending only on the mollifier ρ k , with the function λ(x). Fix δ > 0 to be later chosen, and consider the following perturbation of u k inside a ball
B = B r (x) centered at x ∈ S: u τ,B k (y) := φ r,δ (y − x)u k (y + τ ν u (x)/k) + 1 − φ r,δ (y − x) u k (y) , (3.5) where τ ∈ [−1, 1] and φ r,δ (z) ∈ [0, 1] is a radially symmetric cut-off function of class C ∞ , with compact support in B r (0) and such that φ r,δ (z) = 1 if |z| < (1− δ)r and ∇φ r,δ ∞ < 2 δr . Clearly, u τ,B k
is a smooth function obtained by locally "gluing" a suitable translation of u k with u k itself, it coincides with u k outside B, and satisfies
u τ,B k (x) = u k (x + τ ν u (x)/k) = u * ρ k (x + τ ν u (x)/k) . (3.6)
Its gradient is given by
∇u τ,B k (y) = ∇u k (y) + ∇φ r,δ (y − x) u k (y + v k ) − u k (y) + φ r,δ (y − x) ∇u k (y + v k ) − ∇u k (y) .
where we have set v k = τ k ν u (x). For the sake of simplicity, we also set B δ = B (1−δ)r (x), and
C δ = B \ B δ . We find that B |∇u τ,B k | ≤ B δ |∇u k (y + v k )| dy + C δ |∇u k | + 2 δr C δ |u k (y + v k ) − u k (y)| dy + C δ |∇u k (y + v k ) − ∇u k (y)| dy . (3.7)
Then, owing to the fact that |Du k |(A) ≤ |Du|(A + B 1/k (0)), the first term in the right-hand side of (3.7) can be estimated as follows:
B δ |∇u k (y + v k )| dy ≤ |Du|(B δ,k ) ,(3.8)
where we have set
B δ,k = B δ + B 2/k (0) = B (1−δ)r+2/k (x) .
On observing that |v k | ≤ k −1 , the third term in the right-hand side of (3.7) can be estimated as follows:
2 δr C δ |u k (y + v k ) − u k (y)| dy ≤ 2 kδr |Du|(C δ,k ) ,(3.9)
where C δ,k = C δ + B 2/k (0). Concerning the fourth term in the right-hand side of (3.7), we set ρ k,x (z) = ρ k (z − v k ) and denote by u k,z the average of u on B 2/k (z), so that we obtain
C δ |∇u k (y + v k ) − ∇u k (y)| dy ≤ z∈C δ y∈Ω u(y) ∇ρ k,x (z − y) − ∇ρ k (z − y) dy dz = z∈C δ y∈B 2/k (z) (u(y) − u k,z ) ∇ρ k,x (z − y) − ∇ρ k (z − y) dy dz ≤ ∇ 2 ρ k ∞ k z∈C δ y∈B 2/k (z) |u(y) − u k,z | dy dz ≤ Ck n+1 z∈C δ y∈B 2/k (z) |u(y) − u k,z | dy dz ≤ Ck n z∈C δ |Du| B 2/k (z) dz ,
where we have used the fact that ∇ 2 ρ k ∞ ≤ Ck n+2 and, in the last step, the Poincarè-Wirtinger inequality on B 2/k (z) (note that in this last estimate, as well as in the next ones, we will denote by C a dimensional constant that can possibly change from one line to another). We can push further the estimate by noticing that
z∈C δ |Du| B 2/k (z) dz = χ C δ (z)χ B 2/k (z) (y) d|Du|(y) dz = y∈C δ +B 2/k z χ C δ ∩B 2/k (y) (z) dz d|Du|(y) = y∈C δ +B 2/k |C δ ∩ B 2/k (y)| d|Du|(y) ≤ C k n |Du| C δ + B 2/k = C k n |Du| C δ,k . This leads us to C δ |∇u k (y + v k ) − ∇u k (y)| dy ≤ C|Du| C δ,k . (3.10)
Consequently, if we plug (3.8), (3.9), (3.10) into (3.7), we obtain
B |∇u τ,B k | ≤ |Du|(B δ,k ) + C + 2 kδr |Du| C δ,k ,(3.11)
where C is a constant only depending on the dimension n. Note that as soon as k ≥ (rδ) −1 the inequality improves to
B |∇u τ,B k | ≤ |Du|(B δ,k ) + C|Du| C δ,k , where C is another dimensional constant.
Step two: from local to global.
Here we show how to use the local construction and the estimate (3.11) provided by Step one, to define a sequence of approximations that behaves well on a compact subset of S with large measure. To this aim we must first guarantee the continuity and the uniformity of some quantities that appear in Step one, and then apply the appropriate covering theorem.
By Lusin and Egoroff Theorems, for every η ∈ (0, 1) we can find a compact set S η ⊂ S satisfying the following properties:
(i) H n−1 (S η ) ≥ (1 − η)H n−1 (S); (ii) the functions λ, ν u , u + , u − restricted to S η are continuous; (iii) if we set ∆ η (r) := sup x∈Sη 1 r n Br(x) |u(y) − u x,∞ (y)| dy , we have lim r→0 ∆ η (r) = 0; (iv) there exists 0 < r δ,η < 1 such that |Du|(C δ,k (x, r)) ≤ 6(n − 1)δ |Du|(B r (x)) for all 0 < r < r δ,η , k > (δr) −1 , and x ∈ S η , where C δ,k (x, r)
is defined as in the previous step (here the dependence upon x and r is explicitly written for the sake of clarity).
We remark that showing (i), (ii), and (iii) is standard. As for (iv), we notice that the asymptotic behaviour of |Du|(C δ,k ) for k > (δr) −1 and as r → 0 is
|Du|(C δ,k ) = |D d u|(C δ,k ) + |D j u|(C δ,k ) = 1 + o(1) ω n−1 |u + (x) − u − (x)| (r + 2/k) n−1 − (r(1 − δ) − 2/k) n−1 = 1 + o(1) ω n−1 |u + (x) − u − (x)|r n−1 (1 + 2/(kr)) n−1 − ((1 − δ) − 2/(kr)) n−1 = 1 + o(1) (n − 1)ω n−1 |u + (x) − u − (x)|r n−1 δ [4/(kδr) + 1] ≤ 6(n − 1)δ |Du|(B r (x)) ,(3.12)
Which gives (iv) at once.
Fix now an open set U containing S η , then consider the family F of balls centered in S η and contained in U , with radius r so small that ∆ η (r) < ε. By Vitali-Besicovitch Covering Theorem, we can find a finite and mutually disjoint family of balls
{B i } N i=1 , with B i = B r i (x i ) ∈ F, 0 < r i < r δ,η ,
and N depending on δ, η and ξ, such that
H n−1 S η ∩ N i=1 B i ≥ (1 − η)H n−1 (S η ) ≥ (1 − η) 2 H n−1 (S) ,
where we have used property (i) in the last inequality.
Step three: total variation estimate. We set r 0 = min{r i : i = 1, . . . , N } and consider the sequence u λ k constructed by replacing u k with u τ,B k inside each ball B = B i , as obtained and described in the previous steps. Hereafter we show that the total variation of Du λ k is controlled by that of Du, up to an error that goes to zero as δ → 0 (hence as k → ∞). Indeed, if we set C i δ = B i \ B i δ , we can assume, up to a small perturbation of r i , that |Du|(∂C i δ ) = 0 for all i, hence thanks to the estimate (3.11) we obtain
|Du λ k |(Ω) ≤ |Du k | Ω \ N i=1 B i + N i=1 |Du λ k |(B i ) ≤ |Du k | Ω \ N i=1 B i + N i=1 |Du|(B i δ,k ) + C + 2 kδr i |Du|(C i δ,k ) . (3.13)
Let now assume k ≥ (δr 0 ) −1 . Thanks to (iv) and recalling that the balls B i are mutually disjoint, we have
N i=1 C + 2 kδr i |Du|(C i δ,k ) ≤ (C + 2) N i=1 |Du|(C i δ,k ) ≤ Cδ N i=1 |Du|(B i ) ≤ Cδ|Du|(Ω) . (3.14)
Thanks to Theorem A.2, we have the strict convergence of u k to u on Ω, hence by the lower
semicontinuity of |Du k | restricted to the open set N i=1 B i we obtain lim sup k→∞ |Du k | Ω \ N i=1 B i = |Du|(Ω) − lim inf k→∞ |Du k | N i=1 B i ≤ |Du| Ω \ N i=1 B i ,
hence by selecting k large enough we can enforce
|Du k | Ω \ N i=1 B i ≤ |Du| Ω \ N i=1 B i + δr 0 . (3.15)
By combining (3.13), (3.14), and (3.15) we finally get
|Du λ k |(Ω) ≤ |Du| Ω \ N i=1 B i + δr 0 + N i=1 |Du|(B i ) + Cδ|Du|(Ω) = (1 + Cδ)|Du|(Ω) . (3.16)
This shows that the total variation of Du λ k is arbitrarily close to the total variation of Du, up to choosing k large enough. This will eventually lead to the BV -strict approximation property (see Step five). Arguing in a similar way, we can obtain an analogous upper bound for the area functional. This is due to the coincidence of the singular parts of the area and the total variation functionals (see (2.4)) hence the previous construction leads to an estimate like (3.16) with 1 + |Du λ k | 2 and 1 + |Du| 2 replacing the total variation functionals.
Step four: pointwise closeness of u λ k to u λ on S η .
Let us fix B i = B r i (x i ) for i = 1, . . . , N , and choose x ∈ B i δ ∩ S η , where B i δ = B (1−δ)r i (x i ). We would like to prove that u λ k (x) is close to u λ (x). We have |u λ k (x) − u λ (x)| ≤ |u λ k (x) − u λ (x i )| + |u λ (x i ) − u λ (x)| ≤ |u λ k (x) − u x,∞ * ρ k (x + v k (x))|+ + |u x,∞ * ρ k (x + v k (x)) − u x i ,∞ * ρ k (x i + v k (x i ))|+ + |u x i ,∞ * ρ k (x i + v k (x i )) − u λ (x i )| + |u λ (x i ) − u λ (x)| =: A 1 + A 2 + A 3 + A 4 .
First of all, by (3.6) and (iii) we obtain
A 1 ≤ B 1/k (x+v k (x)) ρ k x + v k (x) − y |u(y) − u x,∞ (y)| dy ≤ k n ρ ∞ B 2/k (x) |u(y) − u x,∞ (y)| dy → 0 as k → ∞ . Then, A 3 = 0 by (3.4), recalling that v k (y) = τ (y) k ν u (y)
, while thanks to (ii) the terms A 2 and A 4 are close to 0 if x is close to x i , which in turn depends on the fact that r i is taken small enough. This shows that, by choosing r i small and k ≥ (δr 0 ) −1 large enough, we can enforce the required pointwise closeness.
Step five: conclusion. Let us fix three positive and infinitesimal sequences ε j , η j , δ j , as well as a monotone sequence A j ⊂⊂ Ω of Borel sets, such that j A j = Ω. Let U j be a sequence of open sets as in Step two, satisfying the extra condition |U j | < 2 −j for all j. For any integer j ≥ 1 we can apply the previous steps with A = A j (Step one) and U = U j (Step two), and select from the initial sequence {u k } k a suitable subsequence, that we do not relabel, whose elements can be locally perturbed according to the procedure described in Step one. We shall perform this construction iteratively, so that the sequence that we extract at the (j + 1)-th stage is also a subsequence of the one obtained at the j-th stage. By diagonal selection we obtain a sequence relabeled as u λ k . Owing to Step four, u λ k converges to u λ in H n−1 -measure on the whole jump set J u as k → +∞, while by the choice of U j we have that u λ k converges to u in L 1 (Ω). Hence, by
Step three, u λ k converges to u in BV (Ω)-strict, which is point (1) of the statement. Similarly, we deduce point (3). Up to a further extraction of a subsequence, we obtain the pointwise convergence H n−1 -a.e. on Ω. Finally, the bound |u λ k (x)| ≤ 1 + 1 k u L ∞ (Ω) for all x ∈ Ω follows immediately from the definition of u λ k in terms of u τ,B k and (3.5), thus concluding the proof of the theorem under the assumption u ∞ < +∞. To obtain the complete proof, we apply the previous steps to the sequence of truncations T m (u), and obtain (T m (u) λ k ) k for each m ∈ N. The diagonal sequence T k (u) λ k can then be easily shown to satisfy all the required properties. We state for later use a simple consequence of Theorem 3.3.
1 + |Du λ k | 2 ⇀ 1 + |Du| 2 in M(Ω).
Proof. The lower semicontinuity on open sets is an immediate consequence of the fact that u λ k → u in L 1 (Ω), thanks to point (1)
Admissible measures and a-priori estimates
From the weak formulation of the PMCM equation, see Definition 1.1, we can immediately obtain a necessary condition for the existence of a solution. We recall that Ω is said to be weakly regular if it is an open bounded set such that H n−1 (∂Ω) = H n−1 (∂ * Ω), or, equivalently,
H n−1 (∂Ω \ ∂ * Ω) = 0. Lemma 4.1. If u ∈ BV (Ω)
is a weak solution to the prescribed mean curvature measure equation, then µ satisfies the following conditions:
(1) |µ| ≪ H n−1 , (2) max{|µ(E 1 )|, |µ(E 1 ∪ ∂ * E)|} ≤ P (E), for any set of finite perimeter E ⊂⊂ Ω,(3)
if in addition Ω is weakly regular, then the previous condition is satisfied for all E ⊂ Ω.
Proof. The vector field T associated with u clearly belongs to DM ∞ (Ω); and so | div T | ≪ H n−1 by [26, Theorem 3.2]. Hence, by (1.5), we obtain (1). Then, (2) is an easy consequence of the Gauss-Green formulas for T ∈ DM ∞ (Ω) and sets of finite perimeter E ⊂⊂ Ω. Indeed, by [8, Theorem 3.2 and Corollary 3.7], there exist interior and exterior normal traces
(T i ·ν E ), (T e ·ν E ) ∈ L ∞ (∂ * E; H n−1 ) satisfying div T (E 1 ) = − ∂ * E (T i · ν E ) dH n−1 , div T (E 1 ∪ ∂ * E) = − ∂ * E (T e · ν E ) dH n−1 .
In addition, by (1.3), we have
T i · ν E L ∞ (∂ * E;H n−1 ) ≤ T L ∞ (E;R n ) ≤ 1, T e · ν E L ∞ (∂ * E;H n−1 ) ≤ T L ∞ (Ω\E;R n ) ≤ 1.
All in all, by (1.5), we get (2). Finally, if we assume Ω to be weakly regular, by [8, Corollary 5.5 and Remark 5.6] we have that the zero extension of T to R n , defined aŝ
T (x) := T (x) if x ∈ Ω 0 if x ∈ R n \ Ω ,
satisfiesT ∈ DM ∞ (R n ) and T L ∞ (R n ;R n ) ≤ 1. Thus, we can remove the assumption that E ⊂⊂ Ω in condition (2), and replace it with E ⊂ Ω.
We observe that these results are very similar to the characterization of essentially bounded divergence measure fields in R n given in [24].
In view of the variational approach to the PMCM equation, and with reference to Lemma 4.1, we shall introduce some definitions concerning measures in M(Ω).
Definition 4.2. A measure µ ∈ M(Ω) is said to satisfy the necessary condition for the PMCM equation if |µ(E 1 )| ≤ P (E) , ∀ E ⊂ Ω measurable, 0 < |E| < |Ω| . (4.1) Then, µ is non-extremal if there exists 0 < L < 1 such that |µ(E 1 )| ≤ L P (E) , ∀ E ⊂ Ω measurable. (4.2) Remark 4.3.
It is easy to notice that if µ ∈ M(Ω) satisfies the necessary condition for the PMCM equation, then |µ(B r (x))| ≤ nω n r n−1 , for any x ∈ Ω, and r > 0 small enough so that Remark 4.4. We notice that, if µ ∈ M(Ω) satisfies (4.2), then one can prove that properties (2) and (3)
sup |µ(U )| P (U ) : U ⊂⊂ Ω, ∂U smooth = min F L ∞ (Ω;R n ) : div F = µ .
Therefore, we can choose F ∈ DM ∞ (Ω) such that div F = µ and F L ∞ (Ω;R n ) ≤ L. Thus, the Gauss-Green formula [8, Theorem 3.2 and Corollary 3.7] immediately yields
µ(E 1 ∪ ∂ * E) = div F (E 1 ∪ ∂ * E) = − ∂ * E (F e · ν E ) dH n−1 ,
for all sets of finite perimeter E ⊂⊂ Ω. Thus, we obtain
|µ(E 1 ∪ ∂ * E)| ≤ F L ∞ (Ω\E;R n ) P (E) ≤ L P (E) , ∀ E ⊂⊂ Ω measurable.
We stress the fact that, in general, it is not possible to relax the condition E ⊂⊂ Ω to E ⊂ Ω, since we do not have a control of the normal trace of a field F ∈ DM ∞ (Ω) on ∂Ω. However, if Ω is weakly regular, then the zero extension of F to R n is a fieldF ∈ DM ∞ (R n ), see [8, Corollary 5.5, Remark 5.6], so that we can integrate by parts on any set of finite perimeter E ⊂ Ω. Therefore, arguing as above under the additional assumption that Ω is weakly regular, we can prove that
max{|µ(E 1 )|, |µ(E 1 ∪ ∂ * E)|} ≤ L P (E)
for all measurable E ⊂ Ω and for any measure µ satisfying (4.2).
Definition 4.5. We say that µ ∈ M(Ω) is admissible if |µ| ∈ BV (Ω) * .
Definition 4.6. We say that µ ∈ M(Ω) if
µ = h L n Ω + γ H n−1 Γ
with h ∈ L q (Ω) for some q > n, γ ∈ L ∞ (H n−1 |Γ ), and Γ ⊂ Ω a Borel set, such that there exist Λ, ρ > 0 for which
H n−1 (Γ ∩ B ρ (x)) ≤ Λρ n−1 ∀ 0 < ρ < ρ and ∀ x ∈ Γ . (4.3)
The reason for requiring (4.3) is that it ensures a continuity property of the upper and lower trace operators from BV to L 1 (Γ; H n−1 ), that will be crucial in what follows. In the next two lemmas we state some standard facts concerning measures in the dual of BV : for the reader's convenience we provide the full proofs. Proof. We start by recalling that ν ≪ H n−1 [24, Theorem 4.2, Theorem 8.1], then we observe that ν(Ω) < +∞, which follows from the hypothesis on ν and the fact that constant functions are in BV (Ω). By Theorem 3.3 there exist two sequences (u 0 k ) and (u 1 k ) (corresponding to the choices λ ≡ 0 and λ ≡ 1) of smooth functions in BV (Ω) converging respectively to u − and u + ν-almost everywhere on Ω. With a little abuse of notation, we shall write u − k = u 0 k and u + k = u 1 k , from this point onwards. For any fixed N ∈ N, the truncation operator T N is Lipschitz, so that the sequences of truncations
T N (|u ± k | converge to T N (|u ± |) ν-almost everywhere on Ω, as k → ∞. By Lebesgue's Dominated Convergence we obtain Ω T N (|u ± |) dν = lim k→∞ Ω T N (|u ± k |) dν ≤ C lim sup k→∞ T N (|u k |) BV (Ω) ≤ C lim sup k→∞ |u k | BV (Ω) ≤ C lim sup k→∞ u k BV (Ω) = C u BV (Ω) .
By monotone convergence we can take the limit as N → ∞ and conclude
Ω |u ± | dν ≤ C u BV (Ω) .
Next, we prove that an estimate similar to that of the previous lemma is true for the Hausdorff measure restricted to Γ, whenever Ω is Lipschitz, and assuming the density bound (4.3).
Γ |u ± | dH n−1 ≤ C u BV (Ω) for all u ∈ BV (Ω). (4.4) In particular, H n−1 Γ ∈ BV (Ω) * .
Proof. Let u ∈ BV (Ω).
Since Ω is a bounded open set with Lipschitz boundary, we know that the zero-extension of u to R n belongs to BV (R n ); and we denote it by u 0 . In particular (see for instance [27,Lemma 5.10.4]), there exists c = c(Ω) > 0 such that
u 0 BV (R n ) ≤ c u BV (Ω) . (4.5)
Since Γ satisfies (4.3), we can apply [2, Theorem 3.86], so that we have
Γ |u ± | dH n−1 ≤ C u 0 BV (R n ) (4.6)
where C > 0 is a constant depending only on H n−1 (Γ), ρ and Λ. Combining (4.5) and (4.6), we get (4.4), and this ends the proof. Then |µ| ∈ BV (Ω) * . In particular, we obtain u ± ∈ L 1 (Ω, |µ|) for all u ∈ BV (Ω), as well as
M [u, λ](x) ∈ L 1 (Ω; |µ|).
Proof. We recall that, if µ is admissible, then
µ = h L n Ω + γ H n−1 Γ
with h ∈ L q (Ω) for some q > n, γ ∈ L ∞ (H n−1 |Γ ), and Γ ⊂ Ω being a Borel set such that there exist Λ, ρ > 0 for which (4.3) holds. Thanks to Lemma 4.8, for all u ∈ BV (Ω) we obtain
Ω u * d|µ| ≤ Ω |u| |h| dx + Γ |u * | |γ| dH n−1 ≤ h L n (Ω) u L n n−1 (Ω) + C γ L ∞ (Γ,H n−1 ) u BV (Ω) ,
where C > 0 is a constant depending only on Ω, H n−1 (Γ), ρ and Λ. By the Sobolev embedding of BV (Ω) into L n n−1 (Ω) we deduce that |µ| ∈ BV (Ω) * , which, by Lemma 4.7, immediately yields u ± , M [u, λ](x) ∈ L 1 (Ω, |µ|). Finally, it is easy to notice that µ ∈ BV (Ω) * , since, arguing as above, we have
Ω u * dµ ≤ Ω |u * | d|µ| ≤ Ω |u| |h| dx + Γ |u * | |γ| dH n−1 ≤ C u BV (Ω) .
Remark 4.10. We notice that, if µ ∈ M(Ω) is such that |µ| ∈ BV (Ω) * , then we have also µ ∈ BV (Ω) * . Indeed, we see that
Ω u * dµ ≤ Ω |u * | d|µ| ≤ Ω |u| * d|µ| ≤ C |u| BV (Ω) ≤ C u BV (Ω) .
We underline that, in general, the converse is not true, as shown for instance in [24, Proposition 5.1]. Proof. Let λ : Ω → [0, 1] be fixed. Take N > 0 and let T N (u) λ k be the smooth approximation of T N (u) given by Theorem 3.3. We start by noticing that, thanks to Lemma 4.8, we know that condition (2b) implies (2a). Hence, without loss of generality, we can assume M [u, λ](x) ∈ L 1 (Ω; |µ|). Then, we observe that as k → ∞
T N (u) λ k (x) → T N (u) λ (x)
for |µ|-a.e. x ∈ Ω, since |µ| ≪ H n−1 and |T N (u) λ k (x)| ≤ N for every x ∈ Ω and k ≥ 1. Therefore we obtain
Ω T N (u) λ dµ = lim k→∞ Ω T N (u) λ k dµ,(4.9)
by Lebesgue's dominated convergence theorem with respect to |µ|, since the constant N is a summable majorant, given that |µ|(Ω) < ∞. We notice now that T N (u) λ k ∈ C ∞ (Ω), so that the superlevel and sublevel sets {T N (u) λ k > t} and {T N (u) λ k < t} are open sets with smooth boundary for L 1 -a.e. t ∈ R, by Morse-Sard lemma [22,Lemma 13.15]. In particular, we deduce that, for L 1 -a.e. t ∈ R,
{T N (u) λ k > t} 1 = {T N (u) λ k > t} and {T N (u) λ k < t} 1 = {T N (u) λ k < t}.
Then by using (4.9), the layer-cake representation, (4.7), the coarea formula, Theorem 3.3, the fact that Tr ∂Ω T N (u) λ k = Tr ∂Ω T N (u) H n−1 -almost everywhere on ∂Ω, and finally Lemma A.3, we get
Ω T N (u) λ dµ = lim k→∞ ∞ 0 {T N (u) λ k >t} dµ dt − 0 −∞ {T N (u) λ k <t} dµ dt ≤ lim sup k→∞ N 0 µ({T N (u) λ k > t}) dt + 0 −N µ({T N (u) λ k < t}) dt ≤ L lim sup k→∞ N −N |Dχ {T N (u) λ k >t} |(Ω) dt + N 0 ∂Ω Tr ∂Ω (χ {|T N (u) λ k |>t} ) dH n−1 dt ≤ L lim sup k→∞ ∞ −∞ |Dχ {T N (u) λ k >t} |(Ω) dt + +∞ 0 ∂Ω Tr ∂Ω (χ {|T N (u) λ k |>t} ) dH n−1 dt ≤ L lim k→∞ |DT N (u) λ k |(Ω) + ∂Ω |Tr ∂Ω (T N (u) λ k )| dH n−1 = L |DT N (u)|(Ω) + ∂Ω |Tr ∂Ω (T N (u))| dH n−1 ≤ L |Du|(Ω) + ∂Ω |Tr ∂Ω (u)| dH n−1 ,
since |DT N (u)| ≤ |Du| and |Tr ∂Ω (T N (u))| ≤ Tr ∂Ω (|u|). Hence, we obtain (4.8) by passing to the limit as N → +∞ again thanks to the Lebesgue's dominated convergence theorem, employing Remark 4.12. We stress that we could not directy apply the Cavalieri representation formula to Ω T N (u) λ dµ, since, a priori, we do not know whether
{T N (u) λ > t} = {T N (u) λ > t} 1 or {T N (u) λ > t} = {T N (u) λ > t} 1 ∪ ∂ * {T N (u) λ > t}.
Therefore, we would not be able to apply the assumption (4.7) and proceed with the proof. Instead, we see here the usefulness of Theorem 3.3 in ensuring that, for L 1 -a.e. t ∈ R,
{(T N (u)) λ k > t} 1 = {(T N (u)) λ k > t}, ∂ * {(T N (u)) λ k > t} = ∂{(T N (u)) λ k > t}, {(T N (u)) λ k < t} 1 = {(T N (u)) λ k < t}, ∂ * {(T N (u)) λ k < t} = ∂{(T N (u)) λ k < t}.
We conclude this section with a result concerning the pairing between a BV function and a field whose divergence is admissible, which is an extension of [10, Proposition 4.4] in our framework. and
|(F, Du) λ | ≤ 1 + |Du| 2 − 1 − |F | 2 L n on Ω. (4.11) Finally, if u ∈ BV (Ω) ∩ L ∞ (Ω) and {u λ k } k∈N ⊂ C ∞ (Ω) ∩ BV (Ω)
is the approximating sequence of Theorem 3.3, then we get the following convergence result:
(F · ∇u λ k ) L n ⇀ (F, Du) λ in M(Ω). (4.12)
Proof. By Lemma 4.7, we know that, if div F is admissible in the sense of Definition 4.5, then M [u, λ] ∈ L 1 (Ω; | div F |). Hence we can assume this weaker hypothesis, without loss of generality. Thanks to Proposition 3.1, we see that the truncation T N (u) for N > 0 satisfies
(T N (u)) λ (x) → u λ (x) as N → +∞ and |(T N (u)) λ (x)| ≤ M [u, λ](x) for | div F |-a.e. x ∈ Ω
Hence, we can exploit Lebesgue's Dominated Convergence Theorem with respect to the measure | div F | to deduce that u λ ∈ L 1 (Ω; | div F |). Therefore, [10, Lemma 3.2 and Proposition 4.4] imply that div(uF ), (F, Du) λ ∈ M(Ω) and that (4.10) holds true, thanks to the fact that F L ∞ (Ω;R n ) ≤ 1.
In order to prove (4.11), we first need to prove (4.12) for u ∈ BV (Ω) ∩ L ∞ (Ω). Thanks to (4.10), for all k ∈ N, we have
div(u λ k F ) = u λ k div F + (F · ∇u λ k )L n on Ω. Hence, Theorem 3.3 implies that div(u λ k F ) (Ω) ≤ ϕ L ∞ (Ω) 1 + 1 k u L ∞ (Ω) | div F |(Ω) + F L ∞ (Ω;R n ) sup k∈N ∇u λ k L 1 (Ω;R n ) .
Therefore, we conclude that {div(u λ k F )} k∈N is bounded sequence in M(Ω), and, since div(u λ k F ) converges to div(uF ) in the sense of distributions, we deduce that it also weakly converges in the sense of Radon measures. In addition, Theorem 3.3 and the Lebesgue theorem with respect to the measure | div F | imply that u λ k div F ⇀ u λ div F in M(Ω). All in all, we see that
(F · ∇u λ k )L n = div(u λ k F ) − u λ k div F ⇀ div(uF ) − u λ div F = (F, Du) λ ,
thus proving (4.12) for u ∈ L ∞ (Ω) ∩ BV (Ω). Now, we exploit Theorem 3.3 in order to construct the smooth approximation T N (u) λ k of the truncation T N (u), for all N > 0. Then, for all φ ∈ C c (Ω) with φ ≥ 0, thanks to (4.12) and Corollary 3.4 we have
Ω φ d(F, DT N (u)) λ + Ω φ 1 − |F | 2 dx = lim k→+∞ Ω φ(F · ∇T N (u) λ k + 1 − |F | 2 ) dx ≤ lim inf k→+∞ Ω φ (F, 1 − |F | 2 ) · (∇T N (u) λ k , 1) dx ≤ lim k→+∞ Ω φ 1 + |∇T N (u) λ k | 2 dx = Ω φ d 1 + |DT N (u)| 2 ≤ Ω φ d 1 + |Du| 2 .
Then we notice that (4.10) holds true for T N (u) and F , so that we get
Ω ϕ d(F, DT N (u)) λ = − Ω T N (u)F · ∇ϕ dx − Ω ϕ(T N (u)) λ d div F
for all ϕ ∈ C 1 c (Ω). In particular, this implies that the family of measures {(F, DT N (u)) λ } N >0 is uniformly bounded in N > 0, so that, up to extracting a subsequence, we may pass to the limit as N → +∞. However, the right hand side clearly converges to
− Ω uF · ∇ϕ dx − Ω ϕu λ d div F = Ω ϕ d(F, Du) λ ,
thanks again to Lebesgue's Dominated Convergence Theorem with respect to the measure | div F |. Hence, we deduce that
(F, DT N (u)) λ ⇀ (F, Du) λ in M(Ω) as N → +∞.
All in all, we can conclude that
Ω φ d(F, Du) λ = lim N →+∞ Ω φ d(F, DT N (u)) λ ≤ Ω φ d 1 + |Du| 2 − Ω φ 1 − |F | 2 dx.
Due to the fact that φ ≥ 0, this implies that
(F, Du) + λ ≤ 1 + |Du| 2 − 1 − |F | 2 L n on Ω.
By an analogous computation, in which the term Ω φ 1 − |F | 2 dx is subtracted instead of added, and φ is replaced with −φ, we obtain
− Ω φ d(F, Du) λ + Ω φ 1 − |F | 2 dx ≤ Ω φ d 1 + |Du| 2 , which implies (F, Du) − λ ≤ 1 + |Du| 2 − 1 − |F | 2 L n on Ω. All in all, we conclude that |(F, Du) λ | ≤ 1 + |Du| 2 − 1 − |F | 2 L n on Ω,
as desired.
5.
Minimizing the capillary-type functional J µ Taking into account Proposition 4.11 and Remark 4.4, we shall assume throughout this section that Ω is a bounded open set with Lipschitz boundary, and that µ is admissible in the sense of Definition 4.5, and satisfies the non-extremality condition (4.2).
We fix a ball B such that Ω ⊂⊂ B, and choose a function φ ∈ W 1,1 (B), then set
S(φ) = {u ∈ BV (B) : u = φ on B \ Ω} .
Let {Ω + , Ω − } be a Borel partition of Ω associated with the Hahn decomposition of µ, i.e., such that the positive and negative parts of µ satisfy µ + = µ |Ω + and µ − = −µ |Ω − . We set λ µ = χ Ω − and define the functional J µ :
BV (B) → [−∞, +∞] as J µ [u] = 1 + |Du| 2 (B) + Ω u λµ dµ if u ∈ S(φ) , +∞ otherwise. (5.1)
Recall that, according to the definition of u λ , we have
u λµ = u − · χ Ω + + u + · χ Ω − .
The functional J µ is not identically +∞. To see this, observe that the function z φ ∈ BV (B) defined by z φ = φ on B \ Ω and z φ = 0 on Ω satisfies z φ ∈ S(φ) and
J µ (z φ ) = 1 + |Dφ| 2 (B \ Ω) + ∂Ω |Tr ∂Ω (φ)| dH n−1 < +∞.
In the next subsections we shall discuss the lower semicontinuity and the coercivity properties of the functional J µ .
5.1.
Coercivity of J µ . Let us start by proving that J µ is coercive on S(φ). Since µ is nonextremal; that is, satisfies (4.2), by Proposition 4.11 applied with λ = λ µ we obtain
Ω u λµ dµ ≤ L |Du|(Ω) + ∂Ω |Tr ∂Ω (u)| dH n−1 ,
hence for all u ∈ S(φ) we get
J µ [u] ≥ |Du|(B) − L |Du|(Ω) + ∂Ω |Tr ∂Ω (u)| dH n−1 ≥ |Du|(B \ Ω) + (1 − L)|Du|(Ω) − L ∂Ω |Tr ∂Ω (u)| dH n−1 + ∂Ω |Tr ∂Ω (u − φ)| dH n−1 ≥ (1 − L) |Du|(Ω) + ∂Ω |Tr ∂Ω (u)| dH n−1 − ∂Ω |Tr ∂Ω (φ)| dH n−1 = (1 − L)|Du 0 |(B) − ∂Ω |Tr ∂Ω (φ)| dH n−1 ,(5.2)
where
u 0 (x) = u(x) if x ∈ Ω , 0 if x ∈ B \ Ω .
5.2.
Lower semicontinuity of J µ . In this subsection we prove the following Theorem 5.1 (Semicontinuity). Let µ ∈ M(Ω) be a measure satisfying the non-extremality assumption (4.2). Then the functional J µ is lower semicontinuous in the topology of L 1 (Ω).
A few comments on the assumptions of Theorem 5.1, before giving the proof, are in order. Ideally, one could argue that it would have been more natural to require, instead of µ ∈ M(Ω), that µ is admissible, i.e. such that its total variation belongs to BV (Ω) * . However, various technical difficulties arise when one consider real-valued measures whose negative and positive parts are "too wrapped together". The biggest one is that we could not anymore rely on the truncation argument based on the key inequality (5.5), that is proved in Lemma 5.3. Thus, the validity of the semicontinuity property under the weaker assumption |µ| ∈ BV (Ω) * is an open problem. We stress, however, that the class M(Ω) is rich enough to cover many interesting cases of measures with nontrivial singular parts.
In the next two lemmas we prove some crucial estimates. The first is an upper bound of the Lebesgue measure of the sub-levels and the super-levels of any function v ∈ S(φ) in terms of the function φ, the constant L, the energy value J µ (v), and the truncation parameter M .
Lemma 5.2. Let µ be non-extremal, and let v ∈ S(φ). Then for every M > 0 we have
{x ∈ Ω : |v(x)| > M } ≤ 1 C B (1 − L)M J µ (v) + ∂Ω |Tr ∂Ω (φ)| dH n−1 , (5.3)
where L ∈ (0, 1) is the constant appearing in the definition of non-extremality, and where C B denotes the Poincaré constant of B.
Proof. Let v 0 be the zero-extension of v outside Ω. By (5.2) we have
J µ (v) ≥ (1 − L)|Dv 0 |(B) − ∂Ω |Tr ∂Ω (φ)| dH n−1 ,
hence by Poincaré's inequality on B combined with Chebichev's inequality we obtain
J µ (v) ≥ C B (1 − L) Ω |v 0 | dx − ∂Ω |Tr ∂Ω (φ)| dH n−1 ≥ C B (1 − L)M |{x ∈ Ω : |v(x)| > M }| − ∂Ω |Tr ∂Ω (φ)| dH n−1 ,
whence the conclusion follows.
In the next lemma we consider a local (upper) truncation v of v ∈ S(φ) and we bound J µ ( v) from above in terms of J µ (v), up to a suitably estimated error.
v(x) = min(M, v(x)) if x ∈ U, v(x) otherwise.
Then we have
J µ ( v) ≤ J µ (v) + 2L 2 C B (1 + L)(1 − L) 2 M J µ (v) + ∂Ω |Tr ∂Ω (φ)| dH n−1 + (1 + L) ∂U (v − M ) + dH n−1 + M µ({v > M } ∩ U ) . (5.4)
In particular, if µ + (U ) = 0 then
J µ ( v) ≤ J µ (v) + 2L 2 C B (1 + L)(1 − L) 2 M J µ (v) + ∂Ω |Tr ∂Ω (φ)| dH n−1 + (1 + L) ∂U (v − M ) + dH n−1 (5.5)
Proof. First we notice that, by Theorem 3.3, we can assume v ∈ C ∞ (Ω) without loss of generality. Then (5.4) follows directly from
J µ ( v) ≤ J µ (v) + 2L 2 1 − L 2 {v > M } + (1 + L) ∂U (v − M ) + dH n−1 + M µ({v > M } ∩ U ) , (5.6)
thanks to Lemma 5.2. For the proof of (5.6) we observe that
J µ ( v) = 1 + |D v| 2 (B) + Ω v dµ = 1 + |Dv| 2 (B \ U ) + ∂U (v − M ) + dH n−1 + U 1 + |∇ v| 2 dx + Ω v dµ = 1 + |Dv| 2 (B \ U ) + ∂U (v − M ) + dH n−1 + U 1 + |∇v| 2 dx − {v>M }∩U 1 + |∇v| 2 dx + {v > M } ∩ U + Ω v dµ = J µ (v) + ∂U (v − M ) + dH n−1 − {v>M }∩U 1 + |∇v| 2 − 1 dx − {v>M }∩U (v − M ) dµ = J µ (v) + ∂U (v − M ) + dH n−1 + N .
Next we observe the following, elementary fact:
∀ a > 0, 1 + a 2 − 1 > La ⇔ a > 2L 1 − L 2 .
Consequently, we can set w = v − M and, after noticing that ∇w = ∇v almost everywhere on {v > M }, we obtain
N ≤ −L {v>M, |∇w|> 2L 1−L 2 }∩U |∇w| − {v>M }∩U w dµ = −L {v>M }∩U |∇w| + L {v>M, |∇w|≤ 2L 1−L 2 }∩U |∇w| − {v>M }∩U w dµ ≤ 2L 2 1 − L 2 {v > M } ∩ U − L {v>M }∩U |∇w| − {v>M }∩U w dµ .
At this point we can use the coarea formula on the first integral and the Cavalieri representation of the second integral, obtaining
N ≤ 2L 2 1 − L 2 {v > M } ∩ U − +∞ M L P ({v > t}; U ) + µ({v > t} ∩ U ) dt + M µ({v > M }) = 2L 2 1 − L 2 {v > M } ∩ U − +∞ M L P ({v > t} ∩ U ) + µ({v > t} ∩ U ) dt + L ∂U (v − M ) + dH n−1 + M µ({v > M }) ≤ 2L 2 1 − L 2 {v > M } ∩ U + L ∂U (v − M ) + dH n−1 + M µ({v > M }) ,
where in the last inequality we also used (4.2), so that (5.4) follows. Then (5.5) immediately follows as soon as µ + (U ) = 0, because in this case the last term in (5.4) is negative or zero.
We remark that Lemma 5.3 can be equivalently stated for a lower local truncationv. Since, as noticed above, the absolutely continuous part of µ does not create any problem for the continuity of the functional J µ , from this point onwards we shall focus on the "purely singular" case µ = γH n−1 Γ; that is, we shall assume h = 0.
Proof of Theorem 5.1. First we note that, thanks to Theorem 3.3, any v ∈ S(φ) can be approximated by a sequence of functions {v h } h ⊂ S(φ) ∩ C ∞ (Ω) in such a way that J µ (v) = lim h J µ (v h ). Therefore, in order to check the lower semicontinuity of J µ we can fix a function u ∈ S(φ) and a sequence of functions v h ∈ S(φ) ∩ C ∞ (Ω), such that v h → u in L 1 (Ω) as h → ∞. Then, we argue by contradiction, that is, we assume there exists ρ > 0 such that J µ (v h ) ≤ J µ (u) − Kρ for all h, and for a positive constant K that will be chosen at the end of the argument. The fact that µ ∈ M(Ω) allows to reduce the proof to an estimate either on Γ − , or on Γ + . For such an estimate, it will be crucial to apply (5.5) on a suitably chosen neighbourhood U of, say, Γ − , satisfying some properties:
• the closure of U is disjoint from Γ + ;
• H n−1 (U ∩ J u ) = 0; • v h converges to u in L 1 (∂U ).
To this aim we use coarea formula with the distance function from Γ − and use the assumption that v h converges to u in L 1 (Ω). Therefore, for h large enough we can set
v h (x) = min(M, v h (x)) if x ∈ U, v h (x) otherwise,
and deduce from (5.5) that
J µ ( v h ) ≤ J µ (v h ) + C M + 2 ∂U ( u − M ) + dH n−1 ,
for all h ≥ h 0 , and for some h 0 and C depending only on the choice of u, U and L < 1. Then, we note that by choosing M large enough and only depending on u and U (hence, uniformly with respect to h) the term
C M + 2 ∂U ( u − M ) + dH n−1
can be made arbitrarily small. In conclusion, for every ρ > 0 there exist M > 0 and h 0 , such that
J µ ( v h ) ≤ J µ (v h ) + ρ ∀ h ≥ h 0 .
This key fact allows us to truncate the functions v h from above in the neighbourhood U of, say, Γ − , up to adding a small amount of energy, but with the great advantage of preventing v h to concentrate and blow-up on smaller and smaller portions of Γ − , as h → ∞. Clearly the same argument works for Γ + , but in this case we must truncate from below and considerv h (as noticed above). Without loss of generality, we shall only prove the estimate for Γ − , as the one for Γ + is analogous. Indeed, under our assumptions on Γ ± , the problem can be decoupled and we shall assume without loss of generality that Γ + = ∅ for more simplicity.
Thus, we may directly take
v h ∈ W 1,1 (Ω \ ∂U ) ∩ BV (Ω), v h ≤ M on U ⊃ Γ − , and J µ (v h ) ≤ J µ (u) − (K − 1)ρ for all h.
At the same time, we can truncate the function u above M on U , so that by the continuity of J µ with respect to convergence in BV -norm of the M -truncations (as M → +∞) we can find M large enough with the property J µ ( u) ≥ J µ (u) − ρ. All in all, this implies that we can take from the very beginning u, v h ≤ M on U and J µ (v h ) ≤ J µ (u) − (K − 2)ρ for all h (indeed, the truncation above M of v h still converges to the truncation above M of u in L 1 (Ω)).
Since
1 + |Du| 2 (U ) = U 1 + |∇u| 2 dx + |D s u|(U )
we can choose U with Lebesgue measure so small that
1 + |Du| 2 (U ) ≤ |Du|(Γ − ) + ρ .
Assuming that 1 + |Du| 2 (∂U ) = 0 (this is true for most choices of U ), and since lim inf
h 1 + |Dv h | 2 (B \ U ) ≥ 1 + |Du| 2 (B \ U )
by the lower semicontinuity of the area functional, we can assume that
1 + |Dv h | 2 (B \ U ) ≥ 1 + |Du| 2 (B \ U ) − ρ
for h large enough. Then, up to possibly choosing a bigger h 0 , we have for all h ≥ h 0
|Dv h |(U ) − |Du|(Γ − ) + Γ − (v h − u + ) + dµ ≤ 1 + |Dv h | 2 (U ) − 1 + |Du| 2 (U ) + ρ + Γ − (v h − u + ) + dµ ≤ 1 + |Dv h | 2 (B) − 1 + |Du| 2 (B) + 2ρ + Γ − (v h − u + ) + dµ ≤ J µ (v h ) − J µ (u) + 2ρ ≤ −(K − 4)ρ . (5.7)
In the next step, we cover a sufficiently large portion of Γ − , say Γ 1 , by a finite family of small and mutually disjoint balls contained in U and centered on Γ 1 , such that for the remaining portion Γ 2 = Γ − \ Γ 1 we have M |µ|(Γ 2 ) ≤ ρ, |Du|(Γ 2 ) < ρ, and Γ 2 |u + | d|µ| < ρ. Moreover, on each ball B i = B(x i , r i ) we require that (i) the limit function u is very close to its asymptotic profile (either the jump profile (u − (x i ), u + (x i )), or the constant profile u(x i ), depending on whether x i ∈ Γ − ∩ J u or x i ∈ Γ − \ J u ); (ii) for h large enough, the function v h is very close to the asymptotic profile of u on B i (this follows from the L 1 convergence of v h to u).
Therefore we have
|Du|(Γ − ) ≤ |Du|(Γ 1 ) + ρ and Γ − (v h − u + ) + dµ = Γ 1 (v h − u + ) + dµ + Γ 2 (v h − u + ) + dµ ≥ Γ 1 (v h − u + ) + dµ − Γ 2 |u + | d|µ| − M |µ|(Γ 2 ) ≥ Γ 1 (v h − u + ) + dµ − 2ρ , which combined with (5.7) implies m i=1 |Dv h |(B i ) − |Du|(B i ) + Γ 1 ∩B i (v h − u + ) + dµ ≤ |Dv h |(U ) − |Du|(Γ 1 ) + Γ 1 (v h − u + ) + dµ ≤ |Dv h |(U ) − |Du|(Γ − ) + ρ + Γ − (v h − u + ) + dµ + 2ρ ≤ −(K − 7)ρ . (5.8)
We stress that the family B 1 , . . . , B m does not depend on v h , but only on the function u, the measure µ, and the open set U . At this point we can exploit properties (i) and (ii). By (i) we can take ε > 0 such that ε|µ|(Γ − ) ≤ ρ and require that the truncation of u on B i above u + (x i ) + ε/2 i retains a high percentage of the whole |Du|(B i ); that is,
|Du|(B i ∩ {u > u + (x i ) + ε/2 i }) < 2 −i−1 ρ, (5.9)
so that we can neglect the sum over i. At the same time, in accordance with (i) we can assume that
|{x ∈ B i , u(x) > t}| ≤ |B i |/2, ∀ t > u + (x i ) + ε/2 i .
We consider the following sequence of truncations of v h :
v h (x) = min(u + (x i ) + ε/2 i , v h (x)) if x ∈ B i for some i, v h (x) otherwise. We notice that v h (x) → u in L 1 (B i ) for all i, where u(x) = min(u + (x i ) + ε/2 i , u(x)) if x ∈ B i for some i, u(x) otherwise.
Hence, by lower semicontinuity we have that
m i=1 |D v h |(B i ) − |D u|(B i ) ≥ − ρ 2 (5.10)
for h large enough. All in all, combining (5.9) and (5.10) we obtain
m i=1 |Du|(B i ) ≤ m i=1 |D u|(B i ) + 2 −i−1 ρ ≤ m i=1 |D v h |(B i ) + ρ (5.11)
for h large enough. In parallel, the integral term can be estimated as follows:
Γ 1 ∩B i (v h − u + ) + dµ ≥ Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + + (u + − u + (x i ) − 2 −i ε) + dµ ≥ Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ+ − M u + (x i )+2 −i ε P ({u > t}; B i ) dt+ + M u + (x i )+2 −i ε P ({u > t}; B i ) + µ({u > t} ∩ B i ) dt ≥ Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ − 2 −i ρ − M u + (x i )+2 −i ε H n−1 ({u > t} ∩ ∂B i ) dt + M u + (x i )+2 −i ε P ({u > t} ∩ B i ) + µ({u > t} ∩ B i ) dt,
where in the last inequality we employed the coarea formula and (5.9). Now, by the Poincarétrace inequality (2.6) in B i we have that
H n−1 ({u > t} ∩ ∂B i ) ≤ C P T P ({u > t}; B i ) ,
hence we obtain
Γ 1 ∩B i (v h − u + ) + dµ ≥ Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ − 2 −i ρ + − C P T M u + (x i )+2 −i ε P ({u > t}; B i ) dt = Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ − 2 −i ρ + − C P T |Du|(B i ∩ {u > u + (x i ) + 2 −i ε}) ≥ Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ − (1 + C P T )2 −i ρ (5.12)
We notice that
|Dv h |(B i ∩ {v h > u + (x i ) + 2 −i ε}) = |Dv h |(B i ) − |D v h |(B i ).
Hence, combining (5.11), (5.12) and (5.8)
implies that m i=1 |Dv h |(B i ∩ {v h > u + (x i ) + 2 −i ε}) + Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ ≤ m i=1 |Dv h |(B i ) − |Du|(B i ) + Γ 1 ∩B i (v h − u + ) + dµ + (2 + C P T )ρ ≤ −(K − 7)ρ + (2 + C P T )ρ = −(K − 9 − C P T )ρ .
In addition, the left-hand side of this chain of inequalities can be estimated from below as follows:
m i=1 |Dv h |(B i ∩ {v h > u + (x i ) + 2 −i ε}) + Γ 1 ∩B i (v h − u + (x i ) − 2 −i ε) + dµ = m i=1 M u + (x i )+2 −i ε (P ({v h > t}; B i ) + µ({v h > t} ∩ B i )) dt ≥ m i=1 M u + (x i )+2 −i ε (P ({v h > t} ∩ B i ) + µ({v h > t} ∩ B i )) dt − m i=1 ∂B i (v h − u + (x i ) − 2 −i ε) + dH n−1 ≥ −ρ ,
because the first sum in the penultimate inequality is nonnegative by the non-extremality assumption, and concerning the second sum we can assume that for every i = 1, . . . , m we have u + = u H n−1 -almost everywhere on ∂B i , and v h → u in L 1 (∂B i ) as h → ∞ (this can be granted up to a possibly small perturbation of the radius r i , with a globally and arbitrarily small effect on the estimate, i.e., up to choosing h large enough). Therefore we conclude that
(K − 10 − C P T )ρ ≤ 0 ,
which clearly yields a contradiction if we choose K = 11 + C P T , for instance. This concludes the proof of the lower semicontinuity of J µ .
We dispose now of all the necessary tools for granting existence of a minimizer of J µ on S(φ).
Theorem 5.4 (Existence of minimizers).
Assume that µ ∈ M(Ω) satisfies the non-extremality assumption (4.2). Then the functional J µ admits a minimizer in S(φ).
Proof. We can apply the Direct Method of the Calculus of Variations, owing to the coercivity property shown in Section 5.1 and to Theorem 5.1. Indeed, note that for any 0 < C < +∞ the set
{u ∈ S(φ) : J µ [u] ≤ C} is closed in L 1 (B). Indeed, if {u h } h is a sequence of functions in S(φ) such that J µ [u h ] ≤ C < +∞
for all h, and u h → u in L 1 (B) (or, equivalently, in L 1 (Ω)) as h → ∞, by (5.2) and the lower semicontinuity of the total variation of the gradient we obtain u ∈ BV (B). Therefore, by Proposition 4.11 (and in particular thanks to the trace inequality (4.6)) we obtain that u ± ∈ L 1 (Ω; |µ|). One then concludes by means of Theorem 5.1. Finally, by Poincaré's inequality on B and by compactness in BV (B), we infer that the set {u ∈ S(φ) : J µ [u] ≤ C} is compact, so that there exists a solution to min{J µ [u] : u ∈ S(φ)}.
Existence of weak solutions of (PMCM) via convex duality
Thanks to Theorem 5.4, we know that a minimizerū ∈ BV (B) of the functional J µ exists. In this section we show the existence of a measurable vector field T such that the pair (ū, T ) is a weak solution of the PMCM equation (see Definition 1.1). Using tools of convex analysis, we will see that T can be determined as an element of the dual of the space M(Ω; R n ) of vector-valued Radon measures endowed with the total variation topology, and that its action on a suitable subspace of distributional gradients of BV functions is given precisely by the pairing (T, Du) λ .
Preliminaries of convex optimization and duality.
To begin with, we recall some standard notation and facts from convex analysis (with reference to the monography [12] for most of them). Let V be a locally convex topological vector space, and denote by V * its topological dual. The duality pairing between v * ∈ V * and v ∈ V is denoted as v, v * . We say that a function H : V → R belongs to Γ 0 (V ) if there exists a family {v * α } α (with α belonging to some index set, that we do not specify) of elements of the dual space V * and a corresponding family of real numbers {β α } α , such that
H(v) = sup α v, v * α + β α ,
and moreover the image of H is not reduced to the set {±∞}. It is well-known that any H ∈ Γ 0 (V ) is convex and lower semicontinuous. The Legendre transform (or polar) of H is the function
H * : V * → R defined as H * (v * ) = sup v∈V v, v * − H(v) .
The subdifferential of H at v is the (possibly empty) set
∂H(v) = {u * ∈ V * : H(w) ≥ H(v) + w − v, u * for all w ∈ V }.
Associated with H ∈ Γ 0 (V ) we consider the corresponding minimization problem P and set
inf P := inf v∈V H(v) . (6.1)
This is also called the primal problem. In order to obtain a dual formulation of the problem, we need to consider a family of convex and lower semicontinuous perturbations of H defined by means of an auxiliary space Y . More precisely, let Y be a topological vector space and let Y * denote its dual. For more simplicity, the duality pairing between p ∈ Y and p * ∈ Y * is again denoted as p, p * . Assume that we have a function Φ : 0) for all v ∈ V , and consider for any fixed p ∈ Y the perturbed problem
V × Y → R belonging to Γ 0 (V × Y ), such that H(v) = Φ(v,inf P p := inf v∈V Φ(v, p) . (6.2)
The dual problem to P is denoted by P * , and defined as the following maximization problem:
sup P * := sup p * ∈Y * −Φ * (0, p * ) , (6.3) where Φ * (v * , p * ) = sup (v,p)∈V ×Y v, v * + p, p * − Φ(v, p) .
An immediate property relating the primal and the dual problems is the following inequality:
sup P * ≤ inf P ,
which follows immediately from
−Φ * (0, p * ) = − sup v,p p, p * − Φ(v, p) = inf v,p Φ(v, p) − p, p * ≤ inf v Φ(v, 0) = inf P .
We set
κ(p) = inf v∈V Φ(v, p) .
We remark that κ is convex (see [12] Definition 6.1 (Normal Problem). The primal problem P is called normal if κ(0) is finite and κ is lower semicontinuous at 0. Similarly, the dual problem P * is called normal if κ * (0) is finite and κ * is lower-semicontinuous at 0 (here κ * denotes the Legendre transform of κ).
Proposition 6.2. The following are equivalent: (i) P is normal;
(ii) P * is normal;
(iii) inf P = sup P * ∈ R. Definition 6.3 (Stable problem). The primal problem P is stable if κ(0) is finite and ∂κ(0) = ∅.
Proposition 6.4. The following are equivalent: (i) P is stable;
(ii) P is normal and P * admits a solution.
The next proposition provides a useful stability criterion for the primal problem P.
Proposition 6.5. Let Φ(v, p) be convex, inf P = κ(0) be finite, and κ(p) be continuous at p = 0. Then P is stable. Now we assume that H(v) = I(v, Λv), where I : V × Y → R and Λ : V → Y is linear and continuous. Let also Λ * : Y * → V * denote the adjoint of Λ, defined via the property v, Λ * p * = Λv, p * for v ∈ V and p * ∈ Y * . Clearly the primal problem P corresponds to inf v∈V I(v, Λv) .
We perturb the primal problem by setting
Φ(v, p) = I(v, Λv − p) so that we obtain Φ * (0, p * ) = sup p p, p * − inf v I(v, Λv − p) = sup q sup v Λv − q, p * − I(v, q) = sup v,q v, Λ * p * + q, −p * − I(v, q) = I * (Λ * p * , −p * ) .
In conclusion, the dual problem P * in this specific case is given by
sup p * ∈Y * −I * (Λ * p * , −p * ) .
Next we state the key result that allows to characterize the solutions of the primal and the dual problems.
Proposition 6.6. The following are equivalent:
(i)ū solves P,p * solves P * , and sup P * = inf P;
(ii)ū ∈ V andp * ∈ Y * verify the extremality relation
I(ū, Λū) + I * (Λ * p * , −p * ) = 0 , (6.4)
that is to say, (Λ * p * , −p * ) ∈ ∂I(ū, Λū).
We now further specialise this setting to the case H(v) = I(v, Λv) = F(v) + G(Λv), for some functions F : V → R and G : Y → R. In this case, assuming all the necessary convexity and lower semicontinuity properties of the functions F and G, we have that P * corresponds to
sup p * ∈Y * −F * (Λ * p * ) − G * (−p * ).
Moreover the extremality relation (6.4) is equivalent to the two relations F(ū) + F * (Λ * p * ) = ū, Λ * p * (6.5) and G(Λū) + G * (−p * ) = − ū, Λ * p * . (6.6) These last two relations correspond to Λ * p * ∈ ∂F(ū) and −p * ∈ ∂G(Λū), respectively. 6.2. Optimality relations. In order to recover the weak formulation we recall that the jump part of the distributional gradient ofū is of the form (ū + −ū − )νū H n−1 Jū. Then we shall define the functional F (see (6.10)) in such a way that it is finite on functions u ∈ BV (Ω) such that the trace of u on ∂Ω equals that ofū, and moreover ν u = νū H n−1 -almost everywhere on J u ∩ Γ. Then we let V = BV (Ω) endowed with the strong BV topology. We then let Y = M(Ω; R n ) endowed with the total variation topology. As L 1 (Ω; R n ) is continuously injected into Y , the dual space Y * is a subspace of L ∞ (Ω; R n ). We define a particular set V ⊂ V by setting V := {u ∈ V : Tr ∂Ω (u) = Tr ∂Ω (ū) in L 1 (∂Ω; H n−1 ), H n−1 (J u \ Jū) = 0, ν u = νū H n−1 -a.e. on J u ∩ Γ}.
(6.7)
We notice that this definition can be relaxed by assuming, instead of the last two conditions, that the vector field νū admits an extension N to the whole Γ, so that one requires ν u = N H n−1 -almost everywhere on J u ∩ Γ.
We show now that V is a closed and convex subset of V .
Lemma 6.7. V is a closed and convex subset of V .
Proof. Let (u k ) k∈N ⊂ V be such that u k → u with respect to the strong topology of BV (Ω). Clearly, u ∈ BV (Ω), so we need to check the conditions in the definition of V , (6.7). By the continuity of the trace operator, we have
Tr ∂Ω (u k ) − Tr ∂Ω (u) L 1 (∂Ω;H n−1 ) ≤ C u k − u BV (Ω) → 0 as j → +∞.
However, by definition of V (6.7), we have Tr ∂Ω (u k ) = Tr ∂Ω (ū) for all k ∈ N, and so we conclude that Tr ∂Ω (u) = Tr ∂Ω (ū) in L 1 (∂Ω; H n−1 ). Then, the strong convergence implies that
|D j u k − D j u|(B) → 0 as k → +∞ (6.8)
for all Borel sets B ⊆ Ω. In particular, by choosing B = Ω \ Jū, we get
Ju\Jū |u + − u − | dH n−1 = |D j u|(Ω \ Jū) = lim k→+∞ |D j u k |(Ω \ Jū) = lim k→+∞ Ju k \Jū |u + k − u − k | dH n−1 = 0 by (6.7)
. This implies that either H n−1 (J u \ Jū) = 0 or |u + − u − | = 0 H n−1 -a.e. on J u \ Jū. However, if u + (x) = u − (x), then x / ∈ J u , so that the second case is not possible. Finally, we need to prove that ν u = νū H n−1 -a.e. on J u ∩ Γ. By applying (6.8) to B = J u ∩ Γ we get
|D j u k − D j u|(J u ∩ Γ) = Ju∩Γ (u + k − u − k )ν u k − (u + − u − )ν u dH n−1 = Ju∩Γ (u + k − u − k )νū − (u + − u − )ν u dH n−1 → 0,
which is possible if and only if ν u = νū H n−1 -a.e. on J u ∩ Γ. Indeed, one can argue by contradiction assuming that there exist δ > 0 and a set K ⊂ J u ∩ Γ with H n−1 (K) > 0, such that u + (x) − u − (x) > δ and |ν u (x) − νū(x)| > δ whenever x ∈ K. Up to extracting a subsequence and replacing K with an H n−1 -equivalent set, we can assume that for all x ∈ K,
f k (x)νū(x) → f (x)ν u (x) as k → ∞, (6.9) where f k (x) = u + k (x) − u − k (x) and f (x) = u + (x) − u − (x).
Since f k ≥ 0 and f > δ on K, by (6.9) one infers that f k (x) → f (x) for all x ∈ K, and consequently that |νū(x) − ν u (x)| = 0 for all x ∈ K, a contradiction. This ends the proof.
We define F : V → R as
F(u) = Ω u λ dµ if u ∈ V +∞
otherwise.
(6.10) and G : Y → R as
G(p) = 1 + |p| 2 (Ω) = sup Ω g 0 dx + Ω g · dp ,
where the supremum is computed among pairs (g 0 , g) ∈ C 0 c (Ω)×C 0 c (Ω; R n ) such that g 2 0 +|g| 2 ≤ 1 on Ω.
Proposition 6.8. Under the previous assumptions, F ∈ Γ 0 (V ).
Proof. We only need to prove that F is not identically ±∞, is convex, and is lower semicontinuous in V . The first property is an obvious consequence of Proposition 4.11. To prove the convexity, we fix t ∈ (0, 1) and u 1 , u 2 ∈ V such that F(u 1 ), F(u 2 ) < +∞, which means u 1 , u 2 ∈ V . Defining
u t = tu 1 + (1 − t)u 2 we have u ± t = tu ± 1 + (1 − t)u ± 2 µ-almost everywhere on Γ because ν u 1 = ν u 2 µ-almost everywhere on Γ ∩ J u 1 ∩ J u 2 ,
thanks to (6.7) and the fact that |µ| ≪ H n−1 . Therefore we obtain
Γ u λ t dµ = Γ − u + t dµ + Γ + u − t dµ = t Γ − u + 1 dµ + (1 − t) Γ − u + 2 dµ + t Γ + u − 1 dµ + (1 − t) Γ + u − 2 dµ = t Γ u λ 1 dµ + (1 − t) Γ u λ 2 dµ .
This implies that
Ω u λ t dµ = t Ω u λ 1 dµ + (1 − t) Ω u λ 2 dµ ,
which gives in particular the convexity of F. Finally, the lower semicontinuity is a consequence of the continuity of the integral Ω u λ dµ with respect to convergence in BV norm.
Finally, we let Λ = D the distributional derivative, which is linear and continuous from V to Y . We then have for every u ∈ V
J µ [u] = G(Du) + F(u) so that Φ(u, p) = G(Du − p) + F(u) belongs to Γ 0 (V × Y ).
Moreover we have the following result. Proposition 6.9. Let κ(p) = inf u∈V Φ(u, p), then κ(0) is finite, and κ is continuous at 0.
Proof. The first claim is obvious, as κ(0) = J µ [ū]. The lower semicontinuity at 0 can be proved as follows. Since Y is a normed space, we can consider a sequence {p k } k ⊂ Y such that |p k |(Ω) → 0 as k → ∞. By definition, there exists a sequence {u k } k ⊂ V such that
κ(p k ) ≥ 1 + |Du k − p k | 2 (Ω) + Ω u λ k dµ − 1/k . Therefore we have κ(p k ) ≥ sup (g 0 ,g) Ω g 0 dx + Ω g · dDu k − Ω g · dp k + Ω u λ k dµ − 1/k ≥ sup (g 0 ,g) Ω g 0 dx + Ω g · dDu k + Ω u λ k dµ − |p k |(Ω) − 1/k = 1 + |Du k | 2 (Ω) + Ω u λ k dµ − |p k |(Ω) − 1/k ≥ κ(0) − |p k |(Ω) − 1/k ,
which implies that lim inf k→+∞ κ(p k ) ≥ κ(0). Finally, the upper semicontinuity follows from
κ(p) ≤ sup (g 0 ,g) Ω g 0 dx + Ω g · dDū − Ω g · dp + Ωū λ dµ ≤ κ(0) + |p|(Ω) ,
and this proves the last claim.
We can now state and prove the main result of this section.
Theorem 6.10. Letū be a minimizer of J µ on S(φ). Then, there exists a vector field T ∈ L ∞ (Ω; R n ) such that the pair (ū, T ) satisfies (1.4).
Proof. By combining Proposition 6.9 with Proposition 6.5 we infer that the primal problem P is stable. Hence the dual problem P * admits a solutionp * , therefore we can apply Proposition 6.6 and deduce the necessary relations (6.5) and (6.6). In order to write them explicitly in the present situation, we must compute F * (D * q * ) and G * (q * ) for q * ∈ Y * . We conveniently consider a Sobolev function φ such that its trace on ∂Ω coincides with the trace ofū, and set
V 0 = −φ + V ⊂ BV 0 (Ω). We have F * (D * q * ) = sup u∈V Du, q * − F(u) = Dφ, q * − Ω φ dµ + sup u∈V 0 Du, q * − Ω u λ dµ = Dφ, q * − Ω φ dµ if Du, q * ≤ Ω u λ dµ for all u ∈ V 0 +∞ otherwise.
Therefore, in order to have F * (D * q * ) finite, we must have − div q * = µ as a necessary condition. Indeed, for all u ∈ V 0 we have
Du, q * ≤ Ω u λ dµ,(6.11)
and so, given that C ∞ c (Ω) ⊂ V 0 , (6.11) holds in particular for all u ∈ C ∞ c (Ω). However, for u ∈ C(Ω), we have u λ = u, and, by considering both u and −u as test functions, (6.11) yields Ω q * · ∇u dx ≤ Ω u dµ and Ω q * · ∇u dx ≥ Ω u dµ, which readily implies − div q * = µ in M(Ω). On the other hand,
G * (q * ) = sup q∈Y q, q * − 1 + |q| 2 (Ω) = sup q∈Y q ac , q * − |L n ⊗ q ac |(Ω) + q s , q * − |q s |(Ω) ,
where q ac and q s denote, respectively, the absolutely continuous and the singular parts of the Radon measure q. In order to proceed with our calculation, we set
q * ⊥ := sup ν, q * : ν ∈ Y ⊥ , |ν|(Ω) ≤ 1 , where Y ⊥ := {ν ∈ Y : ν ⊥ L n } .
Then, by orthogonality of the measures q ac and q s , we obtain
G * (q * ) = sup q∈Y q ac , q * − |L n ⊗ q ac |(Ω) + q s , q * − |q s |(Ω) = sup q∈L 1 (Ω;R n ) Ω q · q * dx − Ω 1 + |q| 2 dx + sup ν∈Y ⊥ ν, q * − |ν|(Ω) = − Ω 1 − |q * | 2 dx if q * L ∞ ≤ 1 and q * ⊥ ≤ 1, +∞ otherwise.
Note that the last step above follows from a pointwise almost everywhere optimisation of the quantity ρ|q * (x)| − 1 + ρ 2 as ρ varies in [0, +∞) (see computations in [12][Chapter V, Lemma 1.1]). At this point we can write the optimality relations for the dual pair (ū,p * ). In particular, (6.5) becomes
Ω (ū λ − φ) dµ + ∇φ,p * = Dū,p * − divp * = µ (6.12)
and (6.6) becomes 1 + |Dū| 2 (Ω) = Ω 1 − |p * | 2 − Dū,p * p * ∞ ≤ 1 and p * ⊥ ≤ 1 (6.13)
Let now T = −p * . In particular we have T L ∞ (Ω;R n ) ≤ 1, div T = µ on Ω, and by (6.12) we deduce
Dū, T = Ω T · ∇φ dx + Ω φ dµ − Ωū λ dµ = Ω d div(φT ) − Ωū λ dµ = Ω d div(ūT ) − Ωū λ dµ − Ω d div((ū − φ)T ) = Ω d div(ūT ) − Ωū λ dµ = (T, Dū) λ (Ω)
becauseū − φ has zero trace on ∂Ω, hence div((ū − φ)T )(Ω) = 0. By plugging this last identity into (6.13) we get
(T, Dū) λ (Ω) = 1 + |Dū| 2 (Ω) − Ω 1 − |T | 2 dx . (6.14)
On the other hand, by Lemma 4.13 we know that
|(T, Dū) λ | ≤ 1 + |Dū| 2 − 1 − |T | 2 L n Ω in M(Ω). (6.15)
Finally, by combining (6.14) with (6.15) we obtain that
(T, Dū) λ = 1 + |Dū| 2 − 1 − |T | 2 L n Ω in M(Ω).
This proves thatū and T verify (1.4), as wanted.
Variational approximation
Here we establish a Gamma-convergence result for the sequence of functionals
J µ j [u] = 1 + |Du| 2 (B) + Ω u(x) µ j (x) dx if u ∈ S(φ) ,
+∞ otherwise, (7.1) to the functional J µ defined in (5.1). Here µ is a non-extremal measure; that is, which satisfies (4.2), and µ j ∈ C ∞ (Ω) is such that µ j L n is non-extremal for all j, and converges to µ in the weak- * sense, as j → ∞.
We start showing that any non-extremal measure µ can be approximated by a sequence of smooth, non-extremal functions µ j in the weak- * sense. The key idea is to use the duality between non-extremal divergence measures and vector fields whose L ∞ norm is strictly smaller than 1. While it is not clear if a direct mollification of µ would in general preserve the nonextremality property (even assuming µ of compact support in Ω), we shall instead rely on the fact that the Anzellotti-Giaquinta-type regularization of a vector field almost preserves its L ∞ norm. Similar ideas can be found for instance in [11]. Proposition 7.1. Let µ ∈ M(Ω) be a non-extremal measure. Then there exists a sequence of functions (µ j ) j ⊂ C ∞ (Ω) such that µ j L n ⇀ µ and µ j L n is non-extremal for j large enough. Moreover, if µ has compact support in Ω, then for any open set Ω ′ ⊂⊂ Ω containing the support of µ one has:
• µ j = ρ j * µ on Ω ′ for j large enough and some mollifier ρ
∈ C ∞ c (B 1 ) with ρ j (x) = δ −n j ρ(x/δ j ) for some sequence δ j ↓ 0, • µ j → 0 uniformly on Ω \ Ω ′ .
Proof. The construction of the sequence µ j starting from the measure µ can be done as follows. Let us fix ε = 1 j and consider the sequences (ζ j,k ) k (a partition of unity of Ω) and ρ j,k = ρ δ j,k (a suitable sequence of mollifiers associated with the partition of unity) as in the proof of Theorem A.2. We recall that thanks to [24,Theorem 4.4], if µ is non-extremal then there exist 0 < L < 1 and a vector field F ∈ L ∞ (Ω; R n ) with F L ∞ (Ω;R n ) ≤ L, such that div F = µ. Then we set
F j = k ρ j,k * (ζ j,k F )
and, finally,
µ j = div F j . (7.2) With reference to the proof of Theorem A.2, we fix ϕ ∈ C ∞ c (Ω) and obtain Ω ϕ µ j dx = − Ω F j · ∇ϕ dx = − +∞ k=1 Ω (ζ j,k F ) * ρ j,k · ∇ϕ dx = +∞ k=1 Ω ζ j,k (ρ j,k * ϕ) dµ + +∞ k=1 Ω ϕ ρ j,k * (F · ∇ζ j,k ) − F · ∇ζ j,k dx −→ Ω ϕ dµ , as j → ∞,
where the final passage to the limit is justified because Ω ϕ ρ j,k * (F · ∇ζ j,k ) − F · ∇ζ j,k dx ≤ ϕ L ∞ (Ω) ρ j,k * (F ·∇ζ j,k )−F ·∇ζ j,k L 1 (Ω) ≤ ϕ L ∞ (Ω) j2 k thanks to the assumptions on ζ j,k and ρ j,k in the proof of Theorem A.2. Note moreover that
µ j = +∞ k=1 ρ j,k * (ζ j,k µ) + +∞ k=1 (ρ j,k * (F · ∇ζ j,k ) − F · ∇ζ j,k ) ,
therefore, when µ has compact support in Ω ′ ⊂⊂ Ω, the first sum belongs to C ∞ c (Ω) and, for j large, coincides with ρ j,1 * µ if restricted to Ω ′ . At the same time, the second sum is uniformly small and tends to 0 as j → ∞. This shows the last part of the statement.
We now prove the non-extremality of µ j for j large enough. Given A ⊂⊂ Ω with ∂A Lipschitz, we have
A µ j dx = ∂A F j · ν A dH n−1 ≤ F j L ∞ (Ω;R n ) P (A).
Now we notice that, for x ∈ Ω k 0 +1 \ Ω k 0 −1 and thanks to the assumptions on ζ j,k and ρ j,k , we have
|F j (x)| ≤ F L ∞ (Ω;R n ) 1 + k 0 +1 k=k 0 −1 |ζ j,k * ρ j,k − ζ j,k | ≤ L(1 + 1/j) .
We thus conclude that F j L ∞ (Ω;R n ) ≤ L < 1 for j large enough, which gives the uniform non-extremality of µ j .
The following theorem proves the Gamma-convergence of J µ j to J µ under suitable assumptions on µ and µ j . For the essential definitions of Gamma-convergence, we refer to [4].
Theorem 7.2. Let µ ∈ M(Ω) be a non-extremal measure. Let µ j be as in the thesis of Proposition 7.1, and let J µ j and J µ be defined as in (7.1) and (5.1), respectively. Then J µ j Γ −→ J µ with respect to the L 1 topology, as j → ∞.
Proof. Let us start from the Γ-lim sup inequality. Fix u ∈ S(φ), then using the density-in-energy of smooth functions (a consequence of Theorem 3.3) we can directly assume u ∈ S(φ) ∩ C ∞ (Ω). Then we fix a cut-off function η ∈ C ∞ c (Ω) and write
J µ j [u] − J µ [u] = Ω u(x) µ j (x) dx − Ω u(x) dµ(x) = Ω η(x)u(x) µ j (x) dx − dµ(x) + Ω (1 − η(x))u(x) µ j (x) dx − dµ(x) .
The first integral tends to zero as j → ∞ by the weak- * convergence, while the second integral can be estimated using the fact that the support of µ is compact, hence we can choose η ≡ 1 on the support of µ and then exploit the property of the sequence µ j of being uniformly infinitesimal in a neighbourhood of ∂Ω. This proves that the difference of the energies tends to 0 as j → ∞, which gives the Γ-lim sup inequality at once. Let us prove the Γ-lim inf inequality. Fix u j , u ∈ S(φ) such that u j → u in L 1 (Ω) and sup j J µ j [u j ] < +∞. Using Theorem 3.3 we can assume that u j ∈ C ∞ (Ω) without loss of generality. With a slight abuse of notation, we denote by µ j also the measure µ j L n . The goal is to prove that lim inf
j J µ j [u j ] − J µ [u] ≥ 0 . (7.3)
For j large enough Proposition 7.1 ensures that µ j = ρ j * µ for some mollifier ρ ∈ C ∞ c (B 1 ) with ρ j (x) = δ −n j ρ(x/δ j ) for some sequence δ j ↓ 0. Hence, we see that
J µ j [u j ] − J µ [u] = 1 + |∇u j | 2 (B) − 1 + |Du| 2 (B) + Ω (ρ j * u j − u λµ ) dµ = 1 + |∇v j | 2 (B) − 1 + |Du| 2 (B) + Ω (v j − u λµ ) dµ + 1 + |∇u j | 2 (B) − 1 + |∇v j | 2 (B)
where v j = η(ρ j * u j ) + (1 − η)u j and η is a cut-off function with compact support in Ω, such that η ≡ 1 on the support of µ. On the one hand, the lower-semicontinuity of J µ proved in Section 5.2 implies that
lim inf j 1 + |∇v j | 2 (B) − 1 + |Du| 2 (B) + Ω (v j − u λµ ) dµ ≥ 0 . (7.4)
On the other hand, for all ψ ∈ C c (B) and ϕ ∈ C 1 c (B; R n ) such that (ψ, ϕ) L ∞ (B;R n+1 ) ≤ 1, we see that
B ψ + v j div ϕ dx = B ψ + η(ρ j * u j ) div ϕ + (1 − η)u j div ϕ dx = B ψ + (ρ j * u j ) div(ηϕ) + u j div((1 − η)ϕ) dx + Ω (u j − ρ j * u j ) ϕ · ∇η dx = B ψ + u j div(ρ j * (ηϕ) + (1 − η)ϕ) dx + Ω (u j − ρ j * u j ) ϕ · ∇η dx ≤ (ψ, ρ j * (ηϕ) + (1 − η)ϕ) L ∞ (B;R n+1 ) 1 + |∇u j | 2 (B)+ + ∇η L ∞ (Ω;R n ) ρ j * u j − u j L 1 (Ω) .
Hence, by taking the supremum over
(ψ, ϕ) ∈ C c (B)×C 1 c (B; R n ) such that (ψ, ϕ) L ∞ (B;R n+1 ) ≤ 1 we get 1 + |∇v j | 2 (B) ≤ (1 + ε j ) 1 + |∇u j | 2 (B) + ∇η L ∞ (Ω;R n ) ρ j * u j − u j L 1 (Ω) .
Thanks to Young's inequality, we get
ρ j * u j − u j L 1 (Ω) ≤ ρ j * u j − ρ j * u L 1 (Ω) + ρ j * u − u L 1 (Ω) + u j − u L 1 (Ω) ≤ (1 + ρ L 1 (B 1 ) ) u j − u L 1 (Ω) + ρ j * u − u L 1 (Ω) → 0.
Hence, we obtain lim inf j→+∞ 1 + |∇v j | 2 (B) ≤ lim inf j→+∞ 1 + |∇u j | 2 (B). (7.5) Thus, by combining the inequalities (7.4) and (7.5) with the superadditivity of the liminf we obtain the proof of (7.3), and thus of the theorem.
Uniqueness and characterisation of T (u)
Hereafter we show that the vector field T (u) associated with a weak solution u of (PMCM) is uniquely determined up to Lebesgue null sets by
T (u) = ∇u 1 + |∇u| 2 L n -a.e. in Ω ,(8.1)
where Du = ∇uL n + D s u is the decomposition of the gradient measure of u. First, we prove the following uniqueness result. Proof. We let θ ∈ (0, 1), and we define T θ := θF + (1 − θ)G. Thanks to (1.3) and (1.5) we clearly have T θ L ∞ (Ω;R n ) ≤ 1 and div T θ = µ on Ω.
In addition, thanks to the linearity in the first component of the pairing and (1.4), we get
(T θ , Du) λ = θ(F, Du) λ + (1 − θ)(G, Du) λ = 1 + |Du| 2 − θ 1 − |F | 2 L n − (1 − θ) 1 − |G| 2 L n in M(Ω)
. Now we notice that, since the function v → 1 − |v| 2 is strictly concave in the unit ball B 1 , then for F = G we obtain
1 − |T θ | 2 ≥ θ 1 − |F | 2 + (1 − θ) 1 − |G| 2 for L n -a.e. x ∈ Ω.
This implies that
(T θ , Du) λ > 1 + |Du| 2 − 1 − |T θ | 2 L n on Ω. (8.2)
However, it is easy to notice that div T θ is admissible in the sense of Definition 4.5, so that (4.11) implies that
(T θ , Du) λ ≤ 1 + |Du| 2 − 1 − |T θ | 2 L n on Ω,
which contradicts (8.2). Thus we conclude that F = G. We recall that, if F ∈ DM ∞ (Ω) and u ∈ BV (Ω) with u λ ∈ L 1 (Ω; | div F |), then (F, Du) λ = (F · ∇u)L n + (F, D s u) λ on Ω .
Hence, if we consider separately the absolutely continuous and singular parts of the measures, we obtain T (u) · ∇u = 1 + |∇u| 2 − 1 − |T (u)| 2 for L n -a.e. in Ω (T (u), D s u) λ = |D s u| on Ω.
In particular, this means that (8.1) holds true. Then, since D s u = D c u + D j u, and
(F, D s u) λ = (F, D c u) * + (F, D j u) λ , we get (T (u), D c u) * = |D c u| on Ω. (T (u), D j u) λ = |D j u| on Ω.
Remark 8.2. We notice that if ∇u = 0 a.e. on Ω, then T = 0 and therefore µ = 0. Consequently, if µ is nontrivial, then the distributional gradient Du cannot contain only jump and/or Cantor parts.
Maximum principle for continuous solutions
Next, we prove a maximum principle in the particular case of solutions to the PMCM equation which are continuous inside the domain Ω. This restriction is unavoidable, as the example of non-uniqueness constructed in Section 10 shows. Instead, the continuity assumption on u implies that (T, Du) λ = (T, Du) * = div(uT ) − u div T for any choice of the Borel map λ. We stress that that this maximum principle holds without a-priori restrictions on the admissible measure µ. Proof. By the assumptions on u i and T i , we have
(T i , Du i ) * = 1 + |Du i | 2 − 1 − |T i | 2 L n on Ω (9.1) for i = 1, 2. Let ε > 0, A ε := {x ∈ Ω : u 2 (x) − u 1 (x) > ε} and ϕ ε := max{u 2 − u 1 , ε}. It is clear that ϕ ε ∈ BV (Ω) ∩ C 0 (Ω) with Dϕ ε = Du 2 − Du 1 on A ε , 0 on Ω \ A ε ,
and that A ε is a set of finite perimeter in Ω for L 1 -a.e. ε > 0. Since the functions u 1 , u 2 are continuous, then Du i has no jump part, and so we get (T i , Du i ) λ = (T i , Du i ) * for any Borel function λ : Ω → [0, 1] and i = 1, 2, and we set (T i , Du i ) := (T i , Du i ) * . Thanks to (4.11) and (9.1), we see that
(T 1 − T 2 , D(u 2 − u 1 )) ≤ 0 (9.2)
Indeed, thanks to the bilinearity of the standard pairing, (9.1) and Theorem 4.13, we conclude that
(T 1 − T 2 , D(u 2 − u 1 )) = −(T 1 , Du 1 ) − (T 2 , Du 2 ) + (T 1 , Du 2 ) + (T 2 , Du 1 ) = (T 1 , Du 2 ) − 1 + |Du 2 | 2 − 1 − |T 1 | 2 + + (T 2 , Du 1 ) − 1 + |Du 1 | 2 − 1 − |T 2 | 2 ≤ 0.
In addition, the continuity of u 1 and u 2 implies that
|Du i |(∂ * A ε ) = |Du i |(A 1 ε \ A ε ) = 0 for i = 1, 2. Since ϕ ε = χ Aε (u 2 − u 1 ) + εχ Ω\Aε ,
the linearity of the standard pairing and Theorem 4.13 imply that [10,Proposition 4.4]). Therefore, we deduce that
(T 1 − T 2 , Dϕ ε ) = (T 1 − T 2 , D(χ Aε (u 2 − u 1 ))) + (T 1 − T 2 , D(εχ Ω\Aε )) = χ Aε (T 1 − T 2 , D(u 2 − u 1 )) + (u 2 − u 1 )(T 1 − T 2 , Dχ Aε ) − ε(T 1 − T 2 , Dχ Aε ) = (T 1 − T 2 , D(u 2 − u 1 )) A ε , since u 2 − u 1 = ε on ∂ * A ε , which is where the pairing (T 1 − T 2 , Dχ Aε ) is concentrated, due to the fact that |(T 1 − T 2 , Dχ Aε )| ≤ 2|Dχ Aε | (see for instance0 ≥ (T 1 − T 2 , D(u 2 − u 1 )) A ε = (T 1 − T 2 , Dϕ ε ).(9.(T 1 − T 2 , Dϕ ε )(Ω k ) = div(T 1 − T 2 ϕ ε )(Ω k ) − Ω k ϕ ε d div(T 1 − T 2 ) = − ∂Ω k ϕ ε Tr i (T 1 − T 2 , ∂ * Ω k ) dH n−1 + Ω k ϕ ε d div(T 2 − T 1 ) ≥ −4εP (Ω k ) + Ω k ϕ ε d(µ 2 − µ 1 ) ≥ −4εP (Ω k ),
thanks to the fact that µ 2 ≥ µ 1 and that
Tr i (T 1 − T 2 , ∂ * Ω k ) L ∞ (∂Ω k ;H n−1 ) ≤ T 1 − T 2 L ∞ (Ω k ;R n ) ≤ T 1 L ∞ (Ω;R n ) + T 2 L ∞ (Ω;R n ) ≤ 2.
All in all, we obtain 0 ≥ (T, Dϕ ε )(Ω k ) ≥ −4εP (Ω k ), so that, by taking the limit as k → +∞, we conclude that
0 ≥ (T, Dϕ ε )(Ω) ≥ −4εP (Ω),
which, thanks to (9.3), implies
0 ≥ (T 1 − T 2 , D(u 2 − u 1 ))(A ε ) ≥ −4εP (Ω)
Finally, by taking the limit as ε → 0, we conclude that
(T 1 − T 2 , D(u 2 − u 1 ))(A 0 ) = 0,
which, combined with (9.2), implies
(T 1 − T 2 , D(u 2 − u 1 )) = 0 on A 0 .
The bilinearity of the pairing yields
(T 2 , Du 1 ) − (T 2 , Du 2 ) = (T 1 , Du 2 ) − (T 1 , Du 2 ) on A 0 ,
and then we exploit (9.1) and Theorem 4.13 to obtain
0 ≥ (T 2 , Du 1 )− 1 + |Du 1 | 2 + 1 − |T 2 | 2 L n = 1 + |Du 2 | 2 − 1 − |T 1 | 2 L n −(T 1 , Du 2 ) ≥ 0 on A 0 .
In particular, this means that both sides of the equation are identically zero on the open set A 0 . Hence, if we add the right hand side to (9.1) in the case i = 1 we get
1 + |Du 1 | 2 + 1 + |Du 2 | 2 − (T 1 , D(u 1 + u 2 )) − 2 1 − |T 1 | 2 L n = 0 on A 0 .
On the other hand, Theorem 4.13 implies that
1 2 (T 1 , D(u 1 + u 2 )) = T 1 , D u 1 + u 2 2 ≤ 1 + D u 1 + u 2 2 2 − 1 − |T 1 | 2 L n on Ω.
All in all, we conclude that
1 + |Du 1 | 2 + 1 + |Du 2 | 2 2 ≤ 1 + D u 1 + u 2 2 2 on A 0 .
Therefore, by convexity, this implies that
Du 1 = Du 2 on A 0 , which means Dϕ 0 = 0 on A 0 , where ϕ 0 := max{u 2 − u 1 , 0} ∈ BV (Ω) ∩ C 0 (Ω).
Thus, ϕ 0 is constant on Ω; however, since Tr ∂Ω (ϕ 0 )(x) = 0 for H n−1 -a.e. x ∈ ∂Ω by assumption, we conclude that ϕ 0 = 0 in Ω, and so u 1 ≥ u 2 in Ω.
As a consequence we obtain the following Corollary 9.2 (Uniqueness for continuous weak solutions). Let µ ∈ M(Ω) be an admissible measure in the sense of Definition 4.5. Let u 1 , u 2 ∈ BV (Ω) ∩ C 0 (Ω) be weak solutions of (PMCM), such that that Tr ∂Ω (u 1 ) = Tr ∂Ω (u 2 ) H n−1 -a.e. on ∂Ω. Then u 1 = u 2 in Ω.
Examples
Since the case n = 1 is essentially trivial (the equation div F = µ on Ω is explicitly solved by u(x) = µ((−∞, x) ∩ Ω) + c, for c ∈ R) we directly consider the case n ≥ 2, and construct/characterise weak solutions with radial symmetry on annuli.
First, we characterise the radial solutions of (PMCM) on annuli of the form Ω = B r 3 \ B r 1 for given 0 < r 1 < r 3 , and where µ = µ 2 H n−1 ∂B r 2 for some r 2 ∈ (r 1 , r 3 ) and µ 2 ∈ R. We thus assume that u ∈ BV (Ω) is a radial solution. We conveniently write r = |x| for x ∈ Ω, and denote by u(r) the one-dimensional profile of the solution u(x) (with a slight abuse of notation). Moreover we denote by ν i the outer unit normal to ∂B r i .
Since µ = 0 on Ω \ ∂B r 2 , the function u is smooth in Ω \ ∂B r 2 and therefore the vector field T , such that the pair (u, T ) satisfies (1.3), (1.4), and (1.5), is given by
T (x) = ∇u(x) 1 + |∇u(x)| 2 , ∀ x ∈ Ω \ ∂B r 2 .
Moreover, by (1.5) coupled with the radial symmetry of u, we infer that
T (x) = γ i x |x| n ∀ x ∈ B r i+1 \ B r i , i = 1, 2 ,(10.1)
for suitable constants γ 1 , γ 2 ∈ R. Since |T (x)| < 1 for all x ∈ Ω \ ∂B r 2 , we infer that
|γ i | ≤ r n−1 i , i = 1, 2 . (10.2)
Then we notice that
div T = Tr i (T, ∂B r 2 ) − Tr e (T, ∂B r 2 ) H n−1 ∂B r 2 = γ 2 − γ 1 r n−1 2 H n−1 ∂B r 2 ,
where Tr i,e (T, ∂B r 2 ) denote the (weak) interior and exterior normal traces of T on ∂B r 2 , for their precise definition we refer the reader to [10,Section 2.4]. Thus by combining the above formula with (1.5), we obtain
γ 2 = γ 1 + µ 2 r n−1 2 . (10.3)
By (10.1), and since Therefore we conclude that
∇u(x) = T (x) 1 − |T (x)| 2 for x ∈ Ω \ ∂B r 2 , we obtain u ′ (r) = γ i r 2n−2 − γ 2 i for r i < r < r i+1 .(T, D j u) λµ = |D j u| = (u + − u − ) H n−1 ∂B r 2 .
At the same time, by [10,Proposition 4.7] we have that
(T, D j u) λµ = λ µ (Tr e (T, J u ) + (1 − λ µ )Tr i (T, J u ) (u + − u − )H n−1 J u ,
where Tr i,e (T, J u ) are the normal traces corresponding to the orientation of J u for which u + > u − . According to the sign of µ 2 we have that λ µ is either 0 (if µ 2 > 0) or 1 (if µ 2 < 0) on ∂B r 2 , hence
µ 2 > 0 =⇒ [Tr i (T, J u ) − 1](u + − u − ) = 0 on ∂B r 2 µ 2 < 0 =⇒ [Tr e (T, J u ) − 1](u + − u − ) = 0 on ∂B r 2 ,
which means that, either u + = u − , or Tr i,e (T, J u ) = 1 on ∂B r 2 . More precisely, if we assume that u(r) jumps across r = r 2 , then we have that . Similarly, we have
µ 2 > 0 =⇒ γ 2 = r n−1 2 if ν u = ν 2 γ 1 = −r n−1 2 if ν u = −ν 2 =⇒ γ 2 = r nµ 2 < 0 =⇒ γ 1 = r n−1 2 if ν u = ν 2 γ 2 = −r n−1 2 if ν u = −ν 2 =⇒ γ 2 = −r n−1 2 and ν u = −ν 2 .
Summing up, if u(r) is discontinuous at r = r 2 then we obtain γ 2 = sign(µ 2 )r n−1 2 and thus by (10.3) we find
γ 1 = (sign(µ 2 ) − µ 2 )r n−1 2 .
This allows us to write the derivative u ′ (r) on (r 1 , r 2 ) ∪ (r 2 , r 3 ) as
u ′ (r) = sign(µ 2 )(1 − |µ 2 |)r n−1 2 r 2n−2 − (1 − |µ 2 |) 2 r 2n−2 2
if r 1 < r < r 2 sign(µ 2 )r n−1 2 r 2n−2 − r 2n−2 2 if r 2 < r < r 3 , so that, since u ′ (r) is defined for all r = r 2 , the condition 1 − r 1 r 2 n−1 ≤ |µ 2 | ≤ 1 + r 1 r 2 n−1 must hold. We remark that the second inequality is also a necessary condition for existence of solutions, while the first inequality says that, in order that u(r) jumps across r 2 , the value of |µ 2 | must be not too small. Moreover, the sign of µ determines whether the jump is "up" or "down", and one has u ′ (r + 2 ) = sign(µ 2 )∞ in the right-limit sense. We also note that the problem can be reduced to the discussion of the case µ 2 > 0, because the other case can be obtained by passing from u to −u. In particular, if we take
1 − r 1 r 2 n−1 < µ 2 < 1 ,(10.4)
we conclude that u(r) can only jump "up" at r = r 2 , and in this case u(r) is monotone increasing on the whole (r 1 , r 3 ).
We now build a more sophisticated example by adding a further component to the measure µ. We redefine the annulus by fixing 0 < r 1 < r 2 < r 3 < r 4 , so that Ω = B r 4 \ B r 1 , then choose µ 2 as in (10.4) and µ 3 = 1 − (r 2 /r 3 ) n−1 . Then, we define µ = µ 2 H n−1 ∂B r 2 + µ 3 H n−1 ∂B r 3 .
At the same time, we add a weak Dirichlet boundary condition at r = r 1 , r 4 , that is, we prescribe boundary values 0 at r = r 1 and h at r = r 4 , with h > 0 to be chosen later. Then we consider a radial solution which additionally minimizes the functional J µ among radial BV functions that vanish on B r 1 and take the constant value h outside B r 4 (of course we implicitly assume that a ball B of radius larger than r 4 has been fixed in order to properly define the functional J µ ). It is then clear that the values of the solution must be within the interval [0, h].
We have J µ [u] = |B \ Ω| + 1 + |∇u| 2 (Ω \ (∂B r 2 ∪ ∂B r 3 )) + nω n r n−1 1 u(r + 1 ) + r n−1 2 u(r − 2 )µ 2 + u(r + 2 ) − u(r − 2 ) + r n−1
3 u(r − 3 )µ 3 + u(r + 3 ) − u(r − 3 ) + r n−1 4 h − u(r − 4 ) .
Thus, if we assume that h − u(r − 4 ) > 0 we obtain for 0 < δ < h − u(r − 4 ) (also thanks to the choices of µ 2 and µ 3 ) From now on, we shall assume (10.5) as an extra condition involving the given radii, which guarantees that the boundary datum at r = r 4 is attained in the classical sense (that is, u(r − 4 ) = h).
With a similar computation we claim that the boundary datum is attained also at r = r 1 . Indeed, if we assume that u(r + 1 ) > 0, we can fix 0 < δ < u(r + 1 ) and build the competitor v δ := u − δχ (r 1 ,r 2 ) , so that by easy computations we obtain J µ [v δ ] − J µ [u] = nω n δ r n−1
2
(1 − µ 2 ) − r n−1 1 < 0 because µ 2 > 1 − (r 1 /r 2 ) n−1 . This is a contradiction with the minimality of u, hence our claim is proved. Now, if u(r) were continuous at both r = r 2 , r 3 , we would deduce that h = u(r − 4 ) − u(r + 1 ) = u(r − 4 ) − u(r 3 ) + u(r 3 ) − u(r 2 ) + u(r 2 ) − u(r + 1 ) ≤ C = C(r 1 , r 2 , r 3 , r 4 ) where the constant C results from a catenoidal bound of the global oscillation of the function on each interval (r i , r i+1 ). This means that if we choose h larger than C, we obtain a contradiction with the simultaneous continuity of u at r = r 2 , r 3 . Thus we have that u(r) jumps either at r = r 2 , or at r = r 3 (and possibly at both radii). Let us assume that u(r) jumps at r = r 2 . In this case, by the above characterization of solutions in B r 3 \ B r 1 we deduce that γ 2 = r n−1 2 and γ 3 = γ 2 + µ 3 r n−1 3 = r n−1 3 , hence the restriction of u(r) to the interval (r 2 , r 3 ) is uniquely determined up to suitable vertical translations, while the restriction of u(r) to the interval (r 3 , r 4 ) is uniquely determined because we have proved that u(r + 4 ) = h. It is then immediate to check that there exists a one-parameter family of solutions obtained by vertically translating the profile u(r) restricted to (r 2 , r 3 ), until it preserves the global monotonicity of u(r) on (r 1 , r 4 ). Moreover, the functional J µ attains the same value on this family of solutions. A similar conclusion is achieved if we assume that u(r) jumps at r = r 3 .
Summing up: under the conditions • 1 − (r 1 /r 2 ) n−1 < µ 2 < 1 and µ 3 = 1 − (r 2 /r 3 ) n−1 , • r n−1 4 ≥ r n−1 1 + r n−1 3 ,
• h > C(r 1 , r 2 , r 3 , r 4 ),
we can produce a one-parameter family of weak solutions to (PMCM) attaining the same Dirichlet boundary datum on ∂Ω. This non-uniqueness phenomenon confirms the optimality of the continuity assumption in Theorem 9.1.
Appendix A. Technical results on the area functional
We recall a simple way to estimate the area functional.
Lemma A.1. For any v ∈ BV (Ω) we have 1 + |Dv| 2 ≤ 1 + |Dv| ≤ √ 2 1 + |Dv| 2 in M(Ω).
Proof. It is enough to notice that, for any Borel set E ⊂ Ω, we get
1 + |Dv| 2 (E) = E 1 + |∇v| 2 dx + |D s v|(E) ≤ E (1 + |∇v|) dx + |D s v|(E) = |E| + |Dv|(E) ≤ √ 2 E 1 + |∇v| 2 dx + |D s v|(E) ≤ √ 2 1 + |Dv| 2 (E).
We state now a refinement of Anzellotti-Giaquinta approximation theorem for BV functions involving also the strict convergence with respect to the area functional.
Theorem A.2. Let u ∈ BV (Ω). Then there exists {u ε } ε>0 ⊂ BV (Ω) ∩ C ∞ (Ω) such that (1) u ε → u in L 1 (Ω) as ε → 0, (2) |Du ε |(Ω) ≤ |Du|(Ω) + 4ε for all ε > 0, and so lim ε→0 |Du ε |(Ω) = |Du|(Ω),
(3) 1 + |Du ε | 2 (Ω) ≤ 1 + |Du| 2 (Ω) + 4ε for all ε > 0, and so lim ε→0 1 + |Du ε | 2 (Ω) = 1 + |Du| 2 (Ω), (4) u ε (x) → u * (x) for H n−1 -a.e. x ∈ Ω as ε → 0, (5) u ε L ∞ (Ω) ≤ (1 + ε) u L ∞ (Ω) for all ε > 0.
Proof. The proof is based on a slight modification of the construction of the approximating sequence in Anzellotti-Giaquinta theorem, for which we refer for instance to [13, Theorem 3, Section 5.2.2].
Fix ε > 0. Given a positive integer m, we set Ω 0 = ∅, define for each k ∈ N, k ≥ 1 the sets
Ω k = x ∈ Ω : dist(x, ∂Ω) > 1 m + k ∩ B(0, k + m)
and then we choose m such that |Du|(Ω \ Ω 1 ) < ε.
We define now Σ k := Ω k+1 \ Ω k−1 . Since {Σ k } k∈N is an open cover of Ω, then there exists a partition of unity subordinate to that open cover; that is, a sequence of functions {ζ k } k∈N such that:
(1) ζ k ∈ C ∞ c (Σ k ); (2) 0 ≤ ζ k ≤ 1;
+∞ k=1 ζ k = 1 on Ω.
Then, we take a standard mollifier ρ and for k ∈ N we choose δ k = δ k (ε) > 0 small enough such that the following conditions hold supp(ρ δ k * (uζ k )) ⊂ Σ k , (A.1)
ρ δ k * (uζ k ) − uζ k L 1 (Ω) < ε 2 k , (A.2) ρ δ k * (u∇ζ k ) − u∇ζ k L 1 (Ω;R n ) < ε 2 k , (A.3) ρ δ k * ζ k − ζ k L ∞ (R n ) < ε 2 k (A.4)
and we define u ε := +∞ k=1 ρ δ k * (uζ k ). We notice that, for any fixed x ∈ Ω, there exists a unique k = k(x) ∈ N such that x ∈ Σ k−1 ∩ Σ k ∩ Σ k+1 , so that, by (A.1),
u ε (x) = 1 j=−1 (ρ δ k+j * (uζ k+j ))(x). (A.5)
Hence, u ε ∈ C ∞ (Ω), since locally there are at most three nonzero terms in the sum. The proof of points (1) and (2) . Arguing in a similar way, for any (ϕ, η) ∈ C 1 c (Ω; R n × R) such that (ϕ, η) L ∞ (Ω;R n ×R) ≤ 1 we obtain Ω η + u ε div ϕ dx = Ω η + u div(ζ 1 (ρ δ 1 * ϕ)) dx + +∞ k=2 Ω u div(ζ k (ρ δ k * ϕ)) dx+ − +∞ k=1 Ω ϕ · (ρ δ k * (u∇ζ k ) − u∇ζ k ) dx ≤ 1 + |Du| 2 (Ω) + +∞ k=2 |Du|(Σ k ) + +∞ k=1 ρ δ k * (u∇ζ k ) − u∇ζ k L 1 (Ω;R n ) ≤ 1 + |Du| 2 (Ω) + 3|Du|(Ω \ Ω 1 ) + +∞ k=1 ε 2 k ≤ 1 + |Du| 2 (Ω) + 4ε.
Together with the lower semicontinuity of the area functional with respect to the L 1 convergence, this implies point (3). Then, since ζ k is continuous for every k and u ∈ BV (Ω), by [2, Corollary 3.80] we have (ρ δ k * (uζ k ))(x) → ζ k (x)u * (x) for H n−1 -a.e. x ∈ Ω as ε → 0, where the H n−1 -negligible set of points depends only on u. Hence, as ε → 0 we get u ε (x) = 1 j=−1 (ρ δ k+j * (uζ k+j ))(x) → u * (x) 1 j=−1 ζ k+j (x) = u * (x) for H n−1 -a.e. x ∈ Ω.
As for point (5), we can assume without loss of generality that u ∈ L ∞ (Ω). By (A.4) and (A.5), for any x ∈ Ω we have where the inequality in the fourth line follows from Fatou's lemma, while the exchange of the integration orders is a consequence of Tonelli's theorem.
|u ε (x)| ≤ 1 j=−1 |(ρ δ k+j * (uζ k+j ))(x)| ≤ u L ∞ (Ω) 1 + 1 j=−1 |(ρ δ k+j * ζ k+j )(x) − ζ k+j (x)| ≤ (1 + ε) u L ∞ (Ω) .
T
(u)(x) · ν A (x) dH n−1 (x) < P (A)
Definition 1. 1 .
1We say that u ∈ BV (Ω) is a weak solution to the prescribed mean curvature measure equation if there exist a Borel vector field T : Ω → R n and a Borel function λ : Ω → [0, 1] such thatT L ∞ (Ω;R n ) ≤ 1, (1.3) (T, Du) λ = 1 + |Du| 2 − 1 − |T | 2 L n on Ω, (1.4) div T = µ on Ω, (1.5)where the last two identities involve scalar Radon measures in M(Ω), and where (T, Du) λ denotes the λ-pairing between T and Du.
Theorem 2 . 1 .≤
21Let Ω ⊂ R n be an open, bounded and connected set with Lipschitz boundary. Assume that G is a linear and continuous functional on BV (Ω), such that G(χ Ω ) = 1. Then there exists a constant C > 0 depending only on Ω, such that for all u ∈ BV (Ω) one has u − G(u) C|Du|(Ω) .
Proposition 3. 1 .
1Let u ∈ BV loc (Ω), λ : Ω → [0, 1] be a Borel function and k > 0. For H n−1 -a.e. x ∈ Ω we set M [u, λ](x) := min λ(x)|u + (x)| + (1 − λ(x))|u − (x)| ,
Corollary 3 . 4 .
34Let λ : Ω → [0, 1] be a given Borel function. Let u ∈ BV (Ω) and {u λ k } k ⊂ C ∞ (Ω) be the approximating sequence of Theorem 3.3. Then we have
of Theorem 3.3; while the upper semicontinuity on compact sets follows from the lower semicontinuity on the open sets and point (3) of Theorem 3.3.
B r (x) ⊂ Ω. In particular, this implies µ ∈ M H (Ω), see [24, Theorem 4.2 and Corollary 4.3].
Lemma 4. 7 .
7Let Ω be an open bounded set, and let ν ∈ BV (Ω) * be a nonnegative measure. Then for any u ∈ BV (Ω) one has u ± ∈ L 1 (Ω, ν), as well as M [u, λ] ∈ L 1 (Ω; ν) (see Proposition 3.1 for the definition of M [u, λ]).
Lemma 4 . 8 .
48Let Ω be a bounded open set with Lipschitz boundary and let Γ ⊂ Ω be a Borel set satisfying (4.3). Then there exists a constant C > 0 depending only on Ω, H n−1 (Γ), ρ and Λ such that
Corollary 4. 9 .
9Let Ω ⊂ R n be open, bounded and with Lipschitz boundary. Let µ ∈ M(Ω).
Proposition 4 . 11 .
411Let Ω be an open bounded set with Lipschitz boundary, and let µ ∈ M(Ω). Consider the following set of assumptions:(1) there exists L > 0 such that, for any Borel set E ⊂ Ω we have|µ(E 1 )| ≤ L P (E); (4.7) (2a) M [u, λ](x) ∈ L 1 (Ω; |µ|);(2b) µ is admissible in the sense of Definition 4.5. Then if we assume (1) and, alternatively, (2a) or (2b), we obtain Ω u λ dµ ≤ L |Du|(Ω) + ∂Ω |Tr ∂Ω (u)| dH n−1 .
u ∈ BV (Ω) and any Borel function λ : Ω → [0, 1].
(3. 2 )
2, (3.3), and the fact that M [u, λ](x) ∈ L 1 (Ω; |µ|).
Theorem 4. 13 .
13Let F ∈ DM ∞ (Ω) such that F L ∞ (Ω;R n ) ≤ 1. Let u ∈ BV (Ω) and λ : Ω → [0, 1] be a Borel function. In addition, we assume that• either the function M [u, λ](x) defined in (3.1) belongs to L 1 (Ω; | div F |); • or div F is admissible in the sense of Definition 4.5. Then we have div(uF ), (F, Du) λ ∈ M(Ω), with div(uF ) = u λ div F + (F, Du) λ on Ω (4.10)
Lemma 5 . 3 .
53Given an open set U ⊂⊂ Ω with Lipschitz boundary and such that |µ|(∂U ) = 0, a function v ∈ S(φ), and M > 0, we define
Theorem 8. 1 (
1Uniqueness of T (u)). Let µ ∈ M(Ω) be an admissible measure in the sense of Definition 4.5 and λ = χ Ω − . Assume that there exists u ∈ BV (Ω) and F, G ∈ DM ∞ (Ω) such that (1.3), (1.4) and (1.5) are satisfied. Then we have F = G.
Therefore, if u is a minimizer of J µ in BV (B), then the map u → T (u) is well posed, where T (u) is an essentially bounded divergence-measure field such that the pair (u, T (u)) λ satisfies (1.3), (1.4) and (1.5). In particular, (1.4) means that (T (u), Du) λ = 1 + |Du| 2 − 1 − |T (u)| 2 L n on Ω.
Theorem 9. 1 .
1Let µ 1 , µ 2 ∈ M(Ω) be admissible measures in the sense of Definition 4.5 such that µ 1 ≤ µ 2 . Let u i ∈ BV (Ω) ∩ C 0 (Ω) and T i ∈ DM ∞ (Ω) such that (1.3), (1.4) and (1.5) are satisfied for i = 1, 2. Assume that Tr ∂Ω (u 1 ) ≥ Tr ∂Ω (u 2 ) H n−1 -a.e. on ∂Ω, then we have u 1 ≥ u 2 in Ω.
Now we exploit (1.4) together with the representation of the pairing (T, Du) λµ , and get 1 + |∇u| 2 L n + |D j u| − 1 − |T | 2 L n = 1 + |Du| 2 − 1 − |T | 2 L n = (T, Du) λµ = T, ∇u L n + (T, D j u) λµ = 1 + |∇u| 2 L n − 1 − |T | 2 L n + (T, D j u) λµ .
− 1 2
1and ν u = ν 2 because the second case is in contrast with the bound |γ 1 | ≤ r
J µ [u + δ] − J µ [u] = nω n δ r
follows in a standard way from (A.1), (A.2) and (A.3) (see for instance [13, Theorem 3, Section 5.2.2])
Lemma A. 3 .TrTr
3Let Ω be an open, bounded set with Lipschitz boundary. Let f ∈ BV (Ω) be a non-negative function, and setE t = {x : f (x) > t} for t > 0. Then +∞ 0 ∂Ω Tr ∂Ω (χ Et ) dH n−1 dt ≤ ∂Ω Tr ∂Ω (f ) dH n−1 . Proof. We have ∂Ω Tr ∂Ω (f ) dH n−∩ B r (x)| Ω∩Br(x) χ Et (y) dy dt dH n−1 ∩ B r (x)| Ω∩Br(x)χ Et (y) dy dt dH n−1 ∂Ω (χ Et )(x) dt dH n−1 ∂Ω (χ Et )(x) dH n−1 (x) dt ,
of Lemma 4.1 hold with L P (E) in place of P (E). Indeed,[24, Theorem 4.4] yields the existence of F ∈ DM ∞ (Ω) such that div F = µ. In addition, thanks to[24, Remark 4.8], we know that
3 )
3Now, since Ω is an open bounded set with Lipschitz boundary, there exists a family of bounded open sets with smooth boundary {Ω k } k∈N such that Ω k ⊂ Ω k+1 , iv)] and the references therein). In particular, there exists k 0 ∈ N such that, for all k ≥ k 0 , we haveε ≤ ϕ ε ≤ 2ε on ∂Ω k .Hence, for all k ≥ k 0 we exploit [7, Theorem 4.2] to obtain+∞
k=0
Ω k = Ω and P (Ω k ) → P (Ω)
(see [5, Proposition 8.2 (i), (iii), (
We anticipate here that µ is non-extremal if there exists 0 < L < 1 such that |µ(A)| ≤ L P (A) for all A ⊂⊂ Ω with smooth boundary. Another equivalent definition of non-extremality will be given in the next section.
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| [] |
[
"Spread Flows for Manifold Modelling",
"Spread Flows for Manifold Modelling"
] | [
"Mingtian Zhang \nUniversity College London\n\n",
"Yitong Sun \nUniversity College London\n\n",
"Chen Zhang \nUniversity College London\n\n",
"Steven Mcdonagh \nUniversity College London\n\n"
] | [
"University College London\n",
"University College London\n",
"University College London\n",
"University College London\n"
] | [] | Flow-based models typically define a latent space with dimensionality identical to the observational space. In many problems, however, the data does not populate the full ambient data space that they natively reside in, rather inhabiting a lower-dimensional manifold. In such scenarios, flow-based models are unable to represent data structures exactly as their densities will always have support off the data manifold, potentially resulting in degradation of model performance.To address this issue, we propose to learn a manifold prior for flow models that leverage the recently proposed spread divergence towards fixing the crucial problem; the KL divergence and maximum likelihood estimation are ill-defined for manifold learning. In addition to improving both sample quality and representation quality, an auxiliary benefit enabled by our approach is the ability to identify the intrinsic dimension of the manifold distribution. | null | [
"https://export.arxiv.org/pdf/2109.14216v2.pdf"
] | 257,102,317 | 2109.14216 | b8c75af5f82b6e7a3cab73cb7f3865349586d021 |
Spread Flows for Manifold Modelling
Mingtian Zhang
University College London
Yitong Sun
University College London
Chen Zhang
University College London
Steven Mcdonagh
University College London
Spread Flows for Manifold Modelling
Huawei Noah's Ark Lab Huawei Noah's Ark Lab Huawei Noah's Ark Lab
Flow-based models typically define a latent space with dimensionality identical to the observational space. In many problems, however, the data does not populate the full ambient data space that they natively reside in, rather inhabiting a lower-dimensional manifold. In such scenarios, flow-based models are unable to represent data structures exactly as their densities will always have support off the data manifold, potentially resulting in degradation of model performance.To address this issue, we propose to learn a manifold prior for flow models that leverage the recently proposed spread divergence towards fixing the crucial problem; the KL divergence and maximum likelihood estimation are ill-defined for manifold learning. In addition to improving both sample quality and representation quality, an auxiliary benefit enabled by our approach is the ability to identify the intrinsic dimension of the manifold distribution.
Introduction
Normalizing flows Rezende and Mohamed (2015) have shown considerable potential for the task of modelling and inferring expressive distributions through the learning of well-specified probabilistic models. Recent progress in this area has defined a set of general and extensible structures, capable of representing highly complex and multimodal distributions, see Kobyzev et al. (2020) for a detailed overview. Specifically, assume an absolutely contin-Proceedings of the 26 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2023, Valencia, Spain. PMLR: Volume 206. Copyright 2023 by the author(s). uous 1 (a.c.) random variable (r.v.) Z with distribution P Z and probability density p Z (z). We can transform Z to get a r.v. X: X = f (Z), where f : R D → R D is an invertible function with inverse f −1 = g, so X has a (log) density function p X (x) with the following form log p X (x) = log p Z (g(x)) + log det ∂g ∂x ,
where log det ∂g ∂x is the log determinant of the Jacobian matrix. We call f (or g) a volume-preserving function if the log determinant is equal to 0. Training of flow models typically makes use of MLE. We assume the data random variable X d with distribution P d is a.c. and has density p d (x). In addition to the well-known connection between MLE and minimization of the KL divergence KL(p d (x)||p X (x)) in X space (see Appendix A for details), MLE is also equivalent to minimizing the KL divergence in Z space, due to the KL divergence invariance under invertible transformations Yeung (2008); Papamakarios et al. (2019). Specifically, we define Z Q : Z Q = g(X d ) with distribution Q Z and density function 2 q(z), the KL divergence in Z space KL(q(z)||p(z)) can be written as − p d (x) log p Z (g(x)) + log det ∂g ∂x dx, (2) with constant term discarded, the full derivation can be found in Appendix A. Since we can only access samples x 1 , x 2 , . . . , x N from p d (x), we approximate the integral by Monte Carlo sampling
KL(q(z)||p(z)) ≈ − 1 N N n=1 log p X (x n ) + const.. (3)
We highlight the connection between MLE and KL divergence minimization in Z space for flow models. The prior distribution p(z) is usually chosen to be a D-dimensional Gaussian distribution.
Further to this, we note that if data lies on a lowerdimensional manifold, and thus does not populate the full ambient space, then the estimated flow model will necessarily have mass lying off the data manifold, which may result in under-fitting and poor generation qualities. Contemporary flow-based approaches may make for an inappropriate representation choice in such cases. Formally, when data distribution P d is singular, e.g. a measure on a low dimensional manifold, then P d or the induced latent distribution Q Z will no longer emit valid density functions. In this case, the KL divergence and MLE in equation 2 are typically not well-defined under the considered flow model assumptions. This issue brings theoretical and practical challenges that we discuss in the next section.
Flow Models for Manifold Data
We assume a data sample x ∼ P d to be a D dimensional vector x ∈ R D and define the ambient dimensionality of P d , denoted by Amdim(P d ), to be D. However for many datasets of interest, e.g. natural images, the data distribution P d is commonly believed to be supported on a lower dimensional manifold Beymer and Poggio (1996). We assume the dimensionality of the manifold to be K where K < D, and define the intrinsic dimension of P d , denoted by Indim(P d ), to be the dimension of this manifold. Figure 1a provides an example of this setting where P d is a 1D distribution in 2D space. Specifically, each data sample x ∼ P d is a 2D vector x = {x 1 , x 2 } where x 1 ∼ N (0, 1) and x 2 = sin(2x 1 ). Therefore, this example results in Amdim(P d ) = 2 and Indim(P d ) = 1.
In flow-based models, function f is constructed such that it is both bijective and differentiable. When the prior P Z is a distribution whose support is R D (e.g. Multivariate Gaussian distribution), the marginal distribution P X will also have support R D and Amdim(P X ) = Indim(P X ) = D. When the support of the data distribution lies on a K-dimensional manifold and K < D, P d and P X are constrained to have different support. That is, the intrinsic dimensions of P X and P d are always different; Indim(P X ) = Indim(P d ). In this case it is impossible to learn a model distribution P X identical to the data distribution P d . Nevertheless, flow-based models have shown strong empirical success in real-world problem domains such as the ability to generate high-quality and realistic images Kingma and Dhariwal (2018). Towards investigating the cause, and explaining this disparity between theory and practice, we employ a toy example to provide intuition for the effects and consequences resulting from model and data distributions that possess differing intrinsic dimensions.
Consider the toy dataset introduced previously; a 1D distribution lying in a 2D space (Figure 1a). The prior density p(z) is a standard 2D Gaussian p(z) = N (0, I Z ) and the function f is a non-volume preserving flow with two coupling layers (see Appendix C.1). In Figure 1b we plot samples from the flow model; the sample x is generated by first sampling a 2D datapoint z ∼ N (0, I Z ) and then letting x = f (z). Figure 1c shows samples from the prior distributions P Z and Q Z . Q Z is defined as the transformation of P d using the bijective function g, such that Q Z is constrained to support a 1D manifold in 2D space, and Indim(Q Z ) = Indim(P d ) = 1. Training of Q Z to match P Z (which has intrinsic dimension 2), can be seen to result in "curling up" of the manifold in the latent space, contorting it towards satisfying a distribution that has intrinsic dimension 2 (see Figure 1c). This ill-behaved phenomenon causes several potential problems for contemporary flow models:
1. Poor sample quality. Figure 1b shows examples where incorrect assumptions result in the model generating poor samples.
2. Low quality data representations. The discussed characteristic that results in "curling up" of the latent space may cause representation quality degradation.
3. Inefficient use of network capacity. Neural network capacity is spent on contorting the distribution Q Z to satisfy imposed dimensionality constraints.
A natural solution to the problem of intrinsic dimension mismatch is to select a prior distribution P Z with the same dimensionality as the intrinsic dimension of the data distribution such that: Indim(P Z ) = Indim(P d ). However, since the Indim(P d ) is unknown, one option is to instead learn it from the data. In the next section, we introduce an approach that enables us to learn Indim(P d ).
Learning a Manifold Prior
Consider a data vector x ∈ R D , then a flow-based model prior P Z is usually given by a D-dimensional Gaussian distribution or alternative simple distribution that is also absolutely continuous (a.c.) in R D . Therefore, the intrinsic dimension Indim(P Z ) = D. To allow a prior to have an intrinsic dimension strictly less than D, we let P Z be a 'generalized Gaussian' 3 distribution GN (0, AA T ), where z ∈ R D and A is a D×D lower triangular matrix with D(D + 1)/2 parameters, such that AA T is constrained to be a positive semi-definite matrix. When AA T has full rank D, then P Z is a (non-degenerate) multivariate Gaussian on R D . When Rank(AA T ) = K and K < D, then P Z will degenerate to a Gaussian supported on a Kdimensional manifold, such that the intrinsic dimension For data (black dots) that lie on a 2D manifold in 3D space, we want to use a model P X with Indim(P X ) = 2. Therefore, we learn a prior P Z that is a 2D Gaussian in 3D space and an invertible function f which maps from P Z to P X .
(a) P d (b) X space (c) Z space
Indim(P Z ) = K 4 . Figure 2 illustrates a sketch of this scenario. In practice, we initialize A to be an identity matrix, thus AA T is also an identity and P Z is initialized as a standard Gaussian. We want to highlight that the classic flow method with a standard Gaussian prior is a special case with A = I in our model and the additional free parameters in matrix A allow the model to capture the manifold property of the target data distribution.
Identification of the Intrinsic Dimension
A byproduct of our model is that the intrinsic dimension of the data manifold can be identified. When the model matches the true data distribution P X = P d , the supports of P X and P d will also have the same intrinsic dimension: Indim(P X ) = Indim(P d ).
The flow function f , and its inverse g = f −1 , are bijective and continuous so f is a diffeomorphism Kobyzev et al. (2020). Due to the invariance of dimension property of diffeomorphisms Lee (2013), the manifold that supports P Z will have the same dimension as the manifold that supports P X thus
Indim(P Z ) = Indim(P X ) = Indim(P d ).(4)
Since the intrinsic dimension of P Z is equal to the rank of the matrix AA T , we can calculate Rank(AA T ) by counting the number of non-zero eigenvalues of the matrix AA T . This allows for identification of the intrinsic dimension of the data distribution as
Indim(P d ) = Rank(AA T ).(5)
Dimension Reduction We would also like to conduct dimension reduction using the learned manifold flow model. For Q Z with Amdim(Q Z )=D and Indim(Q Z )=K, we first conduct an eigen-value decomposition of the D × D matrix AA T : AA T = EΛE T , where E = [e 1 , · · · , e D ] contains all the eigen-vectors and Λ = diag(λ 1 , · · · , λ D ). When Rank(AA T ) = K ≤ D, there exist K eigenvectors with positive eigenvalues. We select the first K eigenvectors and form the matrix E = [e 1 , . . . , e K ] with dimension D×K. We then transform each data sample x ∈ R D into Z space: z = g(x), such that z ∈ R D . Finally, a linear projection is carried out z proj = zE to obtain the lower dimensional representation z proj ∈ R K . This procedure can be seen as a nonlinear PCA, where the nonlinearity is learned by the inverse flow function g.
Sample from the Manifold Flow To sample from the flow model with Indim(P X ), we can first take a sample from a standard K-dimensional Gaussian distribution and let z = E K √ Λ K where E K are the first K eigenvectors and Λ K is the corresponding eigen-values. When K = D, the linear matrix E K √ Λ K will be equivalent to the Cholesky decomposition solution of the matrix AA T . We can then let x = f (z ) as the sample from the model.
Addressing Ill-defined KL divergence
When Rank(AA T ) < D, the degenerate covariance AA T is no longer invertible and we are unable to evaluate the density value of p(z) for a given random vector z. Furthermore, when the data distribution P d is supported on a K-dimensional manifold, Q Z will also be supported on a K-dimensional manifold and no longer has a valid density function. Using equation 2 to train the flow model then becomes impossible as the KL divergence between P Z and Q Z is not well defined 5 . Recent work by Zhang et al. (2020Zhang et al. ( , 2019 proposed a new family of divergence to address this problem, which we introduce below.
Let Z Q and Z P be two random variables with distributions Q Z and P Z , respectively. The KL divergence between Q Z and P Z is not well defined if Q Z or P Z does not have valid density functions. Let K be an a.c. random variable that is independent of Z Q and Z P and has density p K , We define ZP = Z P + K; ZQ = Z Q + K with distributionsP Z and Q Z respectively. ThenP Z andQ Z are a.c. (Durrett, 2019, Theorem 2.1.16) with density functions
q(z) = z p K (z − z)dQ Z , p(z) = z p K (z − z)dP Z . (6)
The spread KL divergence between Q Z and P Z as the KL divergence betweenQ Z andP Z is:
KL(Q Z ||P Z ) ≡ KL(Q Z ||P Z ) ≡ KL (q(z)||p(z)) . (7)
In this work we let K be a Gaussian with diagonal covariance σ 2 Z I to satisfy the sufficient conditions such that KL(·) is a valid divergence (see Zhang et al. (2020) for details) and has the properties:
KL(Q Z ||P Z ) ≥ 0, KL(Q Z ||P Z ) = 0 ⇔ Q Z = P Z . (8)
Since Q Z and P Z are transformed from P d and P X using an invertible function g, we have
Q Z = P Z ⇔ P d = P X .(9)
Therefore, the spread KL divergence can be used to train flow-based models with a manifold prior in order to fit a dataset that lies on a lower-dimensional manifold.
Estimation of the Spread KL Divergence
As shown in equation 7, minimizing KL(Q Z ||P Z ) is equivalent to minimizing KL(q(z)||p(z)), which has two terms
KL(Q Z ||P Z ) = q(z) log q(z)dz Term 1 − q(z) log p(z)dz Term 2 ,(10)
where q(z) and p(z) are defined in equation 6. We next discuss the estimation of this objective.
Term 1: We use H(·) to denote the differential entropy. Term 1 is the denoted as −H(ZQ). For a volume-preserving 5 The KL divergence KL(Q||P) is well defined when Q and P have valid densities with common support Ali and Silvey (1966).
g and an a.c. X d , the entropy H(Z Q ) = H(X d ) is independent of the model parameters and can be ignored during training. However, the entropy H(ZQ) = H(Z Q + K) will still depend on g, see Appendix B.1 for an example. We claim that when the variance of K is small, the dependency between H(ZQ) and the volume preserving function g is weak, thus we can approximate equation 10 by disregarding term 1 and this will not adversely affect the training.
To build intuitions, we assume X d is a.c., so Z Q = g(X d ) is also a.c.. Using standard entropic properties Kontoyiannis and Madiman (2014), we have the following relationship
H(Z Q ) ≤ H(Z Q + K) = H(Z Q ) + I(Z Q + K, K),(11)
where I(·, ·) denotes the mutual information. Since Z Q is independent of function g and if σ 2 Z → 0, then I(Z Q + K, K) → 0 (see Appendix B.2 for a proof), the contribution of the I(Z Q + K, K) term, with respect to training g, becomes negligible in the case of small σ 2 Z . Unfortunately, equation 11 is no longer valid when P d lies on a manifold since Z Q will be a singular random variable and the differential entropy H(Z Q ) is not defined. In Appendix B.3, we show that leaving out the entropy H(ZQ) corresponds to minimizing an upper bound of the spread KL divergence. To further explore the contribution of the negative entropy term, we compare leaving out −H(ZQ) with alternatively approximating −H(ZQ) during training. In Appendix B.4, we discuss an appropriate approximation technique and provide empirical evidence which shows that ignoring −H(ZQ) will not adversely affect the training of our model. Therefore, we make use of volume preserving g and small variance σ 2 Z = 1 × 10 −4 in our experiments. In contrast to other volume-preserving flows, that utilize a fixed prior P Z , our method affords additional flexibility by way of allowing for changes to the 'volume' of the prior towards matching the distribution of the target data. In this way, our decision to employ volume-preserving flow functions does not limit the expressive power of the model, in principle. Popular non-volume preserving flow structures, e.g. affine coupling flow, may also easily be normalized to become volume preserving, thus further extending the applicability of our approach (see Appendix C.1).
Term 2: The noisy prior p(z) is defined to be a degenerate Gaussian N (0, AA T ), convolved with Gaussian noise N (0, σ 2 Z I), and has a closed form density p(z) = N (0, AA T + σ 2 Z I).
Therefore, the log density log p(z) is well defined, we can approximate term 2 using Monte Carlo
q(z) log p(z)dz ≈ 1 N N n=1 log p(z n ),(13)
where q(z) = p(z|z)dQ Z . To sample from q(z), we first get a data sample x ∼ P d , use function g to get z = g(x)
(so z is a sample of Q Z ) and finally samplez ∼ p(z|z). The training details can be found in Algorithm 1.
Algorithm 1 Training Spread Flows
Given: X train = {x 1 , · · · , x N } ∼ P d , N (0, σ 2 Z I) Model: An inverse flow function g, a lower triangular matrix A initialized as an identity matrix.
while not converge do Sample a data batch
X B = {x 1 , · · · , x B } ∈ X train . Transform data z b = g(x b ) for x b ∈ X B . Sample noise b ∼ N (0, σ 2 Z I) for each x b . Add noisez b = z b + b for each latent representation.
Train g θ and A using Equation 13. end while
Experiments
Traditional flows assume the data distributions are a.c. and we extend this assumption such that the data distribution lies on one continuous manifold. We demonstrate both the effectiveness and robustness of our model by considering firstly (1) data distributions that satisfy our continuous manifold assumption: toy 2D and 3D data, the fading square dataset; and secondly (2) distributions where our assumption no longer holds: synthesized and real MNIST Le-Cun (1998)
Synthetic Data
2D toy data We firstly verify our method using the toy data depicted in Figure 1a. Figure 3 shows model samples, the learned prior and the eigenvalues of AA T . We observe that sample quality improves upon those in Figure 1b and the prior has learned a degenerate Gaussian with Indim(P Z ) = 1, matching Indim(P d ). We also highlight that our model, in addition to learning the manifold support of the target distribution, can capture the 'density' allocation on the manifold, see Appendix C.2 for further details.
3D toy data (S-curve) We model the S-curve dataset (Figure 4a), where the data lies on a 2D manifold in a 3D space (Indim(P d )=2), see Appendix C.1 for details. Our model learns a function g to transform P d to Q Z , where the latter lies on a 2D linear subspace in 3D space (see Figure 4c). As discussed in Section 3, we also conduct a linear dimension reduction to generate 2D representations, see Figure 4f. The colormap indicates correspondence between the data in 3D space and the 2D representation. Our method can be observed to successfully: (1) identify the intrinsic dimensionality of the data and (2) project the data into a 2D space that faithfully preserves the structure of the original distribution. In contrast, we find that a flow with fixed Gaussian prior fails to learn such a data distribution and generate meaningful representations, see Figure 4g and 4h.
Fading Square dataset The fading square dataset Rubenstein et al. (2018) was proposed in order to assess model behavior when data distribution and model possess differing intrinsic dimension and therefore affords a relevant test of our work. The dataset consists of 32×32 images with 6×6 grey squares on a black background. The grey scale values are sampled from a uniform distribution with range [0, 1], so Indim(P d )=1. Figure 5a shows samples from the dataset to which we fit our model and additional model details may be found in Appendix C. Figure 5b shows samples from our trained model and Figure 5d shows the first 20 eigenvalues of AA T (ranked from high to low), we observe that only one eigenvalue is larger than zero and the others have converged to zero. This illustrates we have successfully identified the intrinsic dimension of P d . We further carry out the dimensionality reduction process; the latent representation z is projected onto a 1D line and we plot the correspondence between the projected representation and the data in Figure 5e. Pixel values can be observed to decay as the 1D representation is traversed from left to right, indicating the representations are consistent with the properties of the original data distribution. In contrast, we find that the traditional flow model, with fixed 1024D Gaussian prior, fails to learn the data distribution; see Figure 5c.
Synthetic MNIST We further investigate model training using digit images. It may be noted that, for real-world image datasets, the true intrinsic dimension is unknown. Therefore in order to further verify the correctness of our model's ability to identify intrinsic dimension, we first construct a synthetic dataset by fitting an implicit model p θ (x)= δ(x − g(z)p(z))dz to the MNIST, and then sample from the trained model x ∼ p θ (x) in order to generate synthetic training data. The intrinsic dimension of the training data can then be approximated 6 via the dimension of the latent variable Z: Indim(P d )=dim(Z). We construct two datasets with dim(Z)=5 and dim(Z)=10 such that Indim(P d )=5 and Indim(P d )=10, respectively. Since the generator of the implicit model is not constrained to be invertible, a diffeomorphism between data distribution and prior will not hold and will also not necessarily lie on a continuous manifold. We found when the continuous manifold assumption is not satisfied, training instabilities can occur. Towards alleviating this issue, we smooth the data manifold by adding small Gaussian noise (with standard deviation σ x =0.05) to the training data. We note that adding Gaussian noise changes the intrinsic dimension of the training distribution towards alignment with the ambient dimension and therefore we mitigate this undesirable effect by annealing σ x after 2000k iterations with a factor of 0.9 every 10k iteration. Experimentally, we cap a lower-bound Gaussian noise level of 0.01 to help retain successful model training and find the effect of the small noise to be negligible when estimating the intrinsic dimension. In Figure 6, we plot the first twenty eigenvalues of AA T (ranked from high to low) after fitting two synthetic MNIST with dim(P d )={5, 10}. We can find that 5 and 10 eigenvalues are significantly larger than other values, respectively. The remaining non-zero eigenvalues can be attributed to the added small Gaussian noise. Therefore, our method can successfully model complex data distributions and identify the intrinsic dimension when the data distribution lies on a continuous manifold. We also provide model sample comparisons in Figure 7, where we can find our model can achieve better sample quality compared to the traditional flow. In the next section, we apply our model to real image data where this assumption may not be satisfied.
(a) X space (b) Z space (c) Eigenval. (d)
Real World Data
Real MNIST For the real MNIST, it was shown that digits have differing intrinsic dimensions Costa and Hero (2006). This suggests the MNIST may lie on several, disconnected manifolds with differing intrinsic dimensions. Although our model assumes that P d lies on one continuous manifold, it is interesting to investigate the case when this assumption is not fulfilled. We thus fit our model to the original MNIST and plot the eigenvalues in Figure 6 Similar to the MNIST experiment, we add small Gaussian noise (with standard deviation σ x =0.01) for twenty initial training epochs in order to smooth the data manifold and then anneal the noise with a factor of 0.9 for a further twenty epochs. Additional training and network structure details can be found in Appendix C.6.
We aim to empirically verify if the learned model is concentrated on a low-dimensional manifold and examine the validity of the estimated intrinsic dimensions. Following Ross and Cresswell (2021), we study if the data can be reconstructed from a low-dimensional linear manifold specified by AA T . In Figure 9, we plot the learned eigenvalues Λ in log space and observe that eigenvalues have an exponential decay starting at ∼2700, which suggests that the intrinsic dimension of the CelebA is around 2700. For reconstructions, we first calculate the low dimensional representation of image x by transforming it to the latent space with the inverse flow z = g(x) (z ∈ R D ) and then conduct a linear projection z proj = zE k to obtain the representation z proj ∈ R K , where E k contains the first K eigenvectors ranked by the corresponding eigenvalues (see Section 3). To reconstruct x from z proj , we firstly conduct an inverse projectionẑ = z proj E T k and then obtain the recon-
structionx by lettingx = f (ẑ), where f = g −1 .
In Figure 10 we show reconstructions with differing K and may observe that good reconstruction can be obtained when K is greater than 2700. The quality goes down as K decreases, which empirically verifies the intrinsic dimension of the data distribution to be around ∼2700, which is consistent with our model estimation (as shown in Figure 9).
We also compare the sample quality of our model with the recently proposed CEF model Ross and Cresswell (2021), which proposes to learn a flow on manifold distributions. Distinct from our model, where different K values can be selected post training, the CEF work requires an intrinsic model dimension to be pre-specified. We follow Ross and Cresswell (2021); select K = 512 and train the model on the CelebA (32×32) dataset. Our model achieves better FID compared to CEF (see Table 5 in the Appendix for FID comparisons). The visualizations of the samples can be found in Figure
Related Work
Classic latent variable generative models assume that data distributions lie around a low-dimensional manifold, for example, the Variational Auto-Encoder Kingma and Welling (2013) . Therefore, we focus on modelling distributions that lie on a low-dimensional manifold.
The study of manifold learning for nonlinear dimensionality reduction Cayton (2005) and intrinsic dimension estimation Camastra and Staiano (2016) is a rich field with an extensive set of tools. However, most methods commonly do not model the data density on the manifold and are thus not used for the same purpose as the models introduced in our work. There are however a number of recent works that introduce normalizing flows on manifolds that we now highlight and relate to our approach.
Several works define flows on manifolds with prescribed charts. Gemici et al. (2016) generalized flows from Euclidean spaces to Riemannian manifolds by proposing to map points from the manifold M to R K , apply a normalizing flow in this space and then mapping back to M. The technique has since been further extended to Tori and Spheres Rezende et al. (2020). In contrast to our work, these methods require knowledge of the intrinsic dimension K and a parameterization of the coordinate chart of the data manifold.
Several recently proposed manifold flow models can learn the data distribution without providing a chart mapping a priori. M-flow Brehmer and Cranmer (2020) proposed an algorithm that maps the data distribution to a lowerdimensional space with an auto-encoder and uses a flow in the lower-dimensional space to learn data distributions. Rectangular flow Caterini et al. (2021) and Conformal Embedding Flows Ross and Cresswell (2021) propose to directly build injective mappings from low-dimension space to high-dimension space. However, their methods still require that the dimensionality of the manifold is known. In Brehmer and Cranmer (2020), it is proposed that dimensionality can be learned either by a brute-force solution or through a trainable variance in the density function. The brute-force solution is clearly infeasible for data embedded in extremely high dimensional space, as is often encoun-tered in deep learning tasks. Use of a trainable variance is natural and similar to our approach. However, as discussed at the beginning of this paper, without carefully handling the KL or MLE term in the objective, a vanishing variance parameter will result in the wild behaviour of the optimization process since these terms are not well defined.
The GIN model considered in Sorrenson et al. (2020) could recover the low-dimensional generating latent variables by following their identifiability theorem. However, the assumptions therein require knowledge of an auxiliary variable, i.e. the label, which is not required in our model. Behind this difference is the essential discrepancy between the concept of informative dimensions and intrinsic dimensions. The GIN model discovers the latent variables that are informative in a given context, defined by the auxiliary variable u instead of the true intrinsic dimensions. In their synthetic example, the ten-dimensional data is a nonlinear transformation of ten-dimensional latent variables where two out of ten are correlated with the labels of the data and the other eight are not. In this example, there are two informative dimensions, but ten intrinsic dimensions. Nevertheless, our method for intrinsic dimension discovery can be used together with informative dimension discovery methods to discover finer structures of data.
In recent work Tempczyk et al. (2022), the authors made an intriguing observation regarding the impact of adding Gaussian noise N (0, σ 2 I) to a given data distribution. Specifically, they found that the change in log-likelihood, resulting from adding noise with different values of σ is approximately linear in the logarithm of σ, with a proportionality constant that corresponds to the difference between the intrinsic and ambient dimensions of the data. Although both methods involve fitting flows to a noised data distribution, our method for estimating the intrinsic dimension is different: we estimate the intrinsic dimension using the rank of a learned degenerate Gaussian prior. We consider the future study of connections between these approaches may help to afford a deeper understanding of intrinsic dimensionalities with respect to complex data distributions. We find that our model can provide improved sample quality on these manifold datasets.
Limitations and Future Work
In our work, we present a novel manifold flow model that utilizes a learnable generalized Gaussian prior to capture the low-dimensional manifold data distribution and identify their intrinsic dimensions. We demonstrated the benefits of our model in terms of sample generation and representation quality. While our model offers a promising step forward on the modelling of manifold distributions, it is important to note that we rely on a somewhat strong assumption: that the data distribution lies on a continuous manifold. This assumption may not hold for real-world image data, which typically exhibit complex and varied distributions. We conjecture that this limitation can be partially alleviated through the incorporation of prior techniques developed by Cornish et al. (2019); affording relaxation of the continuous manifold assumption and thus enable the flow model to capture data distributions that lie on a set of disconnected manifolds. We leave this to future work. We can see the quality of the reconstruction slightly decreases when K is lower than 2700, which empirically verified that the intrinsic dimension of the data distribution is around 2700.
References
A Maximum Likelihood Estimation and KL divergence
Given data x 1 , x 2 , . . . , x N sampled independently from the true data distribution P d , with density function p d (x), we want to fit the model density p(x) 7 to the data. A popular choice to achieve this involves minimization of the KL divergence where:
KL(p d (x)||p(x)) = p d (x) log p d (x)dx − p d (x) log p(x)dx (14) = − p d (x) log p Z (g(x)) + log det ∂g ∂x dx + const..(15)
Since we can only access samples from p d (x), we approximate the integral by Monte Carlo sampling
KL(p d (x)||p(x)) ≈ − 1 N N n=1 log p(x n ) + const..(16)
Therefore, minimizing the KL divergence between the data distribution and the model is (approximately) equivalent to Maximum Likelihood Estimation (MLE).
When p(x) is a flow-based model with invertible flow function f : Z → X, g = f −1 , minimizing the KL divergence in X space is equivalent to minimizing the KL divergence in the Z space. We let X d be the random variable of the data distribution and define Z Q : Z Q = g(X d ) with density q(z), so q(z) can be represented as
q(z) = δ(z − g(x))p d (x)dx.(17)
Let p(z) be the density of the prior distribution P Z , the KL divergence in Z space can be written as
KL(q(z)||p(z)) = q(z) log q(z)dz Term 1 − q(z) log p(z)dz Term 2 .(18)
Term 1: using the properties of transformation of random variables Papoulis and Pillai (2002, pp. 660), the negative entropy can be written as
q(z) log q(z)dz = p d (x) log p d (x)dx const. − p d (x) log det ∂g ∂x dx.(19)
Term2: the cross entropy can be written as
q(z) log p(z)dz = δ(z − g(x))p d (x) log p(z)dzdx (20) = p d (x) log p(g(x))dx.(21)
Therefore, the KL divergence in Z space is equivalent to the KL divergence in X space
KL(q(z)||p(z)) = KL(p d (x)||p(x)).(22)
We thus build the connection between MLE and minimizing the KL divergence in Z space.
B Entropy
B.1 An example
Assume a 2D Gaussian random variable X with covariance 1 0 0 1 . There exits two volume-perserving flows g with parameters θ 1 and θ 2 (θ 1 = θ 2 ) to make Z 1 = g θ1 (X) and Z 2 = g θ2 (X) be Gaussians with covariance 2 0 0 1 2 and 7 For brevity, we use notation p(x) to represent the model pX (x) unless otherwise specified.
3 0 0 1 3 respectively. In this case, the entropy H(Z 1 ) = H(Z 2 ) = H(X) and does not depend on θ. However, for a given Gaussian variable K, H(Z 1 + K) = H(Z 2 + K) and H(g θ (X) + K) will depend on θ. For example, the distribution of K is a Gaussian with covariance 1 0 0 1 , so Z 1 + K is a Gaussian with covariance 3 0 0 3 2 and Z 2 + K is a Gaussian with covariance 4 0 0 4 3 , so H(Z 1 + K) = H(Z 2 + K). A similar example can be constructed when X is not absolutely continuous.
B.2 Z is an absolutely continuous random variable
For two mutually independent absolutely continuous random variable Z and K, the mutual information between Z + K and K is
I(Z + K, K) = H(Z + K) − H(Z + K|K) (23) = H(Z + K) − H(Z|K) (24) = H(Z + K) − H(Z).(25)
The last equality holds because Z and K are independent. Since mutual information I(Z + K, K) ≥ 0, we have
H(Z) ≤ H(Z + K) = H(Z) + I(Z + K, K).(26)
Assume K has a Gaussian distribution with 0 mean and variance σ 2 Z 8 . When σ 2 Z = 0, K degenerates to a delta function, so Z + K = Z and
I(Z + K, K) = H(Z + K) − H(Z) = 0,(27)
this is because the mutual information between an a.c. random variable and a singular random variable is still well defined, see Yeung (2008, Theorem 10.33). Assume K 1 , K 2 are Gaussian random variables with 0 mean and variances σ 2 1 and σ 2 2 respectively. Without loss of generality, we assume σ 2 1 > σ 2 2 and σ 2 1 = σ 2 2 + σ 2 δ , and let K δ be the random variable of a Gaussian that has 0 mean and variance σ 2 δ such that K 1 = K 2 + K δ . By the data-processing inequality, we have I(Z + K 2 , K 2 ) ≤ I(Z + K 2 + K δ , K 2 + K δ ) = I(Z + K 1 , K 1 ).
Therefore, I(Z + K, K) is a monotonically decreasing function when σ 2 Z decreases and when σ 2 Z → 0, I(Z + K, K) → 0.
B.3 Upper bound of the spread KL divergence
In this section, we show that leaving out the entropy term H(ZQ) in equation 10 is equivalent to minimizing an upper bound of the spread KL divergence.
For singular random variable Z Q = g(X d ) and absolutely continuous random variable K that are independent, we have
H(Z Q + K) − H(K) = H(Z Q + K) − H(Z Q + K|Z Q ) (29) = I(Z Q + K, Z Q ) ≥ 0.(30)
The second equation is from the definition of Mutual Information (MI); the MI between an a.c. random variable and a singular random variable is well defined and always positive, see Yeung (2008, Theorem 10.33) for a proof.
Therefore, we can construct an upper bound of the spread KL objective in equation 10
KL(q||p) = q(z) log q(z)dz −H(Z Q +K) − q(z) log p(z)dz (31) = −H(K) const. − I(Z Q + K, Z Q ) ≥0 − q(z) log p(z)dz.(32)
≤ −H(K)
const.
− q(z) log p(z)dz.
Therefore, ignoring the negative entropy term during training is equivalent to minimizing an upper bound of the spread KL objective, and the gap between the bound and the true objective is I(Z Q + K, Z Q ). Unlike the case when Z Q is a.c., I(Z Q + K, Z Q ) will be close to 0 when the variance of K is sufficiently small, we do not have the same result when Z Q is not absolutely continuous. However, we show that, under some assumptions of the flow function, the gap I(Z Q + K, Z Q ) can be upper bounded.
B.4 Empirical evidence for ignoring the negative entropy
In this section, we first introduce the approximation technique to compute the negative entropy term, and then discuss the contribution of this term during training. The negative entropy of random variable ZQ is
−H(ZQ) = q(z) log q(z)dz,(34)
where
q(z) = z p K (z − z)dQ Z ,(35)
and p K is the density of a Gaussian with diagonal covariance σ 2 Z I. We first approximateq(z) by a mixture of Gaussians
q(z) ≈ 1 N N n=1 N (z; z n , σ 2 Z I) ≡q N (z)(36)
where z n is the nth sample from distribution Q z by first sampling x n ∼ P d and letting z n = g(x n ). We denote the random variable of this Gaussian mixture asẐ Ñ Q and approximate
−H(ZQ) ≈ −H(Ẑ Ñ Q ).(37)
However, the entropy of a Gaussian mixture distribution does not have a closed form, so we further conduct a first order Taylor expansion approximation Huber et al.
(2008) −H(Ẑ Ñ Q ) ≈ 1 N N n=1 logq N (z = z n ),(38)
this approximation is accurate for small σ 2 Z . Finally we have our approximation
−H(ZQ) ≈ 1 N N n=1 logq N (z = z n ).(39)
To evaluate the contribution of the negative entropy, we train our flow model on both low dimensional data (Toy datasets: 2D) and high dimensional data (Fading square dataset: 1024D) by optimization that uses two objectives: (1) ignoring the negative entropy term in equation 7 and (2) approximating the negative entropy term in equation 7 using the approximation discussed above. During training, we keep track of the value of the entropy H(ZQ) (using the approximation value) in both objectives. We let N equal the batch size when approximating the entropy. Additional training details remain consistent with those described in Appendix C.
In Figure 11, we plot the (approximated) entropy value H(ZQ) during training for both experiments. We find the difference between having the (approximated) negative entropy term and ignoring the negative entropy to be negligible. We leave rigorous theoretical investigation on the effects of discarding the entropy term during training to future work.
C Experiments
We conduct all our experiments on one single NVDIA GeForce RTX 2080 Ti GPU.
y 1:d = x 1:d (40) y d+1:D = x d+1:D exp(s(x 1:d )) + t(x 1:d ),(41)
where s : R d → R D−d and t : R d → R D−d are scale and translation functions parameterized by neural networks, is the element-wise product. The log-determinant of the Jacobian of a coupling layer is the sum of the scaling function j s(x 1:d ) j . To make the coupling transform volume preserving, we normalize the output of the scale function, so the i-th dimension of the output iss
(x 1:d ) i = s(x 1:d ) i − 1 D − d j s(x 1:d ) j ,(42)
and the log-determinant of the Jacobian is is (x 1:d ) i = 0. We compare a volume-preserving flow with a learnable prior (normalized s(·)) to a non-volume preserving flow with a fixed prior (unnormalized s(·)). In this way both models have the ability to adapt their 'volume' to fit the target distribution, retaining comparison fairness.
In our affine coupling layer, the scale function s and the translation function t have two types of structure: fully connected net and convolution net. Each fully connected network is a 4-layer neural network with hidden-size 24 and Leaky ReLU with slope 0.2. Each convolution net is a 3-layer convolutional neural network with hidden channel size 16, kernel size 3×3 and padding size 1. The activation is Leaky ReLU with a slope 0.2. The downsampling decreases the image width and height by a factor of 2 and increases the number of channels by 4 in a checkerboard-like fashion Sorrenson et al. (2020);Jacobsen et al. (2018). When multiple convolutional nets are connected together, we randomly permute the channels of the output of each network except the final one. In Table 1, 2, 3, 4, we show the network structures for our four main paper experiments.
C.2 Toy dataset
We also construct a second dataset and train a flow model using the same training procedure discussed in Section 5. Figure 12a shows the samples from the data distribution P d . Each data point is a 2D vector x = [x 1 , x 2 ] where x 1 ∼ N (0, 1) and x 2 = x 1 , so Indim(P d ) = 1. Figure 12f shows that the prior P Z has learned the true intrinsic dimension Indim(P Z ) = Indim(P d ) = 1. We train our model using learning rate 3 × 10 −4 and batch size 100 for 10k iterations. We compare to samples drawn from a flow model that uses a fixed 2D Gaussian prior, with results shown in Figure 12. We can observe, for this simple dataset, flow with a fix Gaussian prior can generate reasonable samples, but the 'curling up' behavior, discussed in the main paper, remains highly evident in the Z space, see Figure 12c.
Manifold density We also plot the 'density allocation' on the manifold for the two toy datasets. For example, for the data generation process x = [x 1 , x 2 ] where x 1 ∼ p = N (0, 1) and x 2 = x 1 , we use the density value p(x = x 1 ) to indicate the 'density allocation' on the 1D manifold. To plot the 'density allocation' of our learned model, we first sample x s uniformly from the support of the data distribution, the subscript 's' here means that they only contain the information of the support. Specifically, since x s = [x s 1 , x s 2 ], we sample x s 1 ∼ p = U(−3σ, 3σ) (σ is the standard deviation of N (0, 1), U is the uniform distribution) and let x s 2 = x s 1 or x s 2 = sin(x s 1 ), depending on which dataset is used. We use the projection procedure that was described in Section 5 to obtain the projected samples z proj s , so z proj s ∈ R Indim(P d ) . We also project the learned prior P Z to R Indim(P d ) by constructing P proj Z as a Gaussian with zero mean and a diagonal covariance contains the non-zeros eigenvalues of AA T . Therefore P proj Z is a.c. in R Indim(P d ) and we denote its density function as p proj (z). We then use the density value p proj (z = z proj s ) to indicate the 'density allocation' at the location of x s on the manifold support. In Figure 13, we compare our model with the ground truth and find that we can successfully capture the manifold density.
Type of block
Number Input shape Affine coupling layer widths Fully connected 2 2 1 → 24 → 24 → 24 → 1
C.3 S-Curve dataset
To fit our model to the data, we use the Adam optimizer with learning rate 5 × 10 −4 , batch size 500 and train the model for 200k iterations. We anneal the learning rate with a factor of 0.9 every 10k iteration.
C.4 Fading Square dataset
To fit the data, we train our model for 20k iterations with a batch size of 100 using the Adam optimizer. The learning rate is initialized to 5×10 −4 and decays with a factor of 0.9 at every 1k iteration. We additionally use an L2 weight decay with factor 0.1. To fit an implicit model to the MNIST dataset, we first train a Variational Auto-Encoder (VAE) Kingma and Welling (2013) with Gaussian prior p(z)=N (0, I). The encoder is q(z|x) = N (µ θ (x), Σ θ (x)) where Σ is a diagonal matrix. Both µ θ and Σ θ are parameterized by a 3-layer feed-forward neural network with a ReLU activation and the size of the two hidden outputs are 400 and 200. We use a Gaussian decoder p(x|z) = N (g θ (z), σ 2 x I) with fixed variance σ x = 0.3. The g θ is parameterized by a 3-layer feed-forward neural network with hidden layer sizes 200 and 400. The activation of the hidden output uses a ReLU and we utilize a Sigmoid function in the final layer output to constrain the output between 0 and 1. The training objective is to maximize the lower bound of the log-likelihood log p(x) ≥ q(z|x) log p(x|z)dz − KL(q(z|x)||p(z)), see Kingma and Welling (2013) for further details. We use an Adam optimizer with a learning rate 1 × 10 −4 and batch size 100 to train the model for 100 epochs. After training, we sample from the model by first taking a sample z ∼ p(z) and letting x = g θ (z). This is equivalent to taking a sample from an implicit model p θ (x) = δ(x − g θ (z))d(z)dz, see Tolstikhin et al. (2017) for further discussion regarding this implicit model construction. In Figure 14, we plot samples from the trained implicit model with dim(z) = 5, dim(z) = 10 and the original MNIST data.
C.5.2 Flow model training
We train our flow models to fit the synthetic MNIST dataset with intrinsic dimensions 5 and 10 and the original MNIST dataset. In all models, we use the Adam optimizer with learning rate 5 × 10 −4 and batch size 100. We train the model for 300k iterations and, following the initial 100k iterations, the learning rate decays every 10k iterations by a factor 0.9. In Figure 15, we plot the samples from our models trained on three different training datasets. In Figure 16, we also plot the samples from traditional flow models that were trained on these three datasets using the same experiment settings. We find that the samples from our models are comparably sharper than those from traditional flow models.
C.6 Color image experiments
Our network for color image datasets is based on an open source implementation 9 , where we modify the Glow Kingma and Dhariwal (2018) structure to make it volume preserving. The volume preserving affine coupling layer is similar to the incompressible flow, however here the s(·) function takes the form:
s(·) = scale * tanh(NN(·))(43)
where 'scale' are channel-wise learnable parameters and NN(·) is the neural network. For the actnorm function y ij = s x ij + b, where the log-determinant for position x ij is sum(log |s|), we normalize the log |s| = log |s| − mean(log |s|), such that sum( log |s|) = 0. We use LU decomposition for the 1×1 convolution, where W = P L(U + diag(s)) with log-determinant log | det(W )| = sum(log |s|). We can then normalize log |s| = log |s| − mean(log |s|) to ensure it is volume preserving. All experiments utilise a Glow architecture consisting of four blocks of twenty affine coupling layers, applying the multi-scale architecture between each block. The neural network in each affine coupling layer contains three convolution layers with kernel sizes 3,1,3 respectively and ReLU activation functions. We use 512 channels for all hidden convolutional layers, the Adam optimizer with learning rate 1e −4 and a batch-size of 100, for all of our experiments. to transform each pixel to a continuous value in [0, 1]. The manifold structure of these image datasets has been less well studied and may be considered more complex than the datasets previously explored in this work; each potentially lying on a union of disconnected and discontinuous manifolds. Similar to Table 5: FID Comparisons of Samples. We report the FID values (lower indicates better quality) of our model with K = [512,2000,2500,3000]. We surprisingly find that when we increase the dimension of the manifold, the FID becomes higher. We hypothesize that the manifold spanned by the leading eigenvectors can capture the images lying in the modes, resulting in a better FID value. We also compare our method to CEF Ross and Cresswell (2021) model with manifold dimension K = 512 and find our method can consistently achieve better FID value.
Model CEF (512) Ours (512) Ours (2000) Ours (2500) the MNIST experiment, we add small Gaussian noise (with standard deviation σ x =0.01) for twenty initial training epochs in order to smooth the data manifold and then anneal the noise with a factor of 0.9 for a further twenty epochs. Figure 17 provides samples from the models that are trained on these three datasets. We find that our models can generate realistic samples and thus provide evidence towards the claim that our volume-preserving flow, with learnable prior model, does not limit the expressive power of the network. Experimentally, in comparison with the Glow model, we do not observe consistent sample quality improvement and conjecture that the intrinsic dimension mismatch problem, pertaining to these color image datasets, is less severe than the previously considered grey-scale images and constructed toy data. The data manifold has large intrinsic dimensionality due to high redundancy and stochasticity in the RGB channels Yu and Zhang (2010). We then conjecture that color image data distributions typically have a relatively large number of intrinsic dimensions. This conjecture is consistent with recent generative model training trends: a latent variable model e.g. VAE usually requires very large latent dimension to generate high-quality images Vahdat and Kautz (2020); Child (2020). For example, in the state-of-the-art image generation work 10 Vahdat and Kautz (2020), a VAE with latent size 153600 is used to fit a CIFAR10 dataset with shape 32×32×3. These considerations lead us to believe that the estimation of intrinsic dimension for natural images forms an important future research direction. Towards tackling this, one proposal would involve using the results obtained from our model to inform the latent dimension of the latent variable models. We leave deeper study of the geometry of image manifolds to future work.
D Empirical Verifications and Comparisons
We compare our model with the recently proposed CEF model Ross and Cresswell (2021), which is also proposed to learn a flow on manifold distribution. Different from our model where different K values can be selected after training the model, in the CEF model, the intrinsic dimension of the model has to be pre-specified. We then choose K = 512 and train the model on the CelebA (32 × 32) dataset. We reuse their open-source code https://github.com/ layer6ai-labs/CEF and the 32×32 model architecture (see paper Ross and Cresswell (2021) for details). In Figure5, we compare the reconstructions and the samples from both models with the same manifold dimension K = 512. We can find the CEF model has a better reconstruction quality than our models, we hypothesize that this is because our model is not trained with an intrinsic dimension K = 512, so the reconstruction quality cannot be guaranteed when we use the latent code with dimension 512. However, we observe our model achieves better sample quality (see Table5 for a FID comparison) compared to CEF. This phenomenon (good reconstruction but bad sample quality) is similar to the latent space mismatch problem in the latent variable models Dai and Wipf (2019); Zhao et al. (2017a), where the aggregate distribution q φ (z) = δ(z − g(x))P d (x) mismatches the prior p(z), (g is the inverse flow function). Since our model can learn the prior and the manifold prior with K = 512 is constructed by the leading eigen-vectors, it can alleviate the prior mismatch problem and has better sample quality.
E License
The Figure 19: Comparisons with CEF model. We compare the reconstructions (a,b,c) and samples (d,c) from our model and CEF model with the same manifold dimension (K = 512), we can see CEF has a better reconstruction quality whereas our model has a better sample quality, also see 5 for FID comparisons and main text for a discussion of the phenomenon.
Figure 1 :
1Samples and latent visualization from a flow based model with a fixed Gaussian prior where the intrinsic dimension is strictly lower than the true dimensionality of the data space.
Figure 2 :
2This figure gives an overview of our method.
, and CelebA Liu et al. (2015a). We use incompressible affine coupling layers, introduced by Sorrenson et al. (2020); Dinh et al. (2016), for toy and MNIST experiments and a volume-preserving variation of the Glow structure Kingma and Dhariwal (2018). See Appendix C for further experimental details.
Figure 3 :Figure 4 :
34(a) shows the samples from P d (blue) and model P X (red). (b) shows the samples from the learned prior P Z (blue) and Q Z (red). (c) shows the eigenvalues of AA T . (d) and (e) show the true and the learned density on the manifold.(a) x ∼ P d (b) Our samples (c) Our representations (d) Our Learned Pz(e) Eigenvalues (f) z proj = zE (g) Trad. Flow samples (h) Trad. Flow representations (a) S-curve data x ∼ P d . (b) Samples generated from our learned model. (c) Latent representation z = g(x), points are lying on a linear subspace. (d) Samples from the learned prior distribution. (e) Eigenvalues of the matrix AA T , we deduce that Indim(P d ) = 2. (f) Our representation after the dimensionality reduction z proj = zE. (g) and (h): Samples and representations from a learned traditional flow model with a fixed Gaussian prior.
Figure 5 :
5(a) and (b). In contrast to Figures 6a, 6b; the gap between eigenvalues predictably exhibits a less obvious step change. However, the values suggest the intrinsic dimension of MNIST lies between 11 and 14. This result is consistent with previous estimations, stating that the intrinsic dimension of MNIST is between 12 and 14 Facco et al. (2017); Hein and Audib-(a) x ∼ P d (b) Our model (c) Traditional flow (d) Eigenvalues of AA T (e) z proj = zE (a) and (b) show samples from P d and our model, respectively. (c) shows a traditional flow based model with a fixed Gaussian prior fails to fit the data distribution and cannot generate valid samples. (d) the first 20 eigenvalues of the matrix AA T . (e) shows the representation after applying dimensionality reduction. See text for further discussion. ert (2005); Costa and Hero (2006), see also Figure 7 for the model samples comparisons. Recent work Cornish et al. (2019) discusses fitting flow models to a P d that lies on disconnected components, by introducing a mixing prior. Such techniques may be easily combined with our method towards constructing more powerful flow models. CelebA To model CelebA, we first centre-crop and resize each image to 32×32×3. Pixels take discrete values from [0, . . . , 255] and we follow a standard dequantization process Uria et al. (2013): x=(x + Uniform(0, 1))/256 to transform each pixel to a continuous value in [0, 1].
18 (our model with K = {2000, 3000}) and Appendix D (comparisons of both models). We also train our model on SVHN Netzer et al. (2011b) and CI-FAR10 Krizhevsky et al. (2009a), see Appendix C for a detailed discussion.
Figure 6 :
6Eigenvalues estimated by our model trained on synthetic MNIST with Indim(P d ) = {5, 10} and real MNIST respectively. sian distribution), therefore the model distribution is absolutely continuous and maximum likelihood learning is thus well defined Loaiza-Ganem et al. (2022); Zhang et al. (2020). However, common distributions such as natural images usually do not have Gaussian observational noise Zhao et al. (2017b)
( a )Figure 7 :
a7Indim(P d ) = 5 (T) (b) Indim(P d ) = 5 (O) (c) Indim(P d ) = 5 (T) (d) Indim(P d ) = 10 (O) (e) Real MNIST (T) (f) Real MNIST (O)We compare the samples from a traditional nonvolume preserving flow with a fixed prior (denoted as T) and our model (denoted as O) on synthetic MNIST with Indim(P d ) = {5, 10} and real MNIST, see Appendix C.5 for experiment details.
Figure 8 :
8Samples from the proposed manifold prior with K = 2000 (left) and K = 3000 (right) respectively.
Figure 9 :Figure 10 :
910Eigen-values in log-space. We can see the eigenvalues has an exponential decay starting around 2700, which indicates that the intrinsic dimension of the CelebA (with size 32 × 32) is around 2700. Reconstructions with different dimensional representations.
Figure 11 :
11Figure (a) and (b) show the (approximated) entropy value using two different training objectives, for two different experiments. C.1 Network architecture for toy and MNIST dataset The flow network we use consists of incompressible affine coupling layers Sorrenson et al. (2020); Dinh et al. (2016). Each coupling layer splits a D-dimensional input x into two parts x 1:d and x d+1:D . The output of the coupling layer is
Figure 12 :
12Gaussian prior: X space (c) Fixed Gaussian prior: Z space (d) Our method: X space (e) Our method: Z space (f) Eigenvalues of AA T (a) shows the samples from the data distribution. (b) and (d) show samples from a flow with a fixed Gaussian prior and our method, (c) and (e) show the latent space in both models. In (f), we plot the eigenvalues of AA T .
Figure 13 :
13(a) and (c) show the ground truth 'density allocation' on the manifold for two toy datasets, (b) and (d) show the 'density allocation' learned by our models.
CelebA Liu et al. (2015b) and SVHN Netzer et al. (2011a) dataset is available for non-commercial research purposes only. We didn't find licenses for CIFAR Krizhevsky et al. (2009b) and MNIST LeCun (1998). The CelebA dataset may contain personal identifications.The code https://github.com/chrischute/glow we made use of is under the MIT License.
Figure 14 :Figure 15 :
1415Training data for the flow model. Figure (a) and (b) are synthetic MNIST samples from two implicit models with latent dimension 5 and 10 respectively. Figure (c) are samples from the original MNIST dataset.(a) dim(z) = 5 (b) dim(z) = 10 (c) MNIST Samples from our methods. Figure (a) and (b) are samples flow models trained on synthetic MNIST data with intrinsic dimension 5 and 10 respectively. Figure (c) are samples from a flow model that trained on original MNIST data.
Figure 16 :Figure 17 : 3000 Figure 18 :
1617300018Samples from traditional non-volume preserving flow models with fixed Gaussian prior. Samples and eigenvalues for three models. In each case we can find approximately half of the eigenvalues converge to 0 but a large number of eigenvalues remain greater than 0.(a) Samples with K = 1000 (b) Samples with K = 2000 (c) Samples with K = 2500 (d) Samples with K = Samples from the model K-dimensional manifolds prior. from our model with K = 512 (c) Reconstruction from CEF with K = 512 (d) Samples from our model with K = 512 (e) Samples from CEF with K = 512
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Table 1 :
1The network structure for toy datasets.Type of block
Number Input shape
Affine coupling layer widths
§ Fully connected
6
3
even: 2 → 24 → 24 → 24 → 1
odd: 1 → 24 → 24 → 24 → 2
Table 2 :
2The network structure for S-Curve dataset. If we denote the input vector of each coupling is x, the function s and t takes x[0] in the first coupling layer and x[1] in the second coupling layer.Type of block
Number Input shape
Affine coupling layer widths
Downsampling
1
(1, 32, 32)
Convolution
2
(4, 16, 16)
2 → 16 → 16 → 4
Downsampling
1
(4, 16, 16)
Convolution
2
(16, 8, 8)
8 → 16 → 16 → 16
Flattening
1
(16, 8, 8)
Fully connected
2
1024
512 → 24 → 24 → 24 → 512
Table 3 :
3The network structure for our fading square dataset experiments. If we denote the input vector of each coupling to be x, the functions s and t take values x[1:2] in the even coupling layer and x[3] in the odd coupling layer.Type of block
Number Input shape
Affine coupling layer widths
Downsampling
1
(1, 28, 28)
Convolution
4
(4, 14, 14)
2 → 16 → 16 → 4
Downsampling
1
(4, 14, 14)
Convolution
4
(16, 7, 7)
8 → 16 → 16 → 16
Flattening
1
(16, 7, 7)
Fully connected
2
784
392 → 24 → 24 → 24 → 392
Table 4 :
4Network structure for synthetic MNIST dataset experiment.
We conduct further experiments on real-world color images: SVHN, CIFAR10 and CelebA. Data samples from SVHN and CIFAR10 are represented by 32×32×3 dimension vectors and for CelebA our pre-processing involves centre-cropping each image and resizing to 32×32×3. Pixels take discrete values from [0, . . . , 255] and we follow a standard dequantization processUria et al. (2013): x = x+Uniform(0,1)256
The distribution is a.c. with respect to the Lebesgue measure, so it has a density function, see Durrett (2019).2 Since we assume X d is a.c., Z Q : Z Q = g(X d ) is also a.c. with a bijiective mapping g(·) and thus Z Q allows a density function.
We use the generalized Gaussian to include the case that the covariance AA T is not full rank.
By the definition of the manifold Lee (2013), the intrinsic dimension of a manifold is equal or smaller than its ambient dimension.
For implicit model: X = g(Z), the intrinsic dimension of the model distribution will be not-greater-than the dimension of the latent variable: Indim(X) ≤ dim(Z) Arjovsky and Bottou (2017b). Further, if strictly Indim(X) < dim(Z) this results in a 'degenerated' case. When we fit the model on a data distribution where Indim(X d ) ≥ dim(Z), we assume that degeneration will not occur during training, resulting in Indim(P d ) ≈ dim(Z). For simplicity, we equate Indim(P d )=dim(Z) in the main text.
The extension to higher dimensions is straightforward.
https://github.com/chrischute/glow
The latent dimension in GAN based models Goodfellow et al. (2014); Arjovsky and Bottou (2017a); Gulrajani et al. (2017) arechosen to be small (e.g. latent variable has dimension 128 for CIFAR10), but the mode-collapse phenomenon, commonly observed in GANs, indicates that those with small latent dimension will underestimate the support of the data distribution.
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"Membership Inference Attacks against Synthetic Data through Overfitting Detection",
"Membership Inference Attacks against Synthetic Data through Overfitting Detection"
] | [
"Boris Van Breugel \nUniversity of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n\n",
"Hao Sun \nUniversity of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n\n",
"Zhaozhi Qian \nUniversity of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n\n",
"Mihaela Van Der Schaar \nUniversity of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n\n"
] | [
"University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n",
"University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n",
"University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n",
"University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute\n"
] | [
"Proceedings of the 26 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2023"
] | Data is the foundation of most science. Unfortunately, sharing data can be obstructed by the risk of violating data privacy, impeding research in fields like healthcare. Synthetic data is a potential solution. It aims to generate data that has the same distribution as the original data, but that does not disclose information about individuals. Membership Inference Attacks (MIAs) are a common privacy attack, in which the attacker attempts to determine whether a particular real sample was used for training of the model. Previous works that propose MIAs against generative models either display low performancegiving the false impression that data is highly private-or need to assume access to internal generative model parameters-a relatively lowrisk scenario, as the data publisher often only releases synthetic data, not the model. In this work we argue for a realistic MIA setting that assumes the attacker has some knowledge of the underlying data distribution. We propose DO-MIAS, a density-based MIA model that aims to infer membership by targeting local overfitting of the generative model. Experimentally we show that DOMIAS is significantly more successful at MIA than previous work, especially at attacking uncommon samples. The latter is disconcerting since these samples may correspond to underrepresented groups. We also demonstrate how DOMIAS' MIA performance score provides an interpretable metric for privacy, giving data publishers a new tool for achieving the desired privacy-utility trade-off in their synthetic data. | 10.48550/arxiv.2302.12580 | [
"https://export.arxiv.org/pdf/2302.12580v1.pdf"
] | 257,205,896 | 2302.12580 | de1f64522b5594a9f572a92f8816cb1f36adcbac |
Membership Inference Attacks against Synthetic Data through Overfitting Detection
Boris Van Breugel
University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute
Hao Sun
University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute
Zhaozhi Qian
University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute
Mihaela Van Der Schaar
University of Cambridge University of Cambridge University of Cambridge University of Cambridge Alan Turing Institute
Membership Inference Attacks against Synthetic Data through Overfitting Detection
Proceedings of the 26 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2023
the 26 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2023Valencia, Spain. PMLR206
Data is the foundation of most science. Unfortunately, sharing data can be obstructed by the risk of violating data privacy, impeding research in fields like healthcare. Synthetic data is a potential solution. It aims to generate data that has the same distribution as the original data, but that does not disclose information about individuals. Membership Inference Attacks (MIAs) are a common privacy attack, in which the attacker attempts to determine whether a particular real sample was used for training of the model. Previous works that propose MIAs against generative models either display low performancegiving the false impression that data is highly private-or need to assume access to internal generative model parameters-a relatively lowrisk scenario, as the data publisher often only releases synthetic data, not the model. In this work we argue for a realistic MIA setting that assumes the attacker has some knowledge of the underlying data distribution. We propose DO-MIAS, a density-based MIA model that aims to infer membership by targeting local overfitting of the generative model. Experimentally we show that DOMIAS is significantly more successful at MIA than previous work, especially at attacking uncommon samples. The latter is disconcerting since these samples may correspond to underrepresented groups. We also demonstrate how DOMIAS' MIA performance score provides an interpretable metric for privacy, giving data publishers a new tool for achieving the desired privacy-utility trade-off in their synthetic data.
INTRODUCTION
Real data may be privacy-sensitive, prohibiting open sharing of data and in turn hindering new scientific research, reproducibility, and the development of machine learning itself. Recent advances in generative modelling provide a promising solution, by replacing the real dataset with a synthetic dataset-which retains most of the distributional information, but does not violate privacy requirements.
Motivation The motivation behind synthetic data is that data is generated from scratch, such that no synthetic sample can be linked back to any single real sample. However, how do we verify that samples indeed cannot be traced back to a single individual? Some generative methods have been shown to memorise samples during the training procedure, which means the synthetic data samples-which are thought to be genuine-may actually reveal highly private information (Carlini et al., 2018). To mitigate this, we require good metrics for evaluating privacy, and this is currently one of the major challenges in synthetic data (Jordon et al., 2021;Alaa et al., 2022). Differential privacy (DP) (Dwork and Roth, 2014) is a popular privacy definition and used in several generative modelling works (Ho et al., 2021;Torkzadehmahani et al., 2020;Chen et al., 2020;Jordon et al., 2019;Long et al., 2019;Wang et al., 2021;Cao et al., 2021). However, even though DP is theoretically sound, its guarantees are difficult to interpret and many works (Rahman et al., 2018;Jayaraman and Evans, 2019;Jordon et al., 2019;Ho et al., 2021) reveal that for many settings, either the theoretical privacy constraint becomes meaningless ( becomes too big), or utility is severely impacted. This has motivated more lenient privacy definitions for synthetic data, e.g. see (Yoon et al., 2020). We take an adversarial approach by developing a privacy attacker model-usable as synthetic data evaluation metric that quantifies the practical privacy risk.
Aim Developing and understanding privacy attacks against generative models are essential steps in creating better private synthetic data. There exist different privacy attacks in machine learning literature-see e.g. (Rigaki and Garcia, 2020)-but in this work we focus on Membership In-ference Attacks (MIAs) (Shokri et al., 2017). The general idea is that the attacker aims to determine whether a particular sample they possess was used for training the machine learning model. Successful MIA poses a privacy breach, since mere membership to a dataset can be highly informative. For example, an insurance company may possess a local hospital's synthetic cancer dataset, and be interested to know whether some applicant was used for generating this dataset-disclosing that this person likely has cancer (Hu et al., 2022). Additionally, MIAs can be a first step towards other privacy breaches, like profiling or property inference (De Cristofaro, 2021).
Previous work in MIA attacks against generative models is inadequate, conveying a false pretense of privacy. In the NeurIPS 2020 Synthetic Data competition (Jordon et al., 2021), none of the attackers were successful at MIA. 1 Similar negative results were found in the black-box results of (Liu et al., 2019;Hayes et al., 2019;Hilprecht et al., 2019;Chen et al., 2019), where additional assumptions were explored to create more successful MIAs. Most of these assumptions (see Sec. 4) rely on some access to the generator, which we deem relatively risk-less since direct access is often avoidable in practice. Nonetheless, we show that even in the black-box setting-in which we only have access to the synthetic data-MIA can be significantly more successful than appears in previous work, when we assume the attacker has some independent data from the underlying distribution. In Sec. 2 we elaborate further on why this is a realistic assumption. Notably, it also allows an attacker to perform significantly better attacks against underrepresented groups in the population (Sec. 5.3).
Contributions This paper's main contributions are the following.
1. We propose DOMIAS: a membership inference attacker model against synthetic data, that incorporates density estimation to detect generative model overfitting. DOMIAS improves upon prior MIA work by i) leveraging access to an independent reference dataset and ii) incorporating recent advances in deep density estimation.
2. We compare the MIA vulnerability of a range of generative models, showcasing how DOMIAS can be used as a metric that enables generative model design choices 3. We find that DOMIAS is more successful than previous MIA works at attacking underrepresented groups in synthetic data. This is disconcerting and strongly motivates further research into the privacy protection of these groups when generating synthetic data.
MEMBERSHIP INFERENCE: FORMALISM AND ASSUMPTIONS
Formalism for synthetic data MIA Membership inference aims to determine whether a given sample comes from the training data of some model (Shokri et al., 2017). Let us formalise this for the generative setting. Let random variable X be defined on X , with distribution p R (X). Let D mem iid ∼ p R (X) be a training set of independently sampled points from distribution p R (X). Now let G : Z → X be a generator that generates data given some random (e.g. Gaussian) noise Z. Generator G is trained on D mem , and is subsequently used to generate synthetic dataset D syn . Finally, let A : X → [0, 1] be the attacker model, that possesses the synthetic dataset D syn , some test point x * , with X * ∼ p R (X), and possibly other knowledge-see below.
Attacker A aims to determine whether some x * ∼ p R (X) they possess, belonged to D mem , hence the perfect attacker outputs A(x * ) = 1[x * ∈ D mem ]. The MIA performance of an attacker can be measured using any classification metric.
Assumptions on attacker access The strictest black-box MI setting assumes the attacker only has access to the synthetic dataset D syn and test point x * . In this work we assume access to a real data set that is independently sampled from p R (X), which we will call the reference dataset and denote by D ref . The main motivation of this assumption is that an attacker needs some understanding of what real data looks like to infer MI-in Sec. 3 we will elaborate further on this assumption's benefits. Similar assumptions have been made in the supervised learning MI literature, see e.g. (Shokri et al., 2017;Ye et al., 2021). This is a realistic scenario to consider for data publishers: though they can control the sharing of their own data, they cannot control whether attackers acquires similar data from the general population. A cautious data publisher would assume the attacker has access to a sufficiently large D ref to approximate p R (X) accurately, since this informally bounds the MIA risk from above. Related MI works (Liu et al., 2019;Hayes et al., 2019;Hilprecht et al., 2019;Chen et al., 2019) consider other assumptions that all require access to the synthetic data's generative model. 2 These settings are much less dangerous to the data publisher, since these can be avoided by only publishing the synthetic data. Individual assumptions of related works are discussed further in Sec. 4. Consider the generative distribution for two representations of X, optimal methods based on Eq. 1 will infer m = 1 for green and m = 0 for red areas. This is problematic; it implies inference of these methods is dependent on the (possibly arbitrary) representation of variable X. Conclusion: it does not make sense to focus on mere density, MIA needs to target local overfitting directly. This requires data from (or assumptions on) the underlying distribution.
DOMIAS
3.1 Rethinking the black-box setting: why D syn alone is insufficient
The most popular black-box setting assumes only access to D syn . This gives little information, which is why previous black-box works (Hayes et al., 2019;Hilprecht et al., 2019;Chen et al., 2019) implicitly assume:
A prev (x * ) = f (p G (x * )),(1)
where A indicates the attacker's MIA scoring function, p G (·) indicates the generator's output distribution and f : R → [0, 1] is some monotonically increasing function. There are two reasons why Eq. 1 is insufficient. First, the score does not account for the intrinsic distribution of the data. Consider the toy example in Figure 2a. There is a local density peak at x = 4, but without further knowledge we cannot determine whether this corresponds to an overfitted example or a genuine peak in the real distribution. It is thus naive to think we can do MI without background knowledge.
Second, the RHS of Eq. 1 is not invariant w.r.t. bijective transformations of the domain. Consider the left and right plot in Figure 1. Given the original representation, we would infer M = 0 for any point around x = 4, whereas in the right plot we would infer M = 1 for the same points. This dependence on the representation is highly undesirable, as any invertible transformation of the representation should contain the same information.
How do we fix this? We create the following two desiderata: i) the MI score should target overfitting w.r.t. the real distribution, and ii) it should be independent of representation.
3.2 DOMIAS: adding knowledge of the real data.
We need to target overfitting directly. We propose the DO-MIAS framework: Detecting Overfitting for Membership Inference Attacks against Synthetic Data.
Let us assume we know the true data distribution p R (X). We change Eq. 1 to:
A DOMIAS (x * ) = f ( p G (x * ) p R (x * ) ),(2)
that is, we weight Eq. 1 by the real data distribution p R (X). 3 Figure 2 shows the difference between DOMIAS and previous work using Eq. 1, by considering the same toy example as in Figure 1. Effectively, Eq. 2 distinguishes between the real and generative distribution, similar in vain to global two-sample tests (e.g. see Gretton et al. (2012); Arora et al. (2019); Gulrajani et al. (2019)). The probability ratio has the advantage that (cf. e.g. probability difference) it is independent of the specific representation of the data:
Theorem 1. Let X G and X R be two random variables defined on X , with distributions p G (X) and p R (X), s.t. p G p R , i.e. p R dominates p G . Let g : X →X , x → g(x) be some invertible function, and define representa-tionsX G = g(X G ) andX R = g(X R ) with respective distributionp G (X) andp R (X). Then p G (X) p R (X) =p G (g(X)) p R (g(X)) , i.e. the same score is obtained for either data representations.
Proof. Without loss of generalisation let us assume continuous variables and almost everywhere continuous g. Using the chain rule, we havep · (g(x)) = p·(x)
|J(x)| with Jacobian (a) Original space (b) Log-transformed space Figure 2: DOMIAS scores are not dependent on the feature representation. This is the same toy example as in Figure 1, where we now assume the bump at x = 4 has been caused by overfitting in the generator, s.t. this part of the space has become overrepresented w.r.t. the original distribution. DOMIAS infers MI by weighting the generative and real distribution, inferring m = 1 (m = 0) for green (red) areas. Note the difference with Figure 1: whereas MI predictions of previous works that use Eq. 2 are dependent on the representation, DOMIAS scores are the same in both domains (Theorem 1).
J(x) = dg dx (x). Hence we see:
p G (g(x)) p R (g(x)) = p G (x)/|J(x)| p R (x)/|J(x)| = p G (x) p R (x) , a.e.
as desired.
DOMIAS does not purport false privacy safety for underrepresented groups Figure 1a pinpoints a problem with previous works: methods that rely on assumption Eq. 1 cannot attack low-density regions. As a result, one might conclude that samples in these regions are safer. Exactly the opposite is true: in Figure 2 we see DOMIAS infers MI successfully for these samples, whatever the representation. This is distressing, as low-density regions may correspond to underrepresented groups in the population, e.g. ethnic minorities. We will explore this further in the experimental section.
Illustrative attacker examples
Any density estimator can be used for approximating p G (X) and p R (X)-e.g. fitting of some parametric family, training a generative model with Monte Carlo Integration, or a deep density estimator. The choice of density estimator should largely depend whether prior knowledge is available-e.g. p R falls in some parametric familyand on the size of the datasets-for a large dataset a more powerful and more flexible density estimator can be used, whereas for little data this is not suitable as it might lead to overfitting. In the experimental section, we illustrate DO-MIAS using the flow-based BNAF (de Cao et al., 2019) density estimator, chosen for its training efficiency. For the ablation study in Sec. 5.2 we also include a Gaussian KDEbased method as a non-parametric alternative.
RELATED WORK
MIAs against generative models Most of the literature on privacy attacks is focused on discriminative models, not generative models. The few works that are concerned with generative models all focus on membership inference (MIA) (Shokri et al., 2017). Here we focus on works that can be applied to our attacker setting, see Table 1. Hayes et al. (2019) propose LOGAN, a range of MIA attacks for both white-box and black-box access to the generative model, including possible auxiliary information. Two attacks can be applied to our setting.
S k G = {x i } k i=1 from generator G and use score A(x * ; G) = min xi∈S k G L 2 (x * , x i )
as an unnormalised surrogate for p G (x * ). They also introduce a calibrated method that uses a reference dataset D ref to train a generative ref-
erence model G ref , giving calibrated score A(x * ; G, k) − A(x * ; G ref , k)
. This can be interpreted as a special case of DOMIAS-Eq. 2-that approximates p R and p G with Gaussian KDEs with infinitesimal kernel width, trained on a random subset of k samples from D ref and D syn . At last, we emphasise that though (Hayes et al., 2019;Chen et al., 2019) consider D ref too, they (i) assume this implicitly and just for one of their many models, (ii) do not properly motivate or explain the need for having D ref , nor explore the effect of n ref , and (iii) their MIAs are technically weak and perform poorly as a result, leading to incorrect conclusions on the danger of this scenario (e.g. Hayes et al. (2019) note in their experiments that their D1 model performs no better than random guessing).
Stronger attacker access assumptions Other methods in (Hayes et al., 2019;Hilprecht et al., 2019;Chen et al., 2019) make much stronger assumptions on attacker access. (Hayes et al., 2019) propose multiple attacks with a subset of the training set known, which implies that there has already been a privacy breach-this is beyond the scope of this work. They also propose an attack against GANs that uses the GANs discriminator to directly compute the MIA score, but discriminators are usually not published. Chen et al. (2019) propose attacks with white-box access to the generator or its latent code, but this scenario too can be easily avoided by not publishing the generative model itself. All methods in (Hilprecht et al., 2019;Chen et al., 2019) assume unlimited generation access to the generator (i.e. infinitely-sized D syn ), which is unrealistic for a real attacker-either on-demand generation is unavailable or there is a cost associated to it that effectively limits the generation size (De Cristofaro, 2021). These methods can still be applied to our setting by sampling from the synthetic data directly.
Tangential work The following MIA work is not compared against. Liu et al. (2019); Hilprecht et al. (2019) introduce co-membership (Liu et al., 2019) or set MIA (Hilprecht et al., 2019) attacks, in which the aim is to determine for a whole set of examples whether either all or none is used for training. Generally, this is an easier attack and subsumes the task of single attacks (by letting the set size be 1). Webster et al. (2021) define the identity membership inference attack against face generation models, which aims to infer whether some person was used in the generative model (but not necessarily a specific picture of that person). This requires additional knowledge for identify-ing people in the first place, and does not apply to our tabular data setting. Hu and Pang (2021) focus on performing high-precision attacks, i.e. determining MIA for a small number of samples with high confidence. Similar to us they look at overrepresented regions in the generator output space, but their work assumes full model access (generator and discriminator) and requires a preset partitioning of the input space into regions. is similar to (Hilprecht et al., 2019), but uses contrastive learning to embed data prior to computing distances. In higher dimensions, this can be an improvement over plain data or simpler embeddings like PCA-something already considered by (Hilprecht et al., 2019). However, the application of contrastive learning is limited when there is no a priori knowledge for performing augmentations, e.g. in the unstructured tabular domain.
On a final note, we like to highlight the relation between MIA and the evaluation of overfitting, memorisation and generalisation of generative models. The latter is a nontrivial task, e.g. see (Gretton et al., 2012;Lopez-Paz and Oquab, 2016;Arora et al., 2017;Webster et al., 2019;Gulrajani et al., 2019). DOMIAS targets overfitting directly and locally through Eq. 2, a high score indicating local overfitting. DOMIAS differs from this line of work by focusing on MIA, requiring sample-based scores. DOMIAS scores can be used for interpreting overfitting of generative models, especially in the non-image domain where visual evaluation does not work.
EXPERIMENTS
We perform experiments showing DOMIAS' value and use cases. In Sec. 5.1 we show how DOMIAS outperforms prior work, in Sec. 5.2 we explore why. Sec. 5.3 demonstrates how underrepresented groups in the population are most vulnerable to DOMIAS attack, whilst Sec. 5.4 explores the vulnerability of different generative modelsshowcasing how DOMIAS can be used as a metric to inform synthetic data generation. For fair evaluation, the same experimental settings are used across MIA models (including n ref ). Details on experimental settings can be found in Appendix A. 4
DOMIAS outperforms prior MIA methods
Set-up We use the California Housing Dataset (Pace and Barry, 1997) and use TVAE (Xu et al., 2019a) to generate synthetic data. In this experiment we vary the number of TVAE training samples |D mem | and TVAE number of training epochs. We compare DOMIAS against LO-GAN 0 and LOGAN D1 (Hayes et al., 2019), MC (Hilprecht et al., 2019), and GAN-Leaks 0 and GAN-Leaks (3) approximates Eq. 1 or 2; (4) by default does not need generation access to generative model-only synthetic data itself. † GAN-leaks calibrated is a heuristic correction to GAN-leaks, but implicitly a special case of Eq. 2.
Name
( Table 1. Figure 3(a) shows the MIA accuracy of DOMIAS and baselines against TVAE's synthetic dataset, as a function of the number of training samples TVAE n mem . For small n mem TVAE is more likely to overfit to the data, which is reflected in the overall higher MIA accuracy. Figure 3(b) shows the MIA accuracy as a function of TVAE training epochs. Again, we see TVAE starts overfitting, leading to higher MIA for large number of epochs.
1) (2) (3)(4)
DOMIAS consistently outperforms baselines
In both plots, we see DOMIAS consistently outperforms baseline methods. Similar results are seen on other datasets and generative models, see Appendix B. Trivially, DO-MIAS should be expected to do better than GAN-Leaks 0 and LOGAN 0, since these baseline methods do not have access to the reference dataset and are founded on the flawed assumption of Eq. 1-which exposes the privacy risk of attacker access to a reference dataset.
Source of gain
Using the same set-up as before, we perform an ablation study on the value of i) DOMIAS' use of the reference set, and ii) the deep density estimator. For the first, we compare using the DOMIAS assumption (Eq. 2) vs the assumption employed in many previous works (Eq. 1). For the latter, we compare the results for density estimation based on the flow-based BNAF (de Cao et al., 2019) versus a Gaussian kernel density estimator-kernel width given by the heuristic from (Scott, 1992). Figure 4 shows the MIA performance as a function of n syn and n ref . Evidently, the source of the largest gain is the use of Eq. 2 over Eq. 1. As expected, the deep density estimator gives further gains when enough data is available. For lower amounts of data, the KDE approach is more suitable. This is especially true for the approximation of p R (the denominator of Eq. 2)-small noise in the approximated p R can lead to large noise in MIA scores. Also note in the right plot that MIA performance goes up with |D syn | across methods due to the better p G approximation; this motivates careful consideration for the amount of synthetic Figure 4: DOMIAS source of gain. Ablation study of DOMIAS on the California Housing dataset, with attack performance as a function of the reference dataset size (left) and the synthetic dataset size (right). We see that the MIA performance of DOMIAS is largely due to assumption Eq. 2 vs. Eq. 1, i.e. the value of the reference dataset. The deep flow-based density estimator delivers gains over the simpler KDE approach when enough samples are available.
data published.
Underrepresented group MIA vulnerability
Set-up We use a private medical dataset on heart failure, containing around 40, 000 samples with 35 mixed-type features (see Appendix A). We generate synthetic data using TVAE (Xu et al., 2019a).
Minority groups are most vulnerable to DOMIAS attack As seen in Sec. 3, the assumption underlying previous work (Eq. 1) will cause these methods to never infer membership for low-density regions. This is problematic, as it gives a false sense of security for these groups-which are likely to correspond to underrepresented groups.
The left side of Figure 5 displays a T-SNE embedding of the Heart Failure dataset, showing one clear minority group, drawn in blue, which corresponds to patients that are on high-blood pressure medication-specifically, Angiotensin II receptor blockers. The right side of Figure 5 shows the performance of different MIA models. DOMIAS is significantly better at attacking this vulnerable group compared to the overall population, as well as compared to other baselines. This is not entirely surprising; generative models are prone to overfitting regions with few samples. Moreover, this aligns well with supervised learning literature that finds additional vulnerability of low-density regions, e.g. (Kulynych et al., 2019;Bagdasaryan et al., 2019). Importantly, most MIA baselines give the false pretense that this minority group is less vulnerable. Due to the correspondence of low-density regions and underrepresented groups, these results strongly urge further research into privacy protection of low-density regions when generating synthetic data.
DOMIAS informs generative modelling decisions
Set-up Again we use the California Housing dataset, this time generating synthetic data using different generative models. We evaluate the quality and MIA vulnerability of GAN, (Goodfellow et DOMIAS quantifies MIA vulnerability Figure 6 presents the DOMIAS MIA AUC against the data quality (in terms of Wasserstein Distance to an independent hold-out set), averaged over eight runs. We see a clear privacy-utility trade-off, with the additive noise model giving a clean baseline. The NeurIPS 2020 Synthetic Data competition (Jordon et al., 2021) concluded that disappointingly, adding noise usually outperformed generative models in terms of the privacy-utility trade-off. Though we find this is true for WGAN-GP, PATE-GAN and CTGAN-which fall on the right side of the additive noise curve-other methods do yield better synthetic datasets.
ADS-GAN is based on WGAN-GP, hence for small λ (the privacy regularizer) it gets a similar score. Increasing λ promotes a higher distance between generated and training data, hence this reduces vulnerability. At first, it also leads to an increase in quality-raising λ leads to lower overfitting-but when λ increases further the generative distribution is distorted to the point that quality is significantly reduced. In contrast to (Hilprecht et al., 2019), we do not find evidence that VAEs are more vulnerable to MIAs Figure 5: DOMIAS is more successful at attacking patients taking high-blood pressure medication. (left) T-SNE plot of Heart Failure test dataset. There is a cluster of points visible in the top right corner, which upon closer inspection corresponds to subjects who take ARB medication. (right, bottom) Attacking accuracy of DOMIAS and baselines on majority and minority group (averaged over 8 runs). DOMIAS is significantly better at attacking the minority group than the general population. Except for GAN-leaks CAL, baselines fail to capture the excess privacy risk to the patients with blood pressure medication. Comparing DOMIAS with Eq. 1 (BNAF) (see Sec. 5.2), we see that the minority vulnerability is largely due to the availability of the reference data. (right, top) Single run attacking success of different MIA methods on these underrepresented samples; correctly inferred membership in green, incorrectly inferred in red.
than GANs. The Pareto frontier is given by the additive noise method, TVAE, NFlow and PrivBayes, hence the best synthetic data model will be one of these, depending on the privacy requirements.
DISCUSSION
DOMIAS use cases DOMIAS is primarily a tool for evaluating and interpreting generative model privacy. The overall DOMIAS attacking success is a metric for MIA vulnerability, and may hence guide generative model design choices-e.g. choosing privacy parameters-or aid evaluation-including for competitions like (Jordon et al., 2021). Since DOMIAS provides a sample-wise metric, its scores can also provide insight into privacy and overfitting of specific samples or regions in space-as seen in Sec. 5.3. Future work may adopt DOMIAS for active privacy protection, e.g. as a loss during training or as an auditing method post-training-removing samples that are likely overfitted.
Underrepresented groups are more vulnerable to MIA attacks Generative models are more likely to overfit lowdensity regions, and we have seen DOMIAS is indeed more successful at attacking these samples. This is distressing, since these regions can correspond to underrepresented groups in the population. Similar results have been found in supervised learning literature, e.g. (Kulynych et al., 2019;Bagdasaryan et al., 2019). Protecting against this vulnerability is a trade-off, as outliers in data can often be of interest to downstream research. It is advisable data publishers quantify the excess MIA risk to specific subgroups.
Attacker calibration In practice, it will often be unknown how much of the test data was used for training. Just like related works, we have ignored this. This challenge is equivalent to choosing a suitable threshold, or suitable f in Eq. 2 and relates closely to calibration of the attacker model, which is challenging for MIA since-to an attacker-usually no ground-truth labels are available. Future work can explore assumptions or settings that could enable calibrated attacks. In Appendix D we include results for high-precision attacks.
High-dimensionality and image data Traditional density estimation methods (e.g. KDE) perform notoriously poorly in high dimensions. Recent years have seen a rise in density estimation methods that challenge this conception. Domain-specific density estimators, e.g. that define density on lower-dimensional embeddings, can be readily used in DOMIAS. We include preliminary results for the highdimensional CelebA image dataset in Appendix B.3.
Training data size We have seen that for large number of training samples, the performance of all attackers goes down to almost 0.5. The same is observed for large generative image models, Appendix B.3. This is reassuring for synthetic data publishers, for whom this indicates a relatively low privacy risk globally. However, global metrics may hide potential high-precision attacks on a small number of individuals, see Appendix D.
Availability of reference dataset DOMIAS assumes the presence of a reference dataset that enables approximating the true distribution p R (X). In case there is not sufficient data for the latter, more prior knowledge can be included in the parametrisation of p R ; e.g. choose p R (X) to lie in a more restrictive parametric family. Even in the absence of any data D ref , an informed prior (e.g. Gaussian) based on high-level statistics can already improve upon related works that rely on assumption Eq. 1-see Appendix C for results. In Appendix E we include further experiments with Figure 6: DOMIAS can be used to quantify synthetic data MIA vulnerability. We plot the synthetic data quality versus DOMIAS AUC for different generative models on the California Housing dataset. There is a clear trade-off: depending on the tolerated MIA vulnerability, different synthetic datasets are best.
distributional shifts between the D ref and D mem , in which we find that even with moderate shifts the use of a reference dataset is beneficial.
Publishing guidelines Synthetic data does not guarantee privacy, however the risk of MIA attacks can be lessened when synthetic data is published considerately. Publishing just the synthetic data-and not the generative modelwill in most cases be sufficient for downstream research, while avoiding more specialised attacks that use additional knowledge. Further consideration is required with the amount of data published: increasing the amount of synthetic data leads to higher privacy vulnerability (Figure 4b and see (Gretton et al., 2012)). Though the amount of required synthetic data is entirely dependent on the application, DOMIAS can aid in finding the right privacy-utility trade-off.
Societal impact We believe DOMIAS can provide significant benefits to the future privacy of synthetic data, and that these benefits outweigh the risk DOMIAS poses as a more successful MIA method. On a different note, we highlight that success of DOMIAS implies privacy is not preserved, but not vice versa. Specifically, DOMIAS should not be used as a certificate for data privacy. Finally, we hope the availability of a reference dataset is a setting that will be considered in more ML privacy work, as we believe this is more realistic in practice than many more popular MIA assumptions (e.g. white-box generator), yet still poses sig- 1. Train density model p R (X) on D ref .
2. Train density model p G (X) on D syn .
Compute
A DOM IAS (x) = p G (x) p R (x) for all x ∈ D test 4. Choose threshold τ , e.g. τ = median{A DOM IAS (x)|x ∈ D test } 5. Inferm = 1, if A DOM IAS (x) > τ, 0, otherwise,
for all x ∈ D test .
A.2 Data
We use the California housing (Pace and Barry, 1997) (license: CC0 public domain) and Heart Failure (private) datasets, see Table 2 and Figure 7 for statistics. All data is standardised. Figure 7: Correlation matrices of features within Housing and Heart Failure datasets. The first feature of the Heart Failure dataset is used for defining the minority group in Section 5.3.
A.3 Experimental settings
All results reported in our paper are based on 8 repeated runs, with shaded area denoting standard deviations. We experiment on a machine with 8 Tesla K80 GPUs and 32 Intel(R) E5-2640 CPUs. We shuffle the dataset and split the dataset into training set, test set, and reference set. The attack performance is computed over a test set consisting of 50% training data (i.e. samples from D mem ) and 50% non-training data. Choices of sizes for those sets are elaborated below.
Experimental Details for Section 5.1 In this section, we experimented on the California Housing Dataset to compare different MIA performance with DOMIAS. For the experiment varying the number of members in the training dataset (i.e. left panel of Figure 3), we use a fixed training epoch 2000, a fixed number of reference example |D ref | = 10000 and a fixed number of generated example |D syn | = 10000. For the experiment varying the number of training epochs of TVAE (i.e. the right panel of Figure 3), we use a fixed training set size |D mem | = 500, a fixed number of reference example |D ref | = 10000 and a fixed number of generated example |D syn | = 10000. Training with a single seed takes 2 hours to run in our machine with BNAF as the density estimator.
In BNAF density estimation, the hyper-parameters we use are listed in Table 3. Our implementation of TVAE is based on the source code provided by (Xu et al., 2019a). Figure 4, we fix the training epoch to be 2000, set n syn = 10000 and n M = 500. In the experiments varying the number of generated data n syn , i.e. results reported in the right panel of Figure 4, we set n ref = 10000, training epoch to be 2000, and n mem = 500. Our implementation of the kernel density estimation is based on sklearn with an automated adjusted bandwidth. Training with a single seed takes 0.5 hours to finish in our machine with the kernel density estimator.
Experimental Details for Section 5.3 Based on results of Section 5.2, the attacking performance on different subgroups can be immediately calculated by adopting appropriate sample weights.
Experimental Details for Section 5.4 In the Additive-Noise baseline curve, results are generated with the following noise values: [0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, 2.3, 2.5, 2.9, 3.5, 3.9]. In the ADS-GAN curve, results are generated with the following privacy parameter λ = [0.2, 0.5, 0.7, 1.0, 1.1, 1.3, 1.5]. In the WGAN-GP we use a gradient penalty coefficient 10.0. All the other methods are implemented with recommended hyper-parameter settings. Training different generative models are not computational expensive and take no more than 10 minutes to finish in our machine. Using a kernel density estimator and evaluating all baseline methods take another 20 minutes, while using a BNAF estimator takes around 1.5 more hours. In Figure 10 we include the results of experiment 5.4 for all attacks, including error bars. Indeed, we see that DOMIAS outperforms all baselines against most generative models. This motivates using DOMIAS for quantifying worst-case MIA vulnerability.
B.3 CelebA image data
We include additional results for membership inference attacks against the image dataset CelebA. Results indicate DO-MIAS is significantly better at attacking this high-dimensional data than baseline methods. Ablation study of DOMIAS on Heart Failure dataset, with attack performance as a function of the reference dataset size (left) and the synthetic dataset size (right). Similar to Section 5.2, we see that the MIA performance of DOMIAS is largely due to assumption Eq.2 vs Eq. 1, i.e. the value of the reference dataset.
Set-up We use CelebA (Liu et al., 2015), a large-scale face attributes dataset with more than 200K celebrity images. We generate a synthetic dataset with 10k examples using a convolutional VAE with a training set containing the first 1k examples, and use the following 1k examples as test set. Then the following 10k examples are used as reference dataset. As training the BNAF density estimator is computational expensive (especially when using deeper models), we conduct dimensionality reduction with a convolutional auto-encoder with 128 hidden units in the latent representation space (i.e. output of the encoder) and apply BNAF in such a representation space. The hyper-parameters and network details we use in VAE are listed in Table 4 and Table 5. Results Figure 11 includes the attacking AUC of DOMIAS and baselines of 8 runs. DOMIAS consistently outperforms other MIA methods, most of which score not much better than random guessing. These methods fail to attack the 128dimensional representations of the data (originally 64 × 64 pixel images), due to most of them using nearest neighbour or KDE-based approaches. On the other hand, DOMIAS is based on the flow-based density estimator BNAF (de Cao et al., 2019), which is a deeper model that is more apt at handling the high-dimensional data.
C HIGH-LEVEL PRIOR KNOWLEDGE
If we have no reference data at all, we can still perform more successful attacks compared to baselines if we have highlevel statistics of the underlying distribution. Effectively, any informed prior can improve upon methods that use Eq. 1; this being a special case of Eq. 2, where one assumes a uniform prior on p R . In this Appendix, we use the Housing dataset and we assume that we only know the mean and standard deviation of the first variable, median income. This is a very realistic setting in practice, since an adversary can relatively easily acquire population statistics for individual features. We subsequently model the reference dataset distribution p ref as a normal distribution of only the age higher-level statisticsi.e. not making any assumptions on any of the other variables, implicitly putting a uniform prior on these when modelling p ref . Otherwise, we use the same training settings as in Experiment 5.1 (left panel Figure 3). In Figure 12. We see that even with this minimal assumption, we still outperform its ablated versions. These results indicate that a relatively weak prior on the underlying distribution without any reference data, can still provide a relatively good attacker model. Hu and Pang (2021) focus on high-precision membership attacks, i.e. can we attack a small set of samples with high certainty. This is an interesting question, since the risk of high-precision attacks may be hidden if one only looks at overall attacking performance. Their work is not applicable to our setting, e.g. they assume full generator and discriminator access. In this section, we show that even in the full black-box setting high-precision MIAs are a serious risk.
D HIGH-PRECISION ATTACKS
D.1 Tabular data
Set-up We assume the same dataset and generative model set-up as in Section 5.3. We study which samples the different methods give the highest score, i.e. mark as most likely to be in D mem . Let D test be a test set consisting for 50% of samples x i in D mem and 50% samples not in D mem , respectively denoted by m = 1 and m = 0. Letm = A(x) be the attacker's prediction, and let S(A, D test , q) = {x ∈ D test |m > Quantile({m i } i , 1 − q)} be the set of samples that are given the q-quantile's highest score by attacker A. We are interested in the mean membership of this set, i.e. the precision (64, 3, 4, 2, 1) Tanh · Figure 11: Attacking performance on CelebA. DOMIAS scores significantly better at attacking image data compared to baselines.
if threshold Quantile({m i |x i ∈ D test }, 1 − q) is chosen. We include results for DOMIAS and all baselines. Results are averaged over 8 runs.
Results In Figure 13 we plot the top-score precision-quantile curve for each method for each MIA method, i.e. P (A, D test , q) = mean({m|x ∈ S(A, D test , q)}) as a function of q. These figures show the accuracy of a high-precision attacker, if this attacker would choose to attack only the top q-quantile of samples. We see that unlike other methods, the precision of DOMIAS goes down almost linearly and more gradually. Though MC and GAN-Leaks are able to find the most overfitted examples, they do not find all-resulting from their flawed underlying assumption Eq. 1 that prohibits them from finding overfitted examples in low-density regions.
D.2 Image data
Let us run the same high-precision attack on the CelebA dataset-see Appendix B.3, including settings. Again, we see that high-precision attacks are more successful when using DOMIAS, see Figure 14 Figure 12: Using DOMIAS with no reference data but high-level statistics of the underlying data. Using just the mean and standard deviation of the population's median income, DOMIAS outperforms its ablated counterparts that are based on Eq.
1.
E DISTRIBUTION SHIFT D ref AND D mem
There may exist a distributional shift between reference and training data. Because DOMIAS is primarily intended as a tool for data publishers to test their own synthetic data vulnerability, it is recommended that testing is conducted with a reference dataset from the same distribution (e.g. a hold-out set): this effectively tests the worst-case vulnerability. Hence, our work focused on the case where there is no shift.
Nonetheless, reference data may not always come from the same target distribution. For example, reference data may come from a different country, or synthetic data may be created by intentionally changing some part of the real data distribution, e.g. to include fairness guarantees (Xu et al., 2019b;van Breugel et al., 2021). Thus, let us assume there is a shift and that the reference data D ref comes fromp R , a shifted version of p R (i.e. the distribution from which D mem is drawn). We give a specific example and run an experiment to explore how this could affect DOMIAS attacking performance.
Let us assume there is a healthcare provider that publishes D syn , a synthetic dataset of patients suffering from diabetes, based on underlying data D mem ∼ p R . Let us assume there is an attacker that has their own data D ref ∼p R , for which some samples have diabetes (A = 1), but others do not (A = 0). We assume that A itself is latent and unobserved (s.t. the attacker cannot just train a classification model) and that there is a shift in the distribution of A (i.e. with a slight abuse of notationp R (A = 1) < 1). Diabetes is strongly correlated with other features X in the data, additionally we assume the actual condition distribution p R (X|A) is fixed across datasets. This implies the reference and membership set distributions can be written respectively as:p R (X) =p R (A = 1)p(X|A = 1) +p R (A = 0)p(X|A = 0) (3) p R (X) = p(X|A = 1)
Since p R (X|A = 1) = p R (X|A = 0) andp R (A = 1) = 1, there is a distributional shift betweenp R and p R . Now let us see how different attackers perform in this setting as a function of the amount of shift. Evidently, since some of the baselines do not use reference data, some attackers will be unaffected, but we should expect DOMIAS performance to degrade. We take the Heart Failure dataset, which indeed has a feature denoting diabetes,. We vary the amount of shift of p R w.r.t. p R , fromp(A = 0) = 0 (no shift), top(A = 0) = 0.8 (a large shift and the original Heart Failure non-diabetes prevalence). Let us assume test data follows the attacker's existing dataset, i.e.p R . This gives Figure 15.
We see performance of DOMIAS degrades with increasing shift, due to it approximating p R withp R , affecting its scores (Eq. 2). However, we see that for low amounts of shift this degradation is minimal and we still perform beter than not using the reference dataset (baseline Eq. 1 (BNAF)). This aligns well with the results from 5.2, Figure 4, that showed that an inaccurate approximation of p R due to few samples is still preferable over not using any reference data. : DOMIAS is better at high-precision attacks than baselines on heart failure dataset. Plotting the top-quantile precision P (A, D test , q) versus q. For example, if the attacker decides to attack only the 20% highest samples, we get DOMIAS is significantly more precise (86.2 ± 5.5%) compared to baselines-LOGAN D0 (51.0 ± 3.9%), LOGAN D1 (72.6 ± 5.3%), MC (74.2 ± 3.0%), GAN-leaks (74.9 ± 3.1%), GAN-Leaks CAL (57.0 ± 4.1%). Additionally included is Eq. 1 (BNAF), the ablation attacker that does not make use of the reference data. We see that the reference data helps DOMIAS attack a a larger group with high precision. Figure 15: Effect of distributional shift on DOMIAS performance. A distributional shift between D mem and D ref degrades attacking performance, but preliminary experiments show that for small to moderate shifts it is still preferable to use reference data even though it is slightly shifted.
Figure 1 :
1Should we infer membership m = 1 for point A?
Figure 3 :
3DOMIAS outperforms baselines. MIA performance of DOMIAS and baselines versus the generative model training set size |D mem | and training time t epochs on the California Housing dataset. We observe how MIA AUC goes up for fewer training samples and long generative model training time, as both promote overfitting. CAL (Chen et al., 2019)-see
al., 2014), WGAN-GPGulrajani et al., 2017), CTGAN and TVAE(Xu et al., 2019a), NFlow(Durkan et al., 2019), PATE-GAN(Jordon et al., 2019), PrivBayes(Zhang et al., 2017), and ADS-GAN(Yoon et al., 2020). As a baseline, we also include the anonymization method of sampling from training data and adding Gaussian noise. For ADS-GAN and the additive noise model, we vary the privacy level by raising the hyperparameter λ and noise variance, respectively. Results for other attackers are found in Appendix B.
repeat the experiments of Section 5.1 and 5.2 on the Heart Failure dataset, seeFigures 8 and 9. Results are noisier, but we observe the same trends as in Sections 5.
Figure 8 :
8DOMIAS outperforms baselines on Heart Failure dataset. MIA performance of DOMIAS and baselines versus the generative model training set size |D mem | and training time t epochs , evaluated on Heart Failure datasets. The same trends are observed as in Section 5.1.
Figure 9 :
9DOMIAS source of gain.
Figure 10 :
10DOMIAS consistently outperforms baseline attackers at attacking the different generative models.
Figure 13
13Figure 13: DOMIAS is better at high-precision attacks than baselines on heart failure dataset. Plotting the top-quantile precision P (A, D test , q) versus q. For example, if the attacker decides to attack only the 20% highest samples, we get DOMIAS is significantly more precise (86.2 ± 5.5%) compared to baselines-LOGAN D0 (51.0 ± 3.9%), LOGAN D1 (72.6 ± 5.3%), MC (74.2 ± 3.0%), GAN-leaks (74.9 ± 3.1%), GAN-Leaks CAL (57.0 ± 4.1%). Additionally included is Eq. 1 (BNAF), the ablation attacker that does not make use of the reference data. We see that the reference data helps DOMIAS attack a a larger group with high precision.
Table 1 :
1Membership Inference attacks on generative models. (1) Underlying ML method (GAN: generative adversarial network, NN: (weighted) Nearest neighbour, KDE: kernel density estimation, MLP: multi-layer perceptron, DE: density estimator); (2) uses D ref ;
Table 2 :
2Dataset statisticsCalifornia Housing Heart Failure
Number of samples
20640
40300
Number of features
8
35
-binary
0
25
-continuous
8
10
(a) Housing
(b) Heart Failure
Table 3 :
3Hyperparameters for BNAF
batch-dim
50
n-layer
3
hidden-dim
32
flows
5
learning rate 0.01
epochs
50
Experimental Details for Section 5.2 In our experiments varying the number of reference data n ref , i.e. results reported
in the left panel of
Table 4 :
4Hyperparameters for VAE
batch size
128
n-layer
5
Optimizer
Adam
learning rate 0.002
Table 5 :
5Architecture of VAE (a) Network Structure for EncoderLayer
Params (PyTorch-Style)
Conv1
(3, 64, 4, 2, 1)
ReLU
·
Conv2
(64, 128, 4, 2, 1)
ReLU
·
Conv3
(128, 256, 4, 2, 1)
ReLU
·
Conv4
(256, 256, 4, 2, 1)
ReLU
·
Linear1
(256 * 4 * 4, 256)
ReLU
·
Linear2
(256, 256)
ReLU
·
Linear3
(256, 128 * 2)
(b) Network Structure for Decoder
Layer
Params (PyTorch-Style)
Linear1
(128, 256)
ReLU
·
Linear2
(256, 256)
ReLU
·
Linear3
(256, 256 * 4 * 4)
ReLU
·
ConvTranspose1
(256, 256, 4, 2, 1)
ReLU
·
ConvTranspose2
(256, 128, 4, 2, 1)
ReLU
·
ConvTranspose3
(128, 64, 4, 2, 1)
ReLU
·
ConvTranspose4
Specifically, none performed better than random guessing in at least half of the datasets.
Though with varying extents, see (Chen et al., 2019)
This work focuses on relative scores, hence we ignore choosing f -see Sec. 6.
Code is available at https://github.com/vanderschaarlab/DOMIAS
AcknowledgementsWe would like to thank the Office of Navel Research UK, who funded this research.: DOMIAS is better at high-precision attacks than baselines on CelebA image data. For example, an attacker could attack only the examples with top 2% scores, and get a precision of P = 65.7 ± 11.6%-much higher than the second-best method LOGAN 0, scoring P = 54.8 ± 6.5%.
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Achieving causal fairness through. Depeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, Xintao Wu, Depeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, and Xintao Wu. Achieving causal fairness through
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] | [] | Although deep neural networks achieve tremendous success on various classification tasks, the generalization ability drops sheer when training datasets exhibit long-tailed distributions. One of the reasons is that the learned representations (i.e. features) from the imbalanced datasets are less effective than those from balanced datasets. Specifically, the learned representation under classbalanced distribution will present the Neural Collapse (N C) phenomena. N C indicates the features from the same category are close to each other and from different categories are maximally distant, showing an optimal linear separable state of classification. However, the pattern differs on imbalanced datasets and is partially responsible for the reduced performance of the model. In this work, we propose two explicit feature regularization terms to learn high-quality representation for class-imbalanced data. With the proposed regularization, N C phenomena will appear under the class-imbalanced distribution, and the generalization ability can be significantly improved. Our method is easily implemented, highly effective, and can be plugged into most existing methods. The extensive experimental results on widelyused benchmarks show the effectiveness of our method. | 10.48550/arxiv.2302.12453 | [
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Inducing Neural Collapse in Deep Long-tailed Learning
Xuantong Liu
The Hong Kong University of Science and Technology
Jianfeng Zhang
Huawei Noah's Ark Lab
Tianyang Hu
Huawei Noah's Ark Lab
He Cao
The Hong Kong University of Science and Technology
Lujia Pan
Huawei Noah's Ark Lab
Yuan Yao
The Hong Kong University of Science and Technology
Inducing Neural Collapse in Deep Long-tailed Learning
Although deep neural networks achieve tremendous success on various classification tasks, the generalization ability drops sheer when training datasets exhibit long-tailed distributions. One of the reasons is that the learned representations (i.e. features) from the imbalanced datasets are less effective than those from balanced datasets. Specifically, the learned representation under classbalanced distribution will present the Neural Collapse (N C) phenomena. N C indicates the features from the same category are close to each other and from different categories are maximally distant, showing an optimal linear separable state of classification. However, the pattern differs on imbalanced datasets and is partially responsible for the reduced performance of the model. In this work, we propose two explicit feature regularization terms to learn high-quality representation for class-imbalanced data. With the proposed regularization, N C phenomena will appear under the class-imbalanced distribution, and the generalization ability can be significantly improved. Our method is easily implemented, highly effective, and can be plugged into most existing methods. The extensive experimental results on widelyused benchmarks show the effectiveness of our method.
INTRODUCTION
Modern deep neural networks have shown the ability to outperform humans on many tasks, such as computer vision, natural language processing, playing games, etc., and keep refreshing state-of-the-art performance for complex classification tasks. However, when the training dataset is class-imbalanced, such as a long-tailed distribution, where a few majority classes occupy most of the training samples while a large number of minority classes own very limited samples, the performance of the model drops off a cliff (Van Horn & Perona, 2017;Buda et al., 2018). These imbalanced distributions are ubiquitous in real-world applications, e.g., fault diagnosis, face recognition, autonomous driving, etc. Therefore, how to improve the discriminative ability of the model trained on the imbalanced dataset has always been a topic of considerable concern.
Some recent studies focus on learning more effective representation to improve the long-tailed recognition ability. Supervised contrastive loss (Khosla et al., 2020) is utilized to learn compact within-class and maximally distant betweenclass representation by introducing uniformly distributed class centers, which leads to improvement in long-tailed performance Cui et al., 2021;Zhu et al., 2022). These characteristics of the feature representation are consistent with those learned from balanced datasets (Graf et al., 2021), where the classification models can spontaneously learn tight and discriminative features. However, contrastive learning is more computationally expensive and requires more iterations to converge than the standard Cross-Entropy (CE) loss.
Meanwhile, the learning behavior of deep classification models in a balanced setting has been investigated both empirically and theoretically (Papyan et al., 2020;Galanti et al., 2021;Han et al., 2022). The Neural Collapse (N C) phenomenon was uncovered by Papyan et al. (2020) when investigating the last-layer embedding, i.e. the feature representation, and the corresponding classifier weights in deep classification models during training. N C shows that the learned features (or embedded vectors) of the same class will collapse to their class centers. Meanwhile, these class centers, after globally centered, as well as the classifier weights, will form a simplex equiangular tight frame (ETF) during the terminal phase of training (TPT), i.e. when the model achieves zero training error. The ETF structure maximizes the between-class variability so as the Fisher discriminant ratio (Fisher, 1936), resulting in an optimal linear separable state for classification. Subsequent studies have found more characteristics of this phenomenon, including the global optimal property (Zhu et al., 2021) and generalization ability (Galanti et al., 2021).
However, on imbalanced datasets, the deep neural networks will exhibit different geometric structures, and some N C phenomena will no longer occur (Fang et al., 2021;Thrampoulidis et al., 2022). The last-layer features of the same class still converge to their class means, but the class means, as well as the classifier weights, are not in the form of ETFs any more. Specifically, compared to majority classes, the learned features of minority classes will have a larger norm, and correspondingly the norm of classifier weights will be smaller (Kang et al., 2019;Fang et al., 2021). Furthermore, as the imbalance level increases, the phenomenon of Minority Collapse may arise, in which both the learned representations and the classifier weights on minority classes will become indistinguishable (Fang et al., 2021). The absence of some N C property partially explains the performance gap between the balanced and imbalanced datasets.
In this paper, we first elaborate that the appearance of N C can help to minimize the generalization error in the imbalanced problem. According to this property, we propose two simple yet effective regularization terms to explicitly induce all the N C phenomena in neural networks trained on imbalanced datasets. The regularization terms can be added to CE loss directly. Compared with supervised contrastive learning, these terms have lower computational cost. Our proposed method not only helps the N C to occur faster for models trained on the balanced datasets, but also drives the N C phenomenon to occur on datasets with imbalanced categories. The resulting model can also obtain better generalization ability and robustness without over-training as in Papyan et al. (2020). Furthermore, our proposed method is orthogonal to most existing methods dealing with long-tailed problems. It thus can be easily plugged into the objective function to obtain further improvements.
In summary, our contributions can be listed below:
• We observe that when training data is imbalanced, the class centers of minority classes move closer to those of the majority classes, making their instances difficult to distinguish.
• We demonstrate that, although some N C phenomena do not naturally exist in an imbalanced case, we can achieve lower generalization error when all N C proprieties hold. Thus we propose two simple yet effective regularization terms to manually induce the N C during imbalanced training.
• We experimentally show that our method can significantly improve the performance in various long-tailed tasks and boost most existing methods.
PROBLEM SETUP
Preliminaries
Let f φ • g θ (·) denote a neural network classifier, where g θ (·)
is a feature extractor and f φ (·) is a linear classifier. We define H = [h 1 , · · · , h n ] T ∈ R n×P to be the output of g θ (·). Here P is the dimension of the latent feature, and n is the training sample size. The weights of f φ are denoted by W = [w 1 , · · · , w K ] ∈ R P ×K , and the corresponding bias vector is b = [b 1 , ..., b K ], where K is the number of classes. h i ∈ R P and y i ∈ {k} K k=1 denote the feature and label of the i-th sample. The label matrix is denoted by Y ∈ R n×K . In the training data, we have n = K k=1 n k , where n k is the sample size of class k. We use || · || F and || · || to denote the Frobenius norm of a matrix and the l 2 -norm of a vector. Definition 1 (Simplex ETF). A simplex ETF is a collection of equal-length and maximally-equiangular vectors. We call a P × K matrix M an ETF if it satisfies
M T M = α K K − 1 I − 1 K − 1 1 K 1 T K(1)
for some non-zero scalar α. Where I is the identity matrix, and 1 K is an all-ones vector.
Let µ k = 1 n k yi=k h i be the center of class k and µ C = 1 K K k=1 µ k be the arithmetic mean of the class centers. In the balanced case, where we have n k = n K for each class, N C will appear during TPT. The phenomena can be formally described by four properties:
• (N C 1 ) Variability collapse. Intra-class variances collapse to zero during the terminal phase of training, i.e., for any sample i from class k, we have
||h i − µ k || = 0(2)
• (N C 2 ) Convergence to simplex ETF. The class centers (after zero-center normalization) converge to the vertices of an ETF, i.e.
cos(µ k − µ C , µ k − µ C ) = − 1 K − 1 ,(3)||µ k − µ C || = ||µ k − µ C ||.(4)
• (N C 3 ) Convergence to self-duality. The weights of linear classifiers are parallel to the corresponded zerocentered class centers, i.e.
w k = α(µ k − µ C ).(5)
• (N C 4 ) Simple decision rule. Given a feature, the lastlayer classifier's behavior is equivalent to the nearest class center (NCC) decision rule, i.e.
arg max k w k , h = arg min k ||h − µ k ||(6)
Neural Collapse and Imbalanced Data
In this section, we first illustrate why N C disappears on imbalanced datasets using mean squared error (MSE) loss. Then we demonstrate that the N C properties will lead to a lower generalization error bound thus we can benefit from it under imbalanced distribution.
The Optimal Classifier Under Imbalanced Distribution
Some recent studies have shown that the test performance of neural networks trained with MSE loss is comparable to those trained with CE loss in classification tasks (Demirkaya et al., 2020;Fang et al., 2021;Hu et al., 2021;Han et al., 2022). Thanks to its tractability, we can use MSE loss to illustrate the absence of the N C phenomenon on imbalanced datasets. For linear classifiers, the MSE loss is
L(H, W ) = 1 2n ||Y − (HW + 1 n b T )|| 2 F .(7)Leth = 1 n n i=1 h i be the global feature mean, Σ T = (H − 1 nh ) T (H − 1 nh ) be the total covariance matrix of H andṀ = [µ 1 −h, ..., µ K −h] ∈ R P ×K .
We can have the closed form of the optimal W and b under the MSE loss as follows.
Proposition 1. (Webb & Lowe, 1990). In general, for fixed features H, the optimal weight matrix and the bias vector that minimize L(H, W ) are
W LS = Σ † TṀ Λ,(8)b LS = 1 n 1 T n Y − µ G W LS ,(9)
where † denotes the Moore-Penrose pseudoinverse, and Λ = diag(n 1 , · · · , n K ) is a diagonal matrix.
From Eq.(9), we can observe that the optimal weight matrix depends on the features and is strongly affected by Λ, i.e. the proportions of classes. Specifically, the classifier weights of the majority classes will have larger norms. The N C phenomena reflect the intimate connection between the last layer features and the classifier weights. Thus, skewed classifiers imply that the features are also biased, and many studies have empirically investigated that the uneven label distribution can lead to an imbalanced feature space (Kang et al., 2020;Fang et al., 2021;Li et al., 2022).
Particularly, Fang et al. (2021) show the Minority Collapse phenomenon that reveals the skewed classifier weights encountering an imbalanced label distribution where majority classes own much more samples than the minority ones. In addition, they theoretically prove that unbiased classifiers can be obtained through over-sampling. However, empirical results show a limited performance improvement or even decline due to the over-fitting of the minority classes (Drummond et al., 2003;Weiss et al., 2007). On the other hand, the classifiers are always better tuned than the learned features (Thrampoulidis et al., 2022). Therefore, in this work, we mainly focus on regularizing the embeddings during training to get non-skewed and representative features. Then we tune a balanced classifier based on our well-learned features.
Importance of N C on Imbalanced Datasets
As we already know that when the training set is classimbalanced, the geometric structure of the classifiers and centered class means are not symmetric, which may introduce some bias in the model and affect the performance of the test set (Kang et al., 2019(Kang et al., , 2020Fang et al., 2021). Recent work indicates that compact within-class representations along with evenly distributed class centers can help learn high-quality representations, and substantial practice confirms this Zhu et al., 2022;Cui et al., 2022). These intuitions lead to similar situations with N C.
In this section, we explain why the N C can be considered favorable representations and can provide reduced generalization errors under long-tailed distributions from the perspective of domain adaptation.
As a standard evaluation approach in long-tailed learning, models are usually tested on balanced datasets. Since the training set is imbalanced, we can regard this scenario as a label shift domain adaptation problem, where the source domain is imbalanced, and the target domain is balanced.
First, the following proposition shows that properties of N C 1 and N C 2 can be approximately preserved in the target domain.
Proposition 2. (Galanti et al., 2021) Let µ S k (resp. µ T k ) and σ S k (resp. σ T k ) be the mean and variance of the representations of class k on the source domain (resp. target domain). For any two different classes, k and k , with probability at (Galanti et al., 2021). Hence, A and B are upper bounded and diminish to zero as n k gets larger. Therefore, we can roughly speak N C 1 and N C 2 can generalize to the target domain.
least 1 − δ over D S , we have σ T k + σ T k 2 µ T k − µ T k 2 ≤ (1 + A 2 ) σ S k + σ S k 2 µ S k − µ S k 2 + B , (10) where A = O( √ log(1/δ)/n k ) ||µ T k −µ T k || , B = O( √ log(1/δ)/n k ) ||µ S k −µ S k || 2 . The ETF geometry of {µ k } K k=1 indicates that the distance µ S k − µ S k achieve maximum value for all k = k . On the other hand, µ T k −µ T k is also lower bounded by µ S k −µ S k
We then illustrate how the existence of N C 1 and N C 2 help to reduce the generalization error. According to Ben-David et al. (2006), for any classifier h, the error on target domain T (h) will be bound by the empirical error on the source domain and the divergence between source and target feature domains plus a constant:
T (h) ≤ˆ S (h) + d H (D Z S , D Z T ) + const,(11)
where d H (D Z S , D Z T ) 1 measures some 'distance' between source and target domains over the feature space Z. Although not exactly the same, substituting d H with the Jensen-Shannon distance d JS (Endres & Schindelin, 2003) will not significantly change the result. Theoretically, minimizing d JS between source and target distributions will reduce the right-hand side of Eq.(11) as well. Let D Z and D Y be the distributions defined over the latent feature space and label space, respectively. As D Y can be induced from D Z from a generative perspective, according to Zhao et al. (2019), we have
d JS (D Z S , D Z T ) ≥ d JS (D Y S , D Y T ),(12)i.e., d JS (D Z S , D Z T ) is the lower bounded by d JS (D Y S , D Y T )
, which is a constant determined by source and target label distributions.
With N C 1 and N C 2 , the distribution over Z collapses to a K-component mixture Dirac distribution. More precisely, we have Pr(Z = h) = Pr(Y = y). In this case,
d JS (D Z S , D Z T ) attains its lower bound d JS (D Y S , D Y T )
, which is the objective of some classical domain adaptation algorithms (Long et al., 2015;Ganin et al., 2017),
LEARNING REPRESENTATION VIA INDUCING NEURAL COLLAPSE
The previous analysis inspires us to induce N C phenomena to imbalanced training. We mainly focus on the core properties, N C 1 and N C 2 , and come up with two corresponding regularization terms.
Feature Regularization
Compact within-class features. N C 1 underlines that the model is seeking to learn compact within-class features by pushing the last-layer embedding to be close to their class centers, which seems natural but actually hard to achieve in practice. Han et al. (2022) decomposed the MSE loss and discovered that the loss in the late training stages is dominated by the 2 -distance between the feature and the corresponding class center. This indicates that although N C 1 is the inevitable trend, it is quite difficult to realize. Therefore, we add explicit regularization to make N C 1 more inclined to appear. Especially, for the class-imbalanced dataset, we consider the inverse ratio of class sizes as weights to avoid excessive force on the majority classes. This indicates the difference between our N C 1 regularization and the center loss (Wen et al., 2016) that pushes all features equally to their class center. Formally, we define the N C 1 regularization as the within-class feature distance, L W , with the formula of
L W = K k=1 yi=k 1 n k ||h i − µ k || 2 2 .(13)
Distinct between-class features. N C 2 shows that with balanced class distribution, all pairs of centered class means tend to form equal-sized angles, implying the maximally separated between-class features. However, under the imbalanced distribution, the class centers of the minority classes are close to the majority ones, leading to indistinguishable features. Therefore, we propose N C 2 regularization to minimize the maximal pairwise cosine similarity between all the centered class means, equivalent to maximizing the minimal pairwise angle. Consider the angular version, the objective of N C 2 regularization is:
max min k =k arccos μ k ,μ k ||μ k || · ||μ k || ,(14)
whereμ k = µ k − µ C . As noted in Wang et al. (2020), updating the average of each vector's maximum cosine is more efficient than just optimizing the global maximum cosine. Therefore, we define the formula for the N C 2 regularization as
L B = − 1 K K k=1 min k ,k =k arccos μ k ,μ k ||μ k || · ||μ k || .(15)
In summary, our proposed feature regularization includes two terms, L W and L B , corresponding to minimize the within-class distance and maximize the between-class discrepancy, respectively. They can be easily coupled with supervised losses with a linear classifier to regularize the penultimate layer embedding. Finally, we have the following loss for training:
L = L sup + λ 1 L W + λ 2 L B .(16)
where L sup denotes the supervised loss, e.g. CE loss and MSE loss. λ 1 and λ 2 are hyperparameters that control the impact of L W and L B .
Occurrence of Neural Collapse
First, we illustrate that L B will lead all pairs of the K class means to have the same cosine equals to − 1 K−1 , with the following proposition. Proposition 3. The minimum of the maximal pair-wise cosine similarity between n vectors is −1 n−1 , which can be reached when the vectors have an equalsized pair-wise angle and zero mean.
Therefore, denoteM = [μ 1 ||μ1|| , ...,μ K ||μ K || ].
Recall that the objective of L B is to minimize the maximal pair-wise cosine similarity of the centered class means, thus with L B , we haveM
TM = K K − 1 I − 1 K − 1 1 K 1 T K .(17)
According to Definition 1,M form a simplex ETF. Furthermore, although L W and L B do not explicitly enforce the centered class means to have an equal norm, we empirically observe this desired result (see the experimental result in Section 4.2.1). Let ||μ 1 || = · · · = ||μ K || = α and M = [μ 1 , · · · ,μ K ], then we havē
M TM = α K K − 1 I − 1 K − 1 1 K 1 T K ,(18)
indicating the centered class means indeed from an ETF. Therefore, with the proposed feature regularization terms L W and L B , N C 1 and N C 2 can happen even when the training set is imbalanced.
In addition, we can prove that with the existence of N C 1 and N C 2 , retrain the classifier with class-balance sampling, the classifier can become parallel with the centered feature mean, indicating the self-duality (N C 3 ). Ultimately, the symmetric structure of the regularized class means brings about an unbiased linear classifier. Proposition 4. Proposition 1+N C 1 +N C 2 +class-balanced sampling can lead to N C 3 .
Proof. With class-balanced sampling, the training label distribution can be regarded as balanced, andṀ =M . Then the optimal re-trained classifier W r is
W r = n K Σ † TṀ ,(19)
with the existence of N C 1 , we have Σ T =ṀṀ T . Thus,
W r = n K (ṀṀ T ) †Ṁ = n K (ṀṀ T ) †ṀṀ T (Ṁ T ) † = n K (Ṁ T ) † ,
with N C 2 which implies thatṀ form a simplex-ETF, thus, (Ṁ T ) † = cṀ for some constant c (Papyan et al., 2020), then we can obtain W r = αṀ , demonstrating the asserted self-duality (N C 3 ).
In conclusion, with the proposed L W and L B , we can obtain compact within-class and distinct between-class representations under imbalanced-class distribution. In line with linear discriminant analysis (LDA) (Fisher, 1936), this provides an optimal solution for the linear classifier.
EXPERIMENTS
Classification and long-tailed recognition
In this section, we conduct various experiments on image classification tasks on both balanced and long-tailed datasets to validate the effectiveness of our method. We denote our approach as NC, indicating the occurrence of the N C phenomena. By default, CE is adopted as L sup .
Experiment Setup
Datasets. Two balanced datasets (CIFAR10 and CI-FAR100) and three long-tailed datasets (CIFAR10-LT, CIFAR100-LT, and ImageNet-LT) are used in our experiments. Following Cao et al. (2019), CIFAR10/100-LT are created by downsampling each class's samples to obey an exponential decay with an imbalance ratio r = 100 and 10. Here r = max{n k }/ min{n k }. ImageNet-LT (Liu et al., 2019), including 115,846 samples and 1,000 categories with size ranging from 5 to 1,280, is generated from the ImageNet-2012 (Deng et al., 2009) dataset using a Pareto distribution with the power value α = 6.
Baselines. In addition to the typical approaches for addressing imbalanced data, such as re-sampling (RS) and re-weighting (RW) in inverse proportion to the class size, the investigation of more conducive methods that decouple representation learning and classifier training, as well as relevant methods inspired by N C, are also carried out.
To be specific, we compare traditional supervised learning methods with DRW (Cao et al., 2019), LWS (Kang et al., 2019), and cRT (Kang et al., 2019), and two recent works, namely BBN (Zhou et al., 2020) and MiSLAS . Our comparison also includes supervised contrastive learning approaches, namely FCL (Kang et al., 2020), KCL (Kang et al., 2020), and TSC . In addition, the comparison involves N C-inspired methods such as ETF classifier+DR and ARB-Loss (Xie et al., 2023).
Implementation details. We mainly follow the common training protocol. In all experiments, we adopt SGD optimizer with the momentum of 0.9, weight decay of 0.005, and train the model for 200 epochs following Alshammari et al. (2022). We utilize mix-up (Zhang et al., 2018) during the representation learning stage for all datasets. For CIFAR10/100(-LT), we use ResNet-32 (He et al., 2016) as the backbone and a multi-step schedule that decays the learning rate as its 0.1 at the 160-th and 180-th epochs with initialization of 0.1. We use 4 GeForce GTX 2080Ti GPUs with a batch size of 128. For ImageNet-LT, we use ResNeXt-50 (Xie et al., 2017) as the backbone and cosine schedule that gradually decays the learning rate from 0.05 to 0. We use 4 Tesla V100 GPUs to train the models with a batch size of 256. We also adopt Randaugment (Cubuk et al., 2020) for ImageNet-LT. We report the average results of three independent trials with different random seeds. Our code is available at https://github.com/Pepper-lll/NCfeature.
The hyperparameters λ 1 and λ 2 need to be adjusted according to the complexity of the datasets. In general, simple datasets with few categories require a small magnitude of feature regularization, while for complex datasets with plenty of categories, we need larger λ 1 and λ 2 . Besides, similar to Li et al. (2022), we also find that it is better to regularize the feature learning from half of the training process for large-scale datasets, i.e., CIFAR100 and ImageNet-LT. Our hyperparameter settings and the epoch number to start feature regularization are summarized in Table 1.
The class centers {µ k } K k=1 are updated in each mini-batch, instead of in the entire training set, which has been proved not efficient in large-scale datasets (Wen et al., 2016). Besides, our regularization terms are better to combine with re-balancing strategies to ensure the matching between classifier weights and class centers. The combination can lead to a remarkable improvement. In our experiments, we choose DRW and cRT as the re-balancing strategies.
Results
Balanced data. As we mentioned before, our method is applicable to both balanced and imbalanced datasets. First, we conduct experiments to validate our model on balanced CIFAR10 and CIFAR100 datasets. Table 2 shows that our method can reduce the generalization error with both CE and MSE loss.
Imbalanced data. Table 3 and 4 present our results on CIFAR10-LT, CIFAR100-LT, and ImageNet-LT. We can find that our method surpasses existing methods on all three datasets. For ImageNet-LT, we further test the accuracy on three groups of classes according to the sample size, in-cluding Many-shot (>100 samples), Medium-shot (20∼100 samples), and Few-shot (<20 samples). The results show that our method can substantially improve the accuracy of the Medium-and Few-shot categories with almost no impact on the accuracy of the Many-shot categories compared to the plain training with CE. Combine with existing approaches. Our regularization terms can be easily plugged into most of the existing algorithms. To validate the effectiveness, in Table 5, we add the proposed regularization terms to three different types of algorithms. We follow their original experiment settings to compare the performance differences before and after adding regularization terms. The results show that our regularization terms can increase the accuracy in all three algorithms.
Discussions
In this section, to verify the correctness and further explore the properties of our method, we show the learned representations, performance robustness, and ablation study on various combinations of loss and regularizations.
Representation Analysis
We extensively analyze the representations learned with our method to explain the advantages relative to the baseline. As for the corresponding analysis of classifiers, we obtained consistent findings with previous studies (Kang et al., 2019) and therefore do not repeat them here.
Maximally separated class centers. We compare the pair-wise angles of the centered class means learned on CIFAR10-LT with vanilla training, re-sampling (RS), reweighting (RW), and the proposed regularization terms in Figure 2. We arrange the class indexes in descending order based on their sizes. Under a long-tailed distribution, the minority class centers move closer to the majority with plain model. In Figure 2(a), the angles between class 8 and 0, class 9 and 1, and class 5 and 3 are around 50 • which is far lower than the optimal angle of 96 • . RS and RW can assist in the acquisition of more distinguishable features, as demonstrated Figure 2(b) and 2(c)). However, with our regularization terms (Figure 2(d)), we can observe that the pair-wise angles between all the class centers remain consistently close to the optimum value. In addition, the significant improvement on the experimental results indicates that the features learned by our method are more generalizable.
Zero-centered class means with the equal norm. Although neither L W nor L B forces the class center to be of equal norm, we can observe it in our experiments, as shown in Figure 3. This result strongly indicates that we can successfully induce N C in imbalanced data.
Robustness
We test the robustness of our method against random noise with different neural networks on CIFAR10/100 and their long-tailed version where the imbalance ratio r = 100.
Here Resnet-32 and ResNet-18 are employed. ResNet-18 is a wider network with the last-layer feature dimension of 512, while ResNet-32 is 64. The models are all trained with DRW. The results are reported in Table 6. We can observe that our regularization terms can improve the robustness for different model capacities.
Ablation Study
We conduct experiments to examine the effectiveness of two regularization terms separately over CE loss and the comparison with center loss. The results, presented in Table 7, demonstrate that each term can significantly improve accuracy individually, and that their combination produces the best results. Meanwhile, L W consistently produces better results than center loss, suggesting that modifying the coefficient is crucial. We can also find that N C 2 property is more useful, implying the importance of sufficiently distant class centers for the long-tail recognition task.
RELATED WORK
Long-tailed recognition
Long-tailed distribution is ubiquitous in the real world, which brings big challenges for most deep learning models. Classical methods dealing with this problem include data re-sampling and loss re-weighting. The former refers to re-sampling the instances to achieve relatively balanced training data, basically including over-sampling (Ando & Huang, 2017;Shelke et al., 2017), under-sampling (Shelke et al., 2017), and class-balanced sampling (Cui et al., 2019). Instead of changing the original data distribution, loss reweighting uses cost-sensitive re-weighting strategies and assigns different weights to instances from different classes according to the sample sizes (Lin et al., 2017;Cui et al., 2019). However, although the re-sampling and re-weighting approaches can improve the performance of minority classes, they may lead to overfitting and hurt the representation learning (Kang et al., 2019).
Recent works also focus on representation learning under long-tailed data distribution. This stream of study mainly Figure 2: Pair-wise angle degree between centered class means trained on CIFAR10-LT. Note that the optimal pair-wise angle for 10 classes is arccos −1 10−1 ≈ 96.4 • . follows a two-stage training scheme that decouples the representation and classifier learning (Kang et al., 2019;Zhong et al., 2021;Li et al., 2022;Kang et al., 2020;Zhu et al., 2022). Kang et al. (2019) observed that a high-quality representation requires fully utilizing the training instances equally, while a re-balancing technique is crucial for an unbiased classifier. On the other hand, some works take advantage of the superior representation learning ability of contrastive loss to extract the feature for deep long-tailed learning; then train a classifier upon the feature extractor with cost-sensitive loss or class-balanced sampling (Kang et al., 2020;Li et al., 2022;Zhu et al., 2022). Supervised contrastive learning shows superiority in representation learning under imbalanced distribution and achieves SOTA for longtailed recognition tasks Zhu et al., 2022;Cui et al., 2022). However, these methods usually converge slowly and require complex network structures compared to traditional supervised learning.
Researchers also explored methods based on the ensemble. They usually utilize multiple models over different data distributions or perform representation learning and classifier training with separate branches (Zhou et al., 2020;Zhu et al., 2022). This kind of approach is generally considered to be orthogonal to the single-model approach described above.
Neural collapse
A recent study (Papyan et al., 2020) discovered the phenomenon named Neural Collapse (N C), stating that the last-layer embedding and classifiers will converge to a sym-metric geometry named simplex Equiangular Tight Frame (ETF) for deep classifiers trained on balanced data. A more precise description of the N C phenomena is delivered in Section 2.1. Subsequent studies indicate that N C will eventually occur, independent of the loss function, the optimizer, batch-normalization, and regularization, as long as the training data exhibits a balanced distribution (Zhu et al., 2021;Han et al., 2022;Kothapalli et al., 2022). Meanwhile, the intrinsic merit of N C has also been revealed, including ensuring global optimality, stronger generalization and robustness, and transferability (Papyan et al., 2020;Zhu et al., 2021;Galanti et al., 2021).
The investigation of N C has also been carried over to the imbalanced data case, where different phenomena are uncovered. Fang et al. (2021) demonstrated that the minority classifiers have smaller pair-wise angles than the majority ones and will even merge together as the imbalance level increases, named Minority Collapse. This phenomenon provides some reason of the performance drop. Thrampoulidis et al. (2022) provides a general frame that is equivalent to ETF for balanced data, and reveals an asymmetric geometry of the last-layer feature and classifiers for imbalanced distribution. Furthermore, the perfect alignment between the class feature means and classifiers vanished under the imbalanced distribution. However, Thrampoulidis et al. (2022) illustrates the general geometry with a special encoding framework and does not discuss whether this geometry with an imbalanced dataset has merit or defect.
Inspired by the N C phenomenon, some researchers have attempted to improve the model's classification ability encountering imbalanced distribution by eliminating Minority Collapse, including fixing the classifier as an ETF and adjusting the CE loss (Xie et al., 2023). Distinct from these works, our work analyzes that obtaining high-quality features is the key to the improvement and thus proposes regularization to guide learning representations.
CONCLUSIONS
In this paper, we argue that the existence of N C is crucial for long-tailed recognition and propose two simple but effective regularization terms to induce the appearance of N C. We empirically show that under the imbalanced data distribution, the class centers of minority classes are close to the majority ones, leading to the overlap among different classes over the feature space and confusion of the classifier. With our method, the deep classification models are able to learn compact within-class and maximally distinct between-class features. Extensive experiments confirm that our method can enhance the generalization power of the deep classification model, especially when the training set is imbalanced. Our method is more efficient than contrastive loss based methods, and we set new state-of-the-art performance for single model based methods on widely used benchmarks.
Our proposed regularization guides the representation learning to be of 'optimal' geometry for classification, which is particularly beneficial for training sets with imbalanced labels. However, the learned geometry is validated empirically and lacks complete theoretical guarantees, leading to manually tuning the related hyperparameters. In the future, we plan to formally analyze the geometry obtained with our regularization and provide some theoretical justification for the choice of hyperparameters.
Proceedings of the 26 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2023, Valencia, Spain. PMLR: Volume 206. Copyright 2023 by the author(s).
Figure 1 :
1Illustration of geometry configuration of the zerocentered class means and classifier weights under (a) balanced dataset, (b) imbalanced dataset, and (c) imbalanced dataset with our method. The arrows represent classifier weights and the stars are the class centers. The size of the circle around the star reflects the variance of the feature from the same class. In (b) and (c), red and blue represent the majority and minority classes, respectively. Note that under imbalanced label distribution, both the centered class means and classifier weights form an asymmetric structure and are no longer parallel.
1
dH(D Z S , D Z T ) denotes the H-divergence between D Z S and D Z T , a precise definition is provided in Ben-David et al. (2010).
Figure 3 :
3The norm of centered class means on a balanced dataset, long-tailed dataset w/ and w/o inducing NC is represented in different colors. Note that the class index is inversely sorted by the sample size.
Table 1 :
1Hyperparameter setting.Dateset
λ1
λ2 start epoch
Table 2 :
2Top-1 test accuracy (%) on the balanced datasets.Method CIFAR10 CIFAR100
CE
93.4
71.8
+NC
93.3
72.1
MSE
91.1
70.7
+NC
91.7
71.9
Table 3 :
3Top-1 test accuracy (%) on CIFAR10-LT and
CIFAR100-LT. The results of the compared methods are
obtained from their respective original papers. The best and
second-best results are marked in bold and underlined.
Method
CIFAR10-LT CIFAR100-LT
imbalance ratio
100
10
100
10
CE
70.4
86.4
38.4
55.7
CE-RS
72.8
87.8
36.7
57.7
CE-RW
74.4
87.9
32.5
58.2
CE-DRW
75.1
86.4
42.5
56.2
LDAM-DRW
77.0
88.2
43.5
58.7
BBNm
79.9
88.4
42.6
59.2
MiSLAS
82.1
90.0
47.0
63.2
KCL
77.6
88.0
42.8
57.6
TSC
79.7
88.7
43.8
59.0
ETF classifier+ DR 76.5
87.7
45.3
-
ARB-Loss
83.3
90.2
47.2
62.1
NC-DRW
81.9
89.8
48.6
63.1
NC-DRW-cRT
82.6
90.2
48.7
63.6
Table 4 :
4Top-1 test accuracy (%) on ImageNet-LT.Methods
Many Medium Few
All
CE
68.2
38.1
5.82 45.3
CE-RS
64.6
42.6
17.8 47.8
CE-RW
52.0
41.4
19.8 42.5
CE-DRW
52.6
45.7
31.5 46.4
CE-cRT
58.8
33.0
26.1 47.3
CE-LWS
57.1
45.2
29.3 47.7
MiSLAS
61.7
51.3
35.8 52.7
FCL
61.4
47.0
28.2 49.8
KCL
62.4
49.0
29.5 51.5
TSC
63.5
49.7
30.4 52.4
ETF classifier+ DR
-
-
-
44.7
ARB-Loss
60.2
51.8
38.3 52.8
NC-DRW
67.1
49.7
29.0 53.6
NC-DRW-cRT
65.6
51.2
35.4 54.2
Table 5 :
5Top-1 test accuracy (%) on real-world long-tail
datasets of our methods combined with others. Note that we
replicated experiments of RIDE with data distributed paral-
lel training and got results with slight differences from Wang
et al. (2021). c10, c100 and iNet are short for CIFAR10,
CIFAR100 and ImageNet respectively.
Method
c10-LT
c100-LT
iNet-LT
LDAM-DRW
77.0
42.0
48.8
+NC
77.1(0.1 ↑ ) 43.2(1.2 ↑ ) 49.5(0.7 ↑ )
Logit Adjust
77.4
43.9
51.1
+NC
78.8(1.4 ↑ ) 44.6(0.7 ↑ ) 53.2(2.1 ↑ )
RIDE (2 experts)
-
46.5
51.9
RIDE (3 experts)
-
47.5
54.2
RIDE (4 experts)
-
48.8
55.2
+NC (2 experts)
-
46.8(0.3 ↑ ) 52.2(0.3 ↑ )
+NC (3 experts)
-
48.1(0.6 ↑ ) 54.8(0.6 ↑ )
+NC (4 experts)
-
49.1(0.3 ↑ ) 56.0(0.8 ↑ )
Table 6 :
6Random Noise Robustness Results. CE ‡ denotes Cross-Entropy loss with the feature regularization L W + L B . CE ‡ 93.1 89.6 76.6 57.5 39.0 95.1 91.8 78.9 57.1 37.4 CE ‡ 72.3 59.0 39.7 23.9 15.1 78.6 67.8 46.9 28.2 16.6Gaussian noise std
0.00 0.10 0.20 0.30 0.40 0.00 0.10 0.20 0.30 0.40
Dataset
Loss
ResNet-32
ResNet-18
CIFAR10
CE
93.3 89.4 76.6 54.7 35.7 94.9 91.5 79.2 56.9 35.9
CIFAR10-LT
CE
77.0 75.0 62.3 45.3 32.5 79.2 75.7 63.5 47.2 33.8
CE ‡ 79.2 75.5 63.2 47.8 35.5 81.0 77.5 66.5 51.4 37.9
CIFAR100
CE
71.8 60.2 40.0 23.9 14.1 78.2 65.9 44.2 25.0 14.3
CIFAR100-LT
CE
42.5 37.2 25.4 16.3 10.2 46.8 41.1 31.6 23.6 17.6
CE ‡ 45.7 39.0 27.2 16.8 11.1 47.2 41.6 31.6 23.3 16.3
(a) vanilla
(b) w/ RS
(c) w/ RW
(d) w/ LW and LB
Table 7 :
7Ablation studies on the effectiveness of each regularization term on CIFAR10/100-LT. Note that we apply DRW for all experiments here.Method
CIFAR10-LT CIFAR100-LT
imbalance ratio
100
10
100
10
CE
75.1
86.4
42.4
56.2
+Centor Loss
78.7
89.1
46.3
61.2
+LW
79.1
88.1
46.9
61.3
+LB
80.1
88.6
47.6
61.7
+Centor Loss &LB 77.5
89.2
46.5
61.4
+LW &LB
81.9
89.8
48.6
63.1
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| [
"https://github.com/Pepper-lll/NCfeature."
] |
[
"Separating the blue cloud and the red sequence using Otsu's method for image segmentation",
"Separating the blue cloud and the red sequence using Otsu's method for image segmentation"
] | [
"Biswajit Pandey \nDepartment of Physics\nVisva-Bharati University\n731235Santiniketan, BirbhumIndia\n"
] | [
"Department of Physics\nVisva-Bharati University\n731235Santiniketan, BirbhumIndia"
] | [
"MNRAS"
] | The observed colour bimodality allows a classification of the galaxies into two distinct classes: the 'blue cloud' and the 'red sequence'. Such classification is often carried out using empirical cuts in colour and other galaxy properties that lack solid mathematical justifications. We propose a method for separating the galaxies in the 'blue cloud' and the 'red sequence' using Otsu's thresholding technique for image segmentation. We show that this technique provides a robust and parameter-free method for the classification of the red and blue galaxies based on the minimization of the interclass variance and maximization of the intra-class variance. We also apply an iterative triclass thresholding technique based on Otsu's method to improve the classification. The same method can also be applied to classify the galaxies based on their physical properties, such as star formation rate, stellar mass function, bulge-to-disk mass ratio and age, all of which have bimodal distributions. | 10.1016/j.ascom.2023.100725 | [
"https://export.arxiv.org/pdf/2211.15642v1.pdf"
] | 254,043,769 | 2211.15642 | 8818c6edd9bb98c5447154de22b43e649c2c0564 |
Separating the blue cloud and the red sequence using Otsu's method for image segmentation
2022
Biswajit Pandey
Department of Physics
Visva-Bharati University
731235Santiniketan, BirbhumIndia
Separating the blue cloud and the red sequence using Otsu's method for image segmentation
MNRAS
0002022Preprint 29 November 2022 Compiled using MNRAS L A T E X style file v3.0methods: statistical -data analysis -galaxies: formation -evolution - cosmology: large scale structure of the Universe
The observed colour bimodality allows a classification of the galaxies into two distinct classes: the 'blue cloud' and the 'red sequence'. Such classification is often carried out using empirical cuts in colour and other galaxy properties that lack solid mathematical justifications. We propose a method for separating the galaxies in the 'blue cloud' and the 'red sequence' using Otsu's thresholding technique for image segmentation. We show that this technique provides a robust and parameter-free method for the classification of the red and blue galaxies based on the minimization of the interclass variance and maximization of the intra-class variance. We also apply an iterative triclass thresholding technique based on Otsu's method to improve the classification. The same method can also be applied to classify the galaxies based on their physical properties, such as star formation rate, stellar mass function, bulge-to-disk mass ratio and age, all of which have bimodal distributions.
INTRODUCTION
The galaxies are the fundamental unit of the large-scale structures of the Universe. They are a collection of stars, interstellar gas, dust and dark matter bound together by gravity. The photometric and spectroscopic observations from different galaxy surveys reveal a wide variation in the physical properties of galaxies. The galaxies come in various shapes, sizes, luminosity, mass, colour, star formation rate and metallicity. Classifying the galaxies based on these physical properties helps us to understand their formation and evolution.
The colour of a galaxy is defined as the ratio of fluxes in two different filters. It is considered one of the fundamental properties of a galaxy that characterizes its stellar population. It is now well known that the observed distribution of galaxy colour is strongly bimodal (Strateva, et al. 2001;Blanton, et al. 2003;Bell, et al. 2003;Balogh, et al. 2004;). The observed colour distribution reveals two distinct peaks corresponding to a 'blue cloud' and a 'red sequence'. The 'blue cloud' predominantly hosts the active star-forming galaxies with younger stellar populations, lower stellar mass and disk-like morphology (Strateva, et al. 2001;Blanton, et al. 2003;Kauffmann et al. 2003;). On the other hand, the galaxies in the 'red sequence' E-mail: [email protected] have higher stellar mass with an older stellar population and bulge-dominated morphology.
The bimodal character of the galaxy colour distribution has important implications for galaxy formation and evolution. The colour bimodality is known to exist out to z = 1 − 2 (Bell, et al. 2004;Brammer et al. 2009). Observations indicate that the number of massive red galaxies has increased steadily since z ∼ 1 (Bell, et al. 2004;Faber et al. 2007). It indicates that the blue galaxies transform into red ones via the quenching of star formation. Such quenching may happen through different physical processes and mechanisms. A sharp decline in star formation rate between z = 1 to present (Madau et al. 1996) also hints towards a significant evolution of the galaxy properties. Such evolution may have played a decisive role in shaping the observed colour bimodality in the present Universe. The colour bimodality also depends on luminosity, stellar mass and environment (Balogh, et al. 2004;Baldry, et al. 2006;Pandey & Sarkar 2020). Any successful model of galaxy formation must be able to reproduce the observed colour bimodality. The semianalytic models of galaxy formation has been widely used in a number of works to explain the observed colour bimodality (Menci, et al. 2005;Driver, et al. 2006;Cattaneo, et al. 2006Cattaneo, et al. , 2007Cameron, et al. 2009;Trayford et al. 2016;Nelson, et al. 2018;Correa, Schaye & Trayford 2019).
The red and blue galaxies are known to have different two-point correlation function (Zehavi, et al. 2011), three-point correlation function (Kayo, et al. 2004), genus (Hoyle et al. 2002), filamentarity (Pandey & Bharadwaj 2006), local dimension (Pandey & Sarkar 2020) and mass function (Drory, et al. 2009;Taylor, et al. 2015). These measurements provide important inputs to the theories of galaxy formation and evolution. One requires an operational definition of the two classes of galaxies in all such studies. Separating the blue and red galaxies is not a trivial task, as no galaxies can be regarded as either truly 'blue' or 'red' based on their colours. The primary motivation for such a classification lies in the observed bimodality of the colour distribution. The galaxies in the 'blue cloud' and the 'red sequence' are usually separated using specific cuts based on empirical arguments. For instance, Strateva, et al. (2001) proposed that the red and blue galaxies can be optimally separated using a colour cut of (u − r) = 2.22. Pandey (2020) propose a fuzzy set theory-based method for classifying the red, blue and transition valley galaxies. Nevertheless, this method also has some arbitrariness in selecting the membership function and the associated parameters. Ideally, it would be most desirable to have a method that can automatically decide an optimal threshold for any given data set. One such method for statistical decision-making is Otsu's thresholding technique (Otsu 1979), originally proposed by Nobuyuki Otsu for image segmentation. It is a parameter-free method for separating the foreground pixels from the background. Over the years, it has found many important applications in remote sensing, robotic mapping and navigation and identifying tumors (Yuan et al. 2016;Prema et al. 2016;Sehgal et al. 2017).
In this Letter, we propose an automated method for separating the galaxies in the 'blue cloud' and the 'red sequence' for any given data set using Otsu's algorithm (Otsu 1979) for image segmentation. We also implement an improved iterative triclass thresholding technique (Cai et al. 2014) based on Otsu's method to classify the red and blue galaxies. It provides a nearly parameter-free method for classifying the red and blue galaxies.
The Letter is organized as follows: We describe the SDSS data in Section 2, explain the method in Section 3 and present the results and conclusions in Section 4.
SDSS DATA
We apply the proposed method to the data from the Sloan Digital Sky Survey (SDSS) (York et al. 2000) which is the largest galaxy survey to date. It has collected the photometric and spectroscopic information of more than one million galaxies and quasars in five wave bands across one-quarter of the entire sky. We use the data from the SDSS DR16 (Ahumada et al. 2020) for the current work. We download the data from the SDSS SkyServer 1 using SQL. We identify a contiguous region between the right ascension 135 • ≤ α ≤ 225 • and the declination 0 • ≤ δ ≤ 60 • and extract the spectroscopic information of all the galaxies with r-band Petrosian magnitude 13.5 ≤ r p < 17.77 within redshift z < 0.3. These cuts provide us with a total 376495 galaxies. We then prepare a volume limited sample by applying a cut to the K-corrected 1 https://skyserver.sdss.org/casjobs/ and extinction corrected r-band absolute magnitude. We apply an absolute magnitude cut of −21 ≥ M r ≥ −23 that corresponds redshift limit 0.041 ≤ z ≤ 0.120. We finally have a total 103984 galaxies in our volume limited sample. A ΛCDM cosmological model with Ω m0 = 0.315, Ω Λ0 = 0.685 and h = 0.674 (Planck Collaboration, et al. 2018) is used for our analysis.
METHOD OF ANALYSIS
3.1 Implementing Otsu's thresholding technique to separate the galaxies in the blue cloud and the red sequence
Otsu's thresholding technique (Otsu 1979) was originally proposed to separate the foreground pixels from the background pixels in a grayscale image based on their intensities. The pixels with intensities greater than a threshold value are marked as foreground, whereas all pixels with intensity less than or equal to the threshold are labelled as background. It converts the input grayscale image into a binary image. The method iterates through all the possible threshold values and measures the spread of the background and foreground pixels for each threshold. The method aims to find the threshold value for which the 'intra-class variance' of the two separate pixel populations is minimum. Interestingly, this maximises the 'inter-class variance' of the foreground and background pixels. An optimal threshold that ensures either of these will provide the best separation of the two classes. The method is most efficient when the histogram of the pixel intensities shows a bimodal nature. The (u − r) colour distribution of the galaxies in the SDSS is strongly bimodal (Strateva, et al. 2001;Balogh, et al. 2004; (Figure 1). It motivates us to use Otsu's thresholding technique for optimally separating the galaxies in the blue cloud and the red sequence.
We adopt Otsu's method for this work and outline the primary steps involved in this classification. We consider all the galaxies in our volume limited sample and calculate the histogram of the (u − r) colour using a specific number of bins. We normalize the histogram of the (u − r) colour by the total number of galaxies N = M i=1 n i where n i is the number of galaxies in the i th colour bin and M is the total number of bins used in the analysis. This will ensure M i=1 p i = 1 where p i = n i N . The resulting probability distribution can be used to calculate the probabilities of class occurrence for the blue cloud and the red sequence. Let us denote the blue cloud and the reds sequence with B and R respectively. If the threshold corresponds to the k th bin then the bins [1, ...., k] belongs to the class B and the class R is represented by the bins [k + 1, ...., M]. The probabilities of the class occurrences for the class B and R can be respectively written as,
P B = k i=1 p i = w(k)(1)
and
P R = M i=k+1 p i = 1 − w(k).(2)
We can use these to calculate the class means for each threshold. They are respectively given by,
µ B = k i=1 c i p i P B = µ k w(k)(3)
and
µ R = M i=k+1 c i p i P R = µ T − µ k 1 − w(k)(4)
where, c i is the (u − r) colour corresponding to the i th bin, µ k = k i=1 c i p i is the mean upto the k th bin and µ T = M i=1 c i p i is the mean of the entire distribution. It may be noted that P B + P R = 1 and µ T = P B µ B + P R µ R for each and every threshold.
Similarly, we can also estimate the class variances as,
σ 2 B = k i=1 (c i − µ B ) 2 p i P B(5)
and
σ 2 R = M i=k+1 (c i − µ R ) 2 p i P R(6)
We can define the classification threshold in two different ways: (i) by minimizing the within-class variance or intra-class variance σ 2 wc or (ii) by maximizing the betweenclass variance or the inter-class variance σ 2 bc . The within-class variance σ 2 wc and the between-class variance σ 2 bc can be respectively written as,
σ 2 wc = P B σ 2 B + P R σ 2 R(7)
and
σ 2 bc = P B P R (µ B − µ R ) 2(8)
The total variance σ 2 T is the sum of the intra-class and inter-class variances,
σ 2 T = σ 2 wc + σ 2 bc(9)
It may be noted that both σ 2 wc and σ 2 bc depend on the chosen threshold, but σ 2 T is independent of the threshold. Otsu's algorithm iteratively searches for the threshold that minimizes the intra-class variance or maximizes the interclass variance. Fortunately, the threshold that minimizes σ 2 wc also maximizes σ 2 bc . So one can choose either Equation 7 or Equation 8 to determine the optimal threshold for the classification. σ 2 wc involves the second-order statistics whereas σ 2 bc is only based on the first-order statics. Consequently, it is easier to calculate the desired threshold using σ 2 bc . In this work, we use both σ 2 wc and σ 2 bc to classify the galaxies in the blue cloud and the red sequence.
Iterative triclass thresholding technique: an improved version of Otsu's method
The primary limitation of Otsu's method is that the class with the larger variance has a greater influence in determining the classification threshold. It may provide suboptimal results when one of the classes has a considerably larger variance. an iterative triclass thresholding technique proposed by Cai et al. (2014).
In this method, we first determine the threshold using the standard Otsu's algorithm. We then separate the galaxies into three classes based on the mean of the two classes. We define the galaxies in the blue cloud as those with an (u − r) colour less than the smaller mean. The galaxies in the red sequence are defined as those having their (u − r) colour greater than the larger mean. The intervening region between the two means are defined as the "to-be-determined." (TBD) class. In the next iteration, the regions already classified as blue cloud and red sequence are kept unchanged. The standard Otsu's method is again applied only to the TBD region to divide it into three classes similarly. We get a new threshold after each iteration. The iteration procedure stops when the difference between two consecutive thresholds is less than a preset value. The TBD region is divided into two classes instead of three at the last iteration. Finally, the classified blue cloud is the logical union of all the regions that are previously identified as the blue cloud throughout the different iterations. The final region consisting of the red sequence is determined identically.
RESULTS AND CONCLUSIONS
We show the probability distribution function of the (u − r) colour for the SDSS galaxies in Figure 1. The (u − r) colour distribution clearly shows a bimodal nature that motivates us to use the Otsu's thresholding technique to classify the two populations associated with the two peaks of the distribution.
We show the probabilities of the class occurrences Figure 2. We find that the within-class variance σ 2 wc has its minimum at a (u − r) colour threshold of 2.22. We also note that the between-class variance σ 2 bc has its maximum at (u − r) = 2.22. Thus σ 2 wc is minimized and σ 2 bc is maximized at exactly the same (u − r) colour threshold.
It may be noted that the results shown in Figure 1 and Figure 2 correspond to a choice of the number of bins n = 50. We also test if the locations of the minimum of σ 2 wc and the maximum of σ 2 bc are sensitive to the choice of the number of bins. We repeat our analysis for n = 100, 500 and 1000 and show σ 2 wc and σ 2 bc as a function of the (u − r) colour threshold for four different choices of the number of bins in the left and right panels of Figure 3. Interestingly, the locations of the minimum of σ 2 wc and the maximum of σ 2 bc are independent of the choice of the number of bins.
We propose Otsu's technique as a parameter-free method for the automated classification of the galaxies in the blue cloud and the red sequence. It is interesting to note that the (u − r) colour threshold predicted by Otsu's method in this work matches with the (u − r) colour separator proposed by Strateva, et al. (2001). Strateva, et al. (2001) proposed this as an empirical cut based on the best fit colour-magnitude relations. We also applied the iterative triclass thresholding technique based on Otsu's method to improve the classification. The threshold values obtained after each iteration in the iterative triclass thresholding scheme are tabulated in Table 1. The iteration is continued until the absolute difference between two consecutive thresholds (|∆|) is smaller than a preset value. We choose |∆| = 10 −3 for the present analysis. It leads to the convergence to an (u − r) colour threshold of 2.301 after six iterations. We note that the modified version of Otsu's method is a nearly parameterfree method that can be used effectively for separating red and blue galaxies.
The distributions of several other galaxy properties, such as the star formation rate, the stellar mass function, the bulge to disk mass ratio and the age (Elbaz et al. 2007;Drory, et al. 2009;Driver, et al. 2006;Zibetti et al. 2017) also exhibit a bimodal nature. Otsu's method can be also applied to classify the galaxies based on each of these galaxy properties. We conclude that Otsu's thresholding technique provides us with a robust and parameter-free method for classifying the galaxies based on the bimodal distributions of certain galaxy properties.
ACKNOWLEDGEMENT
The author would like to acknowledge the financial support from the SERB, DST, Government of India through the project CRG/2019/001110. The author also thanks Suman Sarkar for the help with the SDSS data.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
DATA AVAILABILITY
The data underlying this article are available in https://skyserver.sdss.org/casjobs/.
The datasets were derived from sources in the public domain: https://www.sdss.org/
Figure 1 .
1This shows the (u − r) colour distribution of the SDSS galaxies in our volume limited sample.
Figure 2 .
2The top left panel shows P B and P R and the top right panel shows µ B and µ R as a function of the (u − r) colour threshold. The variances σ 2 B , σ 2 R and σ 2 wc , σ 2 bc are shown as a function of the (u − r) colour threshold in the bottom left and right panels respectively.
Figure 3 .
3The left panel and the right panel of this figure respectively shows the within-class variance σ 2 wc and the between-class variance σ 2 bc as a function of the (u − r) colour threshold for different choices of the number of bins.
(
Equation 1 and Equation 2) as a function of the (u − r) colour threshold for the blue cloud and the red sequence in the top left panel of Figure 2. The associated means and variances (Equation 3, Equation 4, Equation 5 and Equation 6) of the two populations as a function of the (u − r) colour threshold are respectively shown in the top right and bottom left panels of Figure 2. We compute the within-class variance σ 2 wc and the between-class variance σ 2 bc of the two populations using Equation 7 and Equation 8. The results are shown in the bottom right panel of
Several improvements of the standard Otsu's method have been proposed in the literature. Here, we have chosenIteration
µ B
µ R
threshold |∆| threshold
1
1.861672 2.681102
2.22
2
2.074795 2.508090
2.279581
0.059581
3
2.188867 2.412384
2.295775
0.016194
4
2.244947 2.358102
2.298391
0.002615
5
2.273358 2.330187
2.300393
0.002002
6
2.287296 2.316201
2.301204
0.000812
Table 1. This table shows the smaller mean (µ B ), the larger mean
(µ R ), the (u − r) threshold and the absolute difference between two
consecutive thresholds (|∆|) at each iterations of the iterative tri-
class thresholding method. Here we set a critical |∆| = 0.001 and
choose the number of bins to be 50.
MNRAS 000, 1-6 (2022)
This paper has been typeset from a T E X/L A T E X file prepared by the author.
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"Scale-Semantic Joint Decoupling Network for Image-text Retrieval in Remote Sensing",
"Scale-Semantic Joint Decoupling Network for Image-text Retrieval in Remote Sensing"
] | [
"Chengyu Zheng \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Ning Song \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Ruoyu Zhang \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Lei Huang \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Zhiqiang Wei \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Jie Nie \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Chengyu Zheng \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Ning Song \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Ruoyu Zhang \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Lei Huang \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Zhiqiang Wei \nCollege of Information Science and Engineering\nOcean University of China\nChina\n",
"Jie Nie \nCollege of Information Science and Engineering\nOcean University of China\nChina\n"
] | [
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina",
"College of Information Science and Engineering\nOcean University of China\nChina"
] | [
"J. ACM"
] | Image-text retrieval in remote sensing aims to provide flexible information for data analysis and application. In recent years, state-of-the-art methods are dedicated to "scale decoupling" and "semantic decoupling" strategies to further enhance the capability of representation. However, these previous approaches focus on either the disentangling scale or semantics but ignore merging these two ideas in a union model, which extremely limits the performance of cross-modal retrieval models. To address these issues, we propose a novel Scale-Semantic Joint Decoupling Network (SSJDN) for remote sensing image-text retrieval. Specifically, we design the Bidirectional Scale Decoupling (BSD) module, which exploits Salience Feature Extraction (SFE) and Salience-Guided Suppression (SGS) units to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional pattern to yield different scale clues. Besides, we design the Label-supervised Semantic Decoupling (LSD) module by leveraging the category semantic labels as prior knowledge to supervise images and texts probing significant semantic-related information. Finally, we design a Semantic-guided Triple Loss (STL), which adaptively generates a constant to adjust the loss function to improve the probability of matching the same semantic image and text and shorten the convergence time of the retrieval model. Our proposed SSJDN outperforms state-of-the-art approaches in numerical experiments conducted on four benchmark remote sensing datasets. | 10.1145/3603628 | [
"https://export.arxiv.org/pdf/2212.05752v1.pdf"
] | 254,564,360 | 2212.05752 | 03f294544be117add5b1b7bfe8d804cb3fb9ef85 |
Scale-Semantic Joint Decoupling Network for Image-text Retrieval in Remote Sensing
2022. August 2022
Chengyu Zheng
College of Information Science and Engineering
Ocean University of China
China
Ning Song
College of Information Science and Engineering
Ocean University of China
China
Ruoyu Zhang
College of Information Science and Engineering
Ocean University of China
China
Lei Huang
College of Information Science and Engineering
Ocean University of China
China
Zhiqiang Wei
College of Information Science and Engineering
Ocean University of China
China
Jie Nie
College of Information Science and Engineering
Ocean University of China
China
Chengyu Zheng
College of Information Science and Engineering
Ocean University of China
China
Ning Song
College of Information Science and Engineering
Ocean University of China
China
Ruoyu Zhang
College of Information Science and Engineering
Ocean University of China
China
Lei Huang
College of Information Science and Engineering
Ocean University of China
China
Zhiqiang Wei
College of Information Science and Engineering
Ocean University of China
China
Jie Nie
College of Information Science and Engineering
Ocean University of China
China
Scale-Semantic Joint Decoupling Network for Image-text Retrieval in Remote Sensing
J. ACM
37111182022. August 2022111CCS Concepts: • Computer systems organization → Embedded systemsRedundancyRobotics• Net- works → Network reliability Additional Key Words and Phrases: Remote sensing, scale-semantic joint decoupling, image-text retrieval ACM Reference Format:
Image-text retrieval in remote sensing aims to provide flexible information for data analysis and application. In recent years, state-of-the-art methods are dedicated to "scale decoupling" and "semantic decoupling" strategies to further enhance the capability of representation. However, these previous approaches focus on either the disentangling scale or semantics but ignore merging these two ideas in a union model, which extremely limits the performance of cross-modal retrieval models. To address these issues, we propose a novel Scale-Semantic Joint Decoupling Network (SSJDN) for remote sensing image-text retrieval. Specifically, we design the Bidirectional Scale Decoupling (BSD) module, which exploits Salience Feature Extraction (SFE) and Salience-Guided Suppression (SGS) units to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional pattern to yield different scale clues. Besides, we design the Label-supervised Semantic Decoupling (LSD) module by leveraging the category semantic labels as prior knowledge to supervise images and texts probing significant semantic-related information. Finally, we design a Semantic-guided Triple Loss (STL), which adaptively generates a constant to adjust the loss function to improve the probability of matching the same semantic image and text and shorten the convergence time of the retrieval model. Our proposed SSJDN outperforms state-of-the-art approaches in numerical experiments conducted on four benchmark remote sensing datasets.
INTRODUCTION
With the rapid development of satellite sensors, the quantity and quality of remote sensing (RS) data are increasing rapidly. To improve the utilization efficiency of remote sensing data, a large number of researches have been carried out on RS retrieval tasks [6,12,27]. Compare with RS unimodal retrieval, the RS multimodal retrieval has more significance because it can mine richer semantic information and capture a more comprehensive representation. Besides, image and text cross-modal retrieval plays an important role in multimodal retrieval. It has the capability to explore Fig. 1. Here, we illustrated the "soft scale decoupling" and it can be seen that after scale decoupling by CNN and atrous convolution, the large-scale object tennis court in the yellow box still contains the small-scale object tennis court in the green box. This situation will eventually lead to some severe troubles such as in the subsequent fusion process, the repetitive regions are continuously accumulated, thereby bringing a great amount of redundant noise and resulting inferior utilization of multi-scale contents.
the potential correlation between image and text data and realize the complementarity of the two modal data, which has attracted a great amount of attention from more and more researchers.
The RS image-text cross-modal retrieval framework mainly includes three stages, namely feature extraction, feature index, and feature similarity measuring. The goal of the feature extraction stage is to mine more abstract information in data by leveraging deep learning methods, such as that Napoletano et al. [17] exploited the powerful feature representation capability of convolutional neural networks (CNN) to extract high-dimensional feature representations from remote sensing images and texts. Recently, state-of-the-art methods are dedicated to "scale decoupling" and "semantic decoupling" strategies to further enhance the capability of representation. The "scale decoupling" means that objects of different scales are modeled separately, thus more comprehensively considering the different scales of various objects, while the "semantic decoupling" hold the capability of entangling the semantic-mixing features, which can not only remove background noise but also enhance the inner-semantic consistency and intra-semantic discriminability. Due to the complexity and low resolution of RS images, the "scale decoupling" is often used on images rather than text. For example, aiming at the multi-scale characteristics of RS targets, Cheng et al. [3] designed an Asymmetric Multimodal Feature Matching Network (AMFMN) to simultaneously mine small-scale and large-scale feature representations for RS images. With regard to "semantic decoupling", Lee et al. [10] designed the Stacked Cross Attention Network (SCAN), which first calculates a sentence/word representation for each region in the image based on similarity, and then compares it with the global sentence to achieve semantic decoupling of regions. The feature indexing stage is able to embed high-dimensional features into low-dimensional spaces to achieve efficient search and storage. Zou et al. [34] represented a rotation-invariant hash network to speed up the retrieval process based on the rotation invariance of RS targets. The feature similarity measuring determines the matching degree of the two data by computing the similarity between the query data and the database, and related studies [13,14] proposed various loss functions based on similarity measures to train end-to-end RS cross-modal retrieval networks.
Despite the significance and value of the "scale decoupling" and "semantic decoupling" methods in the feature extraction stage, these previous "semantic decoupling" approaches focus on either the disentangling scale or semantic but ignore merging these two ideas in a union model, which extremely limits the performance of cross-modal retrieval models. Besides, the mentioned "semantic decoupling" approach is only considered as a kind of "soft scale decoupling" where the large-scale targets still contain the small-scale targets after the decoupling operation. Here, we illustrated the "soft scale decoupling" in Figure 1 and it can be seen that after scale decoupling by CNN and atrous convolution, the large-scale object tennis court in the yellow box still contains the small-scale object tennis court in the green box. This situation will eventually lead to some severe troubles such as in the subsequent fusion process, the repetitive regions are continuously accumulated, thereby bringing a great amount of redundant noise and resulting inferior utilization of multi-scale contents.
To tackle these downsides, we propose an imploded framework, called Scale-Semantic Joint Decoupling Network (SSJDN) to perform both "scale decoupling" and "semantic decoupling" for RS image-text retrieval. The SSJDN follows the classical retrieval method, which first extracts the features of images and texts, and measures the similarity between the obtained features. Unlike preceding methods, in the image feature extraction stage, we propose a Bidirectional Scale Decoupling (BSD) module, which exploits Salience Feature Extraction (SFE) and Salience-Guided Suppression (SGS) units to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional pattern to yield different scale clues. Besides, we design the Labelsupervised Semantic Decoupling (LSD) module, which first leverages the category semantic labels as prior knowledge to generate category features, and then applies these category features to image features by multiple operations to decoupling semantics and probing significant semantic-related information. What's more, in the feature similarity measuring stage, we design a Semantic-guided Triple Loss (STL) by performing category matching on two modal features and outputting a constant to adjust the loss function, which can improve the probability of matching the same semantic image and text and shorten the convergence time of the retrieval model. Experiments conducted on three RS image-text retrieval datasets demonstrate the effectiveness of the proposed architecture.
The contributions of this paper are as follows:
• We propose an imploded framework SSJDN to perform both "scale decoupling" and "semantic decoupling" for RS image-text retrieval. By applying such a joint decoupling strategy, the feature representation capability can be crucially improved. • We propose a BSD module to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional pattern to exploit distinct clues. Besides, we design an LSD module, where the category semantic labels are leveraged to supervise the network decoupling image semantics and probing significant semantic-related information. Finally, we present an STL module by generating a constant to improve the probability of matching the same semantic image and text and shorten the convergence time of the retrieval model. • We validate the effectiveness of the proposed method on four public RS image-text retrieval datasets to demonstrate the superiority of our approach.
RELATED WORK
Image-text Retrieval of IS
The RS image-text retrieval is defined as, given a query image (or text), the retrieval network needs to find the corresponding labeled text (or image) from the database according to the measure of similarity. The automatic image caption is the earliest derived cross-modal retrieval method and has gradually developed into the mainstream of cross-modal retrieval [23,32,33]. Shi et al. [20] proposed an RS image captioning framework by leveraging fully convolutional networks to express image element attributes and interaction. To provide data assistance for the verification of the caption-based retrieval methods, Lu et al. [16] published the largest dataset, termed RSICD. Hoxha et al. [7] designed the CNN-RNN framework, which integrates with beam-search to output multiple captions for the target image and takes the caption with the highest similarity as the matching result. Aiming at the duplicate data of RSICD, Li et al. [11] explored the overfitting problem in RS image captioning caused by cross-entropy (CE) loss and accordingly propose a new truncated cross-entropy (TCE) loss. Lu et al. [15] performed an active attention framework for more specific caption generation according to the interest of the observer. Although significant progress has been made in caption-based RS image retrieval, there are still some drawbacks, for example, the generated sentence is always coarse which hinders the widespread application of RS multimodal data. Recently, a small amount of RS image-text retrieval research are sprouting. In 2019, Abdullah et al. represented the Deep Bidirectional Triplet Network (DBTN) [1] to solve the RS image-text retrieval problem for the first time and published a new dataset, called TextRS. Aiming at the multiscale and complex semantic characteristics of remote sensing data, the state-of-the-art retrieval methods are dedicated to two aspects, namely "scale decoupling" and "semantic decoupling". We will elaborate on these two strategies as follows.
Scale Decoupling and Semantic Decoupling
Generally, both "scale decoupling" and "semantic decoupling" are applied in the feature extraction stage. In recent years, "scale decoupling" methods have proved to have obvious advantages in the field of vision due to their capability to entirely explore the various objects of images, and have been widely adopted by related research. To restore multi-scale spatial information, the "encoder-decoder" structure, called "U-Net" [19] was proposed. This framework concatenates the obtained low-level features with high-level features by skip connection. Szegedy et al. [22] proposed a multi-scale model based on four parallel inception modules, which used convolution kernels of different scales for feature extraction and pooling. The Atrous Spatial Pyramid Pooling (ASPP) [2] focuses on the receptive field to fuse multi-scale contextual information with different dilated rates, increasing the discriminative power of the resulting feature representation. Undoubtedly, some studies also utilized scale decoupling methods to extract the various object of RS images in the process of RS retrieval. For example, Ye et al. [29] proposed a flexible multiple-feature hashing learning framework for RS image retrieval by taking multi-scale complementary features. Sukhia et al. [21] discussed a content-based RS image retrieval technique using multi-scale, patch-based local ternary pattern and fisher vector encoding. The AMFMN [30] also enhanced the feature representation by integrating multi-scale features into low-level and high-level features. With regard to "semantic decoupling", Lee et al. [ 10] designed the Stacked Cross Attention Network (SCAN), which first calculates a sentence/word representation for each region in the image based on similarity, and then compares it with the global sentence to achieve semantic decoupling of regions. To refine the correspondence between RS images and text, Cheng et al. introduced a Deep Semantic Alignment Network (DSAN) [3] by utilizing a Semantic Alignment Module (SAM) to mine the underlying semantic relationship between RS images and text. Despite the significance and value of the "scale decoupling" and "semantic decoupling" methods, there are still some problems. Firstly, these previous "semantic decoupling" approaches focus on either the disentangling scale or semantic but ignore merging these two ideas in a union model, which extremely limits the performance of cross-modal retrieval models. Besides, the mentioned "semantic decoupling" approach is only considered as a kind of "soft scale decoupling" where the large-scale targets still contain the small-scale targets after the decoupling operation, and this situation will eventually lead to a great amount of redundant noise. Moreover, these "semantic decoupling" methods utilize attention to implicitly mining semantics, which does not consider the application of semantic tags to explicitly mine meaningful information, thus failing to achieve the best semantic decoupling performance.
APPROACH
In this section, we introduce the main parts of our SSJDN approach. Figure 2 illustrates the framework of our approach, which includes four components. (1) Feature Representation, which could extract features of RS images and texts. (2) The Bidirectional Scale Decoupling (BSD) module,
Fig. 2.
The retrieval architecture (SSJDN) is proposed in this paper. Especially, we represent the BSD module to adaptively filter redundant information between different scales, and the LSD module to capture more effective semantic-related semantic information. We also design a STL, which can further enhance the probability of the same class cross-modal data.
which utilizes Salience Feature Extraction (SFE) and Salience-Guided Suppression (SGS) units to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional mode to provide different clues. (3)The Label-supervised Semantic Decoupling (LSD), which applies category semantic labels to supervise the network focusing on predominant semanticrelated features. (4) The Semantic-guided Triple Loss (STL) is designed to increase the retrieval opportunities of the same category cross-modal data. It is worth noting that the former three parts are performed in the feature extraction stage, and the last part is in the feature similarity measuring stage. The entire retrieval process can be divided into the following four steps.
1) enter the query image (or query text) and all the samples of the dataset; 2) input the query image (or query text) and all the samples of the dataset into the trained model to obtain the visual feature (or text feature) of the query image (or query text) and the samples' feature of the dataset;
3) calculate the similarity score between the generated query visual features (or query text features) and all the samples in the dataset; 4) the samples finally are ranked according to the similarity score and returned as the search result.
Next, we will provide details of our proposed model for cross-modal image-text retrieval in remote sensing. First, we introduce the feature representation in Section III-A. Then, we describe in detail our proposed modules in Section III-C, respectively. Finally, we discuss the semantic-guided triple in Section III-D.
Feature Representation
We will summarize the feature representation part from two aspects: image feature representation and text feature representation.
3.1.1 Image Feature Representation. According to the method [2], for an RS image ∈ R × ×3 , we utilze the [5] as feature extractor to train our model. Specifically, the global features can be formulated as:
= ( , ),(1)
where is used to indicate the parameters of Resnet. Subsequently, the with different dilated rates are applied to generate multi-scale feature representations ∈ R ℎ× × , which are defined as:
{ } 3 =1 = ( , ),(2)
where refers to the number of feature scales, and the dilated rates of convolution kernel corresponding to scale 1, 2, and 3 are 6, 12, and 18, respectively. is the parameters of .
Text Feature Representation.
For texts, suppose that there are words in a sentence , each word is encoded into a one-hot vector that indicates the index in the vocabulary, expressed as ∈ R . Then, the one-hot vector is converted into high-dimensional vector by the embedding matrix , which is denoted by a formula = ( ). To retain temporal information in the sentence and reduce the computational cost, the high-dimensional vector is fed into a − ℎ ℎ network [9] to capture the contextual information.
= ( , ),(3)
where ∈ R ′ stands for the high-level text feature and represents the text feature extraction network, termed − ℎ ℎ . refers to the parameter of − ℎ ℎ network.
Bidirectional Scale Decoupling
As mentioned above, considering the multi-scale characteristics of RS images, it is necessary to conduct multi-scale modeling for networks. However, a great amount of redundant information is continuously superimposed due to the "soft scale decoupling". Thus, we hope to use a kind of "hard scale decoupling" approach to increase the availability of multi-scale features. We will implement the "hard scale decoupling" in two steps, extracting potential features on the current scale firstly, and then suppressing cumbersome features at other scales. Based on the above analysis, we propose a BSD consisting of an SFE unit and an SGS unit, as shown in Figure 3. It is worth noting that the BSD is bidirectional (from small to large scale and from large to small scale), which can extract feature representations more comprehensively and reliably.
Salience Feature Extraction.
The SFE unit is proposed to extract potential features on the current scale. In recent years, the Convolutional Block Attention Module (CBAM) [26] has prominent performance in exploring the discriminative information by calculating the attention maps of features. Therefore, we prefer to use the CBAM to construct the SFE unit. Aiming at the different scale features , we first aggregate the channel-wise information of a feature by the average pooling and max pooling operations, and generate two 2D efficient feature descriptor: ( ) ∈ R ℎ× and ( ) ∈ R ℎ× . Then, the feature descriptor passes through a standard convolution layer and sigmoid function to produce the attention map :
= ( ( ( ( ),( ))
)),
where denotes the sigmoid function and (·) represents a convolution layer.
(·, ·) means concatenation of the features. The attention maps are effective in highlighting informative regions of features in different scales.
Salience-Guided Suppression.
After completing salience feature extraction, the SGS unit is created to perform suppression operations on salience features at the current scale, benefiting the other scales features to explore discriminative cues. Specifically, due to the attention map, represents the salience region of the feature , the suppression mask makes full use of to restrain the salience information at the scale :
= B ( ),(5)
where B is a binary mask, which takes the values of the most salience to 0 and others to 1. The suppression masks relieve the coverage effect of on other scales and make different information stand out.
Subsequently, based on SFE and SGS, we build the BSD from two directions. For the sake of simplicity, Figure 3 only shows a unidirectional (from small to large scales) scale decoupling module. First, the attention map and the salience mask are calculated, where the is used to extract the salience features, and the is prepared for suppressing the salience features at other scales. Then, these two maps are applied in the process of generating decoupled feature by way of stepwise suppression. Besides, we also add residual connections [5] to obtain more effective information, and the scale decoupled feature is computed as:
= + if = 1 = ( −1 ) + if = 2, 3(6)
Similarly, the scale decoupled featuresˆin the way of large to small scale is calculated by the following formula:
ˆ= + if = 3 = ( +1 ) + if = 1, 2(7)
Finally, the scale decoupled features andˆare fused by concatenation operations.
= ( 1 , 2 , 3 ,ˆ1,ˆ2,ˆ3),(8)
where ∈ R ℎ× × ′ represents the scale decoupled features of images. It can be seen that the alleviate the redundant information between multi-scale features and enhance the robustness of image representation.
Label-supervised Semantic Decoupling
As discussed above, to moderate the effect of the meaningless semantic features caused by the implicit exploration of attention mechanism, we propose the LSD module to mine effective semantic features. The category semantic labels are utilized to train image and text classification networks to produce category features, which are then applied to two aspects. First, category features are multiplied with the decoupled image features and text features to guide the retrieval network probing significant and reliable semantic-related information. Second, category matching is performed on the category features, determining whether the images and texts belong to the same category for improving the retrieval probability of the same class cross-modal data, which will be detailed in the next subsection.
Category Features Extraction.
We first implement image and text classification separately on each dataset, and fuse the pre-trained classification network into the retrieval network. Specifically, the image classification network with the same structure as under the supervision of the image category semantic labelsˆwith cross-entropy loss to obtain the high-level category feature representation. Besides, to better fine-tune the text feature representation, we utilize the as a text classifier to categorize the sentence based on text category semantic labelsˆ. The category features and of image and text are calculated as follows:
= ( , ′ ) = ( , ),(9)
where ′ and are the parameters of and , respectively.
Category Features Fusion.
After category features extraction, the generated category features are leveraged to guide the decoupled image feature and text feature to mine significant and reliable features. Specifically, we take full advantage of multiplication, which can achieve considerable enhancement of related features in the process of feature combination to compute the final semantic decoupling image and text features ′ and ′ :
′ = ( , ) ′ = ( , ),(10)
where (·, ·) denote multiplication, The ultimate feature ′ and ′ not only capture discriminable multi-scale semantic information, but also highlight the semantic-related reliable knowledge, thereby increasing the accuracy of retrieval network.
Semantic-guided Triple Loss
To implement multi-modal feature alignment, triplet loss is widely used and has become one of the mainstream loss functions in the field of multi-modal feature matching tasks. The purpose of triple loss is to increase the distance between a sample and the corresponding negative sample while reducing the distance between the sample and its positive sample as much as possible. [4] proposes a bidirectional triple loss for text-image matching:
L (F ′ , T ′ ) = ^′ [ − ( ′ , ′ ) + ( ′ ,ˆ′)] + + ^′ [ − ( ′ , ′ ) + (ˆ′, ′ )] + ,(11)
where refers to the margin and [·] + = (·, 0). ( ′ , ′ ) represent the similarity of image and text. The first sum is processed for all negative sentences ′ of a given image ′ , and the second sum considers all negative images ′ of a given a sentence ′ . To save the calculation cost, the loss is usually computed in each batch rather than in all training sets.
However, the data of the same category should be easier retrieved to improve the accuracy of image-text matching. Thus, we perform category matching based on the category features (as mentioned above) to automatically determine whether the input text and image are of the same category. Firstly, the category features are converted into semantic categories and (a total of c categories in the retrieval dataset) of image and text through softmax. Then, we define a constant value to adjust the loss to make it more sensitive to the same category of different modal data. The constant value is expressed as:
= if = 1 others(12)
On the basis of the constant value , we design a semantic-guided triple loss detailed as follows:
L (F ′ , T ′ ) = ^′ [ − ( ′ , ′ ) + ( ′ ,ˆ′)] + + ^′ [ − ( ′ , ′ ) + (ˆ′, ′ )] +(13)
We take the image to text retrieval as an example to implement analysis on semantic-guided triple loss. Since ′ is the positive sample corresponding to and ′ , ′ and ′ belong to the same category, that is to say, is equal to , and the value of is . We restrict to be a value larger than 1, that will be subjected to ablation experiments to evaluate the specific value, to improve the retrieval accuracy of positive samples. Moreover, if ′ andˆ′ are the same class, similarly, the value of is also , but when ′ andˆ′ are different classes, is unequal to , and the value of is still 1, which will not increase the matching rate of negative samples of different classes. In addition, we introduce classification loss here for better understanding of the LSD module. The classification network of image (Resnet) and text (BERT) both apply cross-entropy loss to generate category features under the supervision of image and text semantic labelsˆandˆ.
L ( ,ˆ) = 1 ∑︁ =1 − (ˆ) − (1 − ) (1 −ˆ) L ( ,ˆ) = 1 ∑︁ =1 − (ˆ) − (1 − ) (1 −ˆ) ,(14)
where and are the classification prediction results of RS images and text. refers to the number of training samples. So far, we have introduced our approach, which not only considers the decoupling of multi-scale features but also makes full use of category semantic labels to extract meaningful feature representations.
EXPERIMENTS
To assess the effectiveness of the proposed method, some experiments are conducted. In this section, we first give a short description of the datasets, implementation details, metrics, and baselines. After that, we carry out experiments involving the proposed approach through ablation studies and comparisons with state-of-the-art methods.
Datasets
We perform experiments on four benchmark RS datasets for the cross-modal image-text retrieval: UCMerced-LandUse-Captions, Sydney-Captions, RSICD, and NWPU-RESISC45-Captions.
The UCMerced-LandUse-Captions is build by [18], based on the UCMerced-LandUse dataset [28]. It contains land use images in 21 categories, with 100 images per category. The spatial sizes UCM RSICD RSITMD Sydney Number of categories 21 33 33 7 of the data files are 256 × 256 pixels, and the ground sampling distance (GSD) is 0.3048 m. The research [18] exploit five different sentences to describe every image. The RSICD dataset is used for the remote sensing image captioning task [16]. The spatial sizes of the data files are fixed to 224 × 224 pixels with various resolutions. The total number of remote sensing images is 10921, with five sentences of descriptions per image.
The RSITMD dataset is supplied by [30] for RS cross-modal image-text retrieval. The images in the RSITMD dataset are selected from the RSICD dataset and provide a total of 23715 captions for 4743 images. The RSITMD is more granular and diverse in captions than the RSICD dataset.
The Sydney-Captions dataset is also provided by [18], which is based on the Sydney dataset [31]. The spatial sizes of the data files are 18 000 × 14 000 pixels, and the GSD is 0.5 m. Similar to the UCMerced-LandUse dataset, five different sentences were given to describe each image [18].
In addition, we also conduct statistics on the categories of each dataset, and the calculation results are shown in Table 2. It can be seen from the table that the RSICD and RSITMD datasets have the most scene classes of 33. The Sydney dataset contains fewer scene with 7 categories.
Implementation Details
In this subsection, we mainly introduce the implementation details of the proposed method. The channel ′ of the scale decoupled feature is the same as that of the high-level text feature, which is 512. For the semantic-guided triple loss (Eq. 13), we conduct a large number of ablation experiments and take the value of to be 1.2 and the margin threshold to 0.2. All the models are implemented with PyTorch and the Adam optimizer with a 0.0002 learning rate. We set the batch size of training to 100 and train the model for 70 epochs. All the experiments are conducted on a server with one Tesla V100 SXM2 16GB GPU.
Metrics and Baselines
We conduct two kinds of image-text matching tasks: 1) sentence retrieval, i.e., retrieving groundtruth sentences related to the query image (I2T); and 2) image retrieval, i.e., retrieving ground-truth images related to the query text (T2I). We use the Recall at ( @ , =1,5 and 10) as the evaluation metric, which refers to the ratio of groud truth appearing in the topK results. Besides, we also compute the average of the recall rates of @ raised by Huang et al. [8] to more reasonably evaluate the performance of the model.
As for baselines, six classic image-text retrieval networks are compared, including SCAN [10], CAMP [25] and MTFN [24]. We additionally compare SSJDN with the recently proposed AMFMN [3], which dedicate to RS image-text retrieval.
Performance Comparison
In this section, we compare our method with six baselines on four datasets. The experimental results are listed in Table 1. Due to discrepancy in different datasets, the performance of the model is divergent on different datasets, and we can conduct the following analysis:
• Results on UCM : From the first table of Table 1, we can observe that the retrieval models achieve the best performance due to the superiority of the UCM dataset. For the metric , our approach achieves 7.4 and 5.9 gains compared with the best baseline MTFN and AMFMN, respectively. It is shown that SSJDN improves retrieval efficiency by implementing category matching of images and texts, attaining a competitive performance. • Results on RSITMD: The results on the RSITMD dataset are in the second table of Table 1.
The capabilities of our SSJDN are competitive to some state-of-the-art methods, especially the SCAN-based and CAMP-based models due to the negligence of multi-scale information.
Besides, compared with AMFMN, the proposed method has improved mR by 4.6, which verifies the effectiveness of category guidance and the bidirectional scale decoupling method. • Results on RSICD: The RSICD dataset reduces model robustness due to its blurry image data and the comparison of models on the RSICD dataset is on the third table of Table 1. No matter in the indicator of @1, @5 or @10, our model is superior to other retrieval models, demonstrating the remarkable ability for both I2T and T2I retrieval. The SSJDN achieves an average improvement of 3.7 as shown by mR on the best network AMFMN. The results indicate the availability and feasibility of proposed each modules. • Results on Sydney: It can be observed on the bottom right of Table 1 that the proposed method achieves the best @1 indicator, but is slightly less effective on @5 and @10 since both SCAN and CAMP consider the message passing between cross-modalities. Nonetheless, the overall performance of SSJDN is still prominent, with a score of 54.0 on the mR indicator. The performance on the Sydney dataset strongly illustrates the reliability and robustness of the AMFMN method.
Module Analysis
In this section, we carried out several experiments on the RSITMD data set to further analyze the effectiveness of our model. Specifically, we first explored how each component of our framework affects the image-text retrieval results. We then displayed how the value of influences the retrieval performance. After that, the comparisons of different attention mechanisms are implemented to prove the validity of the BSD module. Finally, we conduct experiments on the different fusion methods between categorical features and image features.
Ablation Studies.
To verify the effectiveness of each module proposed in the paper, we conducted ablation experiments, as reported in Table 3 than w/o LSD, revealing that the label-supervised semantic decoupling module can enhance the representations by mining semantic-related semantic information and boost the model performance. Moreover, the performance drop of w/o BSD can be observed, indicating the vital importance of the bidirectional scale decoupling module as it can capture multi-scale discriminative clues. The w/o all method does not consider multi-scale distinguishing information and category supervision, resulting in inferior accuracy. In general, our proposed model largely exceeds all variants on I2T and T2I retrieval, verifying the effectiveness and complementarity of four modules.
Parameter
Analysis. The hyperparameter is used to adjust the loss function, thereby increasing the retrieval probability of the same category data. We vary from 0.8 to 1.6 with intervals of 0.2 to search the value of that optimizes the performance of the network. From Figure 4, we can observe that the model performs worst when the parameter is 0.8 because the category information has an adverse effect on the model training. If the value of the parameter is 1, it means that the parameter has no action on the network. Moreover, the performance of the model reaches a saturation point when the value is 1.2, and then begins to decline slightly. Based on the above experiments, we choose of 1.2 as the constant to increase the accuracy of retrieval by integrating category information.
Comparison with Different Attention Mechanism.
In this section, we design an experiment to evaluate the performance of different attention approaches, as shown in Table 4. Especially, we select four different methods for combining attention: 1)USD(L2S), noting the utilization of unidirectional (from large to small) scale decoupling module; 2)USD(S2L), indicating the utilization of unidirectional (from small to large) scale decoupling module; 3)MA, representing that the attention maps are computed on image features at each scale; 4)w/o MA, standing for no attention method is employed.
The comprehensive performance of the USD(L2S) and USD(S2L) decreases compared with our method, which indicates that it is necessary to perform bidirectional scale decoupling to explore distinguishing features from different perspectives. Although the MA conducts attention maps to probe salient features in RS images, the results are not satisfactory due to the large amount of redundancy contained in the attention maps. There is no doubt that the metrics @1, @5 and @1 of w/o MA are the lowest in I2T and T2I retrieval because of the lack of mining of salient features. The BSD module can eliminate redundant noise generated by multi-scale feature interaction by utilizing SFE and SGS units and yield encouraging retrieval performance.
Performance of Different Fusion Method.
To better guide image and text feature training by category features, we conducted the following three experiments. 1)Add, referring to the element-wise addition of image (or text) features and category features; 2)Concat, indicating the concatenation; 3)Multi, representing the multiplication;
The experimental results are shown in Figure 5, and we can see that Multi outperforms the other two fusion method on the metrics @1, @5, and @10 in I2T and T2I retrieval. It is concluded through analysis that multiplication further enhances the supervision of category features over image or text features, while Add and Concat simply combine the two features, resulting in inferior model performance than multiplication.
Saliency Mask Visualization
In this section, we visualize the attention map to analyze the function of the BSD module. In Figure 6, the first column is the original RS image, and columns 2 to 5 are attention maps generated by USD from large to small scale, similarly, columns 6 to 9 are the attention maps produced by the small to large scale USD module.
It can be observed from Figure 6 that the BSD module is inclined to focus on the effective region, as shown in the second column of the second image, the attention map concentrates on the "grey house" in the given query sentence. Besides, the attention map in each USD module emphasizes different salient regions by implementing decoupling operations, thereby mining differentiated effective cues. Moreover, there is also a discrepancy in the size of the attention regions of the USD module in two different directions, such as the upper left corner of the attention map in the second column and the sixth column in the third image. It can be seen from these attention maps that the BSD module can filter redundant information generated in multi-scale interaction, improving the performance remarkably of RS image and text retrieval.
Qualitative Results
To qualitatively verify the effectiveness of SSJDN, several typical examples are illustrated from RSITMD in Figure 7 and Figure 8, respectively. For image-to-text retrieval, it can be observed that our model holds the ability to understand abstract short sentences or complex long sentences accurately. For text-to-image retrieval, the proposed model is robust to simple or complex images, which is mainly attributed to BSD, LSD and STL modules to fully exploit reliable and effective information. There are still some erroneous retrieval results, such as the rank-3 sentences of the third image query in Figure 8 and the rank-1 image of the Query (c) in Figure 9. The analysis concluded that these mistakes are caused by the high similarity between remote sensing data. In fact, it is even difficult for humans to distinguish the results, which is also the reason for the generally low accuracy for RS retrieval.
CONCLUSION
We have presented a novel SSJDN for cross-modal RS image retrieval. The main contribution of our method is that we propose a BSD module to adaptively extract potential features and suppress cumbersome features at other scales in a bidirectional pattern to exploit distinct clues. Besides, we utilize category semantic labels as the prior knowledge to generate category features, which are then combined with the original retrieval network to probe more effective class-related information. Finally, we design a semantic-guided triple loss, which tends to match the same class cross-modal data. The qualitative and quantitative results on popular four datasets show that our model achieves state-of-the-art performance in cross-modal retrieval tasks. In particular, in the ablation studies, the evaluation metrics mR of the SSJDN drop by 3.8, 4.9, and 1.8 without BSD, LSD, and STL operations, respectively, illustrating the effectiveness and reliability of the proposed method.
We have verified the effectiveness of the category prior knowledge, and in the future, we plan to build a multi-category label to dynamically explore discriminative information in remote sensing images and texts, making it more interpretable and applicable.
Fig. 3 .
3A illustration of an unidirectional (from small to large scales) scale decoupling (USD) module. The attention map is used to extract the salient features, and the suppression mask is prepared for suppressing the salience features at other scales.
Fig. 4 .
4Performance of image-text retrieval with different scalar values of .
Fig. 5 .
5Performance of image-text retrieval with different fusion method.
Fig. 6 .
6Visualization of salient mask output by the MBD in three RS images, which include three parts: The original RS images, the attention maps from large to small scales and the attention maps from small to large scales.
Fig. 7 .
7Top-3 image-to-text retrieval results on RSITMD dataset. The ground-truth texts are marked with green checks, and the wrong results are indicated by cross marks.
Fig. 8 .
8Top-3 text-to-image retrieval results on RSITMD dataset. The matched images are annotated in green boxes, and the false ones are in red.
Table 1 .
1Comparisons of image-text retrieval experiments on UCM, RSITMD, RSICD and Sydney datasets.UCM dataset
Image-to-Text
Text-to-Image
Approach
R@1 R@5 R@10 R@1 R@5 R@10
mR
SCAN t2i
13.9 45.8
68.6
13.1 50.7
78.2 45.0
SCAN i2t
12.8 47.0
69.1
12.6 46.9
72.7 43.5
CAMP-triplet 11.2 44.3
65.7
9.9
46.1
77.3 42.4
CAMP-bce
15.1 47.2
68.6
11.9 47.2
77.0 44.5
MTFN
10.7 47.6
64.3
14.7 52.7
81.1 45.2
AMFMN
14.5 51.2
67.3
14.5 51.8
80.7 46.7
SSJDN
17.1 56.7
77.1
17.9 60.9
85.9 52.6
RSITMD dataset
Approach
Image-to-Text
Text-to-Image
mR
R@1 R@5 R@10 R@1 R@5 R@10
SCAN t2i
10.1 28.7
38.5
10.6 29.5
44.5 27.0
SCAN i2t
11.3 25.9
40.1
9.8
29.4
41.6 26.4
CAMP-triplet 11.7 28.6
37.1
8.4
26.9
45.0 26.3
CAMP-bce
9.0
24.4
31.2
5.5
23.7
39.2 22.2
MTFN
10.5 27.8
35.3
9.6
31.4
47.1 27.0
AMFMN
10.1 26.7
41.4
10.5 34.8
56.9 30.0
SSJDN
12.2 29.4
44.2
10.8 42.2
68.9 34.6
RSICD dataset
Approach
Image-to-Text
Text-to-Image
mR
R@1 R@5 R@10 R@1 R@5 R@10
SCAN t2i
4.3
11.2
17.6
4.0
16.6
26.6 13.4
SCAN i2t
5.9
12.9
19.7
3.8
16.9
27.1 14.4
CAMP-triplet 5.1
13.0
22.1
4.2
15.3
27.8 14.6
CAMP-bce
4.4
10.2
15.7
2.5
13.4
23.9 11.7
MTFN
5.1
12.6
19.9
4.9
17.8
29.6 14.8
AMFMN
5.5
14.8
23.1
4.0
17.2
31.3 16.0
SSJDN
6.5
19.7
30.1
4.9
20.2
36.5 19.7
Sydney dataset
Approach
Image-to-Text
Text-to-Image
mR
R@1 R@5 R@10 R@1 R@5 R@10
SCAN t2i
19.0 50.7
74.1
17.8 55.9
76.6 49.0
SCAN i2t
20.4 54.2
67.6
16.1 57.6
76.0 48.7
CAMP-triplet 22.8 50.5
75.9
15.3 43.1
70.4 46.3
CAMP-bce
15.6 49.7
71.3
11.6 51.3
76.2 46.0
MTFN
21.7 51.6
69.0
14.1 56.0
78.6 48.5
AMFMN
29.6 55.6
67.6
13.7 60.0
82.7 51.5
SSJDN
30.4 50.8
68.1
20.4 67.5
86.8 54.0
Table 2 .
2Number of categories of UCM, RSITMD, RSICD and Sydney datasets.
Table 3 .
3Comparison with different modules of image-text retrieval.Approach
Image-to-Text
Text-to-Image
R@1 R@5 R@10 R@1 R@5 R@10
w/o all
10.7 23.7
36.1
9.1
33.0
42.5
w/o BSD 12.1 27.4
38.7
8.4
36.6
61.4
w/o LSD 12.0 26.8
38.6
8.9
34.8
56.8
w/o STL 12.2 27.3
42.1
9.4
38.3
67.2
SSJDN
12.2 29.4
44.2
10.8 42.2
68.9
Table 4. Comparison with different attention mechanism of image-text retrieval.
Approach
Image-to-Text
Text-to-Image
R@1 R@5 R@10 R@1 R@5 R@10
w/o MA 12.1 27.4
38.7
8.4
36.6
61.4
MA
11.2 27.9
39.8
10.1 38.9
65.2
USD(L2S) 11.5 27.6
40.9
10.7 39.4
67.9
USD(S2L) 11.6 28.8
42.3
9.9
41.0
68.1
BSD
12.2 29.4
44.2
10.8 42.2
68.9
on R@10 of I2T and T2I for SSJDN, respectively, demonstrating the significance of the semantic-guided triple loss. Besides, our model achieves better results. Specifically, we modify SSJDN with the
following variants: 1)w/o STL, changing the semantic-guided triple loss to triplet loss; 2)w/o LSD,
removing the label-supervised semantic decoupling module, which represent no category semantic
supervision; 3)w/o BSD, eliminating the bidirectional scale decoupling module; 4)w/o all, without
all components, including STL, LSD and BSD modules.
Compared with our model, the performance of w/o STL degrades dramatically. Particularly, It
drops absolutely by 2.1 and 1.7
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"All-in-One: A Highly Representative DNN Pruning Framework for Edge Devices with Dynamic Power Management",
"All-in-One: A Highly Representative DNN Pruning Framework for Edge Devices with Dynamic Power Management"
] | [
"Yifan Gong [email protected] \nNortheastern University\n\n",
"Zheng Zhan [email protected] \nNortheastern University\n\n",
"Pu Zhao [email protected] \nNortheastern University\n\n",
"Yushu Wu \nNortheastern University\n\n",
"Chao Wu \nNortheastern University\n\n",
"Caiwen Ding \nUniversity of Connecticut\n\n",
"Weiwen Jiang \nGeorge Mason University\n\n",
"Minghai Qin \nNortheastern University\n\n",
"Yanzhi Wang [email protected] \nNortheastern University\n\n",
"\nEqual Contribution\n\n"
] | [
"Northeastern University\n",
"Northeastern University\n",
"Northeastern University\n",
"Northeastern University\n",
"Northeastern University\n",
"University of Connecticut\n",
"George Mason University\n",
"Northeastern University\n",
"Northeastern University\n",
"Equal Contribution\n"
] | [] | During the deployment of deep neural networks (DNNs) on edge devices, many research efforts are devoted to the limited hardware resource. However, little attention is paid to the influence of dynamic power management. As edge devices typically only have a budget of energy with batteries (rather than almost unlimited energy support on servers or workstations), their dynamic power management often changes the execution frequency as in the widely-used dynamic voltage and frequency scaling (DVFS) technique. This leads to highly unstable inference speed performance, especially for computation-intensive DNN models, which can harm user experience and waste hardware resources. We firstly identify this problem and then propose All-in-One, a highly representative pruning framework to work with dynamic power management using DVFS. The framework can use only one set of model weights and soft masks (together with other auxiliary parameters of negligible storage) to represent multiple models of various pruning ratios. By re-configuring the model to the corresponding pruning ratio for a specific execution frequency (and voltage), we are able to achieve stable inference speed, i.e., keeping the difference in speed performance under various execution frequencies as small as possible. Our experiments demonstrate that our method not only achieves high accuracy for multiple models of different pruning ratios, but also reduces their variance of inference latency for various frequencies, with minimal memory consumption of only one model and one soft mask. | 10.1145/3508352.3549379 | [
"https://export.arxiv.org/pdf/2212.05122v1.pdf"
] | 254,564,412 | 2212.05122 | 1955793bec2653136818c0fccbd5f43d57d336e4 |
All-in-One: A Highly Representative DNN Pruning Framework for Edge Devices with Dynamic Power Management
Yifan Gong [email protected]
Northeastern University
Zheng Zhan [email protected]
Northeastern University
Pu Zhao [email protected]
Northeastern University
Yushu Wu
Northeastern University
Chao Wu
Northeastern University
Caiwen Ding
University of Connecticut
Weiwen Jiang
George Mason University
Minghai Qin
Northeastern University
Yanzhi Wang [email protected]
Northeastern University
Equal Contribution
All-in-One: A Highly Representative DNN Pruning Framework for Edge Devices with Dynamic Power Management
10.1145/3508352.3549379
During the deployment of deep neural networks (DNNs) on edge devices, many research efforts are devoted to the limited hardware resource. However, little attention is paid to the influence of dynamic power management. As edge devices typically only have a budget of energy with batteries (rather than almost unlimited energy support on servers or workstations), their dynamic power management often changes the execution frequency as in the widely-used dynamic voltage and frequency scaling (DVFS) technique. This leads to highly unstable inference speed performance, especially for computation-intensive DNN models, which can harm user experience and waste hardware resources. We firstly identify this problem and then propose All-in-One, a highly representative pruning framework to work with dynamic power management using DVFS. The framework can use only one set of model weights and soft masks (together with other auxiliary parameters of negligible storage) to represent multiple models of various pruning ratios. By re-configuring the model to the corresponding pruning ratio for a specific execution frequency (and voltage), we are able to achieve stable inference speed, i.e., keeping the difference in speed performance under various execution frequencies as small as possible. Our experiments demonstrate that our method not only achieves high accuracy for multiple models of different pruning ratios, but also reduces their variance of inference latency for various frequencies, with minimal memory consumption of only one model and one soft mask.
INTRODUCTION
As deep neural networks (DNNs) can achieve superior performance compared with traditional methods, they have been applied to a wide range of applications including classification [12], object detection [2], natural language processing [27], and so on recently. Besides, due to the rapid increasing popularity of edge devices such as mobile phones, and tablets, there are ever-increasing demands for deploying DNNs on various resource-limited edge devices.
When deploying DNNs from powerful servers (or workstations) to resource-intensive edge devices, we need to deal with the significant difference of (i) hardware resource and (ii) energy support. The Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. hardware resource usually refers to the available memory, computation units (such as multiplier-accumulator) and so on. Many works [10,13,16,23,28,29,[34][35][36][37][38] investigate how to keep or improve the DNN performance under a more rigid resource constraint (such as limited memory or floating point operations per second (FLOPS)).
Though the resource limitation of edge devices receives much attention from researchers, there are little research efforts devoted to the energy support. The energy support refers to the power or energy (including the execution frequency and voltage) to support the execution of DNNs when performing DNN inference on edge devices. Different from servers or workstations with plenty of power support, edge devices only have a budget of energy (such as battery) and need to adopt certain dynamic strategies to manage the energy usage. For example, if the battery level decreases to 15% (or 10%) on a mobile phone, it usually switches to the energy saving mode and reduces the execution frequency to achieve longer availability. Besides energy saving mode, the dynamic voltage and frequency scaling (DVFS) technique is widely adopted on edge devices to adjust the power and speed settings so that the resource allotment for tasks can be optimized and the power saving can be maximized. In general, the current energy support with dynamic frequencies is unstable for the deployment of DNNs on edge devices.
The unstable energy support leads to unstable DNN performance in terms of inference speed or latency. If the execution frequency switches to a smaller value, the computation-intensive DNN models need more time to finish the computations, incurring larger inference latency. The unstable inference latency not only harms the user experience, but also wastes a lot of hardware resources. DNNs with original real-time inference capability can hardly achieve realtime inference when the execution frequency decreases, incurring stuttering or lag, and poor user experience. Besides, for safetycritical applications on real-time embedded systems like medical monitoring on smart watches, the deadline based scheduling is used to guarantee safety. The deadlines are usually set up based on the worst-case execution time, which could waste many processor resources if the worst-case execution time with low execution frequency is much larger than normal values.
To improve user experience and save hardware resource, it is necessary to make the DNN performance (especially speed performance) as stable as possible under different levels of energy support (especially the execution frequency). Different from the hardware resource limitation which many research efforts focus on during DNN deployment, little attention is paid to the unstable energy support or dynamic frequency management. Our work firstly identify the unstable energy and frequency problem during DNN deployment and propose a framework to deal with the limited hardware resource and unstable energy support. To deal with various execution frequencies, our framework can generate multiple sparse models with different sparsity ratios, such that higher-sparsity models with less computations can run with a smaller frequency, leading to smaller variance of the inference time. To achieve this, we propose parametric pruning together with switchable thresholds and batch normalization. Thus, after training one model and one soft mask, we can use switchable thresholds to convert the soft mask into different binary hard masks of various sparsity ratios to deal with different execution frequencies. Besides, we further adopt switchable batch normalization to improve the accuracy performance of each sparse model. Our memory cost is only one highly representative single model and a soft mask, which is much lower than the cost of training and storing multiple sparse models separately. As a result, our work can deal with the limited hardware resource and unstable energy support on edge devices simultaneously. We summarize our contributions as follows.
• We firstly identify the unstable energy support problem with dynamic frequencies for DNN deployment on edge devices and evaluate the influence of dynamic frequency and power for DNN inference. • To deal with the limited hardware resource and unstable energy support on edge devices, we propose parametric pruning together with switchable threshold and batch normalization to generate one single highly representative model and soft mask, which can represent multiple sparse models to work with different execution frequency. • Our experimental results demonstrate that we can use a minimal memory cost with one model and one rescaled soft mask, to obtain multiple models of various sparsity levels with state-of-the-art classification accuracy compared with baseline methods. Moreover, by using a higher-sparsity model for a lower frequency, we can keep the latency under various frequencies as close as possible and significantly reduce the variance of the inference latency.
RELATED WORK 2.1 Sparse Models
Weight pruning [10,13,14,23,28] is an effective method to reduce model redundancy. Various pruning methods with different sparsity types have been proposed in the literature. Irregular pruning [11] achieves high accuracy but the reduction in parameters cannot be transformed into inference speedup. Structured sparsity [18,22,23,[39][40][41] such as column pruning and row pruning are hardware-friendly, but the coarse-grained pruning nature causes accuracy degradation. Recently proposed fine-grained structured pruning regularities including pattern-based pruning [7,20,24] and block-based pruning [5,6,17,21] preserve high accuracy while maintaining structures that can be exploited by compiler for hardware accelerations. However, most of the sparse model methods focus on the fixed resource scenarios without considering the hardware environment change. The resulting models have one fixed architecture and parameters that only satisfy one sparsity ratio.
Re-configurable DNNs
To obtain multiple models with different sparsity ratios, the most straightforward method is to train multiple models with different sizes using the sparse model methods. However, this causes huge computation burden during training and requires storing multiple models on the device. To decrease the computation and storage cost, recent works consider to train efficient architectures that can be reconfigured at runtime. NS [32] and US-Nets [31] train a model with multiple switches corresponding to different number of channels. This method simply retains the channels in the front of each layer, which neglects the different importance of each channel to the whole model performance. Furthermore, removing entire channels incurs severe accuracy loss. Joslim [4] extends in this direction by further allowing the joint optimization of width configurations and weights. But the method still relies on the coarse-grained structure pruning, incurring accuracy loss. Paper [3] proposes to train a large dense model and queries sub-models upon requirement. However, the method has to store a dense model and needs additional weights to achieve model transformation. LCS [25] learns a compressible subspace to obtain models with different sparsity ratio settings, but the accuracy degradation is non-negligible. RT 3 [26] discovers multiple pattern sets with different sparsity ratio settings to be switched during runtime. The drawback is that the pattern sets have to be stored individually, consuming a large amount of storage, and are too irregular to achieve efficient hardware accelerations.
MOTIVATION
Many deep learning (DL) based applications are deployed on edge devices such as mobile phones. Although many research efforts are devoted to the limited hardware resource on edge devices, little attention is paid to the influence of unstable energy support or dynamic frequency management. As edge devices typically only have a budget of energy with batteries (rather than almost unlimited energy support on servers or workstations), their dynamic power management often changes the execution frequency, leading to highly unstable inference speed performance, especially for computation-intensive DNN models. to transform the soft mask to a hard mask. The hard mask with a higher threshold value is the subset of the hard mask with a lower threshold. During inference, there is no need to store the dense model, only the model weights in the largest compact model needs to be stored. The model can be switched instantaneously according to ℎ when the battery mode changes.
capability may fail to achieve real-time inference and cause stuttering or lag, leading to poor user experience. Besides user experience, many hardware resource can be wasted. For example, for safetycritical applications like medical monitoring on smart watches, the deadline based scheduling is used by the real-time scheduler to guarantee safety. The deadlines are usually set up based on the worst-case execution time, which could waste many processor resources if it is much larger than normal values. To guarantee user experience and save hardware resource, we should keep consistent performance of DL applications (especially speed performance), under various execution frequencies. In this paper, we focus on image classification application as an example. But our method is not only limited to image classification. To keep consistent speed performance under different frequencies, inspired by the model pruning methods to achieve on-mobile real-time inference, we propose to use one model with switchable different pruning masks corresponding to various sparsity ratios. Thus, for higher frequency where the computation could be faster, we can adopt the mask with fewer pruned weights and more computations. For lower frequency where the computation could be slower, we can switch to the mask with more pruned weights and fewer computations. In this way, we may keep the inference latency under different execution frequency as close as possible.
We highlight that it is non-trivial to design such one model with multiple switchable pruning masks corresponding to various sparsity ratios. One naive alternative method is to train multiple pruned models, each model corresponding to one single sparsity ratio. However, DL models usually consume a lot of memory and the available memory on mobile devices is limited. Thus, we choose the method with one single model and multiple masks for this model. There are still several requirements for this method: (i) The memory cost of each mask should be significantly smaller than the whole model to save memory cost. (ii) The accuracy under each mask should be as high as possible. We should keep consistent classification performance after switching to different masks.
Our setting is different from previous pruning work [10,13,14,23,28] as most pruning methods only design one single pruning mask for one model given one sparsity ratio. Specifically, our work designs multiple pruning masks simultaneously for one model and each mask can achieve high classification accuracy under the corresponding sparsity ratio. Compared with methods with only a single mask design, our work has the following advantages: (i) The training cost is saved. After the training, we can obtain multiple sparse masks corresponding to multiple sparsity ratios simultaneously, while the single mask design methods need to run their algorithm multiple times to obtain multiple masks, incurring much larger training cost. (ii) The memory cost is saved. The single mask design methods usually need to fine-tune based on each mask to improve the final accuracy. Thus, each mask is not able to represent the pruned model and each pruned model is stored instead of each mask, incurring much larger memory cost. Moreover, we further adopt switchable threshold and batch normalization to derive one highly representative model and one soft mask so that multiple sparse models can be obtained from them. Thus the memory cost can be further saved.
PROPOSED METHOD
The framework overview of All-in-One in the case of three battery modes is illustrated in Figure 1. Our objective is to design one model and multiple masks where each mask corresponds to one sparsity ratio under a certain battery mode. The model can be easily switched during inference. When the frequency becomes smaller (battery level is lower), the model can switch to a mask with a larger sparsity ratio and becomes more sparse with fewer computations. Thus though the computation may be slower with lower frequency, the inference time with the more sparse model may not change greatly. More specifically, as in the Figure 1, a sparse model with green mask is used when the battery level is high. When the battery level decreases, the framework switches to use the orange mask. As the battery is almost out of power, the mask is switched to the red one. The switch can be achieved instantly without any retraining.
To achieve this, we propose the parametric pruning method to parameterize the pruning. The traditional pruning usually needs the in-differentiable sorting operations, which are unfriendly for the model training especially with multiple sparse masks. With the proposed parametric pruning, we do not need to incorporate sorting operations, improving pruning and training efficiency. As different sparsity ratios cause different mean and variance for the feature map, we further adopt switchable batch normalization (BN) to improve the classification accuracy under each sparsity ratio.
Based on the parametric pruning, we use switchable threshold ℎ to switch among masks of various sparsity ratios during training. When deployed for inference, the soft mask and switchable thresholds can be further re-scaled into low-bit format. Therefore, we do not need to store binary masks for sparsity ratios. Instead, we only need to store one re-scaled mask together with multiple thresholds (each threshold can determine one binary mask based on the re-scaled mask). The re-scaled threshold transforms the re-scaled soft mask into the corresponding hard mask when the battery mode changes. The hard mask with a higher threshold value is the subset of the hard mask with a lower threshold value. For instance, in Figure 1, the mask in red color which corresponds to the low battery mode is the subset of the mask in green color. Furthermore, instead of storing the entire dense model on the mobile device, only the weights in the largest compact model needs to be stored, which significantly reduces the memory cost. Taking Figure 1 as an example, there is no need to store the entire 5×5 weight matrix. Alternatively, the 4×4 weight matrix is stored. Detailed designs are presented in the following sections.
Pruning Schemes
In this work, we adopt fine-grained structured pruning including pattern-based pruning and block-based pruning, to efficiently accelerate the on-mobile inference while maintaining high accuracy. For both of the pruning schemes, the only set of weights to be stored is the largest compact model to accommodate the limited storage resources on the edge devices.
Pattern-based pruning. Pattern-based pruning is a combination of kernel pattern pruning and connectivity pruning, as shown in Figure 2 (a), where grey color represents removed weight. Kernel pattern pruning removes a fixed number of weights in each kernel and the locations of the remaining weights form specific patterns. The total number of pattern styles in the pattern library is limited for hardware accelerations. In our framework, each kernel pattern reserves 4 non-zero weights to match the single-instruction multiple-data (SIMD) architecture of embedded CPU/GPU processors to maximize the hardware throughput. As the sparsity ratio is constant for kernel pattern pruning, connectivity pruning is adopted as the supplementary to kernel pattern pruning for a higher sparsity ratio. Connectivity pruning cuts the connections between certain input and output channels, which is equivalent as removing whole kernels. To provide a re-configurable sparse model with switches, our framework first applies kernel pattern pruning to obtain a compressed model. − 1 different levels of connectivity pruning are further applied to the kernel pattern pruned model so that models of different sparsity are available at runtime.
Block-based pruning. Block-based pruning divides the weight matrix into equal-sized blocks and apply independent column pruning for each block, as shown in Figure 2 (b). To find a re-configurable sparse model, our framework prunes columns in each block with different sparsity ratio settings to obtain well-trained switches.
Parametric Pruning
We first need to make the pruning parametric to avoid the use of in-differentiable sorting operations. To achieve this, we assign importance scores to groups of weights. Let w ∈ R × × × denote the weights for the -th convolution (CONV) layer, with output channels, input channels, and kernels of size × . The output feature of the -th layer is represented as a ∈ R × × × ′ , with channels and × ′ feature size. The operation for the -th layer is represented as a = w ⊙ a −1 , where ⊙ denotes the convolution operation. For pattern-based pruning, as kernel pattern pruning results in a fixed sparsity ratio, a higher sparsity is achieved by pairing with connectivity pruning that removes whole kernels. Therefore, to perform connectivity pruning, for each kernel w ( ) ∈ R × , = 1, · · · , , in the -th layer, we assign an importance score ( ) as a soft mask to weight the importance of each kernel. The importance scores in the -th layer form the soft mask matrix s ∈ R × . For block-based pruning, the weight matrix w is first reshaped to a 2D matrix with size × and divided into equal-sized blocks with size R × , namely,
w = [w ,1 , w ,2 , · · · , w ,
]. An importance score ( ) is assigned for each column in every block . We use each element of s as the pruning indicator for corresponding weights. Larger value of ( ) indicates a more important group of weights that should be preserved while smaller value means that the weights may be removed. In this problem, we would like to achieve level sparsity, i.e., target sparsity ratios. The -th target sparsity ratio is paired with a predefined threshold value ℎ to convert the score ( ) into a binary mask as below,
( ) = 1, ( ) ≥ ℎ 0, ( ) < ℎ ,(1)
where ( ) ∈ {0, 1} is the binarized ( ) . The non-binary s is a soft mask and the binary b is a hard mask. During the inference of model pruning and training, we first obtain the binary mask b from s . Then we apply the binary mask ( ) to the corresponding weights w ( ) following ( ) w ( ) in each layer to achieve kernel or block pruning. ( ) = 1 means that the group of weights is reserved and ( ) = 0 means the group of weights is pruned.
Thus we are able to obtain a binary mask for weights in each CONV layer. The next problem is how to make the soft mask trainable, as the binarization operation is non-differentiable, leading to difficulties for back-propagation. To solve this, we integrate Straight Through Estimator (STE) [1] as shown below,
L ( ) = L ( ) ,(2)
where we can directly pass the gradients through the binarization. STE is originally applied in quantization tasks [19,30] to avoid the non-differentiable issues. If we do not use STE, more complicated strategies may be applied to deal with non-differentiable binary masks such as [8,9]. With binarization and STE, we are able to build a trainable soft mask to incorporate pruning in model training without using sorting operations. The trainable mask has the following advantages: (i) The soft mask can be efficiently trained along with the network parameters via gradient descent optimizers, thus saving training cost compared with [8,9]. (ii) Different from previous methods [11,15,33], which determine the pruning according to the parameter magnitudes, we use the soft mask to serve as the pruning indicator, rather than the parameter magnitudes. Thus pruning is decoupled from the parameter magnitudes.
We can train the model weights and the soft masks simultaneously. As we need to achieve high accuracy for multiple sparsity ratios, one model and one soft mask is not able to achieve this. So we assign each sparsity ratio a corresponding threshold and batch normalization parameters as introduced in Sec. 4.3 and Sec. 4.4, respectively. The classification loss can be expressed as L (W, S, ℎ , B ), where W denotes the model weights, S represents the soft mask, ℎ and B denote the threshold and the BN parameters for the -th sparsity ratio, respectively. Note that W and S are shared between different sparsity ratios when performing training and inference. Besides W and S, each sparsity ratio has its specific threshold and BN parameters.
Switchable Threshold
Given level sparsity or sparsity ratios, we can design masks where each mask can satisfy one sparsity ratio. However, multiple masks still cost a lot of memory. To further reduce the memory cost, based on the parametric pruning, we can only use one soft mask with thresholds (i.e., scales with negligible memory cost). For each sparsity ratio, we can use the soft mask and the corresponding threshold ℎ following Eq. (1) to obtain the binary mask for this sparsity ratio. The illustration of the switchable threshold is shown in Figure. 3. By using a higher threshold value, more weights are removed, which are denoted with the light blue circles. With the switchable threshold, the more sparse model uses a subset of the collection of weights in a more dense model. Get mini-batch of data and label 5: for target MACs C do 6: Switch to the threshold parameter ℎ and the BN parameters B of current target MACs 7: Compute loss L and sparse gradients 8: end for 9: Apply accumulated gradient descent on weights 10: end for
Switchable Batch Normalization
As accumulating model parameters with different sparsity ratio results in different feature mean and variance, the discrepancy across different configurations leads to inaccurate statistics of shared BN layers. Therefore, to maintain high accuracy for each sparsity ratio, we create independent BN parameters for each sparsity ratio, as shown in Figure 4. When inference under the sparsity ratio , only the corresponding BN with parameters B participates into the computation. With the switchable BN, for a model with target sparsity ratios, we need to store set of BN parameters. In most cases, BN layers only have less than 1% of the model size. Besides, the BN runtime cost is also negligible for deployment. Compared to separately trained models or using two sets of weights for learning compressible subspaces [25], the usage of switchable BN is memory efficient while keeping competitive accuracy performance.
Training Method
With the differentiable soft mask, we can train and prune the model via SGD simultaneously with the loss function,
min W,S,B L (W, S, ℎ , B ) + · L (S, ℎ ),(3)
where L is the cross entropy classification loss, and L is the regularization term related to the computation complexity. For simplicity, we take Multiply-Accumulate operations (MACs) as the constraint/target rather than parameter number to estimate ondevice execution cost. can weight the loss and stabilize training. L can be simply defined as ℓ 2 norm between current MACs and target MACs C. For connectivity pruning, L is defined as
L = 4 9 ∑︁ ( ′ ( − ∥b ∥ 0 )) − C 2 ,(4)
where b is obtained by converting the score s into binary values according to Eq. (1), and ∥b ∥ 0 indicates the number of remaining kernels in the -th layer. The constant 4/9 is attributed to kernel pattern pruning that reserves four weights in each 3 × 3 kernel. For block-based column pruning, L is defined as
L = ∑︁ ∑︁ ( ′ ( − ∥b ∥ 0 )) − C 2 ,(5)
where ∥b ∥ 0 represents the number of the remaining columns in the block of the -th layer. Given different battery modes, we could set different target MACs to satisfy different sparsity ratio requirements and form a list of target MACs as C = {C } =1 . With parametric pruning and switchable threshold & BN, we can perform our training following Algorithm 1. With target MACs, in every iteration, for each target MACs, we first switch to the corresponding ℎ and BN parameters, and then compute the loss with the target MACs to obtain gradients through back-propagation. After we collect and accumulate the gradients of all target MACs, we update the weights. Note that the model weights and the soft masks are updated with the gradients accumulated through all target MACs since they are shared between various target MACs. But the BN parameters are updated with the gradients of the single corresponding target MAC, rather than the accumulated gradients, as the BN parameters are switched when moving to another target sparsity ratio.
Cost Analysis
To satisfy the a total of sparsity ratios simultaneously, we only need to store one model, one re-scaled soft mask, re-scaled thresholds, and sets of BN, where the memory cost of thresholds and BN are negligible. With a well-trained soft mask, there is no need to store its precise floating point values during inference. To save the storage cost, the soft mask can be transformed/re-scaled into the low-bit format (named the re-scaled soft mask). For instance, for = 3, the re-scaled soft mask is composed of values of 0, 1, and 2, with element 2 indicating the scores for the most compact model corresponding to the lowest frequency. Besides, since we perform kernel or block pruning and the soft mask corresponds to the number of kernels or columns/rows in the block rather than all parameters in the model, the memory cost of the soft mask is much smaller than the model weights. Our method can greatly reduce the memory cost compared with using models for sparsity ratios.
EXPERIMENT RESULTS
In this section, we compare All-in-One with state-of-the-art reconfigurable DNN methods. The comparison is conducted on both CIFAR-10 and ImageNet datasets. We demonstrate the following: (i) By comparing All-in-One with baseline methods, All-in-One can maintain high accuracy under different sparsity ratio settings with only one set of model weights. (ii) Through real implementations on an off-the-shelf mobile device, All-in-One can mitigate the problem of unstable inference time due to the change of battery level.
Experiment Setting
In order to evaluate whether All-in-One can consistently attain efficient pruned models for tasks with different complexities, we test on two major image classification datasets, i.e., CIFAR-10 and ImageNet. For CIFAR-10, we experiment with Resnet-18, Resnet-20, and Resnet-32. For ImageNet dataset, we experiment on Resnet-18 and MobileNet-v2. We conduct our multi-model training method on an 8× NVIDIA GTX 1080Ti GPU server using Pytorch. During pruning, The SGD optimizer is utilized with a learning rate of 1 × 10 −3 .
All the mobile results are measured on the GPU of an OnePlus 8T mobile phone, which has a Li-Po 4500 mAh battery. The phone itself is equipped with a Qualcomm Snapdragon 865 mobile platform which including a Qualcomm Kryo 585 Octa-core CPU and a Qualcomm Adreno 650 GPU. The same compiler techniques in [24] are applied to optimize the DNN execution.
Accuracy Performance
The accuracy performance on the CIFAR-10 dataset is shown in Table 2. We run our method with pattern-based pruning and blockbased pruning respectively and obtain two sets of accuracy. The results show that All-in-One can maintain high accuracy for all switches with only one set of model weights. It significantly outperforms all other state-of-the-art re-configurable DNN methods on all three models with non-negligible improvements (such as our accuracy above 94% v.s. 92% or lower from baseline methods for Resnet-32). Besides, our method with pattern-based pruning usually achieves higher accuracy than ours with block-based pruning.
The accuracy improvements of All-in-One are mainly attributed to two aspects. The first is the fine-grained structured pruning scheme that removes weight at a finer granularity compared to US-Net [31] and LCS-S [25] that employs coarse-grained structured pruning. The second is the separate BN for each configuration that preserves the distinct mean and variance with different sparsity ratios, which is not considered in RT 3 [26].
We further evaluate on the ImageNet dataset and the results are shown in Table 3. For Resnet-18, both pattern-based pruning and block-based pruning are conducted. For MobileNet-v2, as the computations are mainly from 1×1 CONV layers, which is not suitable for pattern-based pruning, the results using block-based pruning are presented. According to the results, All-in-One again provides the best accuracy performance on both models for each switch in the re-configurable model with non-negligible improvements compared with baseline methods (such as our accuracy above 67% v.s. 65% or below from baselines for the 350M MACs case). In general, All-in-One possesses the ability to maintain high accuracy for each switch with only one set of weights. When the battery level changes, the model can be easily switched with negligible accuracy degradation compared to other re-configurable DNN methods, greatly improving user experience.
Speed Performance
To show that All-in-One can reduce the inference variance when the frequency changes, we show the speed performance comparison of single model and our method. Table 4 and 5 show the results for ImageNet with = 2 and = 3 under various GPU frequencies on mobile devices using Resnet-18. Single model refers to the case with one single model to work with various frequencies.
From Table 4 and 5 we can notice that our structured pruning scheme can achieve inference speedup compared to the dense model on both datasets. Though reducing the latency, using a single sparse model cannot provide stable inference when the frequency changes. Table 5, given three frequencies 305mHz, 442mHz, and 587mHz, the model with 850M MACs is applied for the 587mHz frequency, the model with 480M MACs is used for the 442mHz, and the model with 350M MACs is adopted for the 305mHz.
Combining the results in Table 4 and 5, we have the following observations. (i) A more sparse model runs faster under the same frequency (such as 29.1ms for the model with 850M MACs v.s. 23.5ms for the model with 480M MACs under 442mHz using blockbased pruning in Table 5). (ii) Besides, for the same model, a higher frequency leads to a faster inference speed (such as 21.9ms under 400mHz v.s. 18.6ms under 525mHz for the model with 480M MACs in Table 4). The significant changes of the inference latency under various frequencies lead to large variance of inference latency (such as 29.64 for the single block-pruned model with 850M MACs in Table 5).
To deal with the different execution frequencies, our method uses models of various sparsity levels so that a more sparse model with less computations can be adopted for a lower frequency, leading to a smaller change to the inference latency. As shown in the results, the latency under difference frequencies are very close with our method (such as 21.9ms and 21.6ms under 400mHz and 525mHz, respectively, in Table 4). All-in-One can significantly reduce the variance of the inference latency. For example, In the case of = 2 on ImageNet, our variance with pattern-based pruning is just 0.02, much lower than the single pattern-based sparse model with 480M MACs (7.61) or the sparse model with 350M MACs (5.45), with a reduction rate on the latency variance as high as 381×. In the case of = 3, our variance is 0.76 and 0.97 for block-based and pattern-based pruning, respectively, also significantly smaller than the variance of the single model method.
We further show the variance on the CIFAR-10 dataset using Resnet-32 in Figure. 5, where the red line is the worst-case execution time. All-in-One not only reduces the inference time compared to storing a dense model but also mitigates the inference variance problem due to frequency change.
Memory Cost
We compare the memory cost of our method and other baselines in Table 6. To obtain models of sparsity ratios, our method only need to store one model and one re-scaled soft mask. Since each element of the mask represents a group of weights, the memory cost of a soft mask is much smaller than that of a model. For other methods, the naive method to train multiple models each for a sparsity ratio needs to store models. RT 3 [26] needs to store a model and masks so that each mask can convert the dense model into a sparse model corresponding to a specific sparsity ratio. LCS-U or LCS-S [25] need to one model for the compressible points case, or two models for the compressible lines case, together with a mask. Some other methods may have similar memory cost with ours such as Joslim and US-Net. But as shown in Table 2 and 3, unlike our method with high accuracy performance, they are not able to achieve state-of-the-art accuracy on CIFAR-10 or ImageNet. More specifically, take storing a Resnet-18 model on ImageNet with = 3 sparsity ratios as an example, All-in-One only requires two additional sets of BNs, which accounts for only 0.14% of the total number of parameters, and one set of re-scaled soft mask which accounts for 0.69% of the total number of parameters. The total extra parameters incurred by All-in-One is 0.83%. As All-in-One only needs to store the largest compact model, which is already compressed by 55% of the total parameters, the overall memory cost is saved by 54.17% comparing to saving a single dense Resnet-18. Therefore, All-in-One is memory-efficient.
Ablation Study
We compare the accuracy performance of various ( = 2, 3, or 4) in Figure 6. The model architecture is Resnet-18 and the accuracy is tested on CIFAR-10. Pattern-based pruning is applied. We test the accuracy of the models with 250M and 80M MACs for = 2, models with 250M, 80M, and 40M MACs for = 3, and the models with 250M, 80M, 55M, and 40M MACs for = 4, respectively. We also show the accuracy of the single sparse models which are separately trained for each target MACs.
As shown in Figure 6, generally a lower target MACs leads to a lower accuracy. Different values does not lead to significant changes on the accuracy. For example, the max accuracy difference of various and separately trained models at each target MACs is no larger than 0.5%. In some cases, our method can achieve even higher accuracy than the separately trained models (such as = 4 for the model with 40M MACs). Moreover, though increases, the memory consumption of our method does not increase as we only need to store one model and one soft mask for various .
CONCLUSION
We propose All-in-One, a highly representative pruning framework to work with dynamic power management using DVFS. Extensive experiments and real-world edge device evaluations show that Allin-One maintains high accuracy for each switch with low memory cost, and provide stable inference speed performance. It indicates that our work can deal with the limited hardware resource and unstable energy simultaneously on the widely adopted edge device.
ACKNOWLEDGEMENT
This work is partly supported by the Army Research Office/Army Research Laboratory via grant W911-NF-20-1-0167 to Northeastern University, National Science Foundation CCF-1937500, and CNS-1909172.
ICCAD '22, , San Diego, CA, USA © 2022 Association for Computing Machinery. ACM ISBN 978-1-4503-9217-4/22/10. . . $15.00 https://doi.org/10.1145/3508352.3549379
Figure 1 :
1Illustration of All-in-One framework in the case of three battery modes. Lower battery level requires a more sparse model to reduce the latency under a lower frequency. The framework learns one shared set of model weights and one soft mask. Each battery mode is paired with a threshold value ℎ
FiltersFigure 2 :
2(b) Fine-grained structured pruning (block-based pruning)Channels Filters (a) Fine-grained structured pruning (pattern-based pruning) Fine-grained structured pruning schemes.
Figure 3 :
3Illustration of switchable threshold. Each battery mode corresponds to one threshold. Dark blue circles represent weights participating computation. White circles are weights that do not need to be stored. Light blue circles indicate the weights pruned by ℎ .
Figure 4 :
4Switchable Batch Normalization. Algorithm 1 Pruning with Switchable Batch Normalization Require: Target MACs list C = {C } =1 Require: , : step size hyperparameters 1: Randomly initialize score values S 2: Initialize independent BN parameters for each target MACs 3: for = 1, · · · , do 4:
Figure 5 :
5Speed performance with = 3 of Resnet-32 on CIFAR-10 under various GPU frequencies.
Figure 6 :
6Performance of All-in-One with different switch number on CIFAR-10 using Resnet-18.
Table 1 :
1Frequency/Voltage levels on Adreno 650 GPU of
Qualcomm Snapdragon 865 Chipset in OnePlus T8 platform
N 1
N 2
N 3
N 4
N 5
clk/freq (mHz)
305
400
442
525
587
Vol (mV)
0.47-0.73 0.52-0.79 0.55-0.84 0.58-0.89 0.61-0.90
Table 1
1demonstrates the avail-
Table 2 :
2Accuracy of All-in-One and state-of-the-art reconfigurable DNN methods on CIFAR-10 dataset.Method
Pruning Scheme
MACs under Resnet-18
250M 80M
40M
RT 3 [26]
Unstructured
92.86 91.88
87.10
All-in-One (Block)
Structured
94.98 94.61
94.01
All-in-One (Pattern)
Structured
95.19 94.59
93.89
MACs under Resnet-20
20M 11M
7M
US-Net [31]
Structured
90.77 89.13
87.10
Joslim [4]
Structured
90.99 89.43
87.43
LCS-S [25]
Structured
86.56 83.13
78.75
LCS-U [25]
Unstructured
89.58 87.51
85.29
RT 3 [26]
Unstructured
88.13 86.62
83.83
All-in-One (Block)
Structured
91.83 91.23
90.65
All-in-One (Pattern)
Structured
92.49 90.97
90.42
MACs under Resnet-32
33M 20M
13M
US-Net [31]
Structured
92.06 90.56
89.83
Joslim [4]
Structured
92.26 90.70
89.82
RT 3 [26]
Unstructured
91.04 90.03
86.31
All-in-One (Block)
Structured
94.79 94.55
94.05
All-in-One (Pattern)
Structured
95.07 94.63
94.31
Table 3 :
3Top-1 accuracy performance on ImageNet dataset.Method
Pruning Scheme
MACs under Resnet-18
850M 480M
350M
US-Net [31]
Structured
66.51 63.42
61.49
Joslim [4]
Structured
68.49 64.51
61.82
LCS-S [25]
Structured
57.61 53.70
51.30
LCS-U [25]
Unstructured
68.75 67.51
65.62
RT 3 [26]
Unstructured
66.72 65.39
63.87
All-in-One (Block)
Structured
69.97 68.28
67.02
All-in-One (Pattern)
Structured
70.22 69.28
67.87
MACs under MobileNet-v2
210M 170M
150M
US-Net [31]
Structured
69.71 68.19
67.59
Joslim [4]
Structured
70.60 69.90
69.10
RT 3 [26]
Unstructured
68.12 66.97
65.65
All-in-One (Block)
Structured
70.79 70.01
69.25
Table 4 :
4Speed performance of single model and All-in-One with = 2 under various GPU frequencies.Different from single model, our method can dynamically reconfigure to a suitable model once the frequency switches. For example, in the case of = 3, our method can generate three sparse models with different MACs of 850M, 480M, and 350M on ImageNet. The more sparse model is used for the lower frequency. For example, inMethod
MACs under
Resnet-18
Latency under various
GPU frequencies (ms)
Var. of
latency
Reduct.
rate
400mHz
525mHz
Dense
1820M
33.7
28.1
15.68
-
Single model
(Block)
480M
25.7
21.6
8.41
187×
Single model
(Block)
350M
21.9
18.6
5.45
121×
All-in-One
(Block)
480M, 350M
21.9
21.6
0.045
1×
Single model
(Pattern)
480M
25.9
22.0
7.61
381×
Single model
(Pattern)
350M
22.2
18.9
5.45
272×
All-in-One
(Pattern)
480M, 350M
22.2
22.0
0.02
1×
Table 5 :
5Speed performance of single model and All-in-One with = 3 under various GPU frequencies.Method
MACs under
Resnet-18
Latency under various
GPU frequencies (ms)
Var. of
latency
Reduct.
rate
305mHz 442mHz 587mHz
Dense
1820M
37.5
31.9
26.6
29.71
-
Single model
(Block)
850M
35.7
29.1
24.9
29.64
39.0×
Single model
(Block)
480M
29.7
23.5
18.8
29.89
39.3×
Single model
(Block)
350M
25.1
20.3
15.8
21.63
28.5×
All-in-One
(Block)
850M, 480M,
350M
25.1
23.5
24.9
0.76
1×
Single model
(Pattern)
850M
35.9
29.4
25.5
27.60
28.5×
Single model
(Pattern)
480M
30.1
23.7
19.0
31.04
32.0×
Single model
(Pattern)
350M
25.3
20.6
16.0
21.62
22.3×
All-in-One
(Pattern)
850M, 480M,
350M
25.3
23.7
25.5
0.97
1×
5
0
10
15
20
Dense model results of ResNet-32 on cifar10 (70M MACs)
305mHz
442mHz
587mHz
5
0
10
15
20
All-in-One results of ResNet-32 on cifar10
305mHz
442mHz
587mHz
Table 6 :
6Memory cost of various methods. Stored model num. Stored mask num.Method
Multiple models
N
-
US-Net [31]
1
-
Joslim [4]
1
1
LCS-S [25]
1 or 2
1
LCS-U [25]
1 or 2
1
RT 3 [26]
1
N
All-in-One
1
1
50
75
100
125
150
175
200
225
250
MACs (M)
93.50
93.75
94.00
94.25
94.50
94.75
95.00
95.25
Accuracy (%)
Separately
Ours (N=4)
Ours (N=3)
Ours (N=2)
Dense
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| [] |
[
"Set Theory and Many Worlds",
"Set Theory and Many Worlds"
] | [
"Paul Tappenden [email protected] "
] | [] | [] | The 2022 Tel Aviv conference on the Many Worlds interpretation of quantum mechanics highlighted many differences between theorists. A very significant dichotomy is between Everettian fission (splitting) and Saunders-Wallace-Wilson divergence. For fission, an observer may have multiple futures, whereas for divergence they always have a single future. Divergence was explicitly introduced to resolve the problem of pre-measurement uncertainty for Everettian theory, which is universally believed to be absent for fission. Here, I maintain that there is indeed uncertainty about future observations prior to fission, so long as objective probability is a property of Everettian branches. This is made possible if the universe is a set and branches are subsets with probability measure. A universe which is a set of universes which are macroscopically isomorphic and span all possible configurations of microscopic local beäbles fulfils that role. If objective probability is a property of branches, a successful Deutsch-Wallace decision-theoretic argument would justify the Principal Principle and be part of probability theory rather than being specific to Many Worlds. Any macroscopic object in our environment becomes a set of isomorphs with different microscopic configurations, each in an elemental universe (elemental in the set-theoretic sense). This is similar to Many Interacting Worlds theory but the observer inhabits the set of worlds, not an individual world. An observer has many elemental bodies. | 10.3390/quantum5010016 | [
"https://export.arxiv.org/pdf/2306.03583v1.pdf"
] | 257,337,828 | 2306.03583 | affd6a727f3f633ccf4c201309b81591b57a200b |
Set Theory and Many Worlds
Paul Tappenden [email protected]
Set Theory and Many Worlds
an earlier version was published by Quantum Reports 2 nd March 2023 significant modifications are on pages 8 and 15-17
The 2022 Tel Aviv conference on the Many Worlds interpretation of quantum mechanics highlighted many differences between theorists. A very significant dichotomy is between Everettian fission (splitting) and Saunders-Wallace-Wilson divergence. For fission, an observer may have multiple futures, whereas for divergence they always have a single future. Divergence was explicitly introduced to resolve the problem of pre-measurement uncertainty for Everettian theory, which is universally believed to be absent for fission. Here, I maintain that there is indeed uncertainty about future observations prior to fission, so long as objective probability is a property of Everettian branches. This is made possible if the universe is a set and branches are subsets with probability measure. A universe which is a set of universes which are macroscopically isomorphic and span all possible configurations of microscopic local beäbles fulfils that role. If objective probability is a property of branches, a successful Deutsch-Wallace decision-theoretic argument would justify the Principal Principle and be part of probability theory rather than being specific to Many Worlds. Any macroscopic object in our environment becomes a set of isomorphs with different microscopic configurations, each in an elemental universe (elemental in the set-theoretic sense). This is similar to Many Interacting Worlds theory but the observer inhabits the set of worlds, not an individual world. An observer has many elemental bodies.
Many Faces of Many Worlds
If a Many Worlds interpretation of quantum mechanics is ever to become generally accepted there first has to be agreement on what the Many Worlds interpretation is, which is very far from being the case. There's even dispute about what to call it; are we to think in terms of a single branching world or a partitioning multiplicity of worlds? Some theorists work with the Heisenberg picture and a basic ontology of operators, others work with the Schrödinger picture and a basic ontology of wavefunctions. On both approaches there's scope for arguing that microscopic local beäbles are needed for a satisfactory physical ontology.
Within the diversity of views there's a fundamental dichotomy which I aim to resolve here. It's between the ideas that an observer may have a multiplicity of futures or always has a single future. Everett wrote of splitting in quantum measurement situations and it has generally been accepted that a well-informed observer cannot be uncertain about their future observations prior to Everettian fission. In an attempt to introduce pre-measurement uncertainty, Simon Saunders and David Wallace developed versions of Many Worlds theory that reject the concept of splitting, which was arguably Everett's key idea. There shall be more on this in the following section.
To begin with, I will address the thorny matter of understanding the relationship between probability and uncertainty. This will lead to the conclusion that pre-measurement uncertainty exists for a fission interpretation of branching, where an Everettian observer splits into observers seeing different outcomes. The only reason why that feels counterintuitive is that we've inherited a folk metaphysics which interprets future probabilities as properties of alternative possibilities. It's that which stands in the way of interpreting probabilities as properties of future coexistent actualities. A thought experiment helps to sugar this pill.
Understanding uncertainty as a cognitive state of assigning partial degrees of belief to coexistent futures requires assigning objective probabilities to those futures equal to the absolute squares of their quantum amplitudes. What's needed is an account of how this branch weight can be understood to constitute objective probability. I shall argue that it can do so if understood to be a subset measure. This leads to interpreting the universal wavefunction as being a set of deterministic universes which contain microscopic local beäbles. Objects in our environment become sets of objects which are macroscopically isomorphic but differ in their microscopic configurations. They are extended in configuration space, so to speak.
The result is a set-theoretic metaphysics for quantum mechanics which incorporates Everettian fission and microscopic local beäbles. It opens the way to new physics if the interaction between the universes which are the set-theoretic elements or our universe is the source of phase relations. After discussing spin, separability and locality in the context of this metaphysics, I close with further reflections on Lev Vaidman's World Splitter and its implications.
Probability and Uncertainty
For classical mechanics, all physical processes are regarded as deterministic. The idea of there being a mind-independent, i.e. objective, probability can only be applied to the determination of initial conditions, relegated to an inscrutable past. Probability arises, as in statistical mechanics, from the epistemic condition of ignorance on the part of observers. Lack of Laplacian omniscience as to the exact positions and momenta of particles entails that perfect prediction is impossible, so epistemic probabilities are assigned to fictional possibilities on the basis of statistical evidence. The gathering of that evidence involves the measurement of frequencies which can be regarded as surrogate approximations of epistemic probabilities, given the Law of Large Numbers. Uncertainty about the future is regarded as a mental state which involves the entertaining of partial degrees of belief about future observations equal to the epistemic probabilities assigned to the possibilities of those observations on the basis of the measured frequencies.
In the wake of quantum mechanics came the concept of stochastic physical processes, which are objectively probabilistic. Continuing to employ the metaphysics of possibility, a stochastic analysis of quantum processes with multiple possible outcomes supposes that one of those outcomes will be actualised by virtue of a random selection constrained by the objective probabilities of the possibilities. Those objective probabilities are determined by the Born rule when interpreted as assigning quantum amplitude to the fictional possibilities. As in the case of classical mechanics, stochastic theory interprets uncertainty about the future as the entertainment of partial degrees of belief about alternative possible futures but now the partial degrees of belief are equal to the supposed objective probabilities.
The idea of stochastic processes has widely been accepted as plausible by physicists. It can seem plausible that the half-life of an unstable particle is a mind-independent property of that object. However, an air of mystery surrounds the concept, often referred to as propensity. How can propensity be a property of an object? What's the ontic status of propensity?
Hugh Everett III replaced the concept of a stochastic process by that of a dendritic process. Consider, for example, Lev Vaidman's World Splitter [ i ]. Connecting to the device with a smartphone, you can choose a setup that will initiate a quantum measurement process with six equal-amplitude outcomes, i.e. a quantum die. The concept of a quantum die simply having six outcomes is an idealisation which I'll use for the sake of argument to begin with. Later, I'll consider the implications of abandoning that idealisation.
On "rolling" the quantum die, an Everettian observer fissions into six observers, each seeing one of six different outcomes which are all actual [ ii p. 459]. Where is uncertainty to be found? Presumably, Everett thought that it was nowhere to be found, which is why he first entitled his thesis Wave Mechanics Without Probability. Presumably, the apparent lack of uncertainty didn't bother Everett; after all, what's uncertainty got to do with physics? He was simply suggesting that the histories of quantum processes are typically not linear, they branch. They're partially ordered series of events, not well-ordered series. Everett's world wasn't many, it was one; a single branching world which Saunders once appropriately dubbed the quantum block universe [ iii ]. However, I'll continue to use the term Many Worlds since it's become virtually ubiquitous and is harmless enough so long as it's qualified in ways which will become clear as we go on.
As you roll Vaidman's quantum die, believing that you'll fission, can you really deny being uncertain about what you'll observe? Many theorists have thought so, including Vaidman himself [ iv §3]. In search of pre-measurement uncertainty, others have preferred to replace Everett's concept of fission with those of overlap and divergence, where the body of an observer at a time is one of a multitude of doppelgängers in erstwhile "parallel" worlds
[ v ][ vi ][ vii ]
. An observer has a single future but is uncertain as to what they'll observe because uncertain as to which type of world they inhabit; whether it's a world where outcome A will occur, or outcome B, and so on. The observer is subject to self-location uncertainty. However, inasmuch as that idea is motivated by trying to fill a lacuna left by supposedly absent premeasurement uncertainty for fission, it's unnecessary, as we shall see.
Content to do without pre-measurement uncertainty, Vaidman has kept to the traditional path by following Everett in believing that on rolling the quantum die you'll split into six "successors", each in a different branch and each seeing a different outcome. He writes:
The quantum world splitter lets you enjoy all the possibilities in life with no need to choose. Why choose one, when you can do it all (AT ONCE!) [ i ] (original emphasis)
The idea is that you decide in advance to act on each of six different enjoyable options according to which number is observed after the measurement. An obvious first objection is to ask in what sense it will be "you" acting in those different futures. Each of the six successors is a different observer seeing a different number and it's logically impossible for them all to be the same observer as you. It's clear that a metaphysics of persistence is needed to make sense of Everettian fission even before considering uncertainty.
Vaidman's term successors for post-split observers has generally been used by fission theorists and simply fails to meet this objection concerning personal identity. Note that the problem is avoided for the overlap and divergence interpretations of branching because no splitting occurs. Vaidman asserts that you can "do it all at once" but you are not any of your successors.
What's required is known as stage theory, which was introduced by Ted Sider in 1996 [ viii ]. It was first explicitly applied to Many Worlds theory in [ ix ] and most recently in [ x §2.1]. It's often thought that a persisting object is one-and-the-same thing from moment to moment; that's what could be called the folk metaphysics of persistence. However, it's not necessary to think of persistence like that. One can understand the history of an object as consisting of a series of momentary parts: stages. What Sider recognised is that an object, at any given moment, can be understood to be a stage of its history and that a persisting object can be understood to be one which has a special relationship with the stages which are called its past and future temporal counterparts. A persisting object was its past temporal counterparts and will be its future temporal counterparts. Contrary to folklore, a persisting object (or observer) doesn't have to be one-and-the-same thing from moment to moment after all. If he were to adopt stage theory, Vaidman could say, without fear of contradiction, that you will be each of six different observers, each seeing a different number after you roll the quantum die.
What's the ontic status of non-present stages on this account? That depends on ones view of the ontic status of past and future states of affairs. On the eternalist, "block universe" view, which I suggest is most appropriate for Many Worlds theory, the past and future temporal counterparts of a persisting object will be objects which exist in the past and future of the present object. On non-eternalist views, the present object will bear the temporal relations was and will be to objects which did and will exist.
The Logic of Uncertainty
For stochastic theory, a quantum die involves an objectively probabilistic process with six possible outcomes. One of those outcomes will be actualised randomly and each possibility can be assigned an objective, mind-independent probability. To put it another way, the quantum die has a propensity. The propensity is such that each of the six possible outcomes has an equal probability of being actualised. There's long been an aura of mystery about propensity which I hope to dispel.
For stochastic theory, an observer rolling a quantum die is uncertain about the future for the following reason. As with classical mechanics, uncertainty is understood to be a mental state involving the assignment of subjective probabilities, degrees of belief, to alternative possible futures. Stochastic theorists derive the values for the degrees of belief by appealing to what's become known as the Principal Principle, which is basically the idea that an observer should assign subjective probabilities to possible outcomes equal to what they believe the objective probabilities of those outcomes to be [ xi §2.2]. For stochastic theory, the degrees of belief are guided by what are taken to be objective probabilities, whereas for classical mechanics the degrees of belief are guided by epistemic probabilities arising from ignorance of microstates. According to stochastic theory, an observer is uncertain about the future prior to rolling the quantum die because they assign degrees of belief 1/6 to each of the possible outcomes, whose objective probabilities are 1/6. Vaidman, like other Many Worlds theorists such as David Papineau and Sean Carroll, has followed Everett in understanding the process involved in rolling the quantum die to be dendritic rather than stochastic. All six outcomes occur, each in a different branch of physical reality. Each branch is assigned the same quantum amplitude as is assigned to the possible outcomes of stochastic theory and, since the branches actually exist, quantum amplitude must be a physical property which they possess. The absolute square of quantum amplitude is the quantity which stochastic theorists identify with objective probability and that can seem acceptable when amplitudes are assigned to alternative possibilities, but can it be acceptable when amplitudes are assigned to coexistent actualities? Can objective probability be a property of branches? It's certainly logically possible, for if the objective probability of all the outcomes occurring is 1 then that entails that each of the outcomes will occur but it does not give reason to believe that the objective probabilities of each of those individual outcomes must also be 1. The objective probability of the occurrence of each outcome can be 1/6, contrary to the common belief that if an event will occur then the probability of its future occurrence must be 1. That may seem to involve a contradiction because a well-informed observer must be certain that any particular outcome will occur whilst assigning it an objective probability of 1/6. However, the observer is not required to apply the Principal Principle here, where the future occurrence of the outcomes is concerned.
There is as yet no agreed justification of the Principal Principle; it's used by stochastic theorists simply because it seems self-evident. If you believe that a process has six possible outcomes whose objective probabilities are 1/6, what else can you do but assign a degree of belief of 1/6 to the future occurrence of any particular outcome? However, stochastic theorists are in the habit of applying this idea in the context of multiple futures thought of as alternatives, whereas in the context of the dendritic quantum die the futures are being thought of as coexistent. In that context, the application of the Principal Principle is overruled by logical consequence because, again, if the objective probability of all outcomes occurring is 1 then, necessarily, each outcome will occur, whatever its individual objective probability of occurrence. The observer can assign a subjective probability of 1 to all the outcomes occurring because the objective probability of their combined occurrence is 1. This entails that the observer is certain that each outcome will occur, despite the objective probability of the occurrence of each outcome being 1/6. I should mention in passing that this brings and alternative perspective to the Deutsch-Wallace decision theory argument that observers should assign degrees of belief to future measurement outcomes in accordance with the Born rule [ xii pp. 160-89]. If objective probability can be understood to be a property of future branches, then the decision-theoretic argument, if good, constitutes a justification of the Principal Principle and thus belongs to the philosophy of probability rather than to Many Worlds theory.
In what sense, then, can an observer be uncertain about the future prior to rolling the quantum die? They can be uncertain in the sense of assigning a subjective probability of 1/6 to each of the future observations. The observer will be each of six observers seeing different outcomes whose objective probabilities are 1/6. Applying the Principal Principle, the observer assigns a degree of belief of 1/6 to the future observation of each outcome. They are certain as to what will occur and uncertain as to what they'll observe. Whether the future observations are understood to be alternative possibilities or coexistent actualities is beside the point, uncertainty is the very same thing in both cases. The thrall of a folk metaphysics of alternative possibilities can make this hard to grasp.
Should doubt remain, a thought experiment demonstrates that an observer can believe that they are assigning subjective probabilities to alternative possible future observations whilst they are in fact assigning them to coexistent actual future observations. This involves a set-theoretic metaphysics for physical objects which leads directly to an explanation of how objective probability can be a physical property of Everettian branches.
Many Worlds without Everett
What cosmologists call the observable universe is a finite region of space which is currently estimated to have a radius of about 46 billion lightyears. Since there is as yet no evidence that space is finite, there may be a countably infinite number of such regions which are observationally identical.
Consider an observer who inhabits one of an infinite set of observationally identical universes where quantum dice are, hypothetically, stochastic. On rolling a die, an infinite number of doppelgängers in the set of erstwhile "parallel" universes move in concert and an infinite number of quantum dice are rolled. The set of universes subsequently partitions into six subsets whose measures are necessarily 1/6, the reason being that what it means in stochastic theory for an outcome of a particular type of process to have an objective probability of 1/6 is that the subset measure for that outcome on an infinite set of such processes is 1/6, the appropriately-named probability measure. Now drop the ubiquitous assumption of folk metaphysics that there's a one-to-one relation between observers and doppelgängers. This requires an exercise in what Donald Davidson has called radical interpretation [ xiii ]. The idea is that truth values must be preserved for relevant utterances by an observer on the original interpretation and the alternative. On the original interpretation of the parallel universes setup, a single utterance by an observer is tokened by a single noise made by a single doppelgänger, but on the alternative interpretation a single utterance is tokened by the infinite number of isomorphic noises emitted by each of the doppelgängers. Likewise for intensional acts: on the original interpretation the act of rolling a die is tokened by the movements of a single doppelgänger, whereas on the alternative interpretation the act of rolling a die is tokened by the parallel movements of all the doppelgängers. On the alternative interpretation, a single die is rolled by a single observer; a single die which is constituted by all the parallel dice. This is the unitary interpretation of mind [ xiv §2].
A novel use of set theory is required [ x §4]. Following Willard Quine, physical objects in each individual observable universe are to be construed as self-membered singleton sets which have come to be known to logicians as Quine atoms [ xv p. 31]. Quine spent much energy trying to find a way to do mathematical logic without reference to sets, but failed. Having become resigned to the necessity of sets, he noticed that non-sets could be brought into the set-theoretic fold in a way which is harmless in the sense that it doesn't impair the use of set theory in mathematics. Sets had always been understood to be abstract objects but Quine demonstrated that concrete objects could be construed as sets too. What's required for the unitary interpretation of mind is the hypothesis that any set of Quine atoms has all and only the properties its elements share other than number of elements and value-definiteness. I'll say more about this assumption and its consequences in the next section.
So now there's a single subject whose body is an infinite set of doppelgängers and who rolls a die which is an infinite set of hypothetically stochastic dice. When the single observer rolls the quantum die, each of the doppelgängers which are set-theoretic elements of the observer's body moves isomorphically so that the parallel quantum dice are caused to roll. In each elemental universe the outcome gives rise to sensory input to a doppelgänger so that, as the set of elemental universes partitions into six subsets with different outcomes, so the set of doppelgängers partitions into six subsets with different sensory input. Differences in sensory input give rise to different observations so the single observer fissions into six observers seeing different outcomes. The bodies of the six downstream observers are each an infinite set of doppelgängers whose subset measures relative to the body of the upstream observer are 1/6, i.e. the probability measure.
For this non-Everettian cosmological setup, the single die of the alternative interpretation is not stochastic, it's dendritic. The conclusion must be that an observer can be mistaken when believing that their uncertainty prior to rolling a quantum die derives from there being six alternative possible outcomes which all have an objective probability of 1/6. Their uncertainty can derive from there being six coexistent actual outcomes which all have an objective probability of 1/6.
A Metaphysics for Everettian Fission
According to Everett, the quantum die splits into six dice, each showing a different number, and the observer splits along with it. As he saw it, of course, there can be no probability since there's no uncertainty, thus his pursuit of a back door to probability via typicality.
Everett's key idea was that the concept of a stochastic process could be replaced by that of a dendritic process. To make it fully intelligible, there has to be an account of how a wellinformed observer can be uncertain about future observations in a quantum measurement situation, i.e. observations they will make, together with other nearby observers who have split too, along with the measuring device and the laboratory. We now have an account:
Uncertainty without alternatives
Uncertainty about future observations is the cognitive state of assigning partial degrees of belief to multiple observations; whether those observations are thought of as alternative possibilities or coexistent actualities is irrelevant because the occurrence of a future observation doesn't entail that the objective probability of its occurrence is 1.
If it's useful to our understanding of physics to employ the concept of fission rather than that of stochasticity, then we are free to do so. To be certain that all outcomes will occur entails that each will occur. Therefore we can be certain that any particular outcome will occur whilst believing that the objective probability of its occurrence is 1/6. Assuming the Principal Principle, the observer assigns a degree of belief of 1/6 to the future observation of that outcome, by observers who they and their laboratory colleagues will be. How can the real-world quantum die split in such a way that the objective probability of each of its immediate future temporal counterparts is 1/6? By being an infinite set which partitions into subsets with probability measures of 1/6. The cosmological thought experiment provides the framework for a metaphysics for quantum fission which incorporates a modification of Quine's definition of concrete objects as being self-membered singleton sets:
Concrete Sets
A set of Quine atoms has all and only the properties which its elements share except number of elements and value-definiteness.
That this assumption has an Alice in Wonderland aspect, which has long been associated with quantum mechanics, is not to be denied. It entails that the set of a duck and a rabbit, each weighing a kilo, is a duckrabbit, a warm-blooded creature which is neither mammal nor fowl and has an indefinite number of feet, though no more than four and no less than two, and which weighs a kilo. To see why, consider isomorphic rooms, one on the left and the other on the right, each containing a doppelgänger. Put the duck and the rabbit in identical boxes and introduce them isomorphically to the rooms, the duck on the left and the rabbit on the right. The single observer's body, the set of the two doppelgänger, makes parallel movements resulting in boxes being moved to scales so that the observer reports that the box which is the set of two boxes contains something which weighs a kilo. It's the duckrabbit. When the observer opens their box they split into Lefty who finds a duck and Righty who finds a rabbit. If the set of the duck and the rabbit is a third creature which weighs a kilo, is the unit set of that set an object which weighs a kilo too? And the unit set of that set, and so on? There's no good reason to suppose so. Up until now the unit sets of concrete objects have generally been supposed to be abstract and the Concrete Sets rule doesn't entail that that's not the case; it says nothing about sets of sets of Quine atoms. Set theory is metaphysical Meccano which can be applied to physics in any way we wish, so long as it leads us not into contradiction.
From Metaphysics to Physics
The cosmological thought experiment invokes an infinite set of elemental parallel stochastic universes populated by Quine atoms. However, the whole point of Everett's idea was to replace stochasticity with fission. For Everettian physics, the elemental universes must have deterministic, linear histories with branches emerging as the set partitions. Pilot Wave theory provides possible candidate elemental universes [ x ]. Interacting worlds theory also provides candidate universes with a purely particle ontology [ xvi ][ xvii ][ xviii ], though it may be replaceable by a field ontology [ xix ]. However, both Pilot Wave and Interacting Worlds theories face problems in dealing with relativistic quantum mechanics and involve nonlocality in the sense that there can be causal connections between spacelike-separated events.
An often-vaunted advantage of Many Worlds theory is that it doesn't face those problems. When conceived of, following Everett, as a pure wave theory, all of the physics used by physicists can be recovered, so the story goes. In defence of Many Worlds as a pure wave theory, Wallace has recommended a mathematics-first approach to the ontology of quantum mechanics, which excludes microscopic local beäbles as objects bearing properties [ xx ]. The project of ontic structural realism, which he defends, is an interesting one but I suggest that it's better suited to a proto-spacetime ontology than to quantum mechanics, where stuff happens in spacetime.
As Louis de Broglie once remarked, a Schrödinger wave is supposedly in configuration space but lacks configurations [ xxi p. 381]. There are currently other attempts to fix that by introducing microscopic local beäbles to Many Worlds theory [ xxii ][ xxiii ]. What I've been describing is a metaphysical framework which is independent of whatever physics may actually be involved. Assuming a particle ontology, just for the sake of illustration, this framework entails that any macroscopic object in our environment is a set of objects which are macroscopically isomorphic but differ in their microscopic particle configurations. There's a sense in which we inhabit configuration space. Objects in our environment have spatial extension and they're extended in configuration space too, as are our bodies. In effect, the unitary interpretation of mind, is a consequence of assuming that objects in our environment are extended in configuration space.
As explained in §2.2, recall that the unitary interpretation of mind is the idea that multiple doppelgängers instance a single observer, not multiple observers in qualitatively identical mental states. If your body is understood to be extended in configuration space, in the sense of being a set of bodies that are only anisomorphic at the level of microscopic configurations, then your mental state, now, is instanced by a multiplicity of doppelgängers. You are legion, to adapt a biblical phrase.
In light of this, think about Vaidman's quantum die again. It's an apparatus in a physics lab which is a set of labs including all possible configurations of particles consistent with the Born rule. That's the reason why the set which is the die partitions in the same way that a set of hypothetically stochastic dice would. However, is Vaidman's die an infinite set? That would depend on whether spacetime is continuous. Can the branch subset measures still be identified with objective probabilities if the set of deterministic quantum dice is finite? Perhaps not, in which case perhaps an effective Law of Large Numbers is good enough for very large samples. In any case, given the cosmological setup, if there's a finite number of configurations there can still be a countably infinite set of each configuration until such time as we have evidence that space is finite, because there can be an infinite number of observable universes which are observationally identical.
According to this framework, an unstable particle in our environment would be a set of particles constantly partitioning into a decay subset of increasing measure. An observer with a detector would be constantly splitting into an observer not seeing decay and observers seeing decays at later and later times. The probability of observing decay within a given period would depend on the rate of change of the decay subset measure for that type of particle, i.e. its propensity to decay. Likewise, we are free to hypothesise that the quantum die is a very large or infinite of set of dice which will partition in the same way as a set if stochastic dice would. The subset measures of Vaidman's six downstream dice will be 1/6 relative to the upstream die, because that's the probability measure.
For another illustration of the idea that objects in our environment are extended in configuration space, consider a free electron at any given moment. It's a set of elemental electrons which are in different corresponding positions and have different momenta in the elemental universes. The term elemental here is strictly set-theoretic. Again, our universe is being construed as a set of universes and any object in an observer's environment is a set of objects. A free electron in our universe is a set of elemental electrons which are on different trajectories in the universes which are elements of our universe. Our electron doesn't lack a trajectory, it has an indefinite trajectory. At any moment, the electron has indefinite position and momentum in our non-elemental universe, where objects have a definite position and momentum only if their elements have corresponding positions and momenta in the elemental universes. There's more on this concept of correspondence in §4.1.
The introduction of particles as local beäbles in the way I've described, as being settheoretic elements of particles in our environment, effectively preserves the full structure of the wavefunction and avoids the drawbacks of Pilot Wave and Interacting Worlds theories, as I shall now explain.
The World as Wavefunction
Consider the wavefunction of a free electron understood in terms of the set-theoretic metaphysics. For a pure wave theory, any region of space is assigned a quantum amplitude and the absolute square is taken to yield the probability of finding the electron there if a position measurement is made. There's no account of how an electron can be "spread out" in this way, hence Wallace's appeal to a thingless ontology. However, for the set-theoretic metaphysics the absolute square of amplitude for a spatial region yields a subset measure for the single free electron, which is a set of elemental electrons. Each elemental electron in that subset is at an elemental location which is an element of a location within the given spatial region. There is thus a fully concrete interpretation of the wavefunction in that region for the free electron in an observer's environment, which I shall call an environmental electron. It's not in any sense counterfactual. Every location in that region is a set of elemental locations where elemental electrons may be actually located.
Notice that this appears to invoke a substantivalist interpretation of spacetime. To construe the universal wavefunction as being a set of universes "extended" in configuration space entails that our environmental spacetime is a set of elemental spacetimes. An environmental spatial point is a set of corresponding elemental spatial points, each in an elemental universe, one where there's a particular configuration of microscopic local beäbles, each of which is a Quine atom. That means that Everettian fission involves the fission of spacetime itself. I'll say more about this in the following section.
It's often said that the paradox of superposition is dealt with in Many Worlds theory by understanding superpositional states as being composed of definite states on different branches. Thus Schrödinger's cat is dead on some branches and alive on others (sometimes put as dead in one world and alive in another). However, Everettian theory has only ever given an account of macroscopic superpositions in this way. Mystery still surrounds the concept of microscopic superpositons, hence, again, the motivation for defending a pure wave theory in terms of an ontology that doesn't involve objects bearing properties. The settheoretic metaphysics resolves this problem by construing microscopic superpositions as also being constituted by multiple definite states. Again, the free electron in an observer's environment becomes an extended object, extended in configuration space. It does so by being a set of electrons, each of which is on a different trajectory in a universe which is a settheoretic element of the observer's universe.
But that only provides a momentary snapshot of the electron's wavefunction. There needs to be the dynamics of unitary evolution too; where is that to come from? It strikes me that the most plausible option here is to adopt some version of Interacting Worlds theory. The individual elemental universes which contain the set-theoretic elements of the observer's electron interact in such a way as to generate the unitary dynamics. Here, there's scope for new physics in order to understand how universes separated in configuration space interact. The possibility of such new physics has already been suggested by Interacting Worlds theorists, but what must be stressed is the radically new perspective that the set-theoretic metaphysics brings to Interacting Worlds theory.
All the difference is in how the observer is situated. For extant Interacting Worlds theory the observer is situated within an individual world, which corresponds to what I've been calling an elemental universe. For the set-theoretic metaphysics, the observer is situated in the set of interacting universes; objects in the observer's environment, including their body, are multipleton sets. The observer's universe becomes a set of interacting universes.
In a sense, the observer spans the set of interacting universes. They span the universes in the sense that the mental states of an observer are instanced by a multitude of brains in a multitude of doppelgängers. Each of those brains is a set-theoretic element of the brain to which an observer indexically refers by a tap on the skull. The observer's mental states are instanced by a multitude of brains rather in the way that a single novel is instanced by a multitude of books.
Extant Interacting Worlds theory involves causal nonlocality because particle trajectories in the observer's world are mutually interactive at spacelike separation in virtue of the interactions between worlds. By construing our universe as a set of interacting universes rather than an element of a set, this problem is avoided. Causal locality is preserved for Many Worlds theory, as we shall see.
Being Indefinite
Consider an observer who rolls a quantum die blindfolded. According to Vaidman, the observer will fission into six successors, each on an Everettian branch where the outcomes are different. According to the set-theoretic metaphysics, the body of the observer will partition into six subsets and each subset will have elements which are doppelgängers in the presence of elements of one of the six outcomes. The partitioning of the observer's body will be caused by slight physical effects propagating from the six different post-roll dice, even if those effects are very slight indeed, such as gravitational differences. However, the observer themself won't fission because the doppelgängers aren't different enough to instance distinct perceptual states. The observer doesn't fission because their perceptual apparatus is screened by the blindfold. Post-measurement and pre-observation there'll be a single successor whose body is the set of all the doppelgängers in the six subsets. The environment of that single successor will contain a die with subsets which are six dice displaying different numbers. In other words, the die in the environment of the post-measurement, pre-observation observer will be in an indefinite number-state [ xiv p. 14]. Now consider a terrestrial observer watching the roll of a quantum die on Mars through a powerful telescope. Post-roll on Mars, there'll be no causal influence on the observer's body on Earth for several minutes and so there'll be no relevant partitioning of the observer's body. When light from the roll of the die reaches the observer's eyes, their body will partition into six subsets and then, after retinal states have been processed, there will be six sets of doppelgängers instancing elements of six different perceptual states and the observer will have fissioned. During the intervening few minutes, the quantum die will have been in an indefinite number-state relative to the terrestrial observer but not relative to a Martian observer.
For the set-theoretic metaphysics, an observer cannot fission into observers seeing different outcomes until the observer's body partitions into subsets which are bodies instancing different cognitive states. Note that this has nothing to do with consciousness, it has to do with mental content. It's well established by experimental psychology that we can perceive states of the world around us whilst not being conscious of those perceptions; and we can recall information to consciousness which we were unconscious of knowing a moment earlier. Two distinct observers may be in identical conscious states and yet act differently because of different unconscious mental content.
Necessarily, quantum measurements with multiple outcomes which occur at spacelike separation form an observer are in indefinite outcome states relative to that observer. This casts doubt on the idea that the observation of correlations between spacelike-separated measurements on entangled particles entails nonlocal causation. That conclusion only necessarily follows if measurements always have single, definite outcomes. However, the settheoretic metaphysics construes the observer's universe as a set of universes and within the elemental universes there seems to be nonlocality for the reasons acknowledged by Pilot Wave and Interacting Worlds theorists. So, is non-locality involved on the set-theoretic view after all?
Not necessarily, because apparent nonlocality at the elemental level is not really nonlocality at all. It would be if observers inhabited the individual elemental universes but the whole idea is that they do not. Observers inhabit sets of elemental universes and, at that level, nonlocality would seem to be absent for the reason I've just given. Elemental nonlocality is not nonlocality because elemental locations are not locations. For the set-theoretic metaphysics, there's no reason to suppose that there's causal influence between spacelikeseparated locations, which are locations in an observer's spacetime, a set of elemental spacetimes. This will become clearer with an analysis of EPR-Bell experiments, and what's needed by way of preparation for that is the set-theoretic characterisation of spin and entanglement coming in the following subsections, but first a few words on what may seem a contradiction.
The Concrete Sets rule is that sets of concrete objects have all and only the properties which their elements share other than number of elements and value-definiteness. How then can there be no nonlocality in a set of universes whilst there seems to be a form of nonlocality in the universes which are its elements? The reason is that the locality issue is tied to the concept of value-definiteness, as we've just seen and of which there's more to come. 1
Spin
Spin poses a further challenge to the set-theoretic metaphysics. We have to take a step back. The universe is being construed as a set of elemental universes. An electron only has a location if all its elemental electrons are at corresponding elemental locations. For the sake of argument, consider an electron to be a point-particle. In that case, it's located at a spatial point only if all its elements are at corresponding elemental points.
The correspondence can be thought of in the following way. For an observer at a time, the universe exhibits a definite distribution of objects in space on the surface of the past light cone. The observer's universe at a time is to be construed as the set of universes containing all possible configurations of particles (or fields) consistent with that definite distribution of objects. A particle only has a position in the observer's universe if its elements are all at the same position relative to isomorphic distributions of macroscopic objects in each elemental universe.
The set-theoretic metaphysics interprets objects with indefinite properties as sets of objects with definite properties, so, when it comes to spin, elemental electrons cannot have indefinite spin relative to any axis. An elemental electron must have a definite spin, i.e. up or down, relative to some axis, period. Just as an environmental free electron has an indefinite trajectory whilst the electrons which are its elements follow trajectories, likewise, an environmental electron can have indefinite spin relative to all axes but one whilst the electrons which are its elements simply have definite spin relative to a single axis. I shall continue to italicise these terms to avoid confusion. Observers always make measurements on environmental electrons which are sets of elemental electrons (which are Quine atoms, like all physical objects in an elemental universe). The spin of an elemental electron can't be measured. Quantum measurement is something we do in our universe but not in the elemental universes which are its set-theoretic elements. Bearing that in mind, here's an attempt to provide a set-theoretic metaphysics for spin.
In the spirit of string theory, let an elemental point be baton-like, having an orientation. In that case, an environmental point in the observer's universe will have an orientation too, following the Concrete Sets rule, since all its elements have orientations. Let an observer's environmental point be a set of elemental points with all possible orientations. In that case, an environmental point will have an indefinite orientation. We're in the habit of thinking of spatial points as lacking orientation but now the idea is that a spatial point has indefinite orientation because it's a set of elemental points with different orientations. Again, that's like the idea that a free electron in our environment doesn't lack a trajectory but rather has an indefinite trajectory since the electrons which are its elements are not on corresponding trajectories in each elemental universe. As a point-particle, an elemental electron can be supposed to have an orientation too. Let any elemental electron have an orientation which is exclusively either parallel or orthogonal to the elemental point where it's located. We can adopt the convention that an elemental electron which is parallel is spin-up and an elemental electron which is orthogonal is spin-down. An environmental electron which is x-spin-up can then be construed in the following way. All its elemental electrons which are located at elemental points oriented parallel to the x-axis are spin-up. A little formalism may help.
Let e E be an environmental electron and e e an elemental electron. Likewise, let p E be an environmental spatial point and p e an elemental point. Every elemental point has an orientation, so, for an elemental point oriented parallel to the x-axis, we can write xp e . Every elemental electron is located at an elemental point (e e @p e ) and is either oriented parallel or orthogonal relative to that point, with parallel being spin-up and orthogonal being spin-down. So we can write e e @ up xp e for an elemental x-spin-up electron and e e @ down xp e for an elemental x-spin-down electron. An x-spin-up environmental electron is defined thus: e E (x-spin-up) iff ∀e e [(e e ∊e E )&(e e @xp e )] → [e e @ up xp e ] An x-spin-up environmental electron measured on the z-axis has equal probabilities for being measured spin-up and spin-down. Given the earlier analysis of objective probability in terms of subset measure, this implies that the environmental x-spin-up electron has a subset of elemental electrons which are all at elemental points parallel to the z-axis and, of that subset, the spin-up and spin-down elemental electrons are of equal measure. In other words, the measures of {e e @ up zp e } and {e e @ down zp e } on {e e @zp e } are equal.
As a consequence, an observer measuring an environmental x-spin-up electron on the zaxis will fission into observers whose bodies are of equal measure, one observing an environmental electron which is z-spin-up and the other observing an environmental electron which is z-spin-down. For the post-measurement z-spin-up environmental electron, all its elemental electrons which are at elemental points parallel to the z-axis are spin-up; correspondingly for the post-measurement environmental electron which is z-spin-down.
Why do the post-measurement observers have bodies of equal measure? Recall the cosmological thought experiment with an infinite set of hypothetically stochastic universes. Now think of an equal-chance measurement being made in each universe, i.e. a quantum coin flip. The set of universes will partition into two subsets of equal probability measure where different outcomes occur. For this setup, if the unitary interpretation of mind is adopted, there's a single observer at the outset whose body is a set of bodies (doppelgängers) that partitions into two subsets of equal measure, which are the bodies of the two post-coin-flip observers. Recall also that in §3.1 the set of hypothetical stochastic universes was replaced by a set of Pilot Wave or Interacting Worlds universes, which would partition in the same way as a set of stochastic universes would, i.e. the branch subset measures would take the same values. The reason for this would be that the set of hidden-variable universes would include all possible hidden-variable configurations consistent with the Born rule (corresponding to the assumption of "equilibrium" in Pilot Wave theory). To put it another way, the universal wavefunction is being interpreted as a set of hidden-variable universes which includes all possible configurations and so Everettian branching is construed as the partitioning of a set, where the subset measures just are the outcome probabilities. To repeat, because it's a very counterintuitive, the perspective is that which arises from the unitary interpretation of mind coupled with the idea that environmental objects are sets of elemental object extended in configuration space, so the fissioning of an observer consists in the partitioning of the observer's body into subsets.
To return to spin, let an environmental x-spin-up electron have subsets of elemental electrons at elemental points parallel to all possible orientations. For any orientation ô, the subset of elemental electrons at elemental points parallel to ô has two subsets, namely {e e @ up ôp e } and {e e @ down ôp e }, which are non-elemental spin-up and spin-down electrons, since any set of elemental electrons is an electron. They become the post-measurement environmental electrons if a spin measurement is made on the ô-axis. The measures of those subset electrons relative to {e e @ôp e } are the probabilities for measuring spin-up and spindown on that axis.
This provides a characterisation of spin for the set-theoretic metaphysics, however, before we can apply it to the analysis of EPR-Bell experiments we need a set-theoretic characterisation of entanglement.
Entanglement
A pair of electrons in a singlet state has zero net spin because they have opposite spins. Emitted from a source and collimated, the wavefunction propagates as a sphere with peaked amplitudes in opposite directions. The wave propagates in configuration space but the settheoretic metaphysics provides, at any given moment, a characterisation of the wave as a distribution in 3D space. Both the environmental electrons are sets of elemental electrons. At any region of environmental space at a time (the space in the environment of the observer), there will be subsets of the elemental electrons which are elements of the environmental electrons and which are located at elemental points which are the set-theoretic elements of environmental points in the given environmental region. This is what I meant when I said earlier that the set-theoretic metaphysics completely recovers the structure of the wavefunction. Here, we see an instantaneous reconstruction (relative to some simultaneity hyperplane). The dynamics, which provides the phase aspect of the wave, might be recovered via an Interacting Worlds theory, as I've already suggested.
Consider congruent environmental spatial regions of measurement, A and B, which are equidistant from the source and spacelike-separated. Both environmental regions are sets of elemental regions containing electrons which are elements of each of the two entangled environmental electrons. For both A and B, some elemental points will be the location of one of the elements of one of the two entangled electrons, assuming that no two elemental electrons can be located at the same elemental point. In each environmental region, for every elemental point which is the location of an element of one of the entangled electrons, there will be another elemental point which is the location of an element of the other entangled electron. Since electrons lack haecceity, there's no sense in which the entangled electrons can be permutated, but there are two of them nonetheless, so there are two distinct sets of elemental electrons which are the elements of the entangled environmental electrons.
Furthermore, for each of the two environmental regions there will be elemental points of all orientations which are the locations of electrons that are the elements of the two entangled electrons. Also, for every orientation, there will be two non-elemental electrons which are subsets of elemental electrons of equal measure, because if either electron has its spin measured relative to any axis there are equal probabilities for spin-up and spin-down. One of those non-elemental electrons will be an environmental spin-up electron post-measurement, and the other an environmental spin-down electron. Both the entangled electrons will be equally present in both environmental regions, A and B, so to speak, where the presence of a free electron in an environmental region is construed as its having subsets of elemental electrons which are located at elemental points which are elements of environmental points in that environmental region. So the two entangled environmental electrons are separable because they're two distinct objects. They are two distinct sets of elemental electrons with no elements in common. There's a single wavefunction for two distinct environmental electrons. This analysis clarifies the account I gave in [ x §3] which wrongly assumed that the entangled electrons are non-separable.
Being entangled, the two environmental electrons are causally linked because of their common origin. If one of the electrons is measured ô-spin-up in region A then the other, if measured in region B, must be ô-spin-down and vice versa. To see why this doesn't violate causal locality we now need to think about the EPR-Bell setup.
EPR-Bell
We are to consider Alice and Bob making spin measurements on the singlet state in regions A and B. When Alice makes her measurement on the ô-axis she fissions into Alice UP and Alice DOWN whose bodies occupy the local environmental spatial regions A UP and A DOWN , which are subsets of region A. The set of the elemental points which are the elements of points in A is the fusion of the two distinct non-elemental subsets whose elements are elemental points in A UP and A DOWN. The fissioning of Alice's body involves the fissioning of the environmental spatial region which it occupies. Prior to measurement, Alice inhabited an environmental region which was a set of elemental regions, each in an elemental universe. Post-measurement, Alice UP and A DOWN inhabit two distinct environmental regions which contain elemental points in two distinct subsets of elemental universes.
What distinguishes the environmental regions A UP and A DOWN is that they contain two different environmental electrons and two different subsets of the elemental bodies which are the set-theoretic elements of the bodies of Alice UP and Alice DOWN.. Region A UP contains all the elements of Alice UP 's body and none of the elements of Alice DOWN 's, and vice versa. In Bob's absolute elsewhere, Alice's body has evolved into an indefinite ô-spin-measurement state because it has partitioned into subsets which are the bodies of Alice UP and Alice DOWN , on different branches of the quantum multiverse.
Notice that Alice's measurement doesn't change anything in the set-theoretic structure of region B. Keeping things simple to begin with, let Bob make his measurement on the ô-axis too. He fissions into Bob UP and Bob DOWN in regions B UP and B DOWN . The key point here is that, because of the entanglement, these two subsets of region B have different set-theoretic structures from the regions A UP and A DOWN whilst regions A and B are set-theoretically isomorphic. Necessarily, Alice UP cannot have measured the same electron as Bob UP and Alice DOWN cannot have measured the same electron as Bob DOWN. That's a consequence of the two entangled environmental electrons having been in causal contact at their origin and so having opposite spins relative to any axis. Now Bob's body has evolved into an indefinite ô-spin-measurement state relative to both Alice UP and Alice DOWN and Alice's body has evolved into an indefinite ô-spinmeasurement state relative to both Bob UP and Bob DOWN . These measurement results of the Alices and Bobs can't come into causal contact sooner than half the light-time between regions A and B. To see why, consider Clotilde, halfway along a light path between regions A and B and watching the observers who will be the Alice and Bob who make the measurements. When Clotilde sees the result's of Alice's and Bob's measurements she fissions into Clotilde AliceUP+BobDOWN and Clotilde AliceDOWN+BobUP. As Cai Waegell and Kelvin McQueen put it, "A world containing a Bob and an Alice is only created when the wavefront from Alice's measurement meets the wavefront from Bob's measurement" [ xxiv §6]. However, it's unclear why they use the term wavefront; it's rather a matter of the past light cones of Alice's and Bob's future temporal counterparts coming to overlap.
Things get more complicated if Bob makes his measurement on a different axis from Alice. Alice measuring spin-up on the ô-axis entails that Clotilde AliceUP will see Bob DOWN if Bob measures on the ô-axis. However, as we saw in §4.1, the set-theoretic structure of the environmental region where Bob might measure ô-spin is such that if the measurement had been made on a different axis, the results spin-up and spin-down would have probabilities determined by the subset measures of elements of the environmental electron not measured by Alice UP . Those elemental electrons would be ones located at elemental points oriented parallel to the axis chosen by Bob. So a series of measurements would have to be made on a succession of singlet states for the observers who are Clotilde's future temporal counterparts to gather statistical evidence confirming the predicted Bell correlations.
Beyond Idealisation
With the set-theoretic metaphysics in place, consider a non-idealised version of Vaidman's quantum die. Apart from the six equiprobable outcomes, there will be a plethora of extremely low-amplitude outcomes. Outcomes where "quantum accidents" occur, such as your smartphone transforming into a simulacrum of a salamander rather than displaying one of six numbers. These sorts of future events were also conceivable for classical physics as the result of highly improbably particle trajectories, amusingly illustrated in Bertrand Russell's tale The Metaphysician's Nightmare [ xxv ]. However, on the fission interpretation of Many Worlds theory, all such bizarre events exist in the multiple futures of an observer. Vaidman doesn't take them into account because such events have, as he would put it, very low measure of existence [ xxvi ]. I've effectively argued that Vaidman's "measure of existence" can be strictly identified with objective probability. So bizarre futures should be left out of account when rolling a quantum die because they have ridiculously low probabilities. There's nothing new in that idea.
However, the idea that all those bizarre futures actually exist is not necessarily anodyne. Pause for thought is called for in view of scenarios such as Huw Price's Legless at Bondi [ xxvii p. 382]. More briefly, suppose that you're ill and offered treatment which involves quantum processes with multiple outcomes. There's a high probability that you'll be cured but a low probability that you'll end up much worse off. In a conventional context you may well take the risk, even if a little anxiously. In the fission context, you can be sure that the cured person will know that someone else is suffering because of the decision you took. Is it consolation enough to know that the suffering person will also have been the person who took the decision? It's not obvious that a fission interpretation of Many Worlds is free from moral conundrums. Another interesting case is to be found in [ xxviii ]. But then why should we expect such a profound change of worldview not to have consequences for how we understand the human predicament?
Parting Lines
I've argued that Everett's key idea was to replace the concept of a stochastic process with that of a dendritic process, which is the idea that quantum phenomena induce the splitting of observers and their environments. This ostensibly raises problems which cannot be resolved by physics alone because assumptions rooted in folk metaphysics stand in the way. Observers cannot make predictions and test them unless they persist, but how can an observer persist through fissioning into multiple different observers? Sider's stage theory solves that problem, but it didn't become available until 1996 and remains neglected in the philosophical literature on persistence.
How can an observer be uncertain about future observations whilst believing that multiple outcomes will occur? The folk metaphysics of possibility and actuality stands in the way but logic doesn't. That the objective probability of multiple outcomes occurring is 1 does not entail that the objective probability of each outcome's occurrence is 1. In that case, uncertainty can be understood as assigning partial degrees of belief to multiple future observations without those future observations needing to be alternative possibilities, as has always been thought. They can be coexistent actualities.
How can objective probability be a property of multiple actual outcomes? The proposal which has been described and further developed here involves the hypothesis that individual objects in an observer's environment can be construed as multipleton sets which are macroscopically isomorphic and microscopically anisomorphic because they're constituted by different configurations of local beäbles. Quantum processes induce the partitioning of those sets into macroscopically distinct subsets whose measures are the objective probabilities of outcomes. As a consequence, a single observer's body is a set of doppelgängers, so the belief that there can be multiple copies of observers, which is widely held, should be rejected. A future-oriented account of objective probability is provided by the idea that a single observer, whose body is a set of doppelgängers, fissions into multiple observers whose bodies are subsets of doppelgängers with probability measures. According to stage theory, the premeasurement observer bears the relation will be to each of the post-measurement observers and is uncertain about what will be observed because assigning degrees of belief to future observations equal to the probability measures of the future branches. There's no question as to which post-measurement observer the pre-measurement observer will be, they will be each of them.
This set-theoretic metaphysics provides a framework for a version of the Many Worlds interpretation of quantum mechanics which includes causal locality, separability and Everettian fission (rather than divergence). It provides an account of probability that doesn't appeal to self-location uncertainty and an account of microscopic reality which includes local beäbles. It leaves work to be done on the physics of those beäbles and the way that they participate in the unitary dynamics.
My thanks to Rainer Plaga for raising this point.
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| [] |
[
"Physically Consistent Multiple-Step Data-Driven Predictions Using Physics-based Filters",
"Physically Consistent Multiple-Step Data-Driven Predictions Using Physics-based Filters"
] | [
"Yingzhao Lian ",
"Jicheng Shi ",
"Colin N Jones "
] | [] | [] | Data-driven control can facilitate the rapid development of controllers, offering an alternative to conventional approaches. In order to maintain consistency between any known underlying physical laws and a data-driven decisionmaking process, preprocessing of raw data is necessary to account for measurement noise and any inconsistencies it may introduce. In this paper, we present a physics-based filter to achieve this and demonstrate its effectiveness through practical applications, using real-world datasets collected in a building on theÉcole Polytechnique Fédérale de Lausanne (EPFL) campus. Two distinct use cases are explored: indoor temperature control and demand response bidding. | 10.48550/arxiv.2303.09437 | [
"https://export.arxiv.org/pdf/2303.09437v3.pdf"
] | 257,557,310 | 2303.09437 | 22c01fe7bb6c5c7c9423135909e1df28fc520aa3 |
Physically Consistent Multiple-Step Data-Driven Predictions Using Physics-based Filters
2 Jun 2023
Yingzhao Lian
Jicheng Shi
Colin N Jones
Physically Consistent Multiple-Step Data-Driven Predictions Using Physics-based Filters
2 Jun 2023
Data-driven control can facilitate the rapid development of controllers, offering an alternative to conventional approaches. In order to maintain consistency between any known underlying physical laws and a data-driven decisionmaking process, preprocessing of raw data is necessary to account for measurement noise and any inconsistencies it may introduce. In this paper, we present a physics-based filter to achieve this and demonstrate its effectiveness through practical applications, using real-world datasets collected in a building on theÉcole Polytechnique Fédérale de Lausanne (EPFL) campus. Two distinct use cases are explored: indoor temperature control and demand response bidding.
I. INTRODUCTION
Data-driven control can improve the speed and quality of controller design and deployment via an end-to-end solution from I/O data to a functional controller. However, it is often crucial to ensure that the data-driven control should respect the known physical laws in order to make a meaningful decision. However, due to measurement noise present in the data, a direct use of raw data 1 may lead to incorrect conclusions or predictions. Such inconsistencies were spotted by [1], where minor perturbations in the input were shown to significantly deteriorate prediction accuracy [2].
The incorporation of physical laws in data-driven and machine learning methods has been an active area of research for decades. In fact, this idea has been used to solve partial differential equations since the 1990s [3]. The idea of incorporating a physical rule in a parametric model is referred to as "physics-guided" or "physics-informed" in the literature [4]. This can involve using the physical rule to define the loss function and to confine the model's parameters to a subset that is consistent with known physical rules. Researchers have applied this idea to various architectures, such as enforcing a positive correlation between indoor temperature and heating power consumption in neural networks [5], and using a similar approach in linear parametric models [6]. While the aforementioned methods are important, preprocessing data can be a more direct approach to improve consistency. The methods falling in this category are highly related to robust optimization, where algorithms similar to This work received support from the Swiss National Science Foundation (SNSF) under the NCCR Automation project, grant agreement 51NF40 180545.The first two authors contributed equally. Extended version: https://arxiv.org/abs/2303.09437 (corresponding author: Yingzhao Lian) YL, JS and CNJ are with Automatic Laboratory, EPFL, 1015
Lausanne, Switzerland. {yingzhao.lian, jicheng.shi, colin.jones}@epfl.ch scenario approaches have been successfully employed in natural language processing [7] and computer vision [8].
In this work, we propose a physics-based filter that is tailored to data-driven control schemes based on Willems' fundamental lemma [9]. Willems' fundamental lemma offers a direct characterization of the system responses of lineartime-invariant (LTI) systems given an informative historical dataset. Such a characterization has been used in data-driven methods, and has been deployed in output prediction [10], input reconstruction [11], [12], and in controller design [13], [14], [15], [16], [17]. The main contribution lies in showing that some a priori knowledge can be integrated into Willems' fundamental lemma by robust optimization. The proposed scheme remains a non-parametric prediction structure, which differentiates it from other parametric schemes [5], [6].
In order to present the proposed method with a more intuitive exposition, the idea presented in this paper will be motivated and related to building applications. In the following, the Willems' fundamental lemma and its corresponding prediction problem is reviewed in Section II, after which the physics-based filter is investigated in Section III. The efficacy of the proposed scheme is validated on an indoor temperature control problem and a demand response bidding problem, with data collected from a building on the EPFL campus. Notation: I n ∈ R n×n denotes a n-by-n identity matrix, similarly, we denote the zero matrix by O. 0 and 1 respectively denotes a zero vector and a one vector. blkdiag(A 1 , . . . , A n ) generates a block-diagonal matrix whose diagonal blocks are A 1 , . . . , A n accordingly. x := {x i } T i=1 denotes a sequence of size T indexed by i. x i denotes the measurement of x at time i, and x 1:
L := [x ⊤ 1 , x ⊤ 2 .
. . x ⊤ L ] ⊤ denotes a concatenated sequence of x i ranging from x 1 to x L , and we drop the index to improve clarity if the intention is clear from the context. II. PRELIMINARIES Definition 1: A Hankel matrix of depth L associated with a vector-valued signal sequence s :
= {s i } T i=1 , s i ∈ R ns is H L (s) := s 1 s 2 . . . s T −L+1 s 2 s 3 . . . s T −L+2 . . . . . . . . . s L s L+1 . . . s T .
A linear time-invariant (LTI) system is defined by x i+1 = Ax i + Bu i , y i = Cx i + Du i , dubbed B(A, B, C, D). Its order is n x with n u , n y denoting its input and output dimensions respectively. An L-step trajectory generated by this system is u 1:L y 1:
L := u ⊤ 1 . . . u ⊤ L y ⊤ 1 . . . y ⊤ L ⊤ .
The set of all possible L-step trajectories generated by B(A, B, C, D) is denoted by B L (A, B, C, D). For the sake of consistency, a datapoint coming from the historical dataset is marked by boldface subscript d . Given a sequence of input-output measurements {u d,i , y d,i } i , we call the input sequence persistently exciting of order L if H L (u d ) is full row rank. By building the following stacked Hankel matrix For the sake of consistency, L is reserved for the length of the system responses. A data-driven control scheme has been proposed in [13], [18], where Lemma 1 generates a trajectory prediction. Before introducing the prediction, we state the following assumption to simplify the presentation of this paper:
H L (u d , y d ) := H L (u d ) ⊤ H L (y d ) ⊤ ⊤ ,
Assumption 1: The output measurements y are contaminated by measurement noise, the input measurements u are exact. It is possible to consider noisy input measurements; please refer to [16] for more details. Under Assumption 1, the trajectory prediction problem is defined by:
y pred (u pred ) = H L,pred (y d )g (1a)
g ∈ arg min
g l ,σ l 1 2 σ l 2 + 1 2 g ⊤ l E g g l (1b) s.t. H L,init (y d ) H L,init (u d ) H L,pred (u d ) g l = y init + σ l u init u pred ,
where E g is a user-defined positive definite penalty and u init , y init are t init -step sequences of the measured inputs and outputs preceding the current point in time. Accordingly, u pred , y pred are the corresponding n h -step predictive sequences viewed from the current time step. The matrix H L (y d ) is split into two sub-Hankel matrices:
H L (y d ) = H L,init (y d ) H L,pred (y d ) .
The matrix H L,init (y d ) is of depth t init and the depth of H L,pred (y d ) is the prediction horizon n h such that t init + n h = L. The matrices H L,init (u d ), H L,pred (u d ) are defined similarly. The choice of t init is made to ensure a unique estimation of the initial state; please refer to [10] for more details. This prediction problem (1) predicts n h -step output trajectory y pred for any given predictive input sequence u pred , whose objective in (1b) minimizes a Wasserstein distance upper bound; the interested readers are referred to [16] for more details. Further, recalling the conditions of Willems' fundamental Lemma 1, this prediction problem requires the following assumption: Assumption 2: u d is persistently exciting of order L + n x .
III. MAIN RESULTS
A. Physics-based Filter
As discussed in Section I, the measurement noise presented in {y d } may lead to inconsistent output predictions in (1). Hence, the data preprocessing scheme should modify the data {y d } such that the prediction generated by (1) is consistent with some prior physical rules. Here we focus on the following two rules from building control applications:
• Temperature consistency: The indoor temperature is positively correlated with the power consumption of the heating, cooling and ventilation (HVAC) system. More specifically, if the room is heated by control input u pred , the predicted indoor temperature must be higher than the predicted temperature that is controlled by u pred = 0. • Bidding consistency: Demand response (DR) is a method of managing power demand on the consumption side [19]. If a building is to provide, for example, secondary frequency control services, it tracks an area generation control (AGC) signal provided by the transmission system operator (TSO), while maintaining indoor comfort. Intuitively speaking, the TSO manipulates the building as a slow but large-capacity "battery", and as a result, a higher/lower power consumption than its nominal value relatively "charge/discharge" the "battery". The "capacity" of the battery is accordingly central to its flexibility in the context of DR, which is reflected by the accumulative indoor temperature relative to that operated by the nominal power consumption. Note that the absolute power consumption is still non-negative. The minimal physical rule to ensure a reasonable bidding proposal is therefore the positive correlation between the accumulated indoor temperature and power consumption (i.e. i y pred,i ≥ 0, ∀ u pred ≥ 0).
Drawing inspiration from the discussion above, we can identify the essential components required to define a physics-based filter:
• The convex set Y of trajectories that is aligned with the physical rule, and y pred is consistent if y pred ∈ Y. • The set of control inputs U and initial conditions U init , Y init where the physical rule is imposed.
Recall the aforementioned examples, their mathematical components are defined by (see Remark 1 for more details):
• Temperature consistency:
Y = {y|y ≥ 0} , Y init = 0, U = {u|u ≥ 0} , U init = 0 (2)
• Bidding consistency:
Y = y 1 ⊤ y ≥ 0 , Y init = 0, U = {u|u ≥ 0} , U init = 0 (3)
Accordingly, the physics-based filter is defined by the following robust optimization problem: miñ
y d ỹ d − y d (4a) subject to: ∀ u pred ∈ U, u init ∈ U init , y init ∈ Y init y pred = H L,pred (ỹ d )g ∈ Y (4b) g ∈ arg min g l ,σ l 1 2 σ l 2 + 1 2 g ⊤ l E g g l s.t. H L,init (ỹ d ) H L,init (u d ) H L,pred (u d ) g l = y init + σ l u init u pred .(4c)
This is a bi-level robust optimization problem, which minimizes the perturbation of the offline dataset {y d }. Particu-larly, the post-processed output data {ỹ d } will replace the raw data {y d } in the definition of the prediction problem. The robust constraint enforces that, for any possible predictive input sequence u pred ∈ U, the corresponding output sequence y pred should be consistent with the physical rule in (4b). In the next section, we will show how to convert this problem (4) into a numerically tractable form.
B. Single-level Reformulation
Regardless of the physical rule Y, solving a bi-level optimization can be non-trivial. However, in this case, the physics-based filter (4) can be reformulated into a singlelevel optimization problem:
Lemma 2: The following single-level problem is equivalent to the bi-level problem (4): miñ
y d ỹ d − y d (5a) s.t. ∀ u pred ∈ U, u init ∈ U init , y init ∈ Y init y pred = H L,pred (ỹ d )g ∈ Y M (ỹ d ) g κ(u pred ) = H L,init (ỹ d ) ⊤ y init u init u pred , (5b)
where κ(u pred ) is the dual variable of (4c) and
M (ỹ d ) := H L,init (ỹ d ) ⊤ H L,init (ỹ d ) + Eg H L (u d ) ⊤ H L (u d ) O . (6)
Proof: Note that the lower level problem in (4) is strongly convex, it is therefore equivalent to its KKT system [20,Chapter 4]. By replacing σ l by H L,init (y d )g l −y init , the Lagrangian of the lower level problem is
L(g) = 1 2 H L,init (y d )g − y init 2 + 1 2 g ⊤ E g g + κ ⊤ (H L (u d )g − u init u pred ) ,
where κ(u pred ) is the dual variable of the equality constraint. Hence, we have the stationary condition of the KKT system:
∂L(g) ∂g ⊤ = (H L,init (y d ) ⊤ H L,init (y d ) + E g )g + H L (u d ) ⊤ κ − H L,init (y d ) ⊤ y init = 0 . By recalling the primal feasibility condition H L (u d )g = u ⊤ init u ⊤ pred ⊤ ,
we get the robust KKT matrix M in (6). Remark 1: The physical rules (2) and (3) are defined on the transient response, which is linear with respect to u pred in LTI systems. In the rule of temperature consistency (2), our a priori knowledge requires that if u pred ≥ũ pred , their corresponding transient responses satisfy y pred ≥ỹ pred . By the superposition property, y pred −ỹ pred is the transient response of u pred −ũ pred , which summarizes the rule in (2).
Remark 2: Assumption 2 is not strong in building applications, the stochastic property of the process noise (e.g. solar radiation and outdoor weather) will cause random fluctuation in the closed-loop input trajectory, and the persistent excitation condition is in turn satisfied.
C. Affine Physical Rules
Recall the physical rules mentioned in Section III-A, we are particularly interested in affine physical rules, i.e. Y = {y|H y,pred y ≤ h y,pred }, Y init = {y|H y,init y ≤ h y,init }, U init = {u|H u,init u ≤ h u,init } and U = {u|H u,pred u ≤ h u,pred }. A tractable reformulation for the affine physical rule is stated in the following corollary.
Corollary 3: Consider an affine physical rule. The solution to the physics-based filter (5) is equivalent to the solution to the following problem: min
ν≥O,ỹ d ỹ d − y d(7)s.t. h y,pred ≥ h aug (ỹ d ) ⊤ ν, M aug (ỹ d ) ⊤ λ + H aug ν = H obj . where x = g ⊤ κ ⊤ y ⊤ init u ⊤ init u ⊤ pred ⊤ , H obj (ỹ d ) := H y,pred H L,pred (ỹ d ) O M aug (ỹ d ) := M (ỹ d ) blkdiag(− H L,init (ỹ d ) ⊤ , −I, −I) , H aug := O blkdiag(H y,init , H u,init , H u,pred ) , h aug := h ⊤ y,init h ⊤ u,init h ⊤ u,pred ⊤ .
Proof: The physics-based filter under an affine physical rule is defined by following robust optimization problem: miñ
y d ỹ d − y d s.t. h y,pred ≥ max uinit,yinit u pred H y,pred H L,pred (ỹ d )g s.t H u,pred u pred ≤ h u,pred , H u,init u init ≤ h u,init H y,init y init ≤ h y,init M (ỹ d ) g κ = H L,init (ỹ d ) ⊤ y init u init u pred ,
which can be reformulated into the standard form of LP: miñ
y d ỹ d − y d s.t. h y,pred ≥ max x H obj (ỹ d )x s.t. M aug (ỹ d )x = 0, H aug x ≤ h aug .
By duality of LP [21], the constraint is reformulated to miñ
y d ỹ d − y d s.t. h y,pred ≥ min λ,ν h aug (ỹ d ) ⊤ ν s.t. M aug (ỹ d ) ⊤ λ + H aug ν = H obj , ν ≥ O .
This is sufficient to summarize the proof.
Remark 3: As an optimization problem still must be solved in the proposed scheme, one may question its benefit. We summarize the scenarios in which proposed scheme is advantageous to a parametric system identification approach:
• When the physical rules are defined based on the I/O sequence, such as the passivity [22], the independence/causality between I/O ports [23] and positive/negative correlation (e.g. rules in this work), using the proposed scheme is more intuitive without converting the physical rule to its parametric correspondence. • If a physical rule is defined by a multi-step I/O sequence, a parametric model may involve high-order polynomials on its parameters that is not desirable for numerical solvers. Consider a one dimensional case with y i+1 = ay i + bu i ; the parametric form of the bidding consistency is defined by a high-order polynomial
n h −1 i=0 a i b ≥ 0.
Instead, the dual solved in the proposed method remains bilinear (see Section III-D).
D. Numerical Details
The reformulated single-level problem (5) is still a nonconvex optimization due to the nonlinear equality constraint (5b), where the quadratic term H ⊤ L,init (ỹ d ) H L,init (ỹ d ) in the matrix M (ỹ d ) is numerically less desirable to most optimization solvers. In order to improve the numerical performance, we suggest reformulating the problem (5) as miñ
y d ỹ d − y d (8a) s.t. ∀ u pred ∈ U, u init ∈ U init , y init ∈ Y init y pred = H L,pred (ỹ d )g ∈ Y M sch,1 (ỹ d ) σ g κ = y ⊤ init 0 ⊤ u ⊤ init u ⊤ pred ⊤ (8b) where M sch,1 (ỹ d ) := −I H L,init (ỹ d ) O H L,init (ỹ d ) ⊤ E g H L (u d ) ⊤ O H L (u d ) O .
The equivalence between (5b) and (8b) follows the Schur complement (i.e. inverse Gaussian elimination) [24]. The benefit of using (8b) instead of (5b) is that, the right-hand side of (8b) is independent ofỹ d and the left-hand side is linear with respect to H ⊤ L,init (ỹ d ) instead of quadratic. Even though problem (8) is still non-convex due to the bilinearity induced by the multiplication between H L,init (ỹ d ) and g in (8b), there exist more efficient and reliable numerical optimization algorithms tailored for bilinear problems, such as the McCormick envelope [25] implemented in a recent release of GUROBI 9.0 [26]. Based on our numerical experiment, this reformulation can roughly gain 50% acceleration in the solution time with the same initialization.
In addition to the benefits in numerical efficiency, the reformulation given in (8b) is particularly valuable when using the horizon splitting technique. As reported in [27], horizon splitting can improve long-term prediction accuracy, which is central to the bidding problem in DR. Under a horizon splitting scheme, the predictor given by equation (1) is recursively called to generate a long prediction trajectory by concatenation. Without loss of generality, we explain it by a special case where t init = n h , and a prediction trajectory of 2n h -steps is generated. This prediction is obtained by solving the following optimization problem y pred,1 = H L,pred (y d )g 1
g 1 ∈ arg min g l ,σ l 1 2 σ l 2 + 1 2 g ⊤ l E g g l s.t. H L,init (y d ) H L,init (u d ) H L,pred (u d ) g l = y init + σ l u init u pred,1 (9a)
y pred,2 = H L,pred (y d )g 2 g 2 ∈ arg min g l ,σ l
1 2 σ l 2 + 1 2 g ⊤ l E g g l s.t. H L,init (y d ) H L,init (u d ) H L,pred (u d ) g l = y pred,1 + σ l u pred,1 u pred,2 (9b)
where the predictive input sequence u pred of length 2n h is partitioned into two n h -step sequences, i.e. u pred = u ⊤ pred,1 u ⊤ pred,2 ⊤ . Similarly, we have y pred,1 and y pred,2 .
The predictive component in (9a) composes the initialization component in (9b). The formulation (8b) plays a crucial role in enabling numerically efficient implementation. By utilizing the single level-reformulation provided in Lemma 2, the resulting physics-based filter is defined as follows: miñ
y d ỹ d − y d s.t ∀ u pred ∈ U, u init ∈ U init , y init ∈ Y init y pred (u pred ) = H L,pred (ỹ d ) g ⊤ 1 g ⊤ 2 ⊤ ∈ Y M sch,2 (ỹ d ) σ 1 g 1 κ 1 σ 2 g 2 κ 2 = y init 0 u init u pred,1 0 0 u pred,1 u pred,2 where M sch,2 (ỹ d ) := M sch,1 (ỹ d ) O O − H L,pred (ỹ d ) O O O O M sch,1 (ỹ d ) .
Although the data-driven predictor is recursively called twice, the resulting optimization problem remains bilinear. In general, by applying the inverse Schur complement technique in (8b), the physics-based filter remains bilinear regardless of the number of segments that the predictive trajectory is split into. It is worth mentioning that the reformulation suggested in this section is compatible with the robust counterpart reformulation discussed in Section III-C.
IV. NUMERICAL RESULTS
The dynamics of buildings are generally slow and can be effectively approximated using linear models, where the use of Willems' fundamental lemma is justified by realworld experiments [16]. Though nonlinearity may be present, particularly the bilinearity in valve position control, there is a way to lift the nonlinear term and retain a linear analysis in the controller design [28]. This section validates the efficacy of the proposed method using real-world I/O data collected from a building called the Polydome located on the EPFL campus, which is a 600 m 2 self-standing building accommodating up to 200 people in a single lecture hall. An AERMEC RTY-04 heat pump (HP) is used to control the indoor climate. The dataset used in this study covers 40 days from December 2021 to January 2022 (i.e. the heating season) and includes indoor temperature as the output variable, the HP's electrical power consumption as the controlled input, and outdoor temperature and solar radiation as process disturbances (uncontrolled inputs) with a 15minute sampling time. Interested readers are refered to [16] for more technical details. In the sequel, the proposed method is validated by indoor temperature control and DR service. All the optimization problems are solved by GUROBI with Intel Core i7-1165G7 2.80 GHz processor. The solution time for different case studies are reported in the extended version.
A. Case Study I: Temperature Consistency
When heating is provided, the temperature consistency (2) is enforced by the filter (4). The Hankel matrices are constructed by 384 data points (i.e 4-day data for training) with t init = 6. For comparison, a parametric autoregressive exogenous (ARX) model is also considered where the physical rule is enforced by forcing the ARX weights to be positive.
As the control input is determined based on the predictor, we first run a comparison of prediction accuracy. The result is presented in Table I, where different prediction horizons are considered. Even though the filtered data gives a lower prediction performance than the raw data, this performance loss results in more reasonable decisions during operation with a predictive controller. In particular, consider the following predictive control problem: min
u pred y pred − ref 2 s.t. u pred ∈ [0, 6 kW]
y pred by (4b)&(4c) or (9) This controller tracks a reference temperature while considering indoor temperature constraints, and the open-loop input sequences given by different predictors are shown in Figure 1. The decision from the filtered-data controller maintains a maximal input before the predicted temperature reaches the reference, which is optimal regarding the turnpike property of optimal control [29]. While such optimal decision is also made by the parametric model, its low prediction accuracy leads to an underestimate in temperature response. This may also cause undesired chattering behaviour when the building operates around the constraint. Using two controllers defined by raw data as a comparison, their input sequences are suboptimal as their inputs oscillate between maximal input and null before raising the temperature to the reference. Note that multiple steps in open-loop input might be used in some specific applications, such as multi-building coordination. The sub-optimality observed here could deteriorate the closed-loop performance. On top of the lack of physical consistency, we believe that these two predictors overfit, as our data is collected during the normal operation of the building, and the patterns in the I/O sequences are quite limited even though the persistent excitation condition is satisfied. In this section, we consider the case where buildings are used to provide DR services and hence bidding consistency (3) is used. Due to a much longer prediction horizon (i.e. 24 hours), a lower sampling time, 30 minutes, is used to lower the computational cost. The Hankel matrices are constructed by 384 data points (i.e. 8-day data) with t init = 12, and the parametric model is dropped due to the lack of convergence in its highly non-convex optimization problem.
Similar to the last part, the prediction performance is first tested on the whole dataset with different prediction horizons (see Table II). In accordance with [27], splitting improves long-term prediction accuracy when we compare the results in the last two columns. However, the predictor using filtered data and splitting still gives a slightly lower prediction accuracy in comparison with the predictor generated by raw data with splitting.
These three data-driven predictors are respectively used to solve the following bidding problem: min
γ,Pbaseline − γ (12a) s.t. u pred,i ∈ [0, 6 kW] y pred,i ∈ [y min , y max ] (12b) u pred,i = P baseline + γAGC i , i = 1, 2, . . . , N scen y pred,i by (4b)&(4c) or (9) ,
where a 24-hour-ahead prediction is made within this problem. More specifically, the input flexibility margin γ is maximized with respect to the uncertain AGC signals, whose uncertainty is handled by a scenario approach with N scen historical scenarios. Depending on the 24-hour open-loop input decision u pred , γ determines the primary remuneration from the TSO. Hence, it should be planned and sent to the TSO before the next operational day (i.e 24-hour-ahead). Interested reader are referred to [30] for more technical details. To keep a compact presentation, only the data-driven predictors based on filtered/raw data with splitting are considered in this comparison. We test different comfort ranges for the indoor temperature in Table III, whose initial indoor climate and weather conditions were selected randomly from the real-world dataset. When Problem (12) is infeasible, it is relaxed to a soft-constrained problem by relaxing (12b) and including its violation to the cost (12a) with a large penalty. This is done to facilitate better comparison, particularly when the temperature constraint is overly tight, such as [19, 20.5]. When the constraint is set to [19, 20.5], the problem should be infeasible due to the limited power of the HVAC system (i.e. γ ≈ 0 in the relaxed problem). Capturing such infeasibility is critical to avoid economic loss, and it is achieved by the problem with filtered data. However, due to the inconsistency presented in the raw data, the problem remains feasible when the raw data is directly used. To better visualize how the physical inconsistency takes effect, we plot the control policy at different temperature constraints in Figure 2. As indicated by Figure 2 (b) and Table III, a larger average heating input is applied in the case of y ∈ [19, 20.5] than that in the case of y ∈ [19, 22.5]. However, it predicts a lower average indoor temperature, which is inconsistent with the enforced physical rule. Hence, the γ bid based on raw data is an overestimate, and may cause indoor discomfort or economic loss in the following operational day.
C. Computation time
In this section, we present a summary of the computation times for solving various optimization problems in Case Study I and II, as shown in Tables IV and V. It is worth noting that the predictive controller and bid problem can be solved efficiently with short computation times. Conversely, the filter requires more time, especially when using splitting, but it is only required when updating the data for Hankel matrices. Thus, considering the computing time, the proposed algorithm can be efficiently implemented online in building systems if a suitable data updating frequency is chosen. In two case studies, the sampling times are chosen according to our previous experience and experiments [16], [30]. They were determined by practice based on the time constants of the building. Recently, there is research exploring the effects of the time intervals for model discretization and control sampling in building systems [31], [32]. Sensitivity analysis is commonly used to determine the best choices without theoretical guarantees. Here gives an example of analysis on the prediction error and computation time for 8-day data in Figure 3. As the sampling time increases, the 6-hour prediction error does not change much but the filtering computation time decreases a lot. In fact, it is an interesting future direction to perform more analysis, such as the one for closed-loop control performance by some simulation software.
V. CONCLUSIONS In this paper, a physics-based filter was proposed to enhance data-driven predictors. The scheme enforces a priori physical rules, improving decision-making reliability.
we state Willems' Fundamental Lemma as Lemma 1: [9, Theorem 1] Consider a controllable linear system and assume {u d } T i=1 is persistently exciting of order L + n x . The condition colspan(H L (u d , y d )) = B L (A, B, C, D) holds.
Fig. 1 :
1Open-loop solution of MPC. (a) Filtered data, no split; (b) Raw data, split ; (c) Raw data, no split; (d) Positive ARX model
Fig. 2 :
2Solution of the demand response problem. (a) Filtered data, split; (b) Raw data, split
Fig. 3 :
3Comparison of different choices of sampling times for 6hour ahead bidding filtering (a) Prediction error ; (b) Computation time
TABLE I :
IComparison of the mean absolute error (MAE) over different prediction horizonsPrediction
steps
Hours
ahead
Filtered
no split
Raw
split
Raw
no split
Positive
ARX
6
1.5
0.235
0.226
0.226
0.303
12
4
0.326
0.301
0.299
0.433
18
4.5
0.440
0.392
0.388
0.589
"split": horizon splitting with n h = t init ; "no split": otherwise
B. Case Study II: Bidding Consistency
TABLE II :
IIComparison of the MAE over different prediction steps by three methods. split": horizon splitting with n h = t init ; "no split": otherwisePrediction
steps
Hours
ahead
Filtered
split
Raw
split
Filtered
no split
Raw
no split
12
6
0.367
0.344
0.367
0.344
24
12
0.496
0.476
0.588
0.494
36
18
0.572
0.509
0.739
0.620
48
24
0.608
0.526
0.917
0.780
"
TABLE III :
IIIComparison of bidding [y min , ymax]Filtered data
Original data
γū predȳpred
γū predȳpred
[19, 20.5]
* 0.00
* 3.10
* 20.01
0.94
4.53
19.88
[19, 21.5]
1.82
3.88
20.24
3.02
3.87
20.24
[19, 22.5]
1.94
4.38
20.45
3.76
3.81
20.51
u andȳ indicate the average value
* : from soft-constrained solution
0
10
20
30
40
50
16
18
20
22
Time step t
Indoor temperature
(a)
Output ([19, 20.5])
Input ([19, 20.5])
Output ([19, 22.5])
Input ([19, 22.5])
TABLE IV :
IVComputation time of the optimization problems in Case Study IPrediction
steps
Hours
ahead
Filter
Controller
6
1.5
15.911s
0.008s
12
4
45.532s
0.013s
18
4.5
111.541s
0.020s
"s": second
TABLE V :
VComputation time of the optimization problems in Case Study IIPrediction
steps
Hours
ahead
No Split:
filter
No Split:
bid
split:
filter
split:
bid
12
6
4.59s
0.10s
4.59s
0.10s
24
12
7.02s
0.21s
6.27m
0.28s
36
18
11.41s
1.01s
23.91m
0.73s
48
24
20.39s
2.45s
71.32m
1.42s
"s": second, "m": minute
Raw data in this work indicates the data without preprocessing.
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| [] |
[
"Estimating Density Models with Truncation Boundaries using Score Matching",
"Estimating Density Models with Truncation Boundaries using Score Matching"
] | [
"Song Liu [email protected] \nUniversity of Bristol\nUniversity of Bristol\n\n",
"Takafumi Kanamori [email protected] \nUniversity of Bristol\nUniversity of Bristol\n\n",
"Daniel J Williams [email protected] \nUniversity of Bristol\nUniversity of Bristol\n\n",
"Qiang Liu \nUniversity of Bristol\nUniversity of Bristol\n\n"
] | [
"University of Bristol\nUniversity of Bristol\n",
"University of Bristol\nUniversity of Bristol\n",
"University of Bristol\nUniversity of Bristol\n",
"University of Bristol\nUniversity of Bristol\n"
] | [
"Journal of Machine Learning Research"
] | Truncated densities are probability density functions defined on truncated domains. They share the same parametric form with their non-truncated counterparts up to a normalizing constant. Since the computation of their normalizing constants is usually infeasible, Maximum Likelihood Estimation cannot be easily applied to estimate truncated density models. Score Matching (SM) is a powerful tool for fitting parameters using only unnormalized models. However, it cannot be directly applied here as boundary conditions used to derive a tractable SM objective are not satisfied by truncated densities. In this paper, we study parameter estimation for truncated probability densities using SM. The estimator minimizes a weighted Fisher divergence. The weight function is simply the shortest distance from a data point to the boundary of the domain. We show this choice of weight function naturally arises from minimizing the Stein discrepancy as well as upperbounding the finite-sample estimation error. The usefulness of our method is demonstrated by numerical experiments and a study on the Chicago crime data set. We also show that the proposed density estimation can correct the outlier-trimming bias caused by aggressive outlier detection methods. | null | [
"https://arxiv.org/pdf/1910.03834v5.pdf"
] | 248,266,632 | 1910.03834 | 1f5f4ce5487b91b6110ff531c95a7ca074dbe1be |
Estimating Density Models with Truncation Boundaries using Score Matching
2022
Song Liu [email protected]
University of Bristol
University of Bristol
Takafumi Kanamori [email protected]
University of Bristol
University of Bristol
Daniel J Williams [email protected]
University of Bristol
University of Bristol
Qiang Liu
University of Bristol
University of Bristol
Estimating Density Models with Truncation Boundaries using Score Matching
Journal of Machine Learning Research
232022Submitted 3/21;Tokyo Institute of Technology, RIKEN AIPscore matchingtruncated density estimationunnormalized density modelsstein operatornon-asymptotic analysis
Truncated densities are probability density functions defined on truncated domains. They share the same parametric form with their non-truncated counterparts up to a normalizing constant. Since the computation of their normalizing constants is usually infeasible, Maximum Likelihood Estimation cannot be easily applied to estimate truncated density models. Score Matching (SM) is a powerful tool for fitting parameters using only unnormalized models. However, it cannot be directly applied here as boundary conditions used to derive a tractable SM objective are not satisfied by truncated densities. In this paper, we study parameter estimation for truncated probability densities using SM. The estimator minimizes a weighted Fisher divergence. The weight function is simply the shortest distance from a data point to the boundary of the domain. We show this choice of weight function naturally arises from minimizing the Stein discrepancy as well as upperbounding the finite-sample estimation error. The usefulness of our method is demonstrated by numerical experiments and a study on the Chicago crime data set. We also show that the proposed density estimation can correct the outlier-trimming bias caused by aggressive outlier detection methods.
Introduction
In many applications, we cannot observe the "full picture" of a problem. Instead, our window of observation is limited, so we can only observe a truncated data set. For example, a police department can only monitor crimes within their city's boundary, despite the fact that crimes do not automatically stop at an artificial border. Similarly, geolocation tracking data can only be observed up to the coverage of mobile signal. data sets such as these are skewed representations of actual activities due to truncation. In many cases, these truncation boundaries can be very complex. For example, the boundary of the city of Chicago is a complex polygon (see Figure 1) which cannot be easily approximated by a bounding box or circle.
The key challenge of estimating parameters in truncated densities is that the normalizing constant is not computationally tractable. The normalizing constant ensures that the integration of a density function equals to one over its input domain. As the normalization takes place in an irregular bounded domain in R d , the integration does not have a closed form in general. This creates a computational issue since the classic Maximum Likelihood Estimation (MLE) requires the evaluation of such a normalizing constant. Although the normalizing constant in the likelihood function can be approximated using Monte Carlo methods (Geyer, 1994), it is hard to guarantee the approximation accuracy when the data is in high dimensional space and the truncation domain is complex.
Recent years have seen a new class of estimators, called Score Matching (SM) (Hyvärinen, 2005(Hyvärinen, , 2007Lyu, 2009) rise in popularity. They estimate parameters by minimizing the Fisher-Hyvärinen divergence (Lyu, 2009). This divergence is defined using the difference between the gradients of log model density and log data density. The gradients are taken with respect to the input variable, so the normalizing constant is eliminated and is not involved in the estimation procedure. Thus SM is a natural candidate for estimating truncated density models.
However, the original SM cannot work for estimating these truncated models, as the regularity condition used to derive the tractable objective function is not satisfied. Hyvärinen (2007) and Yu et al. (2019) proposed generalized SM to handle the distributions on the nonnegative orthant R d + . In generalized SM, a weight function is introduced so that the boundary condition required for deriving the tractable objective function is satisfied. Promising results have been observed on high-dimensional non-negative graphical model structure estimation (Yu et al., 2019). When the density is truncated in a dimension-wise manner with a lower and upper bound, this problem is known as a doubly truncated distribution estimation (Turnbull, 1976;Moreira and de Uña-Álvarez, 2012). Though some works have tackled the problem of estimating truncated multivariate Gaussian densities (Daskalakis et al., 2018(Daskalakis et al., , 2019, little work has been done for estimating a wide range of density models with complicated truncation domains. Note that if the truncation domain is simple, it is possible to apply the change of variable rule to our data set so that the truncated density estimation becomes a non-truncated (or doubly truncated) density estimation problem in the new domain. This technique has been widely used in modelling distributions on a sphere (Mardia and Jupp, 2009). However, this technique is not applicable when the truncation domain is complex such as the one in the Chicago Crime data set.
In this paper, we proposed a novel estimator for truncated density models using generalized SM. Our choice of the weight function is the shortest distance from the data point to the truncation boundary. We show that this choice naturally arises from minimizing a Stein Discrepancy and lowering the finite sample estimation error upperbound. We also show that this distance weight function can be easily computed for many complicated domains.
The usefulness of our method is demonstrated by numerical experiments and an application on the Chicago crime data set. Finally we apply this estimator to correct the outlier trimming bias caused by One-class Support Vector Machines.
Problem Formulation
Denote a probability density function parameterized by θ over the domain V ⊂ R d as p θ (x), x ∈ V . Without loss of generality, we can write
p θ (x) :=p θ (x) Z V (θ) , Z V (θ) := Vp θ (x)dx,
wherep θ is an unnormalized model and Z V (θ) is the normalizing constant so that p θ (x) is integrated to 1 over its domain V . The domain V ⊂ R d may be a complicated bounded domain e.g., a polytope. In such cases, Z V (θ) may not have a closed form and cannot be easily evaluated. For example, letp θ be a Gaussian mixture model restricted in a generic polygon, then Z V (θ) does not have a closed form expression. Our task is classic statistical model estimation: Suppose that V is given, we want to estimate the parameter θ in p θ (x) using X q = {x i } n i=1 , which is a set of observed i.i.d. samples from a truncated data generating distribution Q with an unknown probability density function q(x), x ∈ V .
The challenge comes from the fact that we need to obtain the estimator of θ using only the unnormalized density modelp θ (x): It is not straightforward to calculate Z V (θ) for a complicated V . Therefore, MLE, which requires evaluating the normalizing constant, cannot be performed easily. A popular tool for estimating unnormalized densities is Score Matching (SM) proposed by Hyvärinen (2005). We now introduce SM and its extension (Yu et al., 2019), then explain why it cannot be readily used for estimating complicated truncated densities. Notation: The finite set {1, . . . , d} for a positive integer d is denoted by [d]. Given a vector x, let x k denote the k-th element of x. Let ·, · be the standard inner product, and the Euclidean norm of the vector a is denoted as a = a, a . The p -norm of x is denoted by x p . Thus, x = x 2 holds. Let ∂ k for k ∈ [d] be the partial differential operator ∂ ∂x k and ∇ x be (∂ 1 , . . . , ∂ d ) to the function f (x) for x ∈ R d . Similarly, the gradient operator with respect to the parameter θ ∈ Θ ⊂ R r of the statistical model p θ (x) is denoted by ∇ θ = ( ∂ ∂θ 1 , . . . , ∂ ∂θr ). E q [f (x)] stands for the expectation of f (x) with respect to the probability density q(x). To reduce the clutter, we sometimes shorten p θ (x) and q(x) as p θ and q wherever such abbreviations do not lead to confusion. In addition, the log-likelihood log p θ (x) is expressed by θ (x). f L 2 (Q) := E q f (x) 2 1/2 where Q denotes a distribution with density function q(x). Let e i , i = 1, . . . , d be the unit vector along the i-th axis in R d , i.e., e 1 = (1, 0, . . . , 0), etc. We use V to represent the closure of a bounded open set V .
Score Matching and Its Generalization
In this section, we introduce classic SM (Hyvärinen, 2005) and one of its variants (Yu et al., 2019).
Definition 1
The Fisher-Hyvärinen (FH) divergence (Lyu, 2009) between q and p θ is defined as
FH(q, p θ ) := E q [ ∇ x log p θ (x) − ∇ x log q(x) 2 ].
Suppose q(x) > 0 on R d . Then, the FH divergence is non-negative and it vanishes if and only if p θ (x) = q(x) almost surely. When a density model p θ (x) is defined on V = R d , SM finds an estimate of θ by minimizing FH(q, p θ ) over the parameter space θ ∈ Θ ⊂ R r , i.e.,
θ SM := argmin θ FH(q, p θ ) = argmin θ E q ∇ x log p θ 2 − 2E q [ ∇ x log p θ , ∇ x log q ] + C,(1)
where C is a constant independent of θ. The key advantage of SM is that the normalizing constant Z V (θ) is not required when evaluating (1) as ∇ x log p θ (x) = ∇ x logp θ (x). Thus, SM is widely used for estimating "unnormalizable" statistical models. Unfortunately, (1) is not tractable as we cannot directly evaluate the second term of (1) without access to ∇ x log q. Using the integration by parts rule, however, we find that the equality,
E q [ ∇ x log p θ , ∇ x log q ] = − d k=1 E q [∂ 2 k log p θ ]
holds under the smoothness condition of log p θ (x) and log q(x) w.r.t. x and the boundary condition
lim |x k |→∞ q(x)∂ k log p θ (x) = 0, ∀k ∈ [d].(2)
Many density functions defined on R d , such as multivariate Gaussian or Gaussian mixture, satisfy these conditions. See (Hyvärinen, 2005(Hyvärinen, , 2007 for details. Thus, FH-divergence in (1) can be re-written as
FH(q, p θ ) = E q [ ∇ x log p θ 2 ] + 2 d k=1 E q [∂ 2 k log p θ ] + C(3)
where the objective only relies on q through the expectations which can be approximated by the empirical mean over the observed samples X q .
When p θ (x) is defined on the truncated subset in R d such as the non-negative orthant,
R d + := {x ∈ R d |x k ≥ 0, ∀k ∈ [d]},
the boundary condition required to apply the integration by parts rule (i.e., (2)) no longer holds for many density functions such as Gaussian or Gaussian mixtures. To estimate parameters of density functions on the non-negative orthant, Hyvärinen (2007); Yu et al. (2019) introduced generalized SM:
θ GSM := argmin θ FH g (q, p θ ), FH g (q, p θ ) := E q [ g 1/2 • ∇ x log p θ − g 1/2 • ∇ x log q 2 ], = d k=1 E q [g k · (∂ k log p θ ) 2 ] − 2 d k=1 E q [g k · (∂ k log p θ )(∂ k log q)] + C,(4)
where g(x) := (g 1 (x), . . . , g d (x)) ∈ R d is a non-negative valued, continuously differentiable function with g(0) = 0. g 1/2 is the element-wise square root operation applied on g and • is the element-wise product.
Examples of g include g k (x) = x k or g k (x) = max(x k , 1), ∀k ∈ [d] given x ∈ R d + .
The following result can be used to derive the tractable objective function from (4), which appeared in the proof of Theorem 3 in Yu et al. (2019). Here we restate it as a Lemma using our symbols for conveniences.
Lemma 1 Suppose that log q(x) and g(x) are continuously differentiable almost everywhere (a.e.) on R d + and log p θ (x) is twice continuously differentiable with respect to x on R d + . Furthermore, we assume the boundary condition,
lim |x k |→0+,|x k |→∞ g k (x)q(x)∂ k log p θ (x) = 0 for k ∈ [d]. Then, d i=1 E q [g k · (∂ k log p θ )(∂ k log q)] = − d k=1 E q [∂ k (g k ∂ k log p θ )].
The proof can be found in Section A.1 in (Yu et al., 2019) which uses dimension-wise integration by parts. Using Lemma 1, the generalized SM objective (4) has a tractable expression,
FH g (q, p θ ) = d k=1 E q [g k · (∂ k log p θ ) 2 ] + 2 d k=1 E q [∂ k (g k ∂ k log p θ )] + C(5)
where C is a constant independent of θ. One can replace the expectation with the empirical mean over X q to obtain an unbiased estimator of the above objective function. It is straightforward to modify the generalized SM formulation so that it works for doubly truncated distributions. For example, g k (x) = min{x k − a k , b k − x k } can be used to estimate truncated densities on the product space d k=1 (a k , b k ). It is worth pointing out that Lemma 1 is a specification of the divergence theorem such as Green's theorem or Stokes' theorem which usually deals with a bounded domain. Recently, Mardia et al. (2016) studied an SM objective based on Stokes' theorem, for estimating densities on a Riemannian manifold.
We intend to extend the generalized SM to a generic truncated domain. We now show the validity of the objective function (4) when considering a generic domain V . For two differantiable probability densities p(x) and q(x) with respect to the Lebesgue measure µ(·) on V ⊂ R d , the following lemma holds:
Lemma 2 Suppose that V is a connected open subset in R d and that g k (x) > 0 and q(x) > 0, ∀x ∈ V . Then, FH g (q, p) = 0 if and only if p = q for the probability density functions p and q.
Proof First, it is easy to see that p = q =⇒ FH g (p, q) = 0. Second, as FH g (q, p) = 0,
g k (x)q(x) ∇ x log q(x) − ∇ x log p(x) 2 = 0 should hold a.e. with respect to µ on V . Thus, we have ∇ x (log q(x) − log p(x)) = 0 on V . As V is connected, log q(x) − log p(x) should be a constant independent of x on V . Hence, p(x) is proportional to q(x) on V .
As both are probability density functions that should be normalized, we have p = q.
The following theorem states that the minimizer of FH g (q, p θ ) is unique and is indeed the true parameter under a mild identifiability condition.
Theorem 1 Suppose all assumptions stated in Lemma 2 holds. Let
P = {p θ (x) | θ ∈ Θ ⊂ R r } be a set of parametric statistical models on the connected open subset V in R d . If the model p θ is identifiable in the sense that µ({x | p θ (x) = p θ (x)}) > 0 holds for θ = θ , and q = p θ 0 ∈ P, then argmin θ∈Θ FH g (q, p θ ) is unique and is θ 0 .
Proof Given the assumption on q, we have FH g (q, p θ 0 ) = 0. Thus, θ 0 is a minimizer of FH g (q, p θ ). Moreover, Lemma 2 guarantees that FH g (p θ , p θ ) > 0 holds for θ = θ . We can prove this by the contradiction. Suppose FH g (p θ , p θ ) = 0 for θ = θ . Thus, Lemma 2 states p θ (x) = p θ (x) must hold on V . This contradicts µ({x | p θ (x) = p θ (x)}) > 0. Hence, for all θ = θ 0 , FH g (p θ 0 , p θ ) = FH g (q, p θ ) > 0. This leads to the conclusion that the minimizer θ 0 is unique.
Remark 1 When the domain is not connected, FH g (q, p) = 0 implies that p is proportional to q on each connected region. However, for common statistical models, the proportional relationship on each connected region implies that the probability densities are the same. For instance, let us consider the truncated Gaussian model p θ on V that is the disjoint union of V 1 and V 2 , where θ is the mean parameter of the Gaussian distributionp θ on R d . If p θ /p θ 0 is a constant function on a small open subset in V , we see that θ = θ 0 holds. Hence, Theorem 1 holds for such models even when the domain V is not connected.
When using generalized SM on a generic domain V , the bigger challenge is selecting a weight function g. First, we explain an intuitive way to construct g, then we show this intuitive choice is theoretically sound and empirically effective.
Denote the boundary of V as ∂V . We can design a weight function g k taking 0 at ∂V , then hopefully an analogue to Lemma 1 would hold, giving a tractable form of the estimator. For example, we can consider a dist(x, z), which is a distance function between x and z, (a) (b) (c) Figure 2: Examples of g 0 (x) for (a) a circular boundary (b) a triangular boundary and (c) a polygon boundary. Here dist(·, ·) is the Euclidean distance function. and let the weight function g k be
g k = g 0 := min z∈∂V dist(x, z), ∀k ∈ [d], x ∈ V .(6)
i.e., the distance from x to the truncation boundary ∂V . See Figure 2 for examples of g 0 defined on some simple bounded domains V . This weight function is intuitive and can be easily computed for many complex truncation boundaries (See Section 7 for details).
Since g 0 (x) > 0, ∀x ∈ V by construction, Theorem 1 guarantees the minimizer of the generalized SM objective is the true parameter as long as V is a connected open domain and p θ is correctly specified.
However, there are a few major concerns:
1. It is unclear whether letting g k = g 0 would entail a tractable objective function. Lemma 1 is only derived for the non-negative orthant and it assumes g k (x) to be continuous differentiable a.e.. However, the distance weight g 0 is not necessarily differentiable a.e. on V .
2. The efficacy of the distance weight g 0 is unclear. The weight function g k can be any function which satisfies the boundary condition (g taking 0 at ∂V ) and is positive and differentiable on V . It is not immediately clear why using g 0 as the weight function would yield good statistical estimation performance.
In the following sections, we address these two concerns from theoretical and empirical perspectives. For simplicity, we refer to generalized SM with g k = g 0 as truncated SM, or TruncSM for short.
Tractable Truncated Score Matching Objective
Theorem 1 states that when V is a connected open domain, TruncSM is a valid density estimation criterion. Now we will show when V is a special kind of connected open domain, TruncSM objective is computationally tractable.
The key step of deriving a tractable SM objective is to show that the distance weight g 0 is weakly differentiable, so an analogue to Lemma 1 can be proven. Let us formally define the notion of a weakly differentiable function via Sobolev-Hilbert space:
Definition 2 (Sobolev-Hilbert space) : Let L 2 (V ) be the L 2 space on V ⊂ R d endowed with the Lebesgue measure. Then, H 1 (V ) is the Sobolev-Hilbert space space defined by
H 1 (V ) = f ∈ L 2 (V ) f 2 L 2 (V ) + k D k f 2 L 2 (V ) < ∞ ,
where D k is the weak derivative corresponding to ∂ k and f L 2 (V ) = V |f (x)| 2 dx.
In this paper, we focus on a special type of open and connected domain, called Lipschitz domain. Intuitively speaking, a Lipschitz domain V ⊂ R d is a bounded connected open domain whose local boundary is a level set of some Lipschitz function. In engineering applications, most domains are Lipschitz domains. However, it does not include domains with "cracks" (see examples below). All polytopes are Lipschitz domains.
(x 1 , . . . , x d−1 ) such that V ∩ B(x, r) is expressed by {z ∈ B(x, r) |z d > f (z 1 , . . . , z d−1 ) } upon a transformation of the coordinate system if necessary. B(x, r) denotes a d-dimensional ball centered at x with radius r.
The full definition of the weak derivative and more details about Lipschitz domains and Sobolev-Hilbert spaces can be found in Section 7 of (Atkinson and Han, 2005).
We prove the following lemma which states that g 0 defined on a Lipschitz domain is in H 1 (V ), thus is weakly differentiable. The proof can be found in Appendix B.
Lemma 3 Suppose V is a Lipschitz domain, then g 0 ∈ H 1 (V ).
The classic Green's theorem or Stokes' theorem used in Lemma 1 cannot be applied to derive a tractable objective anymore as functions in H 1 (V ) are not differentiable in a classic sense. However, there exists an extension of Green's theorem of weakly differentiable functions (Proposition 7.6.1 (Atkinson and Han, 2005)).
Lemma 4 (Extended Green's Theorem) For the Lipschitz domain
V ⊂ R d , suppose f 1 , f 2 ∈ H 1 (V ), V f 1 (x)∂ k f 2 (x)dx = ∂V f 1 (x)f 2 (x)ν k (x)ds − V f 2 (x)∂ k f 1 (x)dx, ∀k ∈ [d],
where (ν 1 , . . . , ν d ) is the unit outward normal vector on ∂V and ds is the surface element on ∂V . Now we apply Lemma 3 and 4 to obtain a tractable form of TruncSM objective: Theorem 2 Assume V ⊂ R d is a Lipschitz domain. Suppose q, ∂ k log p θ ∈ H 1 (V ) and that for any z ∈ ∂V it holds that
lim x→z q(x)g 0 (x)∂ k log p θ (x)ν k (z) = 0, ∀k ∈ [d],
where x → z takes any point sequence converging to z ∈ ∂V into account. Then, we have
d i=1 E q [g 0 · (∂ k log p θ )(∂ k log q)] = − d k=1 E q [∂ k (g 0 ∂ k log p θ )].
The proof of the above theorem can be found in Appendix C. This indicates that TruncSM indeed has a tractable objective function.
θ TSM := argmin θ FH g 0 (q, p θ ), = argmin θ d k=1 E q [g 0 · (∂ k log p θ ) 2 ] + 2 d k=1 E q [∂ k (g 0 ∂ k log p θ )] = argmin θ d k=1 E q [g 0 · (∂ k log p θ ) 2 ] + 2 d k=1 E q [g 0 ∂ 2 k log p θ ] + 2 d k=1 E q [∂ k g 0 · ∂ k log p θ ],(7)
where each expectation can be approximated using samples from the data set X q ∼ Q. Although (7) is similar to the generalized SM objective (5) (it replaces g k with g 0 ), this result cannot be taken for granted. It highlights an important restriction of TruncSM: V needs to be a Lipschitz domain, and this constraint is required to ensure the weak differentiability of g 0 , and is connected with the Lipschitzness of g 0 . This result also suggests that we may be able to bypass this constraint when using different weight functions in generalized SM. The study along this line can be an interesting future work. Now we turn our focus to the efficacy of TruncSM. In the next section, we show TruncSM is a Minimum Stein Discrepancy Estimator (Barp et al., 2019).
Truncated Score Matching as Minimum Stein Discrepancy Estimator
Maximum Stein Discrepancies are a family of discrepancies which measure the differences between two distributions (Chwialkowski et al., 2016;Liu et al., 2016). For simplicity, we assume that q and θ are smooth in this section.
Definition 4 Given an
f : R d → R d ,
a Stein operator is defined as
T p f := d k=1 {(∂ k log p) · f k + ∂ k f k } ,
where f = (f 1 , . . . , f d ), p is a probability density function and S p := {f | E p [T p f ] = 0} is called a Stein class of p. The Maximum Stein Discrepancy between two densities q and p (Chwialkowski et al., 2016;Liu et al., 2016) is defined as
max f ∈Sp E q [T p f ].
By constructing different Stein classes S p , we obtain different Maximum Stein Discrepancies.
Since our task is estimating a parametric density model p θ , we can consider a density estimator that minimizes this discrepancy, i.e., argmin
θ max f ∈Sp θ E q [T p θ f ].
This estimator has been shown to be effective, robust, and closely related to SM. It is called Minimum Stein Discrepancy Estimator (Barp et al., 2019).
When using the above estimator, the key issue is constructing a Stein class S p θ which is expressive enough to capture subtle differences between q and p θ . In the following theorem, we construct a Stein class with f that is the product of a smooth function and a Lipschitz function. We show the Maximum Stein Discrepancy using this specific Stein class becomes FH divergence weighted by the distance weight g 0 .
First, we introduce a Lemma whose proof can be found in Appendix D.1.
Lemma 5 Let Lip L 0 (V ) be the set of all functions f : V → R • that are L-Lipschitz continuous with respect to dist(·, ·) • satisfies the property that f (x) = 0, ∀x ∈ ∂V then max g k ∈Lip L 0 , ∀k∈[d] FH g (q, p θ ) = L · FH g 0 (q, p θ ). Theorem 3 Let F be a function class such that ∀f ∈ F, f = h • g 1/2 and E q h(x) 2 ≤ 1, where h : V → R d is a smooth function and ∀k, g k ∈ Lip L 0 (V ) then • F is a Stein class of a smooth density q defined on V . • If ∂ k θ is smooth with respect to θ for all k, we have max f ∈F E q [T p θ f ] = L · FH g 0 (q, p θ ).
The proof can be found in Appendix D.2 and it is partly based on the proof of Theorem 2 in Barp et al. (2019). Theorem 3 shows that the TruncSM objective is a Maximum Stein Discrepancy thus θ TSM is a Minimum Stein Discrepancy Estimator. Similarly, we can show how a "capped" distance weight arises by choosing a slightly different family for g k in Theorem 3. We find this capped weight function can be also effective in many tasks (see Section 6.3 and 8.2).
Corollary 1 Let us define Lip
L 0 (V ) := {f ∈ Lip L 0 (V )|f (x) ≤ 1} and a new function class F such that for all f ∈ F, f = h • g 1/2 , E q h(x) 2 ≤ 1, and g k ∈ Lip L 0 (V ), then max f ∈F E q [T p θ f ] = FHḡ L (q, p θ ), whereḡ L = min(1, L · g 0 ).
The proof can be found in Appendix D.3. Yu et al. (2019) proposed to use a weight function g k = min(1, x k ), k = 1, . . . , d for estimating density functions defined on the non-negative orthant. This weight function is the special case ofḡ L when V is the entire non-negative orthant. We refer toḡ L as a "capped weight function". The capped weight function is discussed in the recent paper by Yu et al. (2021) in the context of generalized SM for the unbounded V .
Although Theorem 3 and Corollary 1 show that g 0 andḡ L are natural in the sense that they both give rise to Stein divergences, these choices may not be necessarily efficient in statistical inference tasks. In Section 6 and 8, we show that our choices of weight functions also lead to good statistical inference performance.
In the following section, we investigate the statistical guarantee of TruncSM through finite sample estimation error analysis.
Finite Sample Statistical Guarantee of Generalized Score Matching and Optimality of Truncated Score Matching
To discuss the efficiency of the TruncSM estimator, we first establish a finite sample estimation error bound for generalized SM. Using this result, we show that the TruncSM estimator is a good statistical estimator.
There have been studies on the asymptotic accuracy of generalized SM in, for example, (Yu et al., 2019) and (Barp et al., 2019). However, it is hard to study the impact of weight functions from the asymptotic variance due to its complicated expression.
We now establish an estimation error bound for generalized SM using a generic weight function g. For the convenience of discussion in this section, let us rewrite the generalized SM objective function as
M (θ) = E q [m θ (x)], where m θ (x) := d k=1 (∂ k θ ) 2 + 2∂ 2 k θ =:A k (x;θ) g k + 2∂ k θ =:B k (x;θ) ∂ k g k ,(8)
where we abbreviated log p θ as θ . It holds that FH g (q, p θ ) = M (θ) + C in which C is independent of θ. Given finite samples X q ∼ Q, M (θ) is approximated by the empirical mean:M
(θ) = 1 n n i=1 m θ (x i ).
Thenθ, the minimizer ofM (θ) s.t. θ ∈ Θ ⊂ R r is an M-estimator (van der Vaart, 2000) with the estimation function m θ (x). If θ is the log-likelihood of an exponential family distribution, i.e., θ :
= θ t(x) − log Z(θ), where Z(θ) is the normalizing constant, M (θ) = 1 n n i=1 d k=1 θ ∂ k t(x i )∂ k t(x i ) θ + 2θ ∂ 2 k t(x i ) g k (x i ) + 2θ ∂ k t(x i )∂ k g k ,
which is convex and quadratic with respect to θ. Thus, a uniqueθ can be easily obtained as long as 1
n n i=1 d k=1 g k (x i )∂ k t(x i )∂ k t(x i ) ∈ R r×r is invertible.
However, our theorem below does not assume that p θ has a specific form.
Non-Asymptotic Error Bound
We first establish the non-asymptotic error bound of the generalized SM procedure (8). Let θ * ∈ Θ be the minimizer of M (θ) over Θ. We assume that the optimal parameter θ * is well-separated from other neighbouring parameters in terms of the population objective values:
Assumption 1 Assume that there exists α > 1 such that inf θ: θ−θ * ≥δ M (θ) − M (θ * ) ≥ C g δ α (9)
holds for any small δ > 0. Here, C g is a positive constant that depends on the weight function g such that C ag = aC g for any positive constant a.
Although we mainly focus on the dependency between the convergence rate and g, the constant C g can also depend on the dimensionality of the parameter space r, as we demonstrate in the following example.
Example 1 Let us consider the exponential family
p θ (x) = exp r k=1 t k (x)θ k − φ(θ) , θ ∈ Θ ⊂ R r (10)
such that Θ is bounded, i.e., θ < R. Suppose q = p θ * , guaranteed by Theorem 1 under mild conditions. Then, some calculation yields that M (θ) is a convex quadratic function,
M (θ) − M (θ * ) = (θ − θ * ) d k=1 T k (θ − θ * ), where (T k ) ij = (E θ * [g k (x)∂ k t i (x)∂ k t j (x)]) ij ∈ R r×r . Note that T k is positive semidefinite. We assume that 1 d d k=1 T k T
O holds for a positive definite matrixT ∈ R r×r , where the inequalities are defined in the sense of positive definiteness. A mild assumption is that the eigenvalues ofT does not depend on d. Let λ 1 ≥ · · · ≥ λ r > 0 be eigenvalues ofT , then, we have
M (θ) − M (θ * ) ≥ dλ r θ − θ * 2 .(11)
Hence, C g = dλ r and α = 2 meet Assumption 1. One can also confirm that C ag = aC g for any a > 0.
Assumption 2
The sequenceθ converges to θ * in probability.
The consistency of M -estimator has been well-studied in the statistical literature, thus we do not discuss this in detail. Here we are only interested in the rate that governs the convergence ofθ and its implication on choosing g k . A set of sufficient conditions for proving the consistency ofθ is discussed in Theorem 5.7 of van der Vaart (2000).
We also make continuity assumptions on m θ (x). This is needed to ensure the upperboundedness of the covering number when proving the convergence rate.
Assumption 3 For the function
A k and B k in m θ (x), there existsȦ k ,Ḃ k , ∀k such that |A k (x, θ 1 ) − A k (x, θ 2 )| ≤Ȧ k (x) θ 1 − θ 2 , |B k (x, θ 1 ) − B k (x, θ 2 )| ≤Ḃ k (x) θ 1 − θ 2 .(12)
If A k (x, θ) is differentiable with respect to θ for all k, due to Taylor expansion and Schwarz inequality, we see
|A k (x, θ 1 ) − A k (x, θ 2 )| ≤ sup θ∈Θ ∇ θ A k (x, θ) · θ 1 − θ 2 . Applying the same argument to B k (x, θ), we can see that bothȦ k := sup θ∈Θ ∇ θ A k (x, θ) 2 anḋ B k := sup θ∈Θ ∇ θ B k (x, θ) 2 would satisfy Assumption 3.
Let us define a function Γ(g; A, B) usingȦ k andḂ k :
Γ(g; A, B) := d k=1 (E q [Ȧ 4 k ]E q [g 4 k ]) 1/4 + (E q [Ḃ 4 k ]E q [(∂ k g k ) 4 ]) 1/4 ,(13)
then we have the following non-asymptotic estimation error bound of generalized SM:
Theorem 4 Suppose that Assumptions 1, 2 and 3 hold and that g k , ∂ k g k ,Ȧ andḂ have the fourth order moment under the population distribution q.
Then for δ < CK α · √ r 2 α−1 Γ(g;A,B) Cg we have P θ − θ * ≤ CK α · Γ(g; A, B) δC g · r n 1/(α−1) ≥ 1 − δ,(14)
where C is a universal constant and K α = 2 2α 2 α−1 −1 . The proof can be found in Appendix E.1. In the proof, we use the convergence analysis of the M-estimator according to Section 5.8 in van der Vaart (2000).
Example 2 Let us analyse the estimation error for the exponential family. Following the model definition (10), we can see that α = 2 and Γ g = O(d) 1 . The analysis in Example 1 leads to C g = dλ r . Thus, we have Γg Cg 1 λr and
Pr θ − θ * 2 ≤ C δλ r r n ≥ 1 − δ,
where C is a constant independent of d, r and δ. In the above bound, the probability δ and the sample size n are variables and other parameters such as r and λ r are regarded as a fixed constant.
Theorem 4 shows how the choice of weight g affects the convergence rate of θ − θ * 2 . The only term involving g on the RHS of (14) is Γg Cg . Naturally, we would like to choose a g such that Γg Cg is minimized. Some details will be discussed in Section 6.2 and Appendix F.
Choice of Weight Function
When our model p θ is not correctly-specified, θ * depends on the weight function g. For simplicity, we assume the model is correctly specified and is identifiable, thus Theorem 1 ensures that θ * is unique and p θ * = q under mild conditions. We cannot minimize Γg Cg with respect to g analytically, as we do not know q(x) and the closed form expressions ofȦ k orḂ k . Numerical minimization of Γg Cg can also be cumbersome.
1. For simplicity, let us shorten Γ(g; A, B) as Γg from now on.
Instead, we present a lightweight selection procedure, which eventually gives rise to the distance weight function g 0 by controlling an upperbound of Γg Cg . Suppose there exists a constant Γ G ≥ sup g∈G Γ g , where G is a function family from which g is chosen. We can obtain an upperbound Γ G Cg ≥
Γg
Cg . The choice of G is important as Γ G may not exist or be very large for some G. However, (13) suggests that as long as |∂ k g k | and |g k | are upperbounded andȦ k andḂ k are well behaved, Γ G should exist. Let us consider
G := {g : V → R d |g k ∈ Lip 1 0 (V ), ∀k ∈ [d]}.
We can see ∀g ∈ G, |∂ k g k | ≤ 1 due to the property of Lipschitz function and |g k | is bounded as it is defined over a bounded domain.
Note that we have used Lipschitz functions to construct a Stein class in Section 5, but here, Lipschitz functions emerge from a different context: Suppressing the upperbound of Γ g .
After fixing G, we seek to maximize the denominator C g by choosing an appropriate g ∈ G. Let us introduce the following proposition:
Proposition 1 Given the G defined above, suppose q = p θ * , then inf θ: θ−θ * ≥δ M g 0 (θ) − M g 0 (θ * ) ≥ sup g∈G inf θ: θ−θ * ≥δ M g (θ) − M g (θ * ), where M g (θ) is M (θ) using g as weight function. Proof inf θ: θ−θ * ≥δ M g 0 (θ) − M g 0 (θ * ) = inf θ: θ−θ * ≥δ FH g 0 (θ) − FH g 0 (θ * ) = inf θ: θ−θ * ≥δ sup g∈G FH g (θ) − FH g (θ * ) = inf θ: θ−θ * ≥δ sup g∈G M g (θ) − M g (θ * ) ≥ sup g∈G inf θ: θ−θ * ≥δ M g (θ) − M g (θ * ).
The second equality is due to Lemma 5 and FH g (θ * ) ≡ 0, ∀g. The inequality is due to the max-min inequality.
Proposition 1 shows, when using the distance weight g 0 as the weight function, Assumption 1 should hold for a constant C g 0 = sup g∈G C g . Since g 0 = (g 0 , . . . , g 0 ) is in G (see the proof of Lemma 3), g 0 maximizes C g for all g ∈ G.
As Γ G does not depend on any individual g, g 0 minimizes the upper bound Γ G Cg . Note that we can reach a similar conclusion by replacing G with G := {g ∈ R d |g k ∈ Lip 1 0 (V ), ∀k ∈ [d]} and the minimizer of Γ G Cg would change from distance g 0 to the capped distanceḡ L .
Here we do not claim that g 0 or its capped counterpartḡ L is the best weight function for estimating truncated densities using generalized SM. (13) and (14) suggest that there should be other data-driven choices of g that yield better error bounds. However, g 0 should be an adequate choice without using any information on q, p θ and V , judging from our analysis. Finding a tractable data-driven g that minimizes the estimation error bound is an interesting future work (See Section 9 for more information).
(c) β = −.5 (d) β = 0 (e) β = 1 (f) β = 3 (g) β = −.5 (h) β = 0 (i) β = 1 (j) β = 3
Case Study: Truncated Density in a Unit Ball
In this section, we consider a case where q has a parametric form, V is a unit ball. Although this is a very specific setting, we can see how the relationship between q, V and different choices of L inḡ L would influence the error bound. In what follows, we consider the capped distance weightḡ L (x) = min { Lg 0 (x), 1} which has been introduced in Corollary 1 where the distance weight g 0 is defined using the euclidean distance. Let us define p dist (z) as the probability density of Z = g 0 (X) for X ∼ q. For simplicity, we assume that there exist positive constants b and b such that bz β ≤ p dist (z) ≤ b z β holds for 0 < z ≤ c, where c and β are constants. Note that β should be greater than −1 since the integral of p dist (z) on the interval (0, c) should be bounded.
Example 3 Let V be the unit open
2 ball in R d , i.e., V = {x ∈ R 2 | x < 1}. We can consider a distribution q β (x) ∝ (1 − x ) β on V , where β > −1. Then, we have p dist (z) ∝ (1 − z) d−1 z β , 0 < z ≤ 1. For a small z, we have bz β ≤ p dist (z) ≤ b z β .
See Figure 3 for illustrations of unnormalized q β and p dist (z; β). It can be seen that:
• If q(x) converges to a positive constant as x → ∂V , β = 0.
• If q(x) tends to zero, as x → ∂V , β > 0.
• If q(x) tends to infinity, as x → ∂V , β < 0.
For β > −1 and L ≥ max{1, 1/c}, where c is independent of L, we have the bounds for
Γḡ L Cḡ L : C A 1 − C b ,β L β+1 1/4 + C B c 0 L (1−β)/2 ≤ Γḡ L Cḡ L ≤ C A 1 − c b,β L β+1 1/4 + C B c 1 L (3−β)/4 ,(15)
where c 0 , c 1 , c b,β , C b ,β , C A , C B are positive constants independent of L. The first term of the lower and upper bounds is positive for L ≥ 1. The detailed derivation of this bound is found in Appendix G. Although we assume that p dist takes a specific form in this example, the derivation mostly concerns the behavior of p dist (z), z < c. Thus a slight modification of (15) should cover a different pair of p dist and q β that has the same behavior near the boundary. First, let us consider the condition β > 3 in which case q(x) rapidly goes to zero as x converges to a point on the boundary of V . The upper and lower bounds in (15) converge to C A as L tends to infinity, meaning that a large L does guarantee a reasonable accuracy. When β > 3, q(x) is almost zero around the boundary (see Figure 3). By setting L to a large value, TruncSM withḡ L is essentially the classic SM which is well-suited for non-truncated density estimation.
On the other hand, when q β goes to zero slowly (0 < β ≤ 1) or converges to a constant at the boundary (β ≤ 0), a larger L leads to a larger lower bound, which is undesirable. In particular, when β < 0, increasing L will increase both upper and lowerbound rapidly. This implies a smaller L would yield a better performance when β is small.
Our analysis can also be validated via Figure 3 and Theorem 4: When β is large, p dist is low around the boundary, thus a steepḡ L (i.e., large L) near the boundary would not blow up Γ g (which depends on E q [(∂ k g k ) 4 ] ≤ L 4 ). However, as we reduce β, p dist takes higher values near the boundary. A steepḡ L near the boundary would lead to a large Γ g , hence a larger estimation error.
For the truncated distribution on the bounded domain such as the truncated Gaussian (or Gaussian mixture) model, the probability density p dist (z) is greater than a positive constant near the boundary. The above analysis shows that the capped distance function works efficiently for such truncated probability models. We will empirically validate this analysis in Section 8.2.
Computation of Distance Weight g 0
Comparing to other choices of weight functions, distance weight g 0 (and capped distancē g L ) have an important advantage: the computation of g 0 and its gradient can be efficiently carried out for a properly defined V .
For example, if V and ∂V are expressed by V = {x ∈ R d |u(x) < 0} and ∂V = {x ∈ R d |u(x) = 0} using a function u : V → R, evaluating g 0 (x) can be turned into an optimization problem: g 0 (x) = min z {dist(x, z) | u(z) = 0}. In addition, if Euclidean distance is considered, the gradient of g 0 (x) is simply given by ∇
x g 0 (x) = (x −x)/ x −x , wherex is the minimizer of min z∈∂V x − z .
We only need to evaluate g 0 and ∂ k g 0 exactly once for all x ∈ X q before estimatingθ, since g 0 is model agnostic. This can be advantageous whenp θ is a sophisticated model and the optimization for θ is time-consuming. In contrast, if one uses Monte Carlo methods (e.g. Kannan et al. (1997)) to approximate the normalizing constant Z V (θ) in the likelihood gradient, they are required to update the estimate of Z V (θ) throughout the entire gradient descent procedure. Approximating Z V (θ) using Markov Chain Monte Carlo (MCMC) (Robert and Casella, 2013) can also get less efficient both in terms of statistical accuracy and computation as dimensionality increases. Moreover, some algorithms designed for estimating the truncated multivariate Gaussian likelihood gradient, such as the ones proposed by Daskalakis et al. (2018Daskalakis et al. ( , 2019, do not require exhaustive sampling. They need to evaluate a membership oracle (if x ∈ V or not) for auxiliary samples freshly drawn at each iteration. Membership evaluation can be more efficient than computing g 0 (x) for each x. However, TruncSM only evaluates g 0 for samples in the data set X q once, thus it is a fixed computation cost that does not grow with the number of gradient descent iterations.
There may be a U ⊂ V , for all x ∈ U , the correspondingx is not unique. If so, g 0 will be non-differentiable on U . However, Lemma 3 states that the area in V where g 0 is non-differentiable has a measure zero, thus U also has a measure zero. Therefore, we do not need to worry about such a case in practice. In what follows, we show efficient analytical methods for computing the distance function defined over unit ball, unit cube, convex polytopes and polygons.
• For the Unit Ball, V = {x ∈ R d | x < 1}, the distance function and its gradient:
g 0 (x) = 1 − x , ∇ x g 0 (x) = −x x .
• For the Unit Cube, V = {x ∈ R d | x ∞ < 1}, the distance function and its gradient:
g 0 (x) = 1 − x ∞ , ∇ x g 0 (x) = −e j , j = argmax k |x k |. • For the Convex Polytope V = {x ∈ R d | a t , x + b t < 0, t = 1, .
. . , T }, the distance function is given by
g 0 (x) = min z { x − z | max t∈[T ] { a t , z + b t } = 0} = min t∈[T ] | a t , x + b t | a t(16)
for x ∈ V . The envelope theorem (Milgrom and Segal, 2002) yields ∇ x g 0 (x) = −a t * / a t * , where t * ∈ [T ] is the minimizer of (16). Briec (1997) studied another representation of the minimum distance problem for the convex polyhedral using the extreme points. Note that if V is not a convex set, (16) cannot be used.
• For the Convex Polytope
V = {x ∈ R d | a t , x + b t < 0, t = 1, . . . , T }, let us compute the 1 -based distance function g 0 (x) = min z∈∂V x − z 1 for x ∈ V . It holds that g 0 (x) = max{|α| | x + αe i ∈ V, ∀i} since V is convex. The condition x + αe i ∈ V, ∀i is expressed by α a t , e i ≤ − a t , x − b t for all t and i. Hence, we have g 0 (x) = min i,t s.t. at,e i =0 a t , x + b t a t , e i = min t∈[T ] | a t , x + b t | a t ∞ .(17)
The envelope theorem (Milgrom and Segal, 2002)
yields ∇ x g 0 (x) = −a t * / a t * ∞ for x ∈ V , where t * ∈ [T ]
is the minimizer of the last minimization in (17).
• For the Polygon V in R 2 surrounded by the points p 1 , p 2 , . . . , p T ∈ R 2 , the boundary is given by ∂V = ∪ T t=1 {αp t + (1 − α)p t+1 |α ∈ [0, 1]}, where p T +1 = p 1 . Hence we have
g 0 (x) = min t∈[T ] min α:0≤α≤1 x − αp t − (1 − α)p t+1 .
The minimizer of the inner optimization is α t = min{1, max{0, pt−p t+1 ,x−p t+1 pt−p t+1 2 }}. Hence, we obtain g 0 (x) = min t∈ [T ] x − α t p t − (1 − α t )p t+1 , and ∇ x g 0 (x) = n/ n , where n = x − α t * p t * − (1 − α t * )p t * +1 and where t * ∈ [T ] is the minimizer of g 0 (x).
In these cases, the computation of g 0 can be done in polynomial time with respect to d (if T in convex polytope is a polynomial function of d).
Numerical and Real-world Data Analysis
To demonstrate the efficacy of TruncSM empirically, we conduct a wide range of experiments. In this section, we use the Euclidean metric for g 0 andḡ L . The code/data sets to reproduce our experiments are available at https://github.com/anewgithubname/ Truncated-Score-Matching.
Illustrative Example and Computation Time
In the first experiment, we show an illustrative example comparing TruncSM and Rejection Sampling MLE (RJ-MLE), as well as their computational times. RJ-MLE uses rejection sampling to approximate the intractable normalizing term Z V (θ) and perform maximum likelihood estimation. It can be seen as an example of Monte Carlo MLE (Geyer, 1994). Samples are generated from a Gaussian mixture on R 2 , with centers at
µ 1 = [2, 2], µ 2 = [−2, 2], µ 3 = [−2, −2], µ 4 = [2, −2],
and standard deviations all set to 1. The pre-truncated data set can be seen in Figure 4(a) as black dots. To create a truncated data set, we limit our observation window to be a green polygon region in the middle, thus only samples inside the green polygon (blue points) can be observed. The task is to find all four centers of the mixture model using only blue points. We generate 10,000 samples and only 1417 of which within the truncation boundary can be used for parameter estimation. Our unnormalized density model is a Gaussian mixture model with four components (parametrized by θ 1 , . . . , θ 4 ) and the unit variance-covariance matrix:p θ 1 ,...,θ 4 (x) = 4 i=1 N x (θ i , I). As the polygon boundary ∂V of the non-convex domain V in R 2 consists of line segments, the analytical algorithm in Section 7 is available to compute g 0 and ∇ x g 0 efficiently. We compare with RJ-MLE which uses rejection sampling to approximate the normalizing constant Z V (θ) := Vp θ 1 ,...,θ 4 (x)dx. In this experiment, 500,000 particles are used to approximate Z V (θ) and they are drawn from a bivariate uniform distribution with density U x 1 (−2, 2) · U x 2 (−2, 2). The estimated mixture component centers are plotted as green crosses (RJ-MLE) and red dots (TruncSM) in Figure 4(a). It can be seen that both methods give estimates close to the true mixture centers. However, the computation time for TruncSM and RJ-MLE are 0.35 seconds and 3.35 seconds respectively 2 3 .
It is worth noting that our particle distribution is carefully chosen so that it tightly covers the truncation domain. It ensures the best rejection sampling performance: Overcoverage would increase the computational cost and insufficient coverage would lower the approximation accuracy of Z V (θ). While in a lower dimensional space, the visualization helps, in a higher dimensional space, it is unclear how to come up with a good rejection sampling distribution.
To further investigate the computation time between TruncSM and RJ-MLE, we study the estimation accuracy and computation time (with standard deviation) against the number of particles used for rejection sampling by RJ-MLE. We optimize both objective functions using MATLAB's fminunc function with default settings. From Figure 4(b), we can see that when using a large number of particles to approximate Z V (θ), RJ-MLE can indeed achieve a slightly better performance than TruncSM. However, such a slight improvement of estimation accuracy comes with a significant penalty in computation cost even with an optimal choice of the rejection sampling distribution.
Capped Weight Function
Now we investigate the performance of a capped weight function:ḡ L := min(1, L · g 0 (x)). It can be seen that when L is small,ḡ L will never be capped, thus it is equivalent to g 0 after multiplying a constant, which does not affect the estimator. In this experiment, two different truncation domains are used: a single rectangle and two disjoint rectangles. 1600 samples are drawn from a normal distribution N ([1, 1], I). We monitor the performance of TruncSM usingḡ L as the weight function as V grows in size. We measure the growth of V using a scaling factor b (see Appendix I on how polygons are re-scaled). As shown in Remark 1, the true parameter is identifiable by TruncSM even when the domain consists of two disjoint rectangles. Figure 5 illustrates the truncated data sets as V grows, the estimation performance and the percentage of data points whoseḡ L are capped. Figure 5: The truncated data sets, the estimation performance and the percentage of data points whoseḡ L are capped as V enlarges. b is the scaling multiplier of polygon vertices. It can be seen that as the area of V grows,ḡ L (x) becomes capped for more and more samples in our data set (the bottom right plots in both (a) and (b)). This is expected as the boundary stretches, fewer and fewer points are adjacent to the boundary. In Figure 5(a), we can observe that the performance gap between L = 0.1, L = 10 and L = 100 widens then shrinks: If the truncation domain is small, not many samples are included in the truncation domain. Thus algorithms with all choices of L would suffer. However, as the truncation boundary grows, the difference starts to show: TruncSM with a smaller L has a better performance as we analyzed in Section 6.3. As V grows beyond a certain point, the data set essentially becomes non-truncated, and TruncSM using large L reduces to a classic SM sinceḡ L are always capped at 1. At this time, TruncSM with all choices of L converges to the same level of performance. Figure 5(b) shows a similar story but estimation error with different L converge to different levels as V enlarges: V never covers the center of our data set thus our data sets are always truncated. Again TruncSM with a smaller L gives a better performance as we analyzed in Section 6.3.
2008 Chicago Crime data set
We also test the performance of TruncSM on a real-world truncated density estimation problem: Analyzing the crime occurrences in Chicago. The data set contains locations of homicides that happened in Chicago during 2008. We fit a Gaussian mixture model with two components on this data set. The standard deviations of two components are fixed to the same value, which is roughly the half of the "width" of the city.
In this experiment, we compare TruncSM with vanilla MLE using the non-truncated density model (MLE for short) and RJ-MLE using the truncated density model. In this data set, Chicago city boundary is expressed via a polygon in R 2 , so the distance weight g 0 is calculated using the analytical solution given in Section 7.
The estimated means of two components are plotted on Figure 6. The estimated 95% confidence region is plotted for TruncSM and RJ-MLE as red and black dotted circles respectively. It can be seen that TruncSM, MLE and RJ-MLE all picked centers at the north and south side of the city. However, MLE picked a northern location inside of the city while TruncSM and RJ-MLE picked a location right next to the western border of Chicago.
In this case, TruncSM and RJ-MLE tend to put observed crimes on the decaying slope of a Gaussian density which would better explain the declining rate of crime from the west to the east. MLE, unaware of the truncation, puts the Gaussian center in the middle of the city, while clearly the crimes happen more rarely in the east.
Although all estimators tested in this section solve non-convex optimization problems, in the vast majority of runs with different initializations, we observe only very minor changes in terms of estimated Gaussian centers between different runs. See Appendix J for more discussion on this.
Outlier Over-trimming Compensation
Removing outliers is an important data preprocessing step. If we know the percentage of outliers in our data set, we can adopt methods such as One-class Support Vector Machines (OSVM) (Schölkopf et al., 1999) to remove them. However, it is often impossible to determine the percentage of outliers a priori and setting the outlier percentage aggressively may result in inliers also being removed from the data set.
Consider outlier trimming using OSVM. The method outputs a "decision function" u(x) := jα j φ j (x) which defines a domain V := {x ∈ R d |u(x) > 0}, whereα j is the parameter obtained from OSVM procedure and φ j is a kernel function. In this experiment, we use Gaussian kernel, defined as φ j (x) := exp(− x j − x 2 /2σ 2 ), where j is the j-th datapoint in the contaminated data set. OSVM chooses a V such that a certain proportion (e.g. 80%) of our data set is included in V . We discard any data point that is not in V as "outliers". However, if the specified outlier percentage is larger than it actually is, we will trim inliers from our data set too, and our data set become a truncated data set. Estimation without considering the truncation boundary would lead to biased estimates. This is demonstrated using the following simulated experiment. We sample
x inlier ∼ N 0, 1.5 2 0 0 1 , x outlier ∼ N 3.5 3 , .8 2 0 0 .8 2 .
In total, 500 inlier samples and 50 outlier samples are drawn. The outlier percentage (ν) in OSVM is set to 20%. The data set (black dots), the selected inliers (blue dots) and the truncation domain V given by OSVM is visualized in the top left plot in Figure 7. In this experiment, we allow MATLAB to automatically determine the kernel bandwidth σ according its predefined protocol. On one hand, it can be seen that OSVM does separate the inliers from outliers using a boundary function u(x). On the other hand, some inlier samples are also removed due to the aggressive setting of the outlier proportion. We use the trimmed data set to estimate a truncated multivariate normal distribution p µ,Σ (x) ∝ N (µ, Σ), where Σ is restricted to be a diagonal matrix. In this experiment, TruncSM uses the distance weight g 0 . g 0 and ∇ x g 0 are computed numerically using the constrained optimization described in Section 7. We compare TruncSM with vanilla MLE which does not use any truncation information. The true and estimated 95% confidence regions are visualized by ellipses in Figure 7. The vanilla MLE underestimates the variance of the inlier distribution, while TruncSM accurately recovers the 95% confidence region The confidence region estimated by TruncSM is much closer to the true confidence region. Top right, MLE getting less and less accurate as the percentage of truncated samples (controlled by ν) increases while TruncSM maintaining a good accuracy. Bottom two rows: TruncSM achieves a comparable or better performance than vanilla MLE as the percentage of truncation increases on CIFAR-10 data set.
using the estimated inlier density function. We then perform the same experiment in a much higher dimensional space: The inlier distribution is a 20 dimensional standard normal distribution while the outlier distribution is a 20 dimensional normal distribution with mean 1 and unit variance. Note in this experiment, the truncation boundary (induced by a kernel function) is highly irregular.
In the top right plot of Figure 7, we plot the hold-out likelihood versus different settings of outlier proportion (ν) in OSVM. The likelihood is computed using both vanilla MLE and TruncSM solutions. As we can see, the more aggressive our outlier trimming is, the worse the vanilla MLE performs. On the other hand, TruncSM only drops very slightly in performance as more and more inlier data points are truncated.
We now perform an experiment on a real-world data set CIFAR-10 which contains 10 different classes of 32 by 32 images. To speed up the computation, we reduced the dimension of the data set to 10 using PCA. For each class, we artificially add 10% outliers that are drawn from a normal distribution with mean 1 and unit variance. We use both TruncSM and vanilla MLE to estimate a multivariate Normal distribution for each class in the reduced 10 dimensional space and plot the hold-out likelihood of each method. The hold-out likelihood is plotted at the bottom two rows of Figure 7. For a large ν, we find that TruncSM performs better than MLE. The holdout likelihood shows that the estimation bias induced by the trimmed data can be corrected by TruncSM. Although the advantage of TruncSM is smaller on this data set, we argue that this is a very challenging problem: The data set is hardly Gaussian so the model assumption used by TruncSM is wrong. Nonetheless, in some cases (automobile and truck), TruncSM significantly outperforms MLE.
Conclusions and Future Works
We propose an estimator for truncated statistical models with complex truncation boundaries based on generalized SM. The proposed method uses the shortest distance (or capped distance) from a data point to the truncation boundary as the weight function. Such a choice of weight function naturally arises from minimizing Stein Discrepancy and lowering the estimation error upper bound. The proposed weight function is also computationally favorable for high dimensional truncation domains. Experiments on synthetic data and the Chicago crime data set show promising results. The proposed estimator was later applied to outlier trimming bias correction.
Although the proposed method achieves promising results in truncated density estimation, there are interesting open questions:
• How to choose an optimal weight function g when p θ , q and V are given? As we have seen from (14) in Theorem 4, the statistical estimation error depends on the ratio Γg Cg and both Γ g and C g depends on the weight function g, the density model p θ , the data density q and the truncation boundary V . A natural idea is to choose a g that minimizes
Γg
Cg . Can we find an efficient numerical procedure for such a minimization? • A related question is, how to choose an appropriate distance metric in g 0 ? For example, 1 and 2 distances are both valid choices for g 0 . Again, Theorem 4 suggests that, in terms of upperbounding the statistical estimation error, the answer depends on q, p θ as well as V . However, how would the geometry of V , q and p θ affect the choice of the distance metric? See Appendix H for an empirical comparison between 1 and 2 distances.
• The computation of g 0 for a generic V is not trivial. In particular, when the dimensionality of data is large, computing g 0 can become computationally infeasible. Finding efficient ways of evaluating or approximately evaluating g 0 may help us extend the usage of TruncSM estimator to higher dimensional data sets.
• In many applications, our data set is automatically filtered by some algorithm. Outlier trimming by OSVM is just one example. Suppose we have mixed images of cats and dogs and a binary classifier, we can easily identify the high confidence region of each class. However, the classification step would also create artificial truncation boundaries around the distributions of individual classes. Can we use TruncSM to accurately reconstruct the true underlying distribution using these pre-classified images?
A. Generalized Score Matching
Assume that p(x), q(x) and g(x) take strictly positive values on the domain V ⊂ R d . Suppose that generalized SM objective function with weight g is equal to zero, i.e.,
E q [ g 1/2 (x) • ∇ x log p(x) − g 1/2 (x) • ∇ x log q(x) 2 ] = 0. (A.18)
Then, we have ∇ x (log p(x) − log q(x)) = 0, meaning that p(x) = Cq(x) on (the connected domain) V with a positive constant C. Since both p and q are the probability densities on V , we have C = 1.
B. Proof of Lemma 3
Proof First, we show that g 0 is Lipschitz continuous with respect to the metric dist(·, ·).
∀x a , x b ∈ V, g 0 (x a ) − g 0 (x b ) = min x ∈∂V max x ∈∂V dist(x a , x ) − dist(x b , x ) ≤ max x ∈∂V dist(x a , x ) − dist(x b , x ) ≤ dist(x a , x b ).
The last inequality is due to the triangle inequality. Likewise, g 0 (x b ) − g 0 (x a ) is also bounded above by dist(x a , x b ). Therefore g 0 is a Lipschitz function with Lipschitz constant 1. Rademacher's theorem (Evans and Gariepy, 1992) asserts that a Lipschitz continuous function is differentiable at every point in a Lipschitz domain outside a set of measure zero. Therefore, we can construct
D k g 0 (x) := ∂ k g 0 (x), if g 0 is differentiable, arbitary constant, otherwise (B.19)
We can check that D k g 0 (x) is a valid weak derivative (Definition 7.1.3., (Atkinson and Han, 2005)):
V g 0 (x)∂ k φ(x)dx = − V D k g 0 (x)φ(x)dx,
where φ is any m-times differentiable function on a compact support on V. The equality holds due to the classic integration by parts formula for continuous differentiable functions. g 0 is only non-differentiable over a zero set, so the arbitary constant we set in (B.19) does not affect the outcome of the integration. Since V is a bounded domain and g 0 is 1-Lipschitz, g 0 and D k g 0 are both bounded in terms of the · L 2 (V ) norm. Therefore, g 0 is weakly differentiable and in a Sobolev-Hilbert space.
C. Proof of Theorem 2
Proof In this proof, we apply Theorem 4 to derive a tractable expression for M (θ). Let us consider the second term of M (θ). As q(x) and g 0 (x)∂ k log p θ (x) are functions in the Sobolev-Hilbert space of the first order, the direct application of Theorem 4 leads to
d k=1 V g k (x)[∂ k log p θ (x)][∂ k log q(x)]q(x)dx = d k=1 V g k (x)[∂ k log p θ (x)][∂ k q(x)]dx = d k=1 ∂V g k (x)[∂ k log p θ (x)]q(x)ν k (x)ds − V ∂ k [g k (x)∂ k log p θ (x)]q(x)dx = − d k=1 V ∂ k [g k (x)∂ k log p θ (x)]q(x)dx.
The second equality is ensured by Theorem 4 and the third equality holds from the boundary condition imposed in the theorem.
D. Proofs of Theorem 3 and Corollary 1 in Section 5
D.1 Proof of Lemma 5
max g k ∈Lip L 0 (V ) E q k (∂ k θ − ∂ k log q) 2 g k = L · E q k (∂ k θ − ∂ k log q) 2 g 0 = L · FH g 0 (q, p θ ),
The first equality holds since ∀k, (∂ k θ − ∂ k log q) 2 ≥ 0. Since for all g k ∈ Lip L 0 (V ), g k (z) = 0, ∀z ∈ ∂V ,
g k (x) = g k (x) − g k (x ) ≤ Ld(x, x ), ∀x ∈ ∂V =⇒ g k (x) ≤ L · min z∈∂V d(x, z) = Lg 0 (x), (D.20)
Hence the second equality.
D.2 Proofs of Theorem 3
Proof As we stated in the proof of v Lemma 3, due to Rademacher's theorem (Evans and Gariepy, 1992), any Lipschitz function defined on a Lipschitz domain is in H 1 (V ), hence g k ∈ H 1 (V ) and ∀k, f k ∈ H 1 (V ). Using the fact that f k , q, θ ∈ H 1 (V ), f k (x) = 0, ∀x ∈ ∂V , we can verify that f is in a Stein class of q by applying the same integration by parts techniques used in Section C. Therefore, we can write the Stein discrepancy between p θ and q as:
max f ∈F E q [T p θ f ] = max f ∈F E q [T p θ f − T q f ] .
(D.21)
Now we optimize (D.21) analytically using Theorem 2 in Barp et al. (2019)
max f ∈F E q [T p θ f − T q f ] = max g k ∈Lip L 0 (V ) max h E q k (∂ k θ − ∂ k log q) · g 1 2 k h k = max g k ∈Lip L 0 (V ) E q k (∂ k θ − ∂ k log q) 2 g k .
Applying Lemma 5, we obtain the desired result.
D.3 Proof of Corollary 1
Proof The proof is mostly the same as the proof of Theorem 3. However we need to prove a different version of Lemma 5 for the capped Lip L 0 (V ) family. Lemma 5 states g k (x) ≤ L · min z∈∂V dist(x, z) = g 0 (x) for all g k ∈ Lip L 0 (V ). Further, we know that g k ≤ 1 by definition. Therefore g k (x) ≤ min(1, Lg 0 (x)) =ḡ L . Notice that g L ∈ Lip L 0 (V ). Using the same argument in Section D.1, we can see
max g k ∈Lip L 0 (V ) E q k (∂ k θ − ∂ k log q) 2 g k = FHḡ L (q, p θ ).
Applying this result to the last step in Section D.2, gives the desired result.
E. Proof of Theorem 4
E.1 Proof of Theorem 4
We assume that
E sup θ: θ−θ * ≤δ |G n (m θ − m θ * )| ≤ C g δ β , (E.22) where G n f = 1 √ n n i=1 (f (X i ) − E q [f (X)])
for the independent random variables X 1 , . . . , X n from q, and C g is a positive constant depending on the function g satisfying the same property as C g . Then, the estimation accuracy is given by the following theorem.
Theorem E.5 (Theorem 5.52 in van der Vaart (2000)) Suppose Assumption 1, 2 and (E.22) hold for α > β > 0 and α > 1. Then, for any positive integer K we have
P θ − θ * > 2 K n 1/(2(α−β)) ≤ 2 K(β−α) 2 2α 2 α−1 − 1 C g C g .
Hence, we have θ − θ * = O p (n −1/(2(α−β)) ). Our concern is the relation between the coefficient of the convergence rate and the weight function g.
The upper bound of the expectation (E.22) is closely related to the covering number of the parameter space Θ ⊂ R r . Let us define N [] (ε, F, L 2 (Q)) be the bracketing number of F with the radius ε under the norm of L 2 (Q) and J [] (δ, F, L 2 (Q)) be
J [] (δ, F, L 2 (Q)) = δ 0 log N [] (ε, F, L 2 (Q))dε
Then, the following theorem holds.
Theorem E.6 (Corollary 19.35 of van der Vaart (2000)) Let F be an envelope for the function class
F ⊂ L 2 (Q), i.e., sup f ∈F f (x) ∞ ≤ F (x) for every x ∈ V and suppose E Q [|F (X)| 2 ] < ∞. Then, we have E sup f ∈F |G n (f )| ≤ C J [] ( F L 2 (Q) , F, L 2 (Q)),
where C is a universal constant.
The bracketing number of the parametric model is given by the following proposition.
Proposition 2 (Example 19.6 in van der Vaart (2000)) The bracketing number of the parametrized loss function m θ (x) is given as follows. Let
Θ ⊂ R r be contained in a ball of radius R. Let F = {m θ (x) | θ ∈ Θ} be a function class indexed by Θ. Suppose there exists a functionṁ(x) with ṁ L 2 (Q) < ∞ such that |m θ 1 (x) − m θ 2 (x)| ≤ṁ(x) θ 1 − θ 2 2
for all x ∈ V and θ 1 , θ 2 ∈ Θ. Then, for every ε > 0,
N [] (ε ṁ L 2 (Q) , F, L 2 (Q)) ≤ 1 + 4R ε r .
In our case, we need to evaluate E[sup f ∈F δ |G n (f )|] for
F δ := {m θ (x) − m θ * (x) | θ ∈ Θ, θ − θ * ≤ δ},
where θ * is the minimizer of the expected loss M (θ). The functionṁ(x) in Proposition 2 should satisfy
|(m θ 1 (x) − m θ * (x)) − (m θ 2 (x) − m θ * (x))| = |m θ 1 (x) − m θ 2 (x)| ≤ṁ(x) θ 1 − θ 2 .
Then, Proposition 2 leads to
N [] (ε ṁ L 2 (Q) , F δ , L 2 (Q)) ≤ 1 + 4δ ε r .
The envelope function F δ (x) of F δ is given by F δ (x) =ṁ(x)δ, because
|m θ (x) − m θ * (x)| ≤ṁ(x) θ − θ * ≤ṁ(x)δ.
Hence, we obtain
N [] (ε F δ L 2 (Q) , F δ , L 2 (Q)) = N [] (εδ ṁ L 2 (Q) , F δ , L 2 (Q)) ≤ 1 + 4 ε r ,
and thus,
E[ sup f ∈F δ |G n (f )|] ≤ Cδ ṁ L 2 (Q) √ r 1 0 log 1 + 4 ε dε ≤ C √ rδ ṁ L 2 (Q)
holds, where C and C are universal constants. We find that β in (E.22) is given by β = 1. Let us evaluate the norm ṁ L 2 (Q) . For the function A k and B k in (8), we have the following inequalities
|A k (x, θ 1 ) − A k (x, θ 2 )| ≤Ȧ k (x) θ 1 − θ 2 , |B k (x, θ 1 ) − B k (x, θ 2 )| ≤Ḃ k (x) θ 1 − θ 2 . (E.23)
It is straightforward to see that the followingṁ satisfies the required condition:
m(x) = k {g k (x)Ȧ k (x) + |∂ k g k (x)|Ḃ k (x)}.
Cauchy-Schwarz inequality leads to
g kȦk L 2 (Q) ≤ (E q [Ȧ 4 k ]) 1/4 (E q [g 4 k ]
) 1/4 . The similar inequality holds for Ḃ k ∂ k g k L 2 (Q) . Hence, we have
ṁ L 2 (Q) ≤ Γ(g; A, B) := d k=1 (E q [Ȧ 4 k ]E q [g 4 k ]) 1/4 + (E q [Ḃ 4 k ]E q [|∂ k g k | 4 ]) 1/4 3 . (E.24)
In summary, we have the estimation error bound in Theorem 4.
F. Some More Analysis of Weight Functions
In this subsection, we assume that the statistical model is realizable, i.e., q = p θ * holds. Let U be a subset in the domain V . Under the regularity condition later shown in Appendix F.1, one can find that the constant C g in (9) is given by
C g = min x∈U min k∈[d] g k (x) (F.25)
up to a constant independent of g. As the continuous and non-negative weight function g k (x) can take zero only on the boundary ∂V , C g > 0 holds if U is a closed subset which does not include boundary points of V . Let us consider Γ(g; A, B)/C g in the upper bound in Theorem 4. The following theorem ensures that the minimum function
h(x) = min k∈[d] g k (x) (F.26)
improves the upper bound of the estimation error under some conditions on g k .
V | g k (x) = g k (x), ∃k, k ∈ [d], k = k } is measure zero. We assume that |∂ k g k (x)| ≥ |∂ k g k * (x)|, holds for x ∈ V and k ∈ [d], where k * ∈ [d] is the number such that h(x) = g k * (x). Then, we have Γ(h; A, B) C h ≤ Γ(g; A, B) C g ,
where C g and C h are defined by (F.25).
The proof is found in Appendix F.2.
Example 4 (Bounded Rectangular Domain) For the d-dimensional rectangular
V = d k=1 [0, c k ]
, let us consider the statistical accuracy of the generalized score matching (SM) method (Yu et al., 2019) with g k (x) = min{x k , c k − x k }, and h(x) = min k g k (x). Note that the weight h is nothing but g 0 in (6). According to Theorem F.7, we find that the estimator with the weight h is superior to the estimator with the above g 1 , . . . , g d in the sense of the estimation error bound.
Example 5 (Unit ball under p-norm) Let us define V as the d-dimensional unit ball under p-norm, V = {x ∈ R d | x p ≤ 1}. The weight g k (x) is defined by the distance from x to the boundary of V along the k-th axis. This is expressed by
g k (x) = max{|ε| | x + εe k ∈ V } = min{(1 − x (k) p p ) 1/p − x k , x k + (1 − x (k) p p ) 1/p },
where e k is the unit vector along with the k-th axis, x (k) is the d−1 dimensional vector dropped the k-th element x k from x. Some calculation yields the inequality |∂ k g k (x)| = 1 ≥ |∂ k g (x)| for = argmin k g k (x). Hence, the minimum function h(x) = min k g k (x) improves the error bound of the estimator with the weight g k . One can confirm that h(x) = min z∈∂V x − z 1 holds. Likewise, for the truncated domain
V = {x = (x 1 , . . . , x d ) ∈ R d | x p ≤ 1, x i ≥ c i }, c i ∈ R, one can prove the inequality |∂ k g k (x)| = 1 ≥ |∂ k g (x)|. The above assertions are explained in Appendix F.3.
Under the assumption of Theorem F.7, we find that the homogeneous weight h improves the upper bound of the estimation error for the estimator with g. Furthermore, the weight h corresponds to the minimum 1 distance to the boundary as shown in Example 5. To compare the distance-based homogeneous weights, we need to evaluate the upper bound for each distance. In Appendix H, we show numerical comparison between 1 and 2 distances and numerical evaluation of the theoretical upper bound.
F.1 Derivation of (F.25)
Suppose the true probability q(x) is p θ * (x). Let U be an open subset of V . Then, we have
M (θ) − M (θ * ) = V k g k (x)(∂ k log p θ (x) − ∂ k log p θ * (x)) 2 p θ * (x)dx ≥ U k g k (x)(∂ k log p θ (x) − ∂ k log p θ * (x)) 2 p θ * (x)dx ≥ min x∈U min k {g k (x)} U k (∂ k log p θ (x) − ∂ k log p θ * (x)) 2 p θ * (x)dx.
Suppose that the Hessian matrix of
U k (∂ k log p θ (x) − ∂ k log p θ * (x)) 2 p θ * (x)dx (F.27)
as the function of θ is non-degenerate, there exists a constant C 0 independent of g such that the above integral is bounded below by C 0 θ − θ * 2 . Hence, C g = min x∈U min k {g k (x)} satisfies the required condition.
F.2 Proof of Theorem F.7
From the definition of h, one can find
C g = min x∈U min k g k (x) = min x∈U h(x) = C h .
We evaluate Γ(g; A, B) and Γ(h; A, B). As for the derivative function, the assumption guarantees that ∂ k h(x) = ∂ k g k * (x) and |∂ k g k (x)| ≥ |∂ k g k * (x)|. Hence, we have
Γ(g; A, B) = d k=1 (E q [Ȧ 4 k ]E q [g 4 k ]) 1/4 + d k=1 (E q [Ḃ 4 k ]E q [|∂ k g k | 4 ]) 1/4 ≥ d k=1 (E q [Ȧ 4 k ]E q [(min k g k ) 4 ]) 1/4 + d k=1 (E q [Ḃ 4 k ]E q [|∂ k g k * | 4 ]) 1/4 = Γ(h; A, B).
As a result, we obtain Γ(g;A,B)
Cg ≥ Γ(h;A,B) C h .
F.3 Some Equations in Example 5
Proof of |∂ k g k (x)| ≥ |∂ k g k * (x)|: Without loss of generality, we suppose x = (x 1 , . . . , x d ) ≥ 0. Then, g k (x) = (1 − x (k) p p ) 1/p − x k holds. Firstly, we prove that g (x) < g k (x) leads to x k < x . For the sake of simplicity let us assume k = 1 and = 2. Let us define c p = 1 − x p 3 − · · · − x p d for c ≥ 0. Then, we find that g 2 (x) < g 1 (x) leads to (c p − x p 1 ) 1/p + x 1 < (c p − x p 2 ) 1/p + x 2 . The function f (x) = (c p − x p ) 1/p + x for 0 ≤ x ≤ c is concave, takes the maximum value at x = c/2 1/p (< c) and satisfies f (x) = f ((c p − x p ) 1/p ). When y lies on the interval between x and (c p −x p ) 1/p , f (x) = f ((c p −x p ) 1/p ) ≤ f (y) holds. Suppose x p 1 +x p 2 ≤ c 2 and f (x 1 ) < f (x 2 ) = f ((c p − x p 2 ) 1/p ). If x 1 lies on the interval between x 2 and (c p − x p 2 ) 1/p , f (x 2 ) ≤ f (x 1 ) holds and it is the contradiction. Hence we have x 1 < x 2 . Next, we evaluate the absolute value of the derivatives. When g (x) < g k (x), x k < x holds. For k = , we obtain |∂ k g k (x)| = 1 and |∂ k g (x)| = |x k | p−1 (1 − x ( ) p p ) 1/p−1 ≤ |x | p−1 (|x | p ) 1/p−1 = 1 for p ≥ 1.
Rough sketch of the proof of h(x) = min z∈∂V x−z 1 : Suppose that all the vertexes of the polytope B 1 (x, c) := {z ∈ R d | x − z 1 ≤ c} are included in V . Then, B 1 (x, c) ⊂ V holds as V is convex. Clearly, h(x) is expressed by sup c≥0 {c|B 1 (x, c) ⊂ V }. This is nothing but the 1 -distance from x to the boundary of V .
G. Derivation of (15)
Let us define U by U = {x ∈ V |g 0 (x) ≥ c}. We see that min x∈UḡL (x) = 1 holds from Section F.1. Hence we have Cḡ L = 1. Let us evaluate the fourth order moments ofḡ L : Next, we evaluate the lower bound of the fourth moment of the derivative ∂ k g k . Jensen's inequality yields that
E[|∂ kḡL | 4 ] = L 4 d k=1 g 0 (x)≤1/L |x k − x k | 4 g 0 (x) 4 q(x)dx ≥ dL 4 1 d d k=1 g 0 (x)≤1/L |x k − x k | 2 g 0 (x) 2 q(x)dx 2 = L 4 d g 0 (x)≤1/L q(x)dx 2 ≥ b 2 d(β + 1) 2 L 2(1−β)
Let us define C
A = d k=1 (E q [Ȧ 4 k ]) 1/4 , C B = d k=1 (E q [Ḃ 4 k ]
) 1/4 , c 0 = (b 2 /(d(β + 1) 2 )) 1/4 , and c 1 = (b /(β + 1)) 1/4 . Then, we have In this section, we test different metrics for g 0 : 1 distance dist(x, y) := x − y 1 and 2 distance dist(x, y) := x − y 2 . The data set is generated from N (1 d · 0.5, I d ), where 1 d and I d are the d dimensional all one vector and the d × d identity matrix respectively. The truncation domain is V = {x| x 1 < 1 and x d > 0}. 150 samples are generated. We plot the estimation error θ − θ * and its standard deviation for both choices of distances: 1 and 2 versus the dimensionality d in Figure 8. It can be seen that the Euclidean distance, i.e., 2 distance consistently achieves slightly lower error than the 1 distance.
C A 1 − C b ,β L β+1 1/4 + C B c 0 L (1−β)/2 ≤ Γ(ḡ L ; A, B) Cḡ L ≤ C A 1 − c b,β L β+1 1/4 + C B c 1 L (3−β)/4 .
H. Choice of Weight Function: 1 vs. 2 distance
Let us numerically evaluate the upper bound Γ(h p ; A, B)/C hp for the weight h p = (h p , . . . , h p ), where h p (x) = min z∈∂V x − z p , p = 1, 2. Let "sup U " be the supremum over all open subsets of V . Then, we have sup U min x∈U h 1 (x) = 1/2 and sup U min x∈U h 2 (x) = 1/( √ d + 1). In order to obtain the tight bound, we use C h 1 = 1/2 and C h 2 = 1/( √ d + 1) for our evaluation. ForĀ = max k E q [|Ȧ k | 4 ] 1/4 ,B = max k E q [|Ḃ k | 4 ] 1/4 , and R =Ā/(Ā +B), In this problem setup, one can confirm that the upper bound for p = 2 is slightly smaller than that of p = 1 for all R ∈ [0, 1], as the bound is the linear function of R. The theoretical conclusion verifies the numerical results.
I. Increasing Boundary Size in Section 8.2
When experimenting with different values of L used in capping the distance functionḡ L , an increasing boundary size (shown by values of b in Figure 5) is used in the experiments. The two regions, the square boundary and the disjoint boundary, are created by supplying set(s) of vertices Ω, detailing the locations of the polygonal truncation domain. The variable b controls the boundary size by offsetting the supplied vertices in Ω. For the square, this was
Ω sq := {(−b, −b), (−b, b), (b, b), (b, −b)}.
Increasing b in this scenario leads to the corners of the square boundary being shifted by an equal amount. For the disjoint boundary, two sets of vertices were given for two disjoint domains: Increasing b in this case will enlarge the truncation domain while the "disjoint" section in the middle remains unchanged.
J. Chicago Crime data set with Different Random Initialization
As we mentioned in Section 8.3, both TruncSM and RJ-MLE solve non-convex optimization problems so it is possible for them to return local optima. Therefore, to show the stability of both methods' solutions, in this experiment, we randomly initialize both methods with the initial point µ q + , where µ q is the mean of the data set X q and ∼ N (0, I d · 0.06 2 ). Both methods were executed 500 times and each time, we plot two lightly shaded balls centered at the estimated mixture centers with their radius equal to the standard deviation. One can see from the blue/red shades in Figure 9, both algorithms consistently place centers at northern and southern Chicago, and both balls are roughly centered at locations reported in the Section 8.3. TruncSM also placed a Gaussian center outside of the boundary of Chicago in one of the simulations. This is a very rare event and such a result can be easily ruled out.
Figure 1 :
1Boundary of Chicago, where blue dots are locations of homicides in 2008.
Definition 3 (
3Lipschitz Domain) Let V be an open and bounded domain in R d . We say V is a Lipschitz domain if for any x ∈ ∂V , there exists an r > 0 and a Lipschitz function f
Figure 3 :
3Top row: unnormalized q β (x), where ∂V in Example 3 is illustrated using a red dashed line. Bottom row: unnormalized p dist (z) in Example 3.
Figure 4 :
4of truncated Gaussian mixture centers (b) Computational cost and estimation accuracy comparison for TruncSM (red) and RJ-MLE (blue) Gaussian mixture centers truncated by a polygon
Figure 6 :
6Chicago Crime data set, whose truncation boundary is a polygon. Blue circles are homicide locations. g 0 is visualized at the upper right corner. The estimated component centers of TruncSM, RJ-MLE and MLE are plotted.
Figure 7 :
7Top left: MLE underestimates the 95% confidence region due to the truncation.
pp
dist (z)dz = 1 − C L 1+β , where C is a positive constant such as c b,β := 4b β 2 +6β+5 ≤ C ≤ C b ,β := 4b β 2 +6β+5 . Due to the non-negativity of E[ḡ 4 L ], L 1+β ≥ C should hold.This inequality is guaranteed dist (z)dz. Let us evaluate the second term of Γ(g; A, B). The derivative ∂ kḡL is given by∂ kḡL (xk − x k x − x , g 0 (x) < 1/L, 0, g 0 (x) > 1/L,where x is the minimum solution of min z∈∂V x − z . Hence, we haveE[|∂ kḡL | 4 ] = L 4 g 0 (x)≤1/L |x k − x k | 4 g 0 (x) 4 q(x)dx ≤ L
Figure 8 :
8Estimation accuracy of TruncSM using 1 vs. 2 distance.
Ω
dis1 :={(1 − b, 0.5 − b), (1 − b, 0.5), (1 + b, 0.5), (1 + b, 0.5 − b)}, Ω dis2 :={(1 − b, 1.5), (1 − b, 1.5 + b), (1 + b, 1.5 + b), (1 + b, 1.5)}.
Theorem F.7 Let g = (g 1 , . . . , g d ) and h = (h, . . . , h) be the weight functions, where h is defined by (F.26). Suppose that g k is differentiable on V except a measure zero set and that the set {x ∈
Table 1 :
1Comparison of the weight h p , p = 1, 2. The table shows RdE q [|h p | 4 ] 1/4 + (1 − R) d k=1 E q [|∂ k h p | 4 ] 1/4 /C hp for R = 0.2, 0.8.R = 0.2
R = 0.8
weight \ dim
2
3
4
5
6
2
3
4
5
6
h 1
3.160 4.558 5.939 7.309 8.685 1.569 2.088 2.537 2.975 3.371
h 2
3.073 4.279 5.412 6.496 7.552 1.545 2.012 2.402 2.764 3.081
clearly we have Γ(h p ; A, B) ≤ (Ā +B) RdE q [|h p | 4 ] 1/4 + (1 − R) k=1 E q [|∂ k h p | 4 ] 1/4 . For x ∼ N d (1 d · 0.5, I d ), we numerically evaluate E q [|h p | 4 ] and E q [|∂ k h p | 4 ].Table 1showsRdE q [|h p | 4 ] 1/4 + (1 − R) d k=1 E q [|∂ k h p | 4] 1/4 /C hp of each dimension and weight for R = 0.2, 0.8.d
. For TruncSM, we include the calculation time for both g0 and ∂ k g k . Same below. 3. Our experiments are run on a workstation with a AMD Ryzen 1700 CPU with 32GB memory.
AcknowledgmentsWe thank two anonymous reviewers for their insightful feedbacks. This work was supported by Japan Society for the Promotion of Science under KAKENHI Grant Number 17H00764, 19H04071, and 20H00576. Daniel J. Williams was supported by a PhD studentship from the EPSRC Centre for Doctoral Training in Computational Statistics and Data Science (COMPASS).
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"Wei Wu \nLanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000Lanzhou, GansuChina\n",
"Ze-Zhou Zhang \nLanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000Lanzhou, GansuChina\n",
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"Lanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000Lanzhou, GansuChina"
] | [] | We propose a strategy to modulate the decoherence dynamics of a two-level system, which interacts with a dissipative bosonic environment, by introducing an ancillary degree of freedom. It is revealed that the decay rate of the two-level system can be significantly suppressed under suitable steers of the assisted degree of freedom. Our result provides an alternative way to fight against decoherence and realize a controllable quantum dissipative dynamics. A microscopic quantum system inevitably interacts with its surrounding environment, which generally results in decoherence 1-3 . Such decoherence process is responsible for the deterioration of quantumness and is commonly accompanied by energy or information dissipation. In this sense, how to prevent or avoid decoherence is of importance for any practical and actual quantum technology aimed at manipulating, communicating, or storing information. Furthermore, understanding decoherence in itself is one of the most fundamental issues in quantum mechanics, since it is closely associated with the quantum-classical transition 4 .Up to now, various strategies have been proposed to suppress decoherence. For example, (1) the theory of decoherence-free subspace 5-7 , in which the quantum system undergoes a unitary evolution irrespective of environment's influence; (2) dynamical decoupling pulse technique 8-10 , which aims at eliminating the unwanted system-environment coupling by a train of instantaneous pulses; (3) quantum Zeno effect 11-13 , which can inhibit the decay of a unstable quantum state by repetitive measurements; and (4) the bound-state-based mechanism scheme 14-17 , which can completely suppress decoherence and generate a dissipationless dynamics in the longtime regime. Each method has its own merit and corresponding weakness. For example, one needs to optimize the shapes and the intervals of pulses when using the dynamical decoupling pulse technique. Such optimization requires an elaborate operation as well as a great deal of experience. Here, we propose a simple scheme, which is more practical than the above methods. We believe that any alternative approach would be beneficial for us to achieve a reliable quantum processing in a noisy environment.In this paper, we propose an efficient scheme to obtain a controllable dynamics of a two-level system (TLS), which interacts with a dissipative bosonic environment. An ancillary single-mode harmonic oscillator (HO), which acts as a steerable degree of freedom, is coupled to the TLS to modulate its decoherence dynamics 18-21 . We find the decay of the TLS can be suppressed via adjusting the parameters of the assisted HO. We also demonstrate the single-mode HO can be equivalently replaced by a periodic driving field or a multi-mode bosonic reservoir, which can likewise achieve the effect of decoherence-suppression. Moreover, we numerically confirm our steer scheme can be generalized to a more general quantum dissipative system, in which the TLS-environment coupling is strong and the so-called counter-rotating-wave terms are included.OPEN | 10.1038/s41598-021-86553-z | null | 221,655,124 | 2009.06255 | 9d6b8444872b74a653ef0cfb10f92283d46c7d41 |
Controllable dynamics of a dissipative two-level system
0123456789
Wei Wu
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000Lanzhou, GansuChina
Ze-Zhou Zhang
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000Lanzhou, GansuChina
+ K Ω K B † K B K + K G K (
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000Lanzhou, GansuChina
Controllable dynamics of a dissipative two-level system
012345678910.1038/s41598-021-86553-zScientific Reports | (2021) 11:7188
We propose a strategy to modulate the decoherence dynamics of a two-level system, which interacts with a dissipative bosonic environment, by introducing an ancillary degree of freedom. It is revealed that the decay rate of the two-level system can be significantly suppressed under suitable steers of the assisted degree of freedom. Our result provides an alternative way to fight against decoherence and realize a controllable quantum dissipative dynamics. A microscopic quantum system inevitably interacts with its surrounding environment, which generally results in decoherence 1-3 . Such decoherence process is responsible for the deterioration of quantumness and is commonly accompanied by energy or information dissipation. In this sense, how to prevent or avoid decoherence is of importance for any practical and actual quantum technology aimed at manipulating, communicating, or storing information. Furthermore, understanding decoherence in itself is one of the most fundamental issues in quantum mechanics, since it is closely associated with the quantum-classical transition 4 .Up to now, various strategies have been proposed to suppress decoherence. For example, (1) the theory of decoherence-free subspace 5-7 , in which the quantum system undergoes a unitary evolution irrespective of environment's influence; (2) dynamical decoupling pulse technique 8-10 , which aims at eliminating the unwanted system-environment coupling by a train of instantaneous pulses; (3) quantum Zeno effect 11-13 , which can inhibit the decay of a unstable quantum state by repetitive measurements; and (4) the bound-state-based mechanism scheme 14-17 , which can completely suppress decoherence and generate a dissipationless dynamics in the longtime regime. Each method has its own merit and corresponding weakness. For example, one needs to optimize the shapes and the intervals of pulses when using the dynamical decoupling pulse technique. Such optimization requires an elaborate operation as well as a great deal of experience. Here, we propose a simple scheme, which is more practical than the above methods. We believe that any alternative approach would be beneficial for us to achieve a reliable quantum processing in a noisy environment.In this paper, we propose an efficient scheme to obtain a controllable dynamics of a two-level system (TLS), which interacts with a dissipative bosonic environment. An ancillary single-mode harmonic oscillator (HO), which acts as a steerable degree of freedom, is coupled to the TLS to modulate its decoherence dynamics 18-21 . We find the decay of the TLS can be suppressed via adjusting the parameters of the assisted HO. We also demonstrate the single-mode HO can be equivalently replaced by a periodic driving field or a multi-mode bosonic reservoir, which can likewise achieve the effect of decoherence-suppression. Moreover, we numerically confirm our steer scheme can be generalized to a more general quantum dissipative system, in which the TLS-environment coupling is strong and the so-called counter-rotating-wave terms are included.OPEN
Results
Controllable dissipative dynamics. Let us consider a TLS interacts with a dissipative bosonic environment. To achieve a tunable reduced dynamics of the TLS, we add an ancillary single-mode HO, which serves as a controllable degree of freedom to modulate the dynamical behaviour of the TLS. The whole system can be described as follows (throughout the paper, we set = k B = c = 1) [18][19][20][21] where σ ± ≡ 1 2 (σ x ± iσ y ) with σ x,y,z being the standard Pauli operators, ǫ is the transition frequency of the TLS, a † and a are creation and annihilation operators of the assisted HO with frequency ω 0 , and the parameter g 0 quantifies the coupling strength between the TLS and the HO. b † k and b k are creation and annihilation operators of the kth environmental mode with frequency ω k , respectively, and the TLS-environment coupling strengthes are denoted by g k . The rotating wave approximation has already been made in the TLS-environment interaction term. Generally, it is very convenient to encode the frequency dependence of the interaction strengths in www.nature.com/scientificreports/ the spectral density J(ω) , which is defined by J(ω) ≡ k g 2 k δ(ω − ω k ) . By doing so, the characteristic of the environment is now completely determined by J(ω) . In this work, the spectral density is characterized by the following Lorentzian form where α is a coupling constant, and ω c is a cutoff frequency. Instead of identifying the values of g k and ω k , we indicate the values of α and ω c when dealing with the dissipative dynamics.
(1) H = 1 2 ǫσ z + ω 0 a † a + 1 2 g 0 σ z (a † + a) + k ω k b † k b k + k g k (σ − b † k + σ + b k ),
To obtain the dynamics of the dissipative TLS in an analytical form, we first apply a polaron transformation 22,23 to the original Hamiltonian H as H = e S He −S , where the generator S is defined by S = g 0 2ω 0 σ z (a † − a) . The transformed Hamiltonian can be expressed as where H.c. denotes Hermitian conjugate and ζ ≡ g 0 ω 0 (a † − a) . One can see the last term in the above expression is just a constant, which just induces a trivial dynamical phase and would not influence the reduced dynamical behaviour of the TLS. Thus, we will drop it from now on.
We employ the quantum master equation approach to investigate the reduced dynamics of the TLS. In the polaron representation, the second-order approximate quantum master equation reads 24
where ρ I s (t) ≡ e itH sρ s (t)e −itH s with H s ≡ 1 2 ǫσ z is the reduced density operator in interac- tion picture, H i (t) ≡ e itH 0H i e −itH 0 with H 0 ≡H s +H a +H b , H a ≡ ω 0 a † a , H b ≡ k ω k b † k b k and H i ≡ k g k (σ − b † k e −ζ + H.c.)
is the interaction Hamiltonian in interaction picture. If both the TLS-HO and TLSenvironment couplings are weak, one can safely adopt the Born approximation ρ I tot (τ ) ≃ρ I s (τ ) ⊗ρ a (0) ⊗ρ b (0) . In this paper, we assume ρ a (0) = |0 a ��0 a | and
ρ b (0) = k |0 k b ��0 k b | , where |0 a � ( |0 k b � )
is the Fock vacuum state of the single-mode HO (k-th bosonic environmental mode). The effect of non-Markovianity has been incorporated into the convolution terms. Such convolution terms mean the evolution of ρ s (t) depends on ρ s (τ ) at all the earlier times 0 < τ < t , implying the memory effect from the environment has been considered. It should be emphasized that one can further use the Markov approximation by neglecting retardation in the integration of Eq. (4), namely ρ I s (τ ) is replaced by ρ I s (t) . Our treatment is beyond such over-simplified Markovian approximation.
(t) = k g k [e −it(ǫ−ω k ) σ − b † k e −ζ(t) + H.c.]
, where ζ(t) ≡ e itω 0 a † a ζ e −itω 0 a † a . Substituting this expression of H i (t) into the quantum master equation, namely Eq. (4), we have where S(t − τ ) ≡ �0 a |e ζ(t) e −ζ (τ ) |0 a � is a dynamical modulation function. The exact expression of S(t − τ ) can be derived by making use of the technique of Feynman disentangling of operators 21,25 . One can find where ≡ (g 0 /ω 0 ) 2 is a steerable parameter completely determined by the ancillary HO. The dynamical modulation function S(t − τ ) fully characterizes the influence of the single-mode HO on the reduced dynamics of the dissipative TLS.
Non-equilibrium dynamics of population difference. Starting from Eq. (5), one can extract the equation of motion for the matrix components of the TLS, i.e., ρ I jj ′ (t) ≡ �j|ρ I s (t)|j ′ � with j, j ′ = e, g , where |e� and |g� are the eigenstates of σ z . Meanwhile, due to the fact that ρ I ee (t) =ρ ee (t) , we derived the following integrodifferential equation for ρ ee (t) in Schrödinger picture where C.c. denotes complex conjugate. With the help of spectral density, one can replace the discrete summation in the above equation by a continuous integrand, i.e., k g 2 k e −iω k t → ∞ 0 dωJ(ω)e −iωt . For the Lorentzian spectral density considered in this paper, the integrand can be greatly simplified by extending the integration range of ω from [0, +∞) to (−∞, +∞) . Such approximation has been widely employed in several previous studies 1, 15,26 and is acceptable when the bound state effect can be neglected in the weak TLS-environment coupling regime 15 . Then, we have
(2) J(ω) = 1 π αω c (ω − ǫ) 2 + ω 2 c , (3) H = 1 2 ǫσ z + ω 0 a † a + k ω k b † k b k + k g k (σ − b † k e −ζ + H.c.) − g 2 0 4ω 0 , (4) d dtρ I s (t) = − t 0 dτ Tr ab [H i (t), [H i (τ ),ρ I tot (τ )]] ,(5)d dtρ I s (t) = − t 0 dτ k g 2 k e i(ǫ−ω k )(t−τ ) S(t − τ )[σ + σ −ρ I s (τ ) − σ −ρ I s (τ )σ + ] + H.c. ,(6)S(t − τ ) = e − ∞ l=0 l l! e −ilω 0 (t−τ ) , (7) d dtρ ee (t) = − t 0 dτ e − ∞ l=0 l l! k g 2 k [e i(ǫ−ω k −lω 0 )(t−τ )ρ ee (τ ) + C.(z) = L [f (t)] ≡ ∞ 0 dte −zt f (t) .
After the Laplace transformation, we find ρ ee (z)/ρ ee (0) = [z + µ(z)] −1 , where the Laplace-transformed kernel µ(z) is given by Thus, the expression of population difference in the polaron representation can be obtained via P (t) ≡ Tr s [σ zρs (t)] = 2ρ ee (t) − 1 . Next, we need to transform P (t) back to the original representation. Thanks to the fact [σ z , S] = 0 , the expression of population difference does not change by the polaron transformation, i.e., P(t) =P(t) . Finally, we arrive at
where L −1 denotes inverse Laplace transformation, i.e. L −1 [f (z)] ≡ 1 2πi ς +i∞ ς −i∞ dte zt f (z)
. As long as the initial state is given, the dynamics of P(t) can be fully determined by Eq. (10). In this paper, the inverse Laplace transformation is numerically performed by making use of the Zakian method 27 , which uses a series of weight functions to approximate an arbitrary function's inverse Laplace transform in time domain. It should be stressed that Eq. (10) only works in the regime where both α and are small, namely α/ω c ≪ 1 and ≪ 1 , due to the Born and the second-order master equation approximations.
On the other hand, the sum of l in the expressions of µ(z) in Eq. (9) can be exactly worked out
where F[{x 1 , x 2 , . .
. , x m }, {y 1 , y 2 , . . . , y n }, z] is the generalized hypergeometric function 28 . If the TLS and the single-mode HO is completely decoupled, using Eq. (11), one can easily demonstrate lim →0 µ(z) = 2α/(z + ω c ) . In this special case, the inverse Laplace transformation in Eq. (10)
Decoherence time.
In an approximate treatment, the density matrix components of the TLS commonly exhibit exponential decays, which are governed by the relaxation time T 1 and the dephasing time T 2 describing the evolution of ρ ee (t) and ρ eg (t) , respectively. Thus, the decoherence time T 1,2 roughly reflects the characteristic of dissipative dynamics 29 . Here, we would like to evaluate the expression of the relaxation time T 1 and explore the influence of the assisted HO on the decoherence time. Starting from Eq. (7), one can find where Strictly speaking, the integration in Eq. (13) should be performed with the Bromwich path. However, in an approximate treatment, the Bromwich path can be changed to that on the real axis −∞ < ̟ < ∞ by a transform z = i̟ + 0 +25,30-32 , where 0 + denotes a positive infinitesimal. Under such treatment, we find Using the Sokhotski-Plemelj theorem
we have iµ ± (i̟ + 0 + ) = � ± (̟ ) − iŴ ± (̟ ) , where (8) d dtρ ee (t) = − t 0 dτ αe − ∞ l=0 l l! e −ω c (t−τ ) [e −ilω 0 (t−τ ) ρ ee (τ ) + C.c.]. (9) µ(z) = L 2αe − ∞ l=0 l l! cos(lω 0 t)e −ω c t = 2αe − ∞ l=0 l l! z + ω c (z + ω c ) 2 + l 2 ω 2 0 . (10) P(t) = 2L −1 ρ ee (0) z + µ(z) − 1, (11) µ(z) = 2αe − z + ω c F − iz ω 0 − iω c ω 0 , iz ω 0 + iω c ω 0 , 1 − −(z + ω c ) 2 ω 0 , 1 + −(z + ω c ) 2 ω 0 , ,(12)P 0 (t) ≡ lim →0 P(t) = 2e − 1 2 ω c t cosh 1 2 �t + ω c � sinh 1 2 �t − 1, (13) ρ ee (t) =ρ ee (0) 2πi ς +i∞ ς −i∞ dz e zt z + µ + (z) + µ − (z) ,(14)µ ± (z) = e − ∞ l=0 l l! k g 2 k z ± i(ǫ − ω k − lω 0 ) . (15) ρ ee (t) =ρ ee (0) 2πi +∞ −∞ d̟ e i̟ t ̟ − iµ + (i̟ + 0 + ) − iµ − (i̟ + 0 + ) .(16)1 x ± i0 + = P 1 x ∓ iπδ(x),̟ 0 + iŴ + (̟ 0 ) + iŴ − (̟ 0 ) , where ̟ 0 is deter- mined by ̟ 0 − � + (̟ 0 ) − � − (̟ 0 ) = 0 .
Then, the integration can be worked out by using the residue theorem and the result is ρ ee (t) ≃ρ ee (0)e i̟ 0 t e −[Ŵ + (̟ 0 )+Ŵ − (̟ 0 )]t . In the weak-coupling regime, one can neglect the level shift induced by � ± (̟ ) [30][31][32] , which results in ̟ 0 ≃ 0 . Finally, the expression of T 1 can be further simplified to where G(x, y 1 , y 2 ) is the generalized incomplete gamma function 28 . Accordingly, the approximate expression of population difference is P (t) ≃ 2 exp(−t/T 1 ) − 1 . One can see lim →0 T −1 1 = 2πJ(ǫ) , which reproduces the well-known Wigner-Weisskopf decay rate without invoking the assisted HO 24 .
In Fig. 1, we plot the dynamics of δP(t) ≡ P(t) − P 0 (t) , which can be regarded as a witness to the effectiveness of our scheme. If δP(t) > 0 , i.e., P(t) > P 0 (t) , one can conclude that the decay of the population difference is slowed down when turning on the coupling between the TLS and the assisted HO. From Fig. 1, one can see δP(t) can be increased by enhancing , which means the coherent dynamics of P(t) becomes more and more robust as becomes larger. In this sense, by adjusting the parameters of the ancillary degree of freedom, we can achieve a controllable quantum dissipative dynamics. As comparisons, we also display P (t) − P 0 (t) . One can see from Fig. 1a that the results from the two different methods are in good agreement for the Markovian regime α/ω c → 0 . However, in non-Markovian regime (see Fig. 1b), a deviation is found. We believe such deviation is induced by the non-Markovianity incorporated in our approach. These results demonstrate our steer scheme works well in both Markovian and non-Markovian cases. Moreover, in Fig. 1a, one can observe that the relaxation time can be effectively prolonged by increasing the value of . This result is consistent with our previous numerical simulations. Using the same method, we also find T −1 2 = 1 2 T −1 1 , which means the dephasing time can be lengthened by adjusting the parameter as well. From the analytical expression of the decoherence time, we once again demonstrate the validity of our steer scheme.
Generalizations. Next, we would like to show that the single-mode HO can be equivalently replaced by a periodic driving field or a multi-mode bosonic reservoir. Though the physical properties of these assisted degrees of freedom are completely different, the effect of decoherence-suppression remains unchanged. Moreover, we extend the single-mode-HO-based steer scheme to a more general quantum dissipative system with hierarchical equations of motion (HEOM) approach, in which the counter-rotating-wave terms are included. www.nature.com/scientificreports/ Periodic driving field case. The assisted degree of freedom can be replaced by a periodic driving along the z direction. We can construct the following time-dependent Hamiltonian in which the TLS is engineered by a cosine driving term,
(17) � ± (̟ ) = e − ∞ l=0 l l! k g 2 k ̟ ± (ǫ − ω k − lω 0 ) , Ŵ ± (̟ ) = πe − ∞ l=0 l l! k g 2 k δ[̟ ± (ǫ − ω k − lω 0 )]. (18) ρ ee (t) =ρ ee (0) 2πi +∞ −∞ d̟ e i̟ t [̟ − � + (̟ ) − � − (̟ )] + i[Ŵ + (̟ ) + Ŵ − (̟ )] ,(19)T −1 1 ≃ k=± Ŵ k (0) = 2πe − ∞ l=0 l l! J(ǫ − lω 0 ) = − iα ω 0 e − (− ) − iωc ω 0 (− ) 2iωc ω 0 G − iω c ω 0 , 0, − − G iω c ω 0 , 0, − ,
where A is the driving amplitude and is the driving frequency. The dynamics of the whole system is governed by the Schrödinger equation ∂ t |ψ(t)� = −iH(t)|ψ(t)� . To handle the time-dependent term in the above Schrödinger equation, we apply a time-dependent transformation to |ψ(t)� as |ψ(t)� = e S t |ψ(t)� , where the time-dependent generator is given by S t = i A 2� sin(�t)σ z 33,34 . Then, in the transformed representation, the dynamics of |ψ(t)� is governed by ∂ t |ψ(t)� = −iH(t)|ψ(t)� , where with φ(t) = A � sin(�t) . If the driving frequency is sufficiently high, the time-dependent Hamiltonian H (t) can be approximately replaced a much simpler, undriven effective Hamiltonian 33,34 . To be more specific, using the Jacobi-Anger identity where J n (x) are Bessel functions of the first kind 28 , one can only retain the lowest order term and neglect all the other higher-order terms in e ±iφ(t) , namely, Then, one can obtain an effective interaction Hamiltonian H eff
i (t) = kǧ k (σ − e −iǫt b † k e iω k t + H.c.)
, where the renormalized coupling strength is defined by ǧ k = J 0 (A/�)g k . Compared with that of the undriven case, one can see the periodic driving field actually renormalizes the coupling constant α in the spectral density, i.e., α →α = J 0 (A/�) 2 α . Considering the fact that 0 ≤ J 0 (A/�) 2 ≤ 1 , then α ≤ α . This result is quite similar to the HO assisted case in which the coupling strengthes are renormalized as g 2 k → g 2 k S(t − τ ) (see Eq. 5). Thus, the periodic driving field is able to facilitate a robust coherent dynamics as well. More importantly, due to the fact that the periodic driving technique has been widely used in the experiments of cold atom systems, it is more friendly from experimental perspective. In fact, a similar periodic driving field has been used to control the dynamics of quantum circuits in the recent experiment 34 .
Multi-mode bosonic reservoir case. Our scheme can be also generalized to the case where the assisted degree of freedom is a multi-mode bosonic reservoir. The whole Hamiltonian of the modulated system in this situation is given by where a † j and a j are creation and annihilation operators of the jth assisted bosonic mode with frequency ν j , respectively, the coupling strengths between the TLS and assisted reservoir are characterized by κ j . The spectral density of the assisted reservoir is then defined by ̺(ν) ≡ j κ 2 j δ(ν − ν j ) . Similar to the single-mode HO case, we apply a polaron transformation to Eq. (24) as H = e G He −G , where the generator G is given by Then, the transformed Hamiltonian H is given by
where ξ = j κ j ν j (a † j − a j ) . Assuming ρ ab (0) =ρ a (0) ⊗ρ b (0) with ρ a (0) = j |0 j a ��0 j a | , ρ b (0) = k |0 k b ��0 k b
| and using the same quantum master equation approach displayed in single-mode HO case, one can find where the dynamical modulation function is given by (20) where χ is the coupling constant and η is the cutoff frequency. Then, G(t) has a very simple expression where � = χ/η . Compared with that of Eq. (6), one can see plays the same role with that of . Following the same process exhibited in single-mode case, one can find the expression of population difference P(t) is almost the same with Eq. (10), the only difference is the expression of µ(z) should be replaced by
H(t) = 1 2 ǫσ z + 1 2 A cos(�t)σ z + k ω k b † k b k + k g k (σ − b † k + σ + b k ),(21)H(t) = e S t [H(t) − i∂ t ]e −S t = 1 2 ǫσ z + k ω k b † k b k + k g k [σ − e −iφ(t) b † k + H.c.],(22)e ix sin β = ∞ n=−∞ J n (x)e inβ , (23) exp ± i A � sin(�t) ≃ J 0 A � . (24) H = 1 2 ǫσ z + j ν j a † j a j + 1 2 j κ j σ z (a † j + a j ) + k ω k b † k b k + k g k (σ − b † k + σ + b k ),(25)G = j κ j 2ν j σ z (a † j − a j ). (26) H = 1 2 ǫσ z + j ν j a † j a j + k ω k b † k b k + k g k (σ − b † k e −ξ + σ + b k e ξ ),(27)d dtρ ee (t) = − t 0 dτ k g 2 k [e i(ǫ−ω k )(t−τ ) G(t − τ )ρ ee (τ ) +
In Fig. 2, we display the dynamics of δP(t) in the case where the assisted degree of freedom is a multi-mode bosonic reservoir. One can see the decay of P(t) can be inhibited due to the interplay between the TLS and the additional degrees of freedom. Similar to single-mode HO case, the decay rate can be further reduced by increasing the value of . Our result is in agreement with that of Ref. 35 in which authors use a stochastic dephasing fluctuation to suppress the relaxation processes of two-level and three-level atomic systems. The physical picture behind this phenomenon is the ancillary degree of freedom effectively modifies the property of original environment acting on the TLS, which gives rise to this decoherence-suppression effect. Similar results have been also reported in several previous studies 21, [36][37][38] .
HEOM treatment. We have demonstrated that the decoherence of the TLS can be effectively suppressed by introducing an auxiliary single-mode HO. However, this conclusion is obtained under the weak-coupling and rotating-wave approximations. Going beyond these limitations, we next consider a more general quantum dissipative system Compared with Eq. (1), the counter-rotating-wave terms have been incorporated in the above Hamiltonian.
To handle the reduced dynamics without the rotating-wave approximation, we employ a purely numerical method, the HEOM approach [39][40][41][42][43] , to obtain the exact reduced dynamics of the TLS. The HEOM can be viewed as a bridge connecting the standard Schrödinger equation, which is exact but commonly hard to solve directly, and a set of ordinary differential equations, which can be treated numerically by using the well-developed Runge-Kutta algorithm. Without invoking the Born, weak-coupling and rotating-wave approximations, the HEOM can provide a rigorous numerical result as long as the initial state of the whole system is a system-environment separable state. To realize the traditional HEOM algorithm, it is necessary that the zero-temperature environmental correlation function C(t) = dωJ(ω)e −iωt can be (or at least approximately) written as a finite sum of exponentials 43,44 . The initial-state conditions of the auxiliary operators are given by ρ � ℓ= � 0 (0) = ρ sa (0) and ρ � ℓ� = � 0 (0) = 0 , where � 0 = (0, 0) is a two-dimensional zero vector. For numerical simulations, we need to truncate the number of hierarchical equations for a sufficiently large integer ℓ c , which can guarantee the numerical convergence. All the terms of ρ ℓ (t) with ℓ 1 + ℓ 2 > ℓ c are set to be zero, and the terms of ρ ℓ (t) with ℓ 1 + ℓ 2 ≤ ℓ c form a closed set of differential equations. Technically speaking, the single-mode HO is a ∞-dimensional matrix in its Fock state basis {|0 a �, |1 a �, |2 a �, ...} . Thus, the size of HO should be truncated in practical simulations. In this paper, we approximately regard the HO as a 10 × 10 matrix due to the limitation of our computation resource, and we have checked that the reduced dynamics of the TLS remains unchanged by further increasing the size of the assisted degree of freedom.
(28) G(t) = � j exp � − κ 2 j ν 2 j � ∞ � l=0 1 l! � κ 2 j ν 2 j � l e −ilν j t = exp � j κ 2 j ν 2 j � e −iν j t − 1 � . (29) ̺(ν) = 1 π χν 2 ν 2 + η 2 , (30) G(t) ≃ exp ∞ −∞ dε ̺(ν) ν 2 (e −iνt − 1) = exp (�e −ηt − �) = e −� ∞ l=0 � l l! e −lηt ,(31)µ(z) = 2αe −� ∞ l=0 � l l! 1 z + ω c + lη . (32) H = 1 2 ǫσ z + ω 0 a † a + g 0 σ z (a † + a) + k ω k b † k b k + k g k σ x (b † k + b k ).
Assuming ρ sa (0) = |e��e| ⊗ e −S |0 a ��0 a |e S , the reduced density operator of the TLS is obtained by partially tracing out of the degree of freedom of the HO from ρ � ℓ= � Figure 3 shows our numerical results obtained by the HEOM approach. It is found that the result from Eq. (10) is in qualitative agreement with those of the numerical HEOM method in weak-coupling regime. However, when coupling becomes strong, the counter-rotating-wave terms lead to a deviation. This result is physically understandable, because the the counter-rotating-wave terms are neglectable in weak-coupling case. Moreover, one can clearly see the decay of P(t) is suppressed by switching on the TLS-HO coupling. As increases, the effect of coherence-preservation becomes more noticeable. This result indicates that our steer scheme can be generalized to the non-rotating-wave approximation case, which greatly extends the scope of validity of our steer scheme.
0 (t) , i.e. ρ s (t) = Tr a [ρ E ℓ= E 0 (t)] .
Discussion
In our theoretical scheme, the inclusion of the single-mode HO can considerably protect the quantum coherence, and the value of plays a crucial role in our recipe. How to obtain a relatively large value of is the main difficulty in realizing our control scheme from an experimental perspective. Fortunately, the research of light-matter interaction has made a great progress in experiment. Nowadays, researchers are able to simulate the quantum Rabi model, whose Hamiltonian is described by H Rabi = − 1 2 (�σ x + ǫσ z ) + ω o (a † a + 1 2 ) + gσ z (a † + a) , in the ultra-strong-coupling and the deep-strong-coupling regimes. For example, by making use of a superconducting In conclusion, we propose a strategy to realize a controllable dynamics of a dissipative TLS with the help of an assisted degrees of freedom, which can be a single-mode HO, a periodic driving field or a multi-mode bosonic reservoir. Via adjusting the parameters of the assisted degree of freedom, we find the decoherence rate of the TLS can be significantly suppressed regardless of whether the counter-rotating-wave terms are taken into account. The physical picture behind this phenomenon is because the decays induced by parallel interaction (caused by the assisted degrees of freedom) and perpendicular interaction (intrinsically appeared in the original Hamiltonian) compete with each other, which effectively modifies the decoherence induced by the perpendicular interaction and gives rise to this coherence-preserve effect. Though our results are achieved in a Lorentzian environment at zero temperature, it would be very interesting to generalize our steer scheme to some more general situations by using the HEOM method, which has been extended to explore the dissipative dynamics in finite-temperature environment described by an arbitrary spectral density function 43,44,[46][47][48] . Finally, due to the generality of the dissipative TLS model, we expect our result to be of interest for some applications in quantum optics and quantum information.
(33) d dt ρ � ℓ (t) = − iH × sa − � ℓ · � υ ρ � ℓ (t) + � 2 p=1 ρ � ℓ+� e p (t) + 2 p=1 ℓ p � p ρ � ℓ−� e p (t),(34)H sa = 1 2 ǫσ z + ω 0 a † a + g 0 σ z (a † + a),(35)�X = −i[σ σ σ x , X], � p X = i 2 α(−1) p {σ σ σ x , X} − i 2 α[σ σ σ x , X],
can be analytically done and the expression of P(t) is then given by where � = ω 2 c − 8α . This result reproduces the Eq. (10.51) in Ref. 1 .
Figure 1 .
1(a) δP(t) is plotted as the function of time with different steer parameters: = 0.02 (yellow circles), = 0.05 (magenta stars), = 0.1 (blue diamonds) and = 0.2 (red squares). The purple solid lines are obtained from the Wigner-Weisskopf approximate expression of P (t) − P 0 (t) . The insert curve shows the relation between T 1 and . The initial state of the TLS is |e��e| , other parameters are chosen as ω 0 = 100 cm −1 , ω c = 10 cm −1 and α = 0.15 cm −1 . (b) The same with (a), but ω c = 1.5 cm −1 . Scientific Reports | (2021) 11:7188 | https://doi.org/10.1038/s41598-021-86553-z
Figure 2 .
2δP(t) is plotted as the function of time with different steer parameters: = 0.02 (yellow circles), = 0.05 (magenta stars), = 0.1 (blue diamonds) and = 0.2 (red squares). The initial state of the TLS is |e��e| , other parameters are chosen as η = 30 cm −1 , ω c = 5 cm −1 and α = 0.1 cm −1 . one can easily demonstrate that C(t) = αe −(ω c +iǫ)t for the Lorentzian spectral density considered in this paper. Then, following the procedure shown in Refs.43,44 , one can obtain the following hierarchy equations where ρ � ℓ= � 0 (t) is the reduced density operator of the TLS plus the HO, ρ �ℓ� = � 0 (t) are auxiliary operators introduced in HEOM algorithm, � ℓ = (ℓ 1 , ℓ 2 ) is a two-dimensional index, � e 1 = (1, 0) , � e 2 = (0, 1) , and � υ = (ω c − iǫ, ω c + iǫ) are two-dimensional vectors, two superoperators and p are defined by where σ σ σ x = σ x ⊗ 1 a with 1 a being an identity operator of the HO, [X, Y ] ≡ XY − YX and {X, Y } ≡ XY + YX.
Figure 3 .
3(a) P(t) with different coupling constants: α = 0.0025 cm −1 (purple solid line is the numerical result from HEOM method, purple diamonds are analytical results from Eq. (10)), α = 0.005 cm −1 (blue dashed line is the numerical result from HEOM method, blue squares are analytical results from Eq. (10)) and α = 0.025 cm −1 (red dotdashed line is the numerical result from HEOM method, red circles are analytical results from Eq. (10)). Other parameters are chosen as ǫ = 1.5 cm −1 , = 0.05 , ω 0 = 5 cm −1 and ω c = 0.5 cm −1 . (b) The dynamics P(t) from the HEOM method with different tunable parameters: = 0 (purple solid line), = 0.15 (magenta dotdashed line), = 0.25 (blue dashed line) and = 1 (red dotted line). Other parameters are chosen as α = 0.025 cm −1 , ǫ = 1.5 cm −1 , ω 0 = 2.5 cm −1 and ω c = 0.35 cm −1 . provide a strong support to our steer scheme in realistic physical systems.
Thus, we finally arrive atThe pole of the above integrand can be approximately viewed asScientific Reports |
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H.c.],Scientific Reports |
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Assuming ̺(ν) has a super-Ohmic spectral density with a Lorentz-type cutoff form, i.e.,
© The Author(s) 2021
Author contributionsW.W. proposed the original idea and performed all the numerical simulations. All authors reviewed the manuscript.Competing interestsThe authors declare no competing interests.Additional informationCorrespondence and requests for materials should be addressed to W.W.Reprints and permissions information is available at www.nature.com/reprints.Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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| [] |
[
"Looking forward to photon-coupled long-lived particles II: dark axion portal",
"Looking forward to photon-coupled long-lived particles II: dark axion portal"
] | [
"Krzysztof Jod \nCenter for Theoretical Physics of the Universe\nParticle Theory and Cosmology Group\nInstitute for Basic Science (IBS)\n34126DaejeonKorea\n"
] | [
"Center for Theoretical Physics of the Universe\nParticle Theory and Cosmology Group\nInstitute for Basic Science (IBS)\n34126DaejeonKorea"
] | [] | The dark axion portal is a dimension-5 coupling between an axion-like particle (ALP), a photon, and a dark photon, which is one of the targets of the intensity frontier searches looking for ∼ sub-GeV long-lived particles (LLPs). In this work, we re-examine the limits set by existing detectors such as CHARM and NuCal, and by future experiments such as FASER2, MATHUSLA, and SHiP. We extend previous works by i) considering several mass regimes of the Dark Sector (DS) particles, leading to an extended lifetime regime of the unstable species, ii) including LLPs production occurring in previously neglected vector meson decays that actually dominate the LLP yield, and iii) by implementing secondary LLP production. It takes place by Primakoff-like upscattering of lighter DS species into LLP on tungsten layers of neutrino emulsion detector FASERν2. This process will allow FASER2 to cover a significant portion of the γcτ ∼ 1 m region of the parameter space that is otherwise difficult to cover due to the large (∼ O(100) m) distance between the primary LLP production point and the decay vessel, where LLP decays take place, which is required in typical beam-dumb experiments for SM background suppression. | null | [
"https://export.arxiv.org/pdf/2305.10409v1.pdf"
] | 258,740,748 | 2305.10409 | 3d2e3adf111717fa2e34d9684f43e1dc176b8ae3 |
Looking forward to photon-coupled long-lived particles II: dark axion portal
Krzysztof Jod
Center for Theoretical Physics of the Universe
Particle Theory and Cosmology Group
Institute for Basic Science (IBS)
34126DaejeonKorea
Looking forward to photon-coupled long-lived particles II: dark axion portal
The dark axion portal is a dimension-5 coupling between an axion-like particle (ALP), a photon, and a dark photon, which is one of the targets of the intensity frontier searches looking for ∼ sub-GeV long-lived particles (LLPs). In this work, we re-examine the limits set by existing detectors such as CHARM and NuCal, and by future experiments such as FASER2, MATHUSLA, and SHiP. We extend previous works by i) considering several mass regimes of the Dark Sector (DS) particles, leading to an extended lifetime regime of the unstable species, ii) including LLPs production occurring in previously neglected vector meson decays that actually dominate the LLP yield, and iii) by implementing secondary LLP production. It takes place by Primakoff-like upscattering of lighter DS species into LLP on tungsten layers of neutrino emulsion detector FASERν2. This process will allow FASER2 to cover a significant portion of the γcτ ∼ 1 m region of the parameter space that is otherwise difficult to cover due to the large (∼ O(100) m) distance between the primary LLP production point and the decay vessel, where LLP decays take place, which is required in typical beam-dumb experiments for SM background suppression.
I. INTRODUCTION
The dark axion portal (DAP) has recently been proposed [1,2] as a novel interaction between an axion-like particle (ALP), a dark photon (DP), and photon induced by interactions in the Dark Sector (DS). Such a mechanism can take place, e.g., due to 1-loop processes involving massive dark fermions charged under global Peccei-Quinn symmetry U (1) PQ and gauge groups -U (1) Y (hypercharge), and U (1) Dark -which can be viewed as a generalization of the KSVZ [3,4] axion to DS containing new U (1) Dark gauge group.
Recent works have shown that the dark axion portal may have interesting astrophysical and cosmological implications that differ from the well-studied photophilic ALP. DAP can, e.g., provide a new mechanism for the production of dark matter (DM) [5,6], facilitate cosmological relaxation [7,8], and lead to axion-photondark photon oscillation [9][10][11][12][13], which affects 21 cm observations, supernovae cooling, and light-shining-throughwalls experiments, among other things. Similarly to an ALP, which is one of the main benchmarks of the searches for ∼ sub-GeV feebly-interacting BSM particles [14][15][16], the DAP detection prospects have been investigated in Bfactories, fixed target neutrino experiments, reactor experiments, and beam dumps [17][18][19].
In particular, it was shown that the long-lived particle (LLP) displaced decay signature at CHARM, FASER, MATHUSLA and SHiP is particularly effective in covering the ∼ O(100 m) region of the parameter space. Since then, the FASER experiment, which began collecting data in 2022 at the start of Run 3 of the LHC, has undergone an intensive research and development phase. As a result, a dedicated neutrino emulsion detector FASERν [20,21] has been installed in front of the main detector. Although its main purpose is the detection of collider * [email protected] neutrinos [22], because it is made of tungsten layers, it can also act as a target for the secondary production of LLPs; see [23,24] for its impact within non-minimal scalar, vector, and sterile neutrino portals.
Another development concerning FASER that motivates our analysis is that, as shown in [24], FASER2 will be sensitive to semi-visible two-body LLP decays. The final states are an invisible particle (a neutrino or a DS state) and a single high-energy photon: E γ > 0.1 TeV for FASER2 1 , and E γ > 0.1 TeV for FASERν2. This opens up a possibility to constrain BSM scenarios with such semi-visible LLP decays in FASER2, where the LLP can be produced in either primary [25][26][27] or secondary [24] production processes.
In this work, we study both of them for DAP, extending the results of [18] in multiple directions, such as: considering single photon LLP decays at FASER detectors, taking into account secondary LLP production occurring just in front of the main decay vessel, which allows to cover the regime of shorter LLP lifetimes, and electron scattering signatures; we note that in upcoming work we investigate other BSM scenarios using similar signatures [28,29]. Moreover, we study both LLP candidates -when it consists of a dark axion or a dark photon, and our simulation is adapted to the general case.
The paper is organized as follows. In Sec. II we discuss the physical aspects of DAP that are relevant to the intensity frontier searches. In particular, we identify the region of parameter space corresponding to the long-lifetime regime of a dark photon or a dark axion. In Sec. III we provide the specifics of the LLP production modes and the signatures under consideration, such as: the displaced LLP decays, secondary LLP production, and scattering of DS states with electrons. Our 1 In fact, recent work [25] considered an even lower energy threshold, Eγ > 1-10 GeV. We follow the original thresholds from [24], while results for lower energy thresholds, or other changes, can be easily generated using the modified version of FORESEE. main results are discussed in Sec. IV. We show sensitivity reach for FASER2, MATHUSLA, NuCal, and SHiP in two mass hierarchies, where either the dark photon or the dark axion act as a LLP. For both scenarios, we consider several fixed values of the mass ratio between the two DS species, which correspond to different LLP lifetime regimes. We also compare our results to the case of photophilic ALP. In Sec. V we summarize our study.
II. DARK AXION PORTAL
The interaction Lagrangian of the dark axion portal is [1,2],
L ⊃ g aγγ 4 aF µνF µν ,(1)
where g aγγ is a coupling of mass-dimension -1, and F µν and F µν are the EM and U (1) Dark field strength tensors, respectively. As described in Sec. I, the dark axion portal leads to an interesting range of phenomena that can be distinct from the photophilic ALP. In particular, dark axion portal was proposed [17] as an explanation of the recently rejuvenated (g − 2) µ anomaly [30][31][32][33]. The region of parameter space relevant to such a solution was the long-lived dark photon with ∼ GeV mass. In fact, [17] analyzed the dark photon displaced decays in the past beam dump and neutrino experiments: LSND [34], MiniBooNE [35,36], and CHARM [37] (missing energy searches at BaBar [38,39] and Belle [40] were also considered, and were shown to provide the coverage of the high mass range) to exclude such possibility. On the other hand, an extended dark axion portal, involving also kinetic mixing with SM hypercharge or muon-philic interactions, has been shown to be a viable solution [41,42]. Moreover, such a scenario could be tested in future lepton fixed target experiments, such as NA64e [43,44], NA64µ [45], LDMX [46], and M 3 [47]. These considerations further motivate dedicated sensitivity study of the long-lifetime regime of the DAP at the far-forward region of the LHC.
In the following, we discuss two benchmarks where, in each case, one of the DS species is massless, stable particle. Then only the coupling g aγγ and the LLP mass are free parameters of the model. In Sec. IV we present results for both of these benchmarks, as well as for several additional scenarios in which the masses of the DS states follow a fixed ratio.
For both benchmarks, the lifetime of the unstable, and typically long-lived, particle depends on the width of the two-body decay into a photon and a DS state. The threebody decays into a pair of charged leptons and a DS state are also possible, especially for m 0.1 GeV, but they are phase-space suppressed. As a result, they will contribute to the total decay width typically only at the O(0.01) level -see Fig. 1 in [18]; relevant formulas are given in Appendix B. Production rate σ × (GeV/g γ ) 2
[pb] Figure 1: Dark photon production modes as a function of its mass. The vector meson decays, which were not included in previous works, dominate at both FASER2 and SHiP. The same relationship exists for other mass schemes and for the dark axion production yields.
π η η ρ ω φ J/ψ ψ(2S) Υ(1S) Υ(2S) Υ(3S)
Since FASER detectors are ∼ 400 − 600 m away from the p-p collision point of the LHC, the typical LLP decay lengths they can probe are
d γ 100 m × E 1000 GeV 0.1 GeV m γ 4 7 × 10 −5 g aγγ 2 ,
(2) for the massless dark axion, while for the massless dark photon analogous formula for d a holds for g aγγ = 4 × 10 −5 GeV −1 . We used d = cβτ γ, where γ = E/m is the boost factor of LLP in the LAB frame, β = 1 − 1/γ 2 , and τ = 1/Γ. The decay widths for the two-body final states are [1] (3)
Γ γ →γa = g 2 aγγ 96π m 3 γ 1 − m 2 a m 2 γ 3 , Γ a →γγ = g 2 aγγ 32π m 3 a 1 − m 2 γ m 2 a 3 .
We note that when one DS particle is massless, the lifetime of a is smaller than the lifetime of γ by a factor of 3, coming from the average over dark photon polarization states. The same factor will occur for other pairs of processes in which a and γ are exchanged which will influence our results in Sec. IV.
III. LLP SIGNATURES
In this section, we describe the LLP signatures we use to constrain the DAP, followed by details of the beam dump and LHC experiments under consideration. [27,[52][53][54] a CHARM was placed at a distance of 5 m from the beam axis. b By primary LLP production at the LHC, we mean the Primakoff process in which photons produced in pp collisions hit the iron hadronic absorber TAN located 140 m further converting into a LLP particle; the same is assumed for other vesions of FASER detector. This production mode has been used for photophilic ALP [50] and massive spin-2 portal [51]. c For FASERν2 and FLArE we also take into the account the angular cuts -see tables 1 and 2 from [52]. Table I: Technical parameters of the considered detectors sensitive to LLP decays, secondary LLP production or scattering with electrons. We specify the technical parameters with the references of each experiment used in our simulations. LHC-based detectors are separated from experiments using dedicated proton beams. See Tab. 1 from [51] for a table of experiments sensitive to a massive spin-2 particle decaying dominantly into a photon pair.
A. LLP production
In the ∼1 MeV-1 GeV mass range, dark axion and dark photon are mainly produced from decays of unstable mesons. Compared to photophilic ALP, for which the Primakoff conversion of an on-shell photon into ALP dominates, the result is about an order of magnitude smaller number of LLPs for DAP. Moreover, with regard to meson decays, previous works [17,18] considered only three-body decays of pseudoscalar mesons. As discusssed in [25,55] for dark fermions coupled to the SM via dimension-5 and dimension-6 electromagnetic form factors, the branching ratios of decays of pseudoscalar and vector mesons of mass M into such DS states are approximately proportional to M 2 . As a result, the heaviest vector meson produced in sufficiently large quantities dominate the pseudoscalar mesons contributions. We found agreement with this argument for the DAP, as shown in Fig. 1, where we present contributions of unstable mesons produced at the LHC to the dark photon yield assuming the dark axion is massless; analogous behavior takes place for other mass scenarios.
We used the FORESEE [56] package to implement our model, in particular we used Eq. (A1) and Eq. (A2) describing the branching ratios of vector and pseudoscalar meson decays, respectively, which are used to obtain the resulting LLP yield. For the far-forward LHC detectors such as FASER2 [57][58][59] and Forward Physics Facility (FPF) [53,60,61], which would accommodate multiple detectors adapted to various searches, e.g., FASERν2 [52,61] and FLArE [52], we used the included spectra generated by EPOSLHC [62] and Pythia [63]. For beam dump experiments such as CHARM [37], NuCal [64,65], and SHiP [66,67], we used Pythia to generate the meson spectra, and we extended FORESEE to simulate the production and decay of LLPs taking place in these detectors.
B. Simulation details
After the production of a LLP, the number of events linked to a LLP signature being detected inside the de-tector are [68,69]
N = dEdθ d 2 N dEdθ p(E, θ) q accept. (E, θ),(4)
where the first term denotes the spectrum of the LLP with a energy E and polar angle θ relative to the beamline; p(E) corresponds to the probability of the signature taking place inside the detector, while experimental or simulation-related cuts are described by q accept. (E, θ, φ). a. Primary production Displaced LLP decays resulting from, e.g., proton-target collisions are the main experimental signature in LLP searches [14,15,70]. The experimental signal consists of high-energy SM particles, typically a pair of photons or charged leptons, and the probability of these decays occurring within a detector of length ∆ is
p(E) = e −L/d(E) − e −(L+∆)/d(E) ,(5)
where d(E) represents the LLP decay length in the LAB frame and L corresponds to the distance between the LLP production point and the start of the detector. It is evident that the majority of events arise from sufficiently long-lived species, characterized by d L, resulting only in linear suppression with the decay length: p(E) ∆/d [15,71]. However, for short-lived species, the second term in Eq. (5) can be neglected and p(E) e −L/d . It is therefore clear that the distance L sets the scale of the LLP decay lengths that can be probed in such a way.
In the DAP, the leading two-body decays deposit energy through a single photon, while decays into a DS and e + e − are suppressed, see bottom panels of each plot in Figs. 2 and 3. Despite the additional SM induced background for the single-photon LLP decay, it was shown [24] that FASER2 will be sensitive to it with the same cuts on the deposited energy and number of events as for the two-photon decays; we refer to that work for discussion of the backgrounds.
b. Secondary production Secondary production of LLPs can take place by coherent upscattering of a lighter DS species into the LLP on tungsten layers of neutrino emulsion detector FASERν2; see fig. 1 from [23] for a schematic illustration.
We study the displaced decay of the LLP produced in this way, where the production takes place at FASERν2, while the decay happens at FASER2. As the distance between these two detectors is L 1 m, this production mode could allow to cover a part of the d ∼ L 1 m region of the parameter space. On the other hand, the cross-section for the secondary production results in additional ∝ g 2 aγγ dependence in the number of decays. As a result, we expect secondary production to cover larger values of g aγγ than the ones covered by primary production [23,24].
The probability of secondary LLP production followed by decay inside FASER2 is given by convolution of Eq. (5) with upscattering cross-section [23] p(E) sec. prod.
= 1 L int ∆ 0 e −(xmin−t)/d − e −(xmin+∆−t)/d dt = d m T /(ρ σ(E)) e −(xmin+∆)/d e ∆/d − 1 e∆ /d − 1 ,(6)
where L int = m T /(ρ σ(E)) is the interaction length corresponding to the upscattering of DS species with energy E on nucleus of mass m T inside the material of density ρ and length∆; σ(E) is the upscattering cross-section;
x min is the distance from the beginning of the upscattering material to the beginning of the detector; and the dummy variable t parameterizes the length of the upscattering material. The cross-section for the upscattering process can be obtained in the closed form following the method described for photophilic ALP [72]; also see eq. B1-B3 from [51], while the derivation of the equation below can be found in the included Mathematica notebook,
σ γ N →aN α EM g 2 aγγ Z 2 12 log d 1/a 2 − t max − 2 ,(7)
where a = 111Z This formula differs (it is smaller) from the one for photophilic ALP only by a factor of 2/3, which results from 2 (3) polarization states for photon (dark photon).
At FASERν2, Primakoff production process takes place on tungsten (W), and the formula describing it has the following form:
σ W γ N →aN 48 GeV 2 × g aγγ 1/GeV 2 .
c. Electron scattering FASERν2 and FLArE detectors will be sensitive to DS states scattering with electrons, see [52] for an extensive discussion.
The corresponding probability for such scattering events is simply given by
p(E) scat. = ∆ L int ,(8)
where ∆ is the length of the FASERν2 or FLArE and L int denotes the interaction length of the scattering process, which is described by the following cross-section:
(9) σ γ e − →ae − α EM g 2 aγγ 12 log E max R E min R ,
where E R ≡ E e − is the electron recoil energy, while the differential cross-section, dσ/dE R , needed for angular cuts indicated in table I, can be found in the Mathematica notebook.
IV. RESULTS
We present the sensitivity lines for the LLP signatures within DAP in past and future experiments listed in table I. In the upper rows, the characteristics of beam dump experiments are displayed, whereas the lower rows exhibit the far-forward LHC detectors. We consider two versions of the FASER2 detector -an extension of current FASER detector or new one placed within FPF. We indicated all their relevant properties needed for simulation of LLP signatures discribed in Sec. III. Although, as indicated by similar formulas for decay widths and Primakoff cross-sections for both DAP and photophilic ALP, the main difference between the two portals is the fact that DAP is composed of two DS species and, depending on the mass hierarchy between them, each can serve as a LLP.
In Sec. IV A we discuss the case when dark photon is the LLP, which for massless dark axion was studied in [17][18][19], while in Sec. IV B we consider dark axion as the LLP.
A. Dark photon as the LLP
When m γ > m a , ∼ sub-GeV dark photon is the LLP, and its decay width is described by Eq. (2). The signatures described in Sec. III were simulated in modified version of FORESEE, and the results are shown in Fig. 2.
We consider three mass ratios, m a /m γ , which are fixed as follows: 0 (top), 0.5 (middle), and 0.9 (bottom).
In the first case, we checked that when the dark photon is produced only by the three-body pseudoscalar meson decays, we reproduce the results of [17,18]. Moreover, for the case of massless dark axion (and also in the opposite case of massless dark photon), we denote with the lightgray color the areas that are excluded by astrophysical and cosmological bounds obtained in [13].
The richness of the DAP is indicated by the middle and bottom plots of Fig. 2. They show that when the masses of the two DS particles are comparable, the LLP decay width is suppressed and, as a result, its lifetime is longer, resulting in shifting the significant reach of FASER2 and SHiP towards higher masses. Note that in this scenario the existing bounds, especially from NuCal, are relaxed due to the high energy threshold on the single photon, which is more difficult to meet because of the compressed spectra. On the other hand, FASER2 reach weakens only mildly because of the typical high energy ∼ O(100) s GeV of the produced LLPs.
Another feature of DAP that distinguishes it from the photophilic ALP is that vector meson decays pro-duce a pair of dark photon-dark axion, both of which can travel virtually undisturbed from the production point to FASERν2, which allows for the secondary LLP production by Primakoff-like upscattering 2 . As a result, this production mode allows to cover part of the d = γcτ ∼ 1 m region of the parameter space, see dashed and dash-dotted lines in Figs. 2 and 3. Note that the probability of LLP decay taking place inside the decay vessel in short-lived regime is exponentially suppressed, p(E) e −L/d for d L, hence this region of the parameter space cannot be covered by a detector placed at a significant distance from the LLP production point.
Lastly, the electron scattering signature allows coverage of the low-mass regime and is complementary to the decays of the dark photons produced in both primary and secondary production processes. It should be noted that the electron scattering limit is typically weaker than in the case of secondary production, mainly due to the lack of Z 2 enhancement, cf. Eq. (7) and Eq. (9).
B. Dark axion as the LLP
The results for the opposite mass hierarchy are shown in Fig. 3. The formulas for the LLP production channels are the same as for the case of the dark photon acting as the LLP, while the LLP lifetime is smaller by a factor of 3; see Eq. (3). As a result, the sensitivity lines are shifted towards smaller masses. Moreover, the Primakoff and electron scattering cross-sections are also smaller by a factor of 3, resulting in smaller reach.
We only show one benchmark corresponding to massless dark photon, while other mass scenarios are analogous to the middle and bottom rows of Fig. 2.
V. CONCLUSIONS
In this paper, we have studied the prospects of detecting the dark axion portal in the intensity frontier searches adapted to a diverse set of LLP signatures. The main difference between DAP with negligible kinetic mixing and the photophilic ALP in the ∼ sub-GeV mass regime is that the Primakoff conversion of an on-shell photon into an ALP is no longer possible, and the leading LLP production modes are vector meson decays, yielding approximately an order of magnitude fewer LLPs.
The second difficulty in probing DAP is that the LLP decays semi-visibly, so its energy can only be deposited by a single photon 3 . Such an experimental signature is more challenging than the usual two-photon ALP decay 2 It also increases the number of events from electron scattering by a factor of 2. 3 As Figs. 2 and 3 show, three-body semi-visible LLP decays into a e + e − pair are suppressed by at least 2 orders of magnitude with respect to the leading two-body decays.
because of, among other things, the additional SM induced background. On the other hand, future detectors like FASER2 and SHiP will be able to effectively probe such LLP decays, resulting in coverage of the parameter space similar to the photophilic ALP.
Moreover, secondary LLP production taking place just in front of the decay vessel will allow to cover part of the shorter LLP lifetime regime corresponding to d ∼ 1 m.
Finally, scatterings of either of the DS species with electrons taking place inside FASERν2 or FLArE will allow to probe the low mass regime of the LLP, m 10 MeV. It is complementary to both the LLP displaced decays and missing energy searches, which are restricted to different LLP mass ranges.
ACKNOWLEDGMENTS
This work was supported by the Institute for Basic Science under the project code, IBS-R018-D1.
Appendix A: Meson decays
In this section, we present expressions for the leading dark axion and dark photon production modes, see Fig. 1 for a comparison of their contributions.
Vector meson decays
We give the branching ratio of the two-body decays of vector mesons into a-γ , V (p 0 ) → γ * (p 1 + p 2 ) → a(p 1 ) + γ (p 2 ), which are mediated by an off-shell photon,
BR V →aγ BR V →ee = g 2 aγγ (−M 2 + m 2 a + m 2 γ ) 2 − 4m 2 a m 2 γ 3/2 32πα EM M M 2 − 4m 2 e (M 2 + 2m 2 e )
, (A1) where BR V →e + e − is the branching ratio of the vector meson with mass M decaying into the e + e − pair [73].
We note that in the m γ → 0 limit Eq. (A1) reduces to the result for photophilic ALP, see, e.g., Eq. 11 from [74].
Pseudoscalar and vector meson decays
The subdominant production mode takes place by decays of pseudoscalar mesons into a photon and DS states mediated by an off-shell photon,
P (p 0 ) → γ(p 1 ) + γ * (p 2 + p 3 ) → γ(p 1 ) + a(p 2 ) + γ (p 3 ).
We obtained the same averaged amplitude squared as [17], while below we give the resulting differential branching ratio in a form convenient for Monte Carlo simulation:
dBR P →γaγ dq 2 d cos θ = BR P →γγ × g 2 aγγ 256π 2 m 6 P q 6 m 2 P − q 2 3 (cos(2θ) + 3) (m 2 γ + m 2 a − q 2 ) 2 − 4m 2 γ m 2 a 3/2 ,(A2)
where m P is the pseudoscalar meson mass, q 2 ≡ (p 2 + p 3 ) 2 , and θ is the angle between a(p 2 ) and the off-shell photon in the meson rest frame; BR P →γγ is the branching ratio of pseudoscalar meson decaying into two photons taken for PDG [73].
Appendix B: Three-body decays
Below, we give formulas for the three-body decay widths of a dark photon and a dark ALP in the m γ m a , m l and m a m γ , m l limits, respectively; general form of the differential decay width can be found in the Mathematica notebook included in . Fig. 2. The light-gray areas are excluded by astrophysical and cosmological bounds which were obtained in [13].
(B1) Γ γ →l + l − a = α EM g 2 aγγ 576π 2 m 3 γ 32m 6 l coth −1 m γ m 2 γ − 4m 2 l + m γ m 2 γ − 4m 2 l 26m 2 γ m 2 l − 7m 4 γ + 8m 4 l − 4m 5 γ log 2m l m 2 γ − 4m 2 l + m γ + 12m γ m 4 l log 16m 4 l m γ − m 2 γ − 4m 2 l m 2 γ − 4m 2 l + m γ 5 ,(B2)Γ a →l + l − γ = α EM g 2 aγγ 192π 2 m 3 a 32m 6 l coth −1 m a m 2 a − 4m 2 l + m a m 2 a − 4m 2 l 26m 2 a m 2 l − 7m 4 a + 8m 4 l − 4m 5 a log 2m l m 2 a − 4m 2 l + m a + 12m a m 4 l log 16m 4 l m a − m 2 a − 4m 2 l m 2 a − 4m 2 l + m a 5 . m γ [GeV]
−1/3 /m e , d = 0.164 GeV 2 A −2/3 , m e is the electron mass, Z (A) is the atomic number (weight) of a nucleus, and t max −(m 4 a + m 4 γ )/(4E 2 1 ).
FPFFigure 2 :
2FASER2 (sec., Eaγ > 0.1 TeV) FPF FASERν2 (sec., Eaγ > 1 TeV) FPF FASERν2 (e − scat., dec. out.) FPF FLArE (e − scat.) Sensitivity reach for the dark photon acting as the LLP at the baseline (left) and the Forward Physics Facility (right) location of FASER2. The mass ratio, m a /m γ , is fixed as follows: 0 (top), 0.5 (middle), and 0.9 (bottom). The contour lines for each experiment correspond to the number of events, N ev , as indicated in table I.Lines derived by the missing energy signature at BaBar and Belle were taken from[17].
Figure 3 :
3Same as Fig. 2 but for dark axion acting as the LLP. We only show results for one mass scheme, massless dark photon, as the other results are analogous to the middle and bottom plots of
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| [] |
[
"Detecting floating black holes as they traverse the gas disk of the Milky Way",
"Detecting floating black holes as they traverse the gas disk of the Milky Way"
] | [
"Xiawei Wang \nDepartment of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA\n",
"⋆ \nDepartment of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA\n",
"Abraham Loeb \nDepartment of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA\n"
] | [
"Department of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA",
"Department of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA",
"Department of Astronomy\nHarvard University\n60 Garden St02138CambridgeMAUSA"
] | [
"Mon. Not. R. Astron. Soc"
] | A population of intermediate-mass black holes (BHs) is predicted to be freely floating in the Milky Way (MW) halo, due to gravitational wave recoil, ejection from triple BH systems, or tidal stripping in the dwarf galaxies that merged to make the MW. As these BHs traverse the gaseous MW disk, a bow shock forms, producing detectable radio synchrotron emission from accelerated electrons. We calculate the synchrotron flux to be ∼ 0.01 − 10 mJy at GHz frequency, detectable by JVLA, and ∼ 0.1 − 1 µJy in the infrared, detectable by HST and JWST. The discovery of the floating BH population will provide insights on the formation and merger history of the MW as well as on the evolution of massive BHs in the early Universe. | 10.1093/mnras/stu1842 | [
"https://arxiv.org/pdf/1402.5975v3.pdf"
] | 119,293,172 | 1402.5975 | bb90aa79742ac64d107176e5091049f967ca6ec6 |
Detecting floating black holes as they traverse the gas disk of the Milky Way
2014
Xiawei Wang
Department of Astronomy
Harvard University
60 Garden St02138CambridgeMAUSA
⋆
Department of Astronomy
Harvard University
60 Garden St02138CambridgeMAUSA
Abraham Loeb
Department of Astronomy
Harvard University
60 Garden St02138CambridgeMAUSA
Detecting floating black holes as they traverse the gas disk of the Milky Way
Mon. Not. R. Astron. Soc
0002014arXiv:1402.5975v2 [astro-ph.GA] Printed 26 March 2014 (MN L A T E X style file v2.2) Accepted ... Received...; in original form..Galaxy: disc -black hole physics -radio continuum: ISM -infrared: ISM
A population of intermediate-mass black holes (BHs) is predicted to be freely floating in the Milky Way (MW) halo, due to gravitational wave recoil, ejection from triple BH systems, or tidal stripping in the dwarf galaxies that merged to make the MW. As these BHs traverse the gaseous MW disk, a bow shock forms, producing detectable radio synchrotron emission from accelerated electrons. We calculate the synchrotron flux to be ∼ 0.01 − 10 mJy at GHz frequency, detectable by JVLA, and ∼ 0.1 − 1 µJy in the infrared, detectable by HST and JWST. The discovery of the floating BH population will provide insights on the formation and merger history of the MW as well as on the evolution of massive BHs in the early Universe.
INTRODUCTION
Galaxies grow through accretion and hierarchical mergers. During the final phase of the merger of two central black holes, anisotropic emission of gravitational waves (GW) kicks the BH remnant with a velocity up to a few hundreds km s −1 (Baker et al. 2006;Campanelli et al. 2007;Blecha & Loeb 2008). Additionally, BHs can be ejected from triple systems (Kulkarni & Loeb 2012;Hoffman & Loeb 2007), or result from tidally-stripped cores of dwarf galaxies (Bellovary et al. 2010). For GW recoils, the typical kick velocity is large enough for the BHs to escape the shallow gravitational potential of low-mass galaxies, but smaller than the escape velocity of the MW halo. This is also the case for triple systems as long as the kick velocity is < 500 km s −1 . Consequently, a population of floating BHs formed from mergers of low-mass galaxies are trapped in the region that eventually makes the present-day MW (Madau & Quataert 2004;Volonteri & Perna 2005;Libeskind et al. 2006). Previous studies suggested that more than ∼ 100 floating BHs should be in the halo today, based on a large statistical sample of possible merger tree histories for the MW halo today (O'Leary & Loeb 2009. This population of recoiled BHs is supplemented by BHs from tidally disrupted satellites of the MW (Bellovary et al. 2010). The discovery of this BH population will provide constraints on the formation and merger history of the MW as well as the dynamical evolution of massive BHs in the early Universe.
It has been proposed that a compact cluster of old stars from the original host galaxies is carried by each floating BH (O'Leary & Loeb 2009). In this Letter, we propose an additional observational signature of floating BHs, using the MW gas disk as a detector. As the BHs pass through the MW disk supersonically ⋆ E-mail: [email protected] they generate a bow shock, which results in synchrotron radiation detectable at radio and infrared frequencies.
The paper is organized as follows. In § 2, we discuss the interaction between BHs and the gas in the MW disk. In § 3, we calculate the synchrotron radiation from the bow shocks produced as the BHs cross the MW disk, and discuss the detectability of this radiation. Finally, in § 4, we summarize our results and discuss their implications.
INTERACTION BETWEEN A FLOATING BLACK HOLE AND THE MW DISK GAS
We consider a BH moving at a speed V• relative to the interstellar medium (ISM) of number density n ISM . The effective radius of influence of the moving black hole is given by the Bondi accretion radius:
R Bondi = GM• c 2 s + V 2 • ≈ GM• V 2 • = 0.01 M5 V −2 200 pc ,(1)
where G is Newton's constant, M5 = M•/10 5 M⊙ and V200 = V•/200 km s −1 . The sound speed cs of hydrogen in the ISM is given by cs = (ΓP/ρ) 1/2 = 11.7 T 1/2 4 km s −1 , where Γ = 5/3 is the adiabatic index and T4 = T /10 4 K . In the case of a supersonic shock with velocity V sh ≫ cs, the total mass enclosed within the Bondi radius is given by ∆MISM = 1.3 × 10 −7 M 3 5 V −6 200 n0 M⊙, where n0 = n ISM /10 0 cm −3 . The rate of fresh mass being shocked in the ISM is ∆ṀISM = 3.7 × 10 −9 M 2 5 V −3 200 n0 M⊙ yr −1 . The total kinetic power can be expressed as, where mp is the proton mass.
L kin = 1 2 2πR 2 Bondi n ISM mpV• V 2 • = 4.7 × 10 31 M 2 5 V −1 200 n0 erg s −1 ,(2)
OBSERVATIONAL APPEARANCE
As a floating BH travels through the MW disk supersonically, a bow shock is formed with a half opening angle θ ∼ M −1 (Shu 1992;Kim & Kim 2009), where the Mach number is given by Fig.1). The electrons accelerated in the shock produce non-thermal radiation that can be detected.
M = V•/cs ≈ 17.0 V200 T −1/2 4 (see
Non-thermal spectrum
Single electron
Next, we calculate synchrotron emission from the shock accelerated electrons around the BH. We adopt n0 = 1 and T4 = 1 in the numerical examples that follow. From the Rankine-Hugoniot jump conditions for a strong shock the density of the shocked gas is ns ≈ (Γ + 1) n ISM / (Γ − 1) = 4 n ISM , whereas its temperature is,
Ts = [(Γ + 1) + 2Γ(M 2 − 1)][(Γ + 1) + (Γ − 1)(M 2 − 1)]T /(Γ + 1) 2 M 2 .
The magnetic field can by obtained by assuming a near-equipartition of energy UB = B 2 /8π = ξB ns kTs, where ξB is the fraction of thermal energy carried by the magnetic field. Thus, the magnetic field behind the shock is given by
B ≈ 35 ξ 1/2 B,−1 n 1/2 0 T 1/2 4 µG ,(3)
where ξB,−1 = (ξB/0.1). We adopt ξB,−1 = 1 in what follows in analogy with supernova (SN) remnants (Chevalier 1998). For a single electron with Lorentz factor γ, the peak of its synchrotron radiation is at a frequency νsyn = 4.2 B−5 γ 2 4 GHz, where γ4 = γ/10 4 and B−5 = B/10 −5 G . The total emitted power per unit frequency is given by (Rybicki & Lightman 1979)
P (ν) = √ 2 e 3 B mec 2 F (x) ,(4)
where
F (x) ≡ x ∞ x K 5/3 (ξ) dξ, K 5/3 (x)
is the modified Bessel function of 5/3 order, x = ν/c1Bγ 2 , c1 = √ 6 e/4πmec, c is the speed of light and me, e are the electron mass and charge respectively. The pitch angle is assumed to be uniformly distributed.
The total power from synchrotron emission of a single electron is given by (Rybicki & Lightman 1979
) Psyn = 4 9 r 2 0 cβ 2 γ 2 B 2 = 2.5 × 10 −18 B−5 ν GHz erg s −1 ,(5)
where r0 = e 2 /mec 2 is the classical radius of the electron and ν GHz = (νsyn/GHz). We estimate the cooling time to be t cool =
γmc 2 /Psyn = 5.0×10 7 B −3/2 −5 ν −1/2 GHz yr for V200 = 1.
Since most of the emission is near the head of the Mach cone, we compare the cooling timescale with the dynamical timescale, which is given by t dyn = R Bondi /V• ≈ 53 yr. For the emission frequencies of interest, the cooling time is much longer than the lifetime of the shock.
Power-law distribution of electrons
Next we consider a broken power law distribution of electrons generated via Fermi acceleration:
N (γ) dγ = K0γ −p 1 + γ γ b −1 (γmin γ γmax),(6)
where K0 is the normalization factor in electron density distribution, p is the electron power law distribution index, and γ b , γmin, γmax are the break, minimum and maximum Lorentz factor respectively. The break in the power law is due to synchrotron cooling. The total synchrotron power can be written as,
Lnt = ǫntL kin = γmax γ min PsynN (γ) dγ = 2.3 × 10 30 ǫnt,5 M 2 5 V −1 200 n0 erg s −1 ,(7)
where ǫnt,5 = (ǫnt/5%) is the fraction of electrons accelerated to produce non-thermal radiation. The normalization constant K0 is obtained from the relation K0 = Lnt/ γmax γ min Psynγ −p dγ. Observations imply that the ISM density distribution in the MW disk midplane can be roughly described by the form (Spitzer 1942;Kalberla & Kerp 2009),
n ISM (r, z) = nce −(r−R ⊙ )/Rn sech 2 z √ 2z0(r) ,(8)
where r and z are the radial and vertical coordinates relative to the disk midplane, nc = 0.9 cm −3 , Rn = 3.15 kpc and z0(r) is the scale height at r, given by z0(r) = h0 e (r−R ⊙ )/r 0 with h0 = 0.15 kpc, R⊙ = 8.5 kpc and r0 = 9.8 kpc (Kalberla & Kerp 2009). The gas density and non-thermal luminosity as a function of radius in the MW disk midplane are shown in Figure 2. The electron acceleration time scale is given by tacc = ξaccγmec 3 /eBV 2 • , where ξacc is a dimensionless constant of order unity (Blandford & Eichler 1987). The upper limit of the Lorentz factor γmax can be obtained by equaling the acceleration and cooling timescale of electrons, tacc = t cool , giving
γmax = 3mecV• 2 ξ 1/2 acc B 1/2 e 3/2 = 2.5 × 10 7 V200 B −1/2 −5 ,(9)
Detecting floating black holes 3
where ξacc is assumed to be unity in this calculation. The emission frequency associated with γmax is νmax = 3 γ 2 max eB/4πmec = 4.2 × 10 15 B−5γ 2 max,7 Hz, where γmax,7 = (γmax/10 7 ). The break Lorentz factor can be obtained by equaling the cooling and the adiabatic loss timescale R•/c, giving γ b = 7.0 × 10 12 B −2 −5 M −1 5 V 2 200 and the corresponding frequency ν b = 4.2 × 10 18 B−5γ 2 b,13 GHz, where γ b,13 = (γ b /10 13 ). The value of νmax and ν b is above the frequency range of interest here.
The emissivity and absorption coefficients are given by (Rybicki & Lightman 1979)
jν = c2B γmax γ min F (x)N (γ) dγ ,(10)αν = −c3B 1 ν 2 γmax γ min γ 2 d dγ N (γ) γ 2 F (x) dγ ,(11)
where c2 = √ 2e 3 /4πmec 2 and c3 = √ 2e 3 /8πm 2 e c 2 . From the radiative transfer equation, the specific intensity is given by (Rybicki & Lightman 1979)
Iν = jν αν 1 − e −τν ,(12)
where τν is the optical depth. The synchrotron luminosity and corresponding flux at a distance d = 10 kpc from the observer are plotted in Figure 3.
Emission from the vicinity of the BH
Next we estimate the emission from the vicinity of the BH through a hot accretion flow (Narayan & Yi 1994). The Bondi accretion rate is given byṀ Bondi = 9.1 × 10 17 M 2 5 n0V −3 200 g s −1 (Armitage & Natarajan 1999), and the Eddington accretion rate can be expressed asṀ Edd = L Edd /0.1c 2 = 1.39 × 10 23 M5 g s −1 . We estimate the total luminosity in a radiatively inefficient accretion flow (RIAF) as,
L• = ηṀ c 2 = 5.4 × 10 31 ζ 2 −1 M 3 5 n 2 0 V −6 200 erg s −1 ,(13)
where η ≈ 0.1 Ṁ /0.1Ṁ Edd is the efficiency of converting matter to radiation forṀ 0.1Ṁ Edd (Narayan & McClintock 2008) and ζ =Ṁ /Ṁ Bondi = 10 ζ−1 is the accretion rate in units ofṀ Bondi . The BH accretion would produce X-ray emission which is not expected from the bow shock spectrum in Fig.3. Since L• ∝ M 3 5 , the accretion luminosity from interstellar medium accretion onto stellar mass BHs is negligible compared to our souce (Fujita 1998).
It is possible that an outflow would be formed near the BH. The outflow would produce a shock at a radius Rout, which can be obtained from fṀ =Ṁout = 4πR 2 out n ISM mpVout, where f 1 is the fraction of the inflowing mass channelled into the outflow. This gives, Rout = fṀ 4πn ISM mpVout 1/2 = 6.8 × 10 −4 f 1/2 ζ 1/2
−1 M5V −3/2 200 V −1/2 out,4 pc ,(14)
where Vout,4 = (Vout/10 4 km s −1 ) is the velocity of the outflow. For typical parameters, we find that the outflow would be bounded with Rout R Bondi . The left label of the vertical axis marks synchrotron luminosity per unit frequency (upper panel) or power per log ν (lower panel) while the right one marks the corresponding flux at a distance of d = 10 kpc. The black, blue, red and green lines correspond to power-law indices p = 2.0, 2.2, 2.5, 2.7 respectively in the electron energy distribution. Synchrotron self-absorption is significant at a frequency MHz and the cooling break corresponds to a frequency ∼ 10 18 GHz, which are outside the frequency range of interest.
Observational signatures and detectability
Observationally, the BH emission cone would appear arcshaped, with an angular diameter θ = R Bondi /d = 0.22 d −1 1 M5V −2 200 arcsec, where d1 = (d/10 kpc). The nonthermal radiation should be detectable at both radio and infrared band. At a frequency ν ∼ 1 GHz, the synchrotron flux at a distance of 10 kpc is of order 0.01 − 10 mJy, depending on the choice of p. This flux is detectable with the Jansky Very Large Array (JVLA), which has a complete frequency coverage from 1 − 50 GHz, with a sensitivity of ∼ 5.5 µJy/beam in a 1-hour integration and a signal to noise ratio S/N = 1 at 1 − 2 GHz (Perley et al. 2011). At a frequency ν ∼ 10 14 Hz in the infrared band, the synchrotron flux at a distance of 10 kpc is of order 0.1 − 1 µJy. The limiting sensitivity with S/N = 10 and integration time of 10 4 s is ∼ 10 − 100 nJy for the Hubble Space Telescope (HST), covering a wavelength range of 0.5 − 1.6 µm. The corresonding sensitivity is ∼ 10 nJy for James Webb Space Telescope (JWST), in the wavelength range of 0.6 − 5.0 µm (STScI 2013). Therefore, the infrared signal is detectable by both HST and JWST.
Morphologically, it is possible to distinguish the bow shock emission from other radio sources such as SN remnants or HII regions. The bow shock emission is elongated along the direction of the BH's motion, whereas SN remnants would appear roughly circular on the sky. There are hundreds of cometary HII regions produced by a combination of supersonic motion of an OB-type star through dense gas and ionization of gas down a density gradient (Cyganowski et al. 2003;Immer et al. 2014). The Mach cone's opening angle can be used to distinguish them from the much faster floating BHs. The ongoing survey of the Galactic plane with JVLA (NRAO 2014) has the potential to separate out these HII regions. There are far fewer confusing HII region sources at larger radius in the disk. Other high-velocity sources are pulsar wind nebulae (Gaensler 2005), hyper-velocity stars (Brown et al. 2006) and runaway stars (del Valle & Romero 2012;del Valle et al. 2013). The first type can be distinguished by observing the pulsar as well as its X-ray emission. The last two types produce less synchrotron radiation (del Valle & Romero 2012;del Valle et al. 2013), and thus can be distinguished as well. Globular clusters crossing the MW disk produce another class of contaminants. Their velocity relative to the disk is much larger than the velocity dispersion of their stars, so their Bondi radius is much smaller than their size. Thus, they should not produce significant synchrotron emission. The floating BHs are also embedded in a star cluster, but the cluster size is more compact and its gravity is dominated by the central BH (O'Leary & Loeb 2009.
SUMMARY AND DISCUSSION
If a floating BH happens to pass through the MW disk, then the nonthermal emission from the accelerated electrons in the bow shock around the BH should produce detectable signals in the radio and infrared bands. The radio flux ∼ 0.01 − 10 mJy is detectable by JVLA, while the infrared flux ∼ 0.1 − 1 µJy is detectable by HST and JWST.
The density distribution of floating BHs in the MW has been studied by O' Leary & Loeb (2009 and by Rashkov & Madau (2014). High resolution simulations show that there is a BH of mass ∼ 2 × 10 5 M⊙ within a few kpc from the Galactic center (Rashkov & Madau 2014).
Observations of the Galactic disk can be used to infer n0 and T4. The BH speed V• can then be estimated from the Mach cone angle. The maximum Lorentz factor γmax can be inferred from the peak of the synchrotron spectrum. This, in turn, yields B−5 based on Eq.(3). From the slope of the synchrotron spectrum, the power law index p can be estimated. Finally, with the above parameters constrained, the synchrotron flux can be used to calibrate M•. The above interpretation can be verified by observing the properties of the star cluster carried by the floating BHs (O'Leary & Loeb 2009. The diffuse X-ray emission from the BH and synchrotron emission from the bow shock is supplemented by stellar emission from the star cluster around it. Since the total mass of the star cluster is much smaller than M•, gravity is dominated by the BH, and thus the stars do not effect the bow shock. One can measure M• spectroscopically from the velocity dispersion of the stars as a function of distance from the BH, and verify consistency with the synchrotron flux estimate.
Figure 1 .
1Sketch of the bow shock geometry around a BH crossing the gaseous MW disk.
Figure 2 .
2Gas density, n ISM (z = 0), and non-thermal luminosity in units of 10 30 ǫ nt,5 M 2 5 V −1 200 n 0 in the midplane of the MW disk.
Figure 3 .
3Synchrotron power and flux from non-thermal electrons accelerated by the bow shock of floating BHs, in units of M 2 5 , for n 0 = 1, V 200 = 1, Lnt = 3.0 × 10 30 erg s −1 , B −5 = 3.5, γ min ∼ 1 and γmax ∼ 1.0 × 10 7 . The upper panel shows synchrotron flux while the lower panel shows the corresponding power.
c 2014 RAS, MNRAS 000, 1-4
ACKNOWLEDGEMENTSWe thank James Guillochon, Mark Reid and Lorenzo Sironi for helpful comments on the manuscript. We thank Piero Madau for providing the data from Via Lactea II simulation. This work was supported in part by NSF grant AST-1312034.
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| [] |
[
"Quantum Phase Transitions in periodically quenched systemś",
"Quantum Phase Transitions in periodically quenched systemś"
] | [
"A Sáiz \nDepartamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain\n",
"J Khalouf-Rivera \nDepartamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain\n\nDepartamento de Ciencias Integradas y Centro de Estudios Avanzados en Física\nMatemática y Computación\nUniversidad de Huelva\n21071HuelvaSpain\n",
"J M Arias \nDepartamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 1065E-41080SevillaSpain\n\nInstituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nFuentenueva s/n18071GranadaSpain\n",
"P Pérez-Fernández \nDepartamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain\n\nInstituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nFuentenueva s/n18071GranadaSpain\n",
"J Casado-Pascual \nFísica Teórica\nUniversidad de Sevilla\nApartado de Correos 106541080SevillaSpain\n"
] | [
"Departamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain",
"Departamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain",
"Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física\nMatemática y Computación\nUniversidad de Huelva\n21071HuelvaSpain",
"Departamento de Física Atómica\nMolecular y Nuclear\nFacultad de Física\nUniversidad de Sevilla\nApartado 1065E-41080SevillaSpain",
"Instituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nFuentenueva s/n18071GranadaSpain",
"Departamento de Física Aplicada III\nEscuela Técnica Superior de Ingeniería\nUniversidad de Sevilla\nSevillaSpain",
"Instituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nFuentenueva s/n18071GranadaSpain",
"Física Teórica\nUniversidad de Sevilla\nApartado de Correos 106541080SevillaSpain"
] | [] | The concept of quantum phase transition (QPT) encompasses a variety of phenomena that occur in quantum systems exhibiting two (or more) possible symmetries. The transition from one symmetry to another is achieved by continuously varying a control parameter present in the timeindependent Hamiltonian of the system. Here, an alternative approach that provides surprisingly identical results is proposed by using a time-periodic Hamiltonian that switches discontinuously between the various symmetries of the system. This approach opens new avenues to study QPTs using Floquet engineering. | null | [
"https://export.arxiv.org/pdf/2302.00382v1.pdf"
] | 256,459,881 | 2302.00382 | 7ea1cb25bd4a9acf6c6b9d6cabbafb8a2f205a6e |
Quantum Phase Transitions in periodically quenched systemś
A Sáiz
Departamento de Física Aplicada III
Escuela Técnica Superior de Ingeniería
Universidad de Sevilla
SevillaSpain
J Khalouf-Rivera
Departamento de Física Aplicada III
Escuela Técnica Superior de Ingeniería
Universidad de Sevilla
SevillaSpain
Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física
Matemática y Computación
Universidad de Huelva
21071HuelvaSpain
J M Arias
Departamento de Física Atómica
Molecular y Nuclear
Facultad de Física
Universidad de Sevilla
Apartado 1065E-41080SevillaSpain
Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada
Fuentenueva s/n18071GranadaSpain
P Pérez-Fernández
Departamento de Física Aplicada III
Escuela Técnica Superior de Ingeniería
Universidad de Sevilla
SevillaSpain
Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada
Fuentenueva s/n18071GranadaSpain
J Casado-Pascual
Física Teórica
Universidad de Sevilla
Apartado de Correos 106541080SevillaSpain
Quantum Phase Transitions in periodically quenched systemś
The concept of quantum phase transition (QPT) encompasses a variety of phenomena that occur in quantum systems exhibiting two (or more) possible symmetries. The transition from one symmetry to another is achieved by continuously varying a control parameter present in the timeindependent Hamiltonian of the system. Here, an alternative approach that provides surprisingly identical results is proposed by using a time-periodic Hamiltonian that switches discontinuously between the various symmetries of the system. This approach opens new avenues to study QPTs using Floquet engineering.
The concept of quantum phase transition (QPT) encompasses a variety of phenomena that occur in quantum systems exhibiting two (or more) possible symmetries. The transition from one symmetry to another is achieved by continuously varying a control parameter present in the timeindependent Hamiltonian of the system. Here, an alternative approach that provides surprisingly identical results is proposed by using a time-periodic Hamiltonian that switches discontinuously between the various symmetries of the system. This approach opens new avenues to study QPTs using Floquet engineering.
Introduction.-Matter can exist in different structural states or phases that have different properties. The transitions between phases are characterized by rapid changes in the properties of the system. The study and characterization of phase transitions and critical phenomena is of great physical importance due, among other things, to the appearance of certain universal behaviors [1,2]. Typically, the change from one phase to another is governed by a control parameter (temperature or other appropriate variable) and the phase transition is characterized by an order parameter that is zero in one phase and nonzero in the other. The way in which this change in the order parameter occurs makes it possible to classify the phase transitions [1,2].
More recently, the phenomenon of phase transitions has been extended to the quantum regime, giving rise to the so-called quantum phase transitions (QPTs) [3][4][5]. For this, a physical system that can exhibit two different symmetries, S 1 and S 2 , is usually considered. These symmetries are represented by two Hamiltonians, H 1 and H 2 , respectively, that correspond to different structures and properties of the system. To study the transition from one of the symmetries to the other, a Hamiltonian of the form
H = ξH 1 + (1 − ξ)H 2(1)
is considered, where ξ is a dimensionless parameter in the range ξ ∈ [0, 1] that serves as a control parameter. For ξ = 0 and ξ = 1 the system exhibits the symmetries S 2 and S 1 , respectively, while for intermediate values of ξ the symmetry is not well-defined and there is a competition between both symmetry phases. Various quantities, both static and dynamic, have been used to study and characterize the phase transition that occurs when the control parameter ξ varies between 0 and 1. Specifically, it has been shown that certain properties of the ground state undergo abrupt changes for given ntermediate crit-ical values of ξ, giving rise to the so-called Ground-State Quantum Phase Transitions (GSQPTs) [6]. Excited-State Quantum Phase Transitions (ESQPTs) have also been reported, characterized by the appearance of divergences in the level density at certain excitation energy values (see, e.g., Ref. [7] and references therein). Obviously, these divergences appear in the thermodynamic limit, where the number of particles tends to infinity. However, for small systems with finite number of particles, precursors of these divergences can be observed. In recent years, QPTs have been analyzed using timeperiodic Hamiltonians and Floquet theory [8][9][10][11][12][13][14]. However, most of the works have focused on the study of a new phenomenon, called Floquet Dynamical Quantum Phase Transitions (FDQPTs), characterized by the appearance of recurrent nonanalytical behaviors of certain quantities over time. It is important to note that this phenomenon is different from the GSQPTs and ESQPTs mentioned above, which have been studied mainly using time-independent Hamiltonians. The question that naturally arises is whether GSQPTs and ESQPTs can also be analyzed using time-periodic Hamiltonians and Floquet theory. In this work, we show that the answer is yes, opening new avenues to study GSQPTs and ES-QPTs using Floquet engineering.
Description of the model and Floquet theory.-Let us consider a system characterized by a time-periodic Hamiltonian with period T defined on 0 ≤ t < T by
H(t) = H 2 if t ∈ [0, t 0 ) H 1 if t ∈ [t 0 , T ),(2)
and extended periodically for all t (see Fig. 1). As in the traditional QPTs, H 1 and H 2 are time-independent Hamiltonians with symmetries S 1 and S 2 , respectively. The parameter t 0 (quench time) denotes the amount of time per period that the Hamitonian remains with the symmetry S 2 , and will be used as a control parameter. arXiv:2302.00382v1 [quant-ph] 1 Feb 2023 Obviously, for t 0 = 0 and t 0 = T the Hamiltonian is time-independent and equals to H 1 and H 2 , respectively. Note that in the present case, the Hamiltonian has a well-defined symmetry at all time instant, and the periodic transitions between symmetries S 1 and S 2 are discontinuous. By contrast, in the traditional QPTs the transition from S 2 to S 1 is continuous, since it is carried out by continuously varying the control parameter ξ from 0 to 1. Furthermore, the Hamiltonian in Eq. (1) only has a well-defined symmetry when the parameter ξ takes the endpoints 0 or 1. Despite these important differences, surprisingly, the time-periodic Hamiltonian in Eq. (2) shows a very similar behavior to that observed in the traditional QPTs [Eq. (1)], as is discussed below.
t 0 T T + t 0 2T 2T + t 0 3T H 2 H 1 t H(t)
Due to the periodicity of the Hamiltonian in Eq. (2), Floquet theory [15] can be used to analyze the time evolution of the system. According to Floquet's theorem, the Schrödinger equation corresponding to the Hamiltonian (2) possesses a complete set of solutions of the form |Ψ j (t) = e −i j t/ |Φ j (t) , with j ∈ {0, · · · , D − 1}, D is the dimension of the state space. The kets |Φ j (t) are T -periodic functions of time and are known as Floquet modes. The quantities j are called quasienergies and can (and will) be taken to lie within the first Brillouin zone [−π /T, π /T ). Henceforth, we will assume that the quasienergies are labeled so that j+1 ≥ j .
The calculation of the Floquet modes |Φ j (t) and the quasienergies j requires only the knowledge of the timeevolution operator U (t, 0) from t = 0 to t ∈ [0, T ]. Since the Hamiltonian in Eq. (2) is piecewise time-independent, U (t, 0) can be explicitly evaluated, yielding
U (t, 0) = e −itH2/ if t ∈ [0, t 0 ) e −i(t−t0)H1/ e −it0H2/ if t ∈ [t 0 , T ).(3)
The first step to calculate |Φ j (t) and j is to obtain the eigenvectors and eigenvalues of the operator
U (T, 0) = e −i(T −t0)H1/ e −it0H2/ .(4)
Since this operator is unitary, it possesses an orthonormal basis of eigenvectors. These eigenvectors provide the Floquet modes at t = 0, i.e., the kets |Φ j (0) . If λ j is the eigenvalue of U (T, 0) associated with the eigenvector |Φ j (0) , the corresponding quasienergy is given by
j = i log(λ j )/T,(5)
where log denotes the principal value of the logarithm. Finally, the Floquet mode at any time t can be obtained extending periodically the function
|Φ j (t) = e it j / U (t, 0) |Φ j (0) ,(6)
with U (t, 0) given by Eq. (3). It is worth mentioning that the quasienergy j can be split into a dynamical and a geometrical contribution as
j =Ē j − ϕ j /T , wherē E j = 1 T T 0 dt Φ j (t)|H(t)|Φ j (t)(7)
is the mean energy, and
ϕ j = i T 0 dt Φ j (t)| d dt |Φ j (t)(8)
is the geometric phase of the Floquet mode. From Eqs. (3), (6), and (7), it is easy to see that the mean energy is given bȳ
E j = t 0 T Φ j (0)|H 2 |Φ j (0) + T − t 0 T Φ j (0)|e it0H2/ H 1 e −it0H2/ |Φ j (0) .(9)
As will be seen below, the quantities j ,Ē j , and ϕ j can be used to characterize the QPTs and ESQPTs described in this work. Physical realization.-In order to realize the ideas discussed in the preceding text, we will use a simple two-level model: the Lipkin-Meshkov-Glick (LMG) model [16][17][18]. This model was originally proposed in the field of Nuclear Physics to validate many-body approximation methods and has subsequently been widely used in many different areas. In the field of phase transitions, the standard LMG Hamiltonian can be understood as a one-dimensional N -site spin-1/2 lattice with infinite interaction range [19,20]. The Hamiltonian can be written as Eq. (1) with
H 1 = − S 2 x S 2 and H 2 = S z 2S ,(10)
where = 1 is assumed and collective spin operators S β = N i=1 s i,β , for β = x, y, z, are introduced. Constant contributions to the Hamiltonian have been dropped. In this scheme, S = N/2 is the maximum collective spin. Concerning symmetries, the LMG Hamiltonian presents a u(2) algebraic structure and possesses two dynamical symmetries: u(2) ⊃ u(1) and u(2) ⊃ so(2) [21]. The symmetries u(1) and so (2) Results.-The GSQPT and ESQPT appearing in the LMG model have been studied previously by varying the control parameter ξ in Eq. (1) (see Ref. [7] and references therein). This is depicted in Fig. 2.a), where the excitation energies E n −E 0 of the time-independent LMG Hamiltonian in Eqs. (1) and (10) are plotted with solid blue lines (even n) and dashed orange lines (odd n) as a function of the control parameter ξ, which is varied continuously from ξ = 0 [symmetry u(1)] to ξ = 1 [symmetry so (2)]. Two phases are clear: in one [harmonic u(1) symmetry limit] the excitation energies are non-degenerated and equally spaced, while in the other [so(2) symmetry limit] they are degenerated in pairs. For the GSQPT the transition is at ξ = 0.2 in the large-N limit [22,23]. The calculations presented here are for S = N/2 = 10, thus finite-N effects are important. In addition, for the bro-ken symmetry phase a separatrix line of higher density of states is clearly observed, which marks the ESQPT. Different observables have been used to characterize both GSQPT and ESQPT always for the time-independent LMG model [24].
In order to determine if the GSQPT and ESQPT can also be identified using Floquet theory with the quench time t 0 as the control parameter, the time-periodic LMG Hamiltonian defined by Eqs. (2) and (10) is now considered. In this case, instead of the energies E n , the Floquet quasienergies, n , and the mean energies of the Floquet modes,Ē n , have been computed using Eqs. (5) and (9), respectively. The results for the differences n − 0 andĒ n −Ē 0 as a function of the dimensionless quench time t 0 /T are depicted in Figs. 2.b) and 2.c), respectively, with the same line code as in Fig. 2.a). Remarkably, Figs. 2.b) and 2.c) are almost indistinguishable from Fig. 2.a), showing that almost the same information about the GSQPT and ESQPT is obtained for the time-periodic system as for the time-independent Hamiltonian. Although the values for the quasienergies and the mean energies shown in Figs. 2.b) and 2.c) are very similar to each other, they are not identical. To confirm this, the dependence on t 0 /T of the geometric phases of the Floquet modes, ϕ n = (Ē n − n )T / , is depicted in Fig. 2.d). For the sake of clarity only the modes n = 0, 1, 8, 9 16, and 17 have been plotted. As can be seen, although the geometric phases of the Floquet modes are zero for t 0 = 0 and t 0 = T , they are generally Fig. 2.b). This is shown even more clearly in Fig. 2.e), where the differences n+1 − n (solid lines and left ordinate axis) and ϕ n+1 − ϕ n (dashed lines and right ordinate axis) are plotted versus the dimensionless quench time t 0 /T for n = 0, 8, and 16. From the above results, it seems clear that the information about the QPT contained in the time-independent Hamiltonian in Eq. (1) is retained by the time-periodic Hamiltonian in Eq. (2) and reflected in the quasienergies, the mean energies, and the geometric phases of the Floquet modes. The question then arises whether it is possible to find an order parameter for the GSQPT and a good marker for the ESQPT. To this end, here we propose the real part of the two-time correlator defined as
f j = Φ j (0)|S x (T )S x (0)|Φ j (0) ,(11)
where S x (t) corresponds to the operator S x expressed in the Heisenberg picture. In Fig. 3, the time-periodic LMG Hamiltonian defined by Eqs. (2) and (10) is again considered but now for N = 50. The dependence of the quasienergy differences n − 0 on the dimensionless quench time t 0 /T is depicted in Fig. 3.a) with a color scale that indicates the value of the real part of the two-time correlator in Eq. (11). In particular, for the Floquet mode corresponding to the lowest quasienergy in the first Brillouin zone [horizontal line at n − 0 = 0 in Fig. 3.a)], the dependence on t 0 /T of the real part of this correlator is depicted in Fig. 3.b).
As can be seen, the quantity Re(f 0 ) behaves as an order parameter, since it is zero in one phase and nonzero in the other, marking clearly the GSQPT around t 0 /T = 0.2. The possibility of detecting the ESQPT using the quantity Re(f n ) is explored in Fig. 3.c). For this purpose, three vertical lines are plotted in Fig. 3.a), which mark different quench times, specifically, t 0 /T = 0.1 (dashed red line), 0.4 (dot-dashed violet line), and 0.6 (dotted orange line). The dependence of Re(f n ) on the quasienergy difference n − 0 along these three vertical lines is depicted in Fig. 3.c). For t 0 /T = 0.1 (red dots), Re(f n ) is a smooth function of n − 0 , which is consistent with the fact that for this quench time the system is at the same phase for any value n − 0 [see Fig. 3.a)]. By contrast, for t 0 /T = 0.4 (violet crosses) and t 0 /T = 0.6 (orange triangles), abrupt changes are observed around n − 0 = 0.18 and n − 0 = 0.42, respectively, which are precisely the values at which the ESQPTs occur in Fig. 3.a). Therefore, we conclude that the real part of the two-time correlator in Eq. (11) can be used as a marker for both GSQPT and ESQPT, and provides relevant information on the phase diagram of the time-independent Hamiltonian in Eq. (1) using the time-periodic Hamiltonian in Eq. (2).
Conclusions.-In this work, QPTs in a timeindependent system are proposed to be studied by using a time-periodic Hamiltonian. This Hamiltonian exhibits two different symmetries and periodically switches from one symmetry to the other. Using the LMG model and through simple calculations in the framework of Floquet theory (mostly analytical except for a matrix diagonalization), we show that the relevant information about the phase diagram of the time-independent system is retained by the time-periodic Hamiltonian. In particular, the GSQPT and ESQPT are clearly manifested on the quasienergies, mean energies, and geometric phases of the Floquet modes. Furthermore, the real part of the two-time correlator in Eq. (11) is shown to be a good marker not only for the GSQPT, but also for the ESQPT. Consequently, this type of time-dependent Hamiltonians with periodic quenches can be used to study and characterize the phase diagram and critical points of timeindependent systems. This finding opens up new promising avenues for the use of Floquet machinery in the field of QPT.
This work was partially supported by the Consejería de Economía, Conocimiento, Empresas y
FIG. 1 .
1Sketch of the time-periodic Hamiltonian in Eq. (2). The quench time t0 can be varied and serves as a control parameter.
FIG. 2 .
2play, respectively, the role of S 2 Panel a) shows the dependence of the excitation energies En − E0 on the control parameter ξ for the time-independent Hamiltonian in Eq. (1), with H1 and H2 given by the LMG model in Eq. (10). The solid blue lines correspond to even n and the dashed orange lines to odd n. Panels b), c), d), and e) refer to the time-periodic Hamiltonian in Eq. (2), with H1 and H2 given by Eq. (10). Panels b) and c) depict, respectively, the differences of quasienergies, n − 0, [calculated using Eq. (5)] and of mean energies,Ēn −Ē0, [calculated using Eq. (9)] as a function of the dimensionless quench time t0/T , with the same line code as in panel a). Panel d) shows the geometric phases of the Floquet modes n = 0, 1, 8, 9, 16, and 17 as a function of t0/T . Panel e) depicts the differences n+1 − n (solid lines and left ordinate axis) and ϕn+1 − ϕn (dashed lines and right ordinate axis) as a function of t0/T for n = 0, 8, and 16. These differences are highlighted in panels b) and d). In all panels N = 20.and S 1 in our formalism, and are realized by the Hamiltonians H 2 and H 1 in Eq. (10), respectively. Each of these symmetries is linked to a different structure (phase) of the system.With the purpose of applying Floquet machinery to the LMG model, the Hamiltonian in Eq. (2) is considered, with H 1 and H 2 given by Eq.(10). Under this theoretical framework, the evolution operator U (T, 0) in Eq. (4), the Floquet modes in Eq. (6), the geometric phase of the Floquet modes in Eq.(8), and the mean energy of the Floquet modes in Eq. (9) can be obtained by analytical expressions. The only numerical task is the diagonalization of the evolution operator U (T, 0).
FIG. 3 .
3Panel a) shows the same as Fig. 2.b) but for N = 50 and using a color scale to indicate the value of the real part of the two-time correlator in Eq. (11). Panel b) shows the real part of this two-time correlator as a function of the dimensionless quench time t0/T for the Floquet mode with the lowest quasienergy in the first Brillouin zone. Panel c) shows the real part of this two-time correlator as a function of the quasi-energies n − 0 for fixed quench times t0/T = 0.1 (red dots), t0/T = 0.4 (violet crosses), and t0/T = 0.6 (orange triangles). These values of the quench time are highlighted in panel a) with vertical red dashed, violet dot-dashed, and orange dotted lines, respectively.nonzero for intermediate values of t 0 . Interestingly, the geometric phases of the Floquet modes become degenerate in pairs at approximately the same values of t 0 as the quasienergies, following the values along the separatrix line in
) under Groups FQM-160, FQM-177 and under projects P20-00617, P20-01247 and US-1380840; also through the projects PID2019-104002GB-C21, PID2019-104002GB-C22, and PID2020-114687GB-I00 funded by MCIN/AEI/10.13039/50110001103 and "ERDF A way of making Europe. Universidad de la Junta de Andalucía (SpainUniversidad de la Junta de Andalucía (Spain) under Groups FQM-160, FQM-177 and under projects P20- 00617, P20-01247 and US-1380840; also through the projects PID2019-104002GB-C21, PID2019- 104002GB-C22, and PID2020-114687GB-I00 funded by MCIN/AEI/10.13039/50110001103 and "ERDF A way of making Europe".
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| [] |
[
"University 4 Peking University 5 Intel Labs 6 MIT 7 Gaoling School of Artificial Intelligence",
"University 4 Peking University 5 Intel Labs 6 MIT 7 Gaoling School of Artificial Intelligence"
] | [
"Chengliang Zhong \nXi'an Research Institute of High\nTech\n\nTHUAI\nTsinghua University\n\n",
"Peixing You ",
"Xiaoxue Chen ",
"Hao Zhao [email protected] ",
"Fuchun Sun [email protected] \nTHUAI\nTsinghua University\n\n",
"Guyue Zhou ",
"Xiaodong Mu \nXi'an Research Institute of High\nTech\n",
"Chuang Gan ",
"Wenbing Huang "
] | [
"Xi'an Research Institute of High\nTech",
"THUAI\nTsinghua University\n",
"THUAI\nTsinghua University\n",
"Xi'an Research Institute of High\nTech"
] | [] | Detecting 3D keypoints from point clouds is important for shape reconstruction, while this work investigates the dual question: can shape reconstruction benefit 3D keypoint detection? Existing methods either seek salient features according to statistics of different orders or learn to predict keypoints that are invariant to transformation. Nevertheless, the idea of incorporating shape reconstruction into 3D keypoint detection is under-explored. We argue that this is restricted by former problem formulations. To this end, a novel unsupervised paradigm named SNAKE is proposed, which is short for shape-aware neural 3D keypoint field. Similar to recent coordinate-based radiance or distance field, our network takes 3D coordinates as inputs and predicts implicit shape indicators and keypoint saliency simultaneously, thus naturally entangling 3D keypoint detection and shape reconstruction. We achieve superior performance on various public benchmarks, including standalone object datasets ModelNet40, KeypointNet, SMPL meshes and scene-level datasets 3DMatch and Redwood. Intrinsic shape awareness brings several advantages as follows.(1) SNAKE generates 3D keypoints consistent with human semantic annotation, even without such supervision. (2) SNAKE outperforms counterparts in terms of repeatability, especially when the input point clouds are down-sampled.(3) the generated keypoints allow accurate geometric registration, notably in a zero-shot setting. Codes are available at https://github.com/zhongcl-thu/SNAKE. | 10.48550/arxiv.2206.01724 | [
"https://export.arxiv.org/pdf/2206.01724v2.pdf"
] | 249,375,353 | 2206.01724 | f429af55cb885ec179bcd7908ac418923556eed1 |
University 4 Peking University 5 Intel Labs 6 MIT 7 Gaoling School of Artificial Intelligence
Chengliang Zhong
Xi'an Research Institute of High
Tech
THUAI
Tsinghua University
Peixing You
Xiaoxue Chen
Hao Zhao [email protected]
Fuchun Sun [email protected]
THUAI
Tsinghua University
Guyue Zhou
Xiaodong Mu
Xi'an Research Institute of High
Tech
Chuang Gan
Wenbing Huang
University 4 Peking University 5 Intel Labs 6 MIT 7 Gaoling School of Artificial Intelligence
3
Detecting 3D keypoints from point clouds is important for shape reconstruction, while this work investigates the dual question: can shape reconstruction benefit 3D keypoint detection? Existing methods either seek salient features according to statistics of different orders or learn to predict keypoints that are invariant to transformation. Nevertheless, the idea of incorporating shape reconstruction into 3D keypoint detection is under-explored. We argue that this is restricted by former problem formulations. To this end, a novel unsupervised paradigm named SNAKE is proposed, which is short for shape-aware neural 3D keypoint field. Similar to recent coordinate-based radiance or distance field, our network takes 3D coordinates as inputs and predicts implicit shape indicators and keypoint saliency simultaneously, thus naturally entangling 3D keypoint detection and shape reconstruction. We achieve superior performance on various public benchmarks, including standalone object datasets ModelNet40, KeypointNet, SMPL meshes and scene-level datasets 3DMatch and Redwood. Intrinsic shape awareness brings several advantages as follows.(1) SNAKE generates 3D keypoints consistent with human semantic annotation, even without such supervision. (2) SNAKE outperforms counterparts in terms of repeatability, especially when the input point clouds are down-sampled.(3) the generated keypoints allow accurate geometric registration, notably in a zero-shot setting. Codes are available at https://github.com/zhongcl-thu/SNAKE.
Introduction
2D sparse keypoints play a vital role in reconstruction [34], recognition [23] and pose estimation [45], with scale invariant feature transform (SIFT) [20] being arguably the most important pre-Deep Learning (DL) computer vision algorithm. Altough dense alignment using photometric or featuremetric losses is also successful in various domains [2,38,8], sparse keypoints are usually preferred due to compactness in storage/computation and robustness to illumination/rotation. Just like their 2D counterparts, 3D keypoints have also drawn a lot of attention from the community in both pre-DL [13,37] and DL [16,1,40] literature, with various applications in reconstruction [47,43] and recognition [28,36].
However, detecting 3D keypoints from raw point cloud data is very challenging due to sampling sparsity. No matter how we obtain raw point clouds (e.g., through RGB-D cameras [42], stereo [4], or LIDAR [10]), they are only a discrete representation of the underlying 3D shape. This fact drives us to explore the question of whether jointly reconstructing underlying 3D shapes helps 3D UKPGAN-like methods predict saliency scores for P . Using chamfer distance, it reconstructs P coordinates based on the saliency scores and latent features. (c) Our SNAKE formulation predicts saliency probabilities and shape indicators simultaneously for each continuous query point q instead of discrete point clouds P . Sub-networks used for keypoint detection and reconstruction are shown in yellow and red, although they have different formulations. Here, the occupied points are those on the input surface. keypoint detection. To our knowledge, former methods have seldom visited this idea. Traditional 3D keypoint detection methods are built upon some forms of first-order (e.g., density in intrinsic shape signature [46]) or second-order (e.g., curvature in mesh saliency [15]) statistics, including sophisticated reformulation like heat diffusion [35]. Modern learning-based methods rely upon the idea of consistency under geometric transformations, which can be imposed on either coordinate like USIP [16] or saliency value like D3Feat [1]. The most related method that studies joint reconstruction and 3D keypoint detection is a recent one named UKPGAN [40], yet it reconstructs input point cloud coordinates using an auxiliary decoder instead of the underlying shape manifold.
Why is this promising idea under-explored in the literature? We argue the reason is that former problem formulations are not naturally applicable for reconstructing the underlying shape surface. Existing paradigms are conceptually illustrated in Fig. 1. USIP-like methods directly output keypoint coordinates while UKPGAN-like methods generate saliency values for input point clouds. In both cases, the representations are based upon discrete point clouds. By contrast, we reformulate the problem using coordinate-based networks, as inspired by the recent success of neural radiance fields [22,18,31] and neural distance fields [24,33]. As shown in Fig. 1-c, our model predicts a keypoint saliency value for each continuous input query point coordinate q(x, y, z).
A direct advantage of this new paradigm is the possibility of tightly entangling shape reconstruction and 3D keypoint detection. As shown in Fig. 1-c, besides the keypoint saliency decoder, we attach a parallel shape indicator decoder that predicts whether the query point q is occupied. The input to decoders is feature embedding generated by trilinearly sampling representations conditioned on input point clouds P . Imagine a feature embedding at the wing tip of an airplane, if it can be used to reconstruct the sharp curvature of the wing tip, it can be naturally detected as a keypoint with high repeatability. As such, our method is named as shape-aware neural 3D keypoint field, or SNAKE.
Shape awareness, as the core feature of our new formulation, brings several advantages. (1) High repeatability. Repeatability is the most important metric for keypoint detection, i.e., an algorithm should detect the same keypoint locations in two-view point clouds. If the feature embedding can successfully reconstruct the same chair junction from two-view point clouds, they are expected to generate similar saliency scores. (2) Robustness to down-sampling. When input point clouds are sparse, UKPGAN-like frameworks can only achieve reconstruction up to the density of inputs. In contrast, our SNAKE formulation can naturally reconstruct the underlying surface up to any resolution because it exploits coordinate-based networks. (3) Semantic consistency. SNAKE reconstructs the shape across instances of the same category, thus naturally encouraging semantic consistency although no semantic annotation is used. For example, intermediate representations need to be similar for successfully reconstructing different human bodies because human shapes are intrinsically similar.
To summarize, this study has the following two contributions:
• We propose a new network for joint surface reconstruction and 3D keypoint detection based upon implicit neural representations. During training, we develop several self-supervised losses that exploit the mutual relationship between two decoders. During testing, we design a gradient-based optimization strategy for maximizing the saliency of keypoints.
• Via extensive quantitative and qualitative evaluations on standalone object datasets Model-Net40, KeypointNet, SMPL meshes, and scene-level datasets 3DMatch and Redwood, we demonstrate that our shape-aware formulation achieves state-of-the-art performance under three settings: (1) semantic consistency; (2) repeatability; (3) geometric registration.
2 Related Work 3D Keypoint Detector As discussed in the introduction, 3D keypoint detection methods can be mainly categorized into hand-crafted and learning-based. Popular hand-crafted approaches [46,32,30] employ local geometric statistics to generate keypoints. These methods usually fail to detect consistent keypoints due to the lack of global context, especially under real-world disturbances, such as density variations and noise. USIP [16] is a pioneering learning-based 3D keypoint detector that outperforms traditional methods by a large margin. However, the detected keypoints are not semantically salient, and the number of keypoints is fixed. Fernandez et al. [9] exploit the symmetry prior to generate semantically consistent keypoints. But this method is category-specific, limiting the generalization to unseen categories and scenes. Recently, UKPGAN [40] makes use of reconstruction to find semantics-aware 3D keypoints. Yet, it recovers explicit coordinates instead of implicit shape indicators. As shown in Fig. 1, different from these explicit keypoint detection methods, we propose a new detection framework using implicit neural fields, which naturally incorporates shape reconstruction.
Implicit Neural Representation Our method exploits implicit neural representations to parameterize a continuous 3D keypoint field, which is inspired by recent studies of neural radiance fields [18,22,31] and neural distance fields [24,33,17,44]. Unlike explicit 3D representations such as point clouds, voxels, or meshes, implicit neural functions can decode shapes continuously and learn complex shape topologies. To obtain fine geometry, ConvONet [26] proposes to use volumetric embeddings to get local instead of global features [21] of the input. Recently, similar local geometry preserving networks show a great success for the grasp pose generation [12] and articulated model estimation [11]. They utilize the synergies between their main tasks and 3D reconstruction using shared local representations and implicit functions. Unlike [11,12] that learn geometry as an auxiliary task, our novel losses tightly couple surface occupancy and keypoint saliency estimates.
Method
This section presents SNAKE, a shape-aware implicit network for 3D keypoint detection. SNAKE conditions two implicit decoders (for shape and keypoint saliency) on shared volumetric feature embeddings, which is shown in Fig. 2-framework. To encourage repeatable, uniformly scattered, and sparse keypoints, we employ several self-supervised loss functions which entangle the predicted surface occupancy and keypoint saliency, as depicted in the middle panel of Fig. 2. During inference, query points with high saliency are further refined by gradient-based optimization since the implicit keypoint field is continuous and differentiable, which is displayed in Fig. 2-inference.
Network Architecture
Point Cloud Encoder As fine geometry is essential to local keypoint detection, we adopt the ConvONets [26], which can obtain local details and scale to large scenes, as the point cloud encoder denoted f θen for SNAKE. Given an input point cloud P ∈ R N ×3 , our encoder firstly processes it with the PointNet++ [27] or alternatives like [49]) to get a feature embedding Z ∈ R N ×C1 , where N and C 1 are respectively the number of points and the dimension of the features. Then, these features are projected and aggregated into structured volume Z ∈ R C1×H×W ×D , where H, W and D are the number of voxels in three orthogonal axes. The volumetric embeddings serve as input to the 3D UNet [6] to further integrate local and global information, resulting in the output G ∈ R C2×H×W ×D , where C 2 is the output feature dimension. More details can be found in the Appendix A.
Filter
(b) Keypoint Coordinate Optimization
Gradients
| − |
Object-scale Keypoints
Scene-scale Keypoints
Input Query
Surface & Sparsity Losses
Repeatability Loss
Inference Result We use an implicit network to decode the surface occupancy and keypoint saliency probability simultaneously. Green arrows indicate the mutual relationships between the geometry and saliency field. Through marching cubes and non-maximum suppression (NMS), it could respectively recover the shape and detect keypoints from the input. Loss functions for keypoint filed: Three loss functions try to make the generated keypoint repeatable, located on the underlying surface, and sparse. Inference: We design a gradient-based optimization method to extract keypoints from the saliency field. Result: The object-scale and scene-scale keypoints after inference are displayed.
Shape Implicit Decoder As shown in the top panel of Fig. 2, each point q ∈ R 3 from a query set Q is encoded into a C e -dimensional vector q e via a multi-layer perceptron that is denoted the positional encoder f θpos , i.e. q e = f θpos (q). Then, the local feature G q is retrieved from the feature volume G according to the coordinate of q via trilinear interpolation. The generated q e and G q are concatenated and mapped to the surface occupancy probability P rob o (q|P ) ∈ [0, 1] by the occupancy decoder f θo , as given in Eq. (1). If q is on the input surface, the P rob o (q|P ) would be 1, otherwise be 0. In our formulation, the points inside the surface are also considered unoccupied.
f θo (q e , G q ) → P rob o (q|P ) (1) Keypoint Implicit Decoder Most of the process here is the same as in shape implicit decoder, except for the last mapping function. The goal of keypoint implicit decoder f θs is to estimate the saliency of the query point q conditioned on input points P , which is denoted as P rob s (q|P ) ∈ [0, 1] and formulated by: f θs (q e , G q ) → P rob s (q|P ).
(2) Here, saliency of the query point q is the likelihood that it is a keypoint.
Implicit Field Training
The implicit field is jointly optimized for surface occupancy and saliency estimation by several selfsupervised losses. In contrast to former arts [12,11] with a similar architecture that learn multiple tasks separately, we leverage the geometry knowledge from shape field to enhance the performance of keypoint field, as shown in the green arrows of Fig. 2. Specifically, the total loss is given by:
L = L o + L r + L m + L s ,(3)
where L o encourages the model to learn the shape from the sparse input, L r , L m and L s respectively help the predicted keypoint to be repeatable, located on the underlying surface and sparse.
Surface Occupancy Loss
The binary cross-entropy loss l BCE between the predicted surface occupancy P rob o (q|P ) and the ground-truth label P rob gt o is used for shape recovery. The queries Q are randomly sampled from the whole volume size H × W × D. The average over all queries is as follows:
L o = 1 |Q| q∈Q l BCE P rob o (q|P ), P rob gt o (q|P ) ,(4)
where |Q| is the number of queries Q.
Repeatability Loss Detecting keypoints with high repeatability is essential for downstream tasks like registration between two-view point clouds. That indicates the positions of keypoint are covariant to the rigid transformation of the input. To achieve a similar goal, 2D keypoint detection methods [29,7,45] enforce the similarity of corresponding local salient patches from multiple views. Inspired by them, we enforce the similarity of local overlapped saliency fields from two-view point clouds. Since the implicit field is continuous, we uniformly sample some values from a local field to represent the local saliency distribution. Specifically, as shown in the top and the middle part of Fig. 2, we build several local 3D Cartesian grids {Q i } n i=1 with resolution of H l × W l × D l and size of 1/U . We empirically set the resolution of Q i to be almost the same as the feature volume G. As non-occupied regions are uninformative, the center of Q i is randomly sampled from the input. Then, we perform random rigid transformation T on the P and Q i to generate T P and T Q i . Similar to [29], the cosine similarity, denoted as cosim, is exploited for the corresponding saliency grids of Q i and T Q i :
L r = 1 − 1 n i∈n cosim P rob s (Q i |P ), P rob s (T Q i |T P ) .(5)
Surface Constraint Loss As discussed in [16], 3D keypoints are encouraged to close to the input. They propose a loss to constrain the distance between the keypoint and its nearest neighbor from the input. Yet, the generated keypoints are inconsistent when given the same input but with a different density. Thanks to the shape decoder, SNAKE can reconstruct the underlying surface of the input, which is robust to the resolution change. Hence, we use the surface occupancy probability to represent the inverse distance between the query and the input. As can be seen in Fig. 2-(surface constraint), we enforce the saliency of the query that is far from input P close to 0, which is defined as
L m = 1 |Q| q∈Q 1 − P rob o (q|P ) · P rob s (q|P ).(6)
Sparsity Loss Similar to 2D keypoint detection methods [29], we design a sparsity loss to avoid the trivial solution (P rob s (Q|P )=0) in Eq.( 5)( 6). As can be seen in Fig. 2, the goal is to maximize the local peakiness of the local saliency grids. As the sailency values of non-occupied points are enforced to 0 by L m , we only impose the sparsity loss on the points with high surface occupancy probability. Hence, we derive the sparsity loss with the help of decoded geometry by
L s = 1 − 1 n i∈n max P rob s (Q 1 i |P ) − mean P rob s (Q 1 i |P ) ,(7)
where
Q 1 i = {q|q ∈ Q i , P rob o (q|P ) > 1 − thr o }, thr o ∈ (0, 0.5]
is a constant, and n is the number of grids. It is noted that the spatial frequency of local peakiness is dependent on the grid size 1/U , see section 4.4. Since the network is not only required to find sparse keypoints, but also expected to recover the object shape, it would generate high saliency at the critical parts of the input, like joint points of a desk and corners of a house, as shown in the Fig. 2-result.
Explicit Keypoint Extraction
The query point q whose saliency is above a predefined threshold thr s ∈ (0, 1) would be selected as a keypoint at the inference stage. Although SNAKE can obtain the saliency of any query point, a higher resolution query set results in a high computational cost. Hence, as shown in Fig. 2-inference, we build a relatively low-resolution query sets Q infer which are evenly distributed in the input space and further refine the coordinates of Q infer by gradient-based optimization on this energy function:
E(Q infer , P ) = 1 |Q infer | q∈Q infer 1 − P rob s (q|P ).(8)
Specifically, details of the explicit keypoint extraction algorithm are summarized in Alg. 1.
Algorithm 1 Optimization for Explicit Keypoint Extraction
Require: P, Q infer , f θen , f θpos , f θo , f θs . Hyper-parameters: λ, J, thr o , thr s . Get initial P rob o (Q infer |P ) according to Eq.( 1). Filter to get new query set
Q infer = {q|q ∈ Q infer , P rob o (q|P ) > 1 − thr o }. for 1 to J do Evaluate energy function E(Q infer , P ).
Update coordinates with gradient descent:
Q infer = Q infer − λ∇ Q infer E(Q infer , P ). end for Sample final keypoints Q k = {q|q ∈ Q infer , P rob s (q|P ) > thr s }.
Experiment
In this section, we evaluate SNAKE under three settings. First, we compare keypoint semantic consistency across different instances of the same category, using both rigid and deformable objects. Next, keypoint repeatability of the same instance under disturbances such as SE (3) transformation, noise and downsample is evaluated. Finally, we inspect the point cloud registration task on the 3DMatch benchmark, notably in a zero-shot generalization setting. Besides, an ablation study is done to verify the effect of each design choice in SNAKE. The implementation details and hyper-parameters for SNAKE in three settings can be found in the Appendix B.
Semantic Consistency
Datasets The KeypointNet [41] dataset and meshes generated with the SMPL model [19] are utilized. KeypointNet has numerous human-annotated 3D keypoints for 16 object categories from ShapeNet [3]. The training set covers all categories that contain 5500 instances. Following [40], we evaluate 630 unseen instances from airplanes, chairs, and tables. SMPL is a skinned vertex-based deformable model that accurately captures body shape variations in natural human poses. We use the same strategy in [40] to generate both training and testing data.
Metric Mean Intersection over Union (mIoU) is adopted to show whether the keypoints across intra-class instances have the same semantics or not. For KeypointNet, a predicted keypoint is considered the same as a human-annotated semantic point if the geodesic distance between them is under some threshold. Due to the lack of human-labeled keypoints on SMPL, we compare the keypoint consistency in a pair of human models. A keypoint in the first model is regarded semantically consistent if the distance between its corresponding point and the nearest keypoint in the second model is below some threshold.
Ours
Human Annotation
Evaluation and Results
We compare SNAKE with random detection, hand-crafted detectors: ISS [46], Harris-3D [32] and SIFT-3D [30], and DL-based unsupervised detectors: USIP [16] and UKPGAN [40]. As USIP has not performed semantic consistency evaluations, we train the model with the code they provided. We follow the same protocols in [40] to filter the keypoints via NMS with a Euclidean radius of 0.1. Quantitative results are provided in Fig. 5-(a,e). SNAKE obtains higher mIoU than other methods under most thresholds on KeypointNet and SMPL. Qualitative results in Fig. 3 show our keypoints make good alignment with human annotations. Fig. 4 provides qualitative comparisons of semantically consistent keypoints on rigid and deformable objects. Owing to entangling shape reconstruction and keypoint detection, SNAKE can extract aligned representation for intra-class instances. Thus, our keypoints better outline the object shapes and are more semantically consistent under large shape variations. As shown in the saliency field projected slices, we can get symmetrical keypoints, although without any explicit constraint like the one used in [40]. Here, a projected slice is obtained by taking the maximum value of a given field along the projection direction.
Repeatability
Datasets ModelNet40 [39] is a synthetic object-level dataset that contains 12,311 pre-aligned shapes from 40 categories, such as plane, guitar, and table. We adopt the official dataset split strategy. 3DMatch [43] and Redwood [5] are RGB-D reconstruction datasets for indoor scenes. Following [16], we train the model on 3DMatch and test it on Redwood to show the generalization performance. The training set contains around 19k samples and the test set consists of 207 point clouds.
Metric We adopt the relative repeatability proposed in USIP [16] Figure 6: Visualization of keypoints under some disturbances on object-level [39] and scene-level [5] datasets compared to hand-crafted [46] and explicit representation based [16] methods. Downsample rate is 8x and the Gaussian noise scale (σ) is 0.06. The shape reconstruction via marching cubes for our occupancy field is also given. Visualization of repeatability can be found in the Appendix C.3.
distance to the nearest keypoint in the other point cloud is below a threshold . Relative repeatability means the number of repeatable points divided by the total number of detected keypoints.
Evaluation and Results Random detection, traditional methods and USIP are chosen as our baselines.
Since UKPGAN does not provide pre-trained models on these two datasets, we do not report its results in Fig. 5 but make an additional comparison on KeypointNet, which is illustrated in the next paragraph.
We use NMS to select the local peaky keypoints with a small radius (0.01 normalized distance on ModelNet40 and 0.04 meters on Redwood) for ours and baselines. We generate 64 keypoints in each sample and show the performance under different distance thresholds , downsample rates, and Gaussian noise scales. We set a fixed of 0.04 normalized distance and 0.2 meters on the ModelNet40 and Redwood dataset when testing under the last two cases. As shown in Fig. 5-(b,f), SNAKE outperforms state-of-the-art at most distance thresholds. We do not surpass USIP on Redwood in the lower thresholds. Note that it is challenging to get higher repeatability on Redwood because the paired inputs have very small overlapping regions. Fig. 5-(c,d,g,h) show the repeatability robustness to different downsample rates (d.r.) and Gaussian noise N (0, σ) levels. SNAKE gets the highest repeatability in most cases because the shape-aware strategy helps the model reason about the underlying shapes of the objects/scenes, which makes keypoints robust to the input variations. Fig. 6 provides visualization of object-level and scene-level keypoints of the original and disturbed inputs. SNAKE can generate more consistent keypoints than other methods under drastic input changes.
We have tried to train UKPGAN (official implementation) on ModelNet40 and 3DMatch datasets from scratch but observed divergence under default hyper-parameters. As such, we provide a new experiment to compare their repeatability on the KeypointNet dataset, on which UKPGAN provided a pre-trained model. We randomly perform SE(3) transformation on the test point clouds to generate the second view point clouds. Then, we select top-32 salient keypoints with NMS (radius=0.03) in each sample and report the keypoint repeatability under different distance thresholds , downsample rates, and Gaussian noise scales. The results are summarized in Table 1, 2, which show that SNAKE achieves significant gains over UKPGAN in most cases. More discussions can be found in the Appendix C.1. Figure 7: (a) SNAKE fails to predict semantically consistent keypoints without the occupancy decoder.
(b) Saliency field slice with a different grid size of (1/U ) 3 . (c) The impact of the optimization step.
Zero-shot Point Cloud Registration
Datasets We follow the same protocols in [40]
Evaluation and Results
As baselines, we choose random detection, ISS, SIFT-3D, UKPGAN, and D3Feat. Note that D3Feat is a task-specific learning-based detector trained on the 3DMatch dataset, thus not included in this zero-shot comparison. Ours and UKPGAN are trained on the synthetic object dataset KeypointNet only. The results are reported under different numbers of keypoints (i.e., 2500, 1000, 500, 250, 100). The NMS with a radius of 0.05m is used for D3Feat, UKPGAN, and ours. As shown in Table 3, SNAKE outperforms other methods consistently under three metrics. For registration recall and inlier ratio, we achieve significant gains over UKPGAN and other traditional keypoint methods. Notably, when the keypoints are high in numbers, SNAKE even outperforms D3Feat which has seen the target domain. Local shape primitives like planes, corners, or curves may be shared between objects and scenes, so our shape-aware formulation allows a superior generalization from objects to scenes.
Ablation Study
Loss Function Table 4 reports the performance w.r.t. designs of loss functions. (Row 1) If the surface occupancy decoder is removed, the surface constraint cannot be performed according to Eq.( 6), so they are removed simultaneously. Although the model could detect significantly repeatable keypoints on ModelNet40 [39], it fails to give semantically consistent keypoints on KeypointNet [41]. Fig. 7-a shows that SNAKE is unable to output symmetric and meaningful keypoints without the shape-aware technique. That indicates the repeatability could not be the only criterion for keypoint detection if an implicit formulation is adopted. (Row 2-4) Each loss function for training keypoint field is vital for keypoint detection. Note that the model gives a trivial solution (0) for the saliency field and cannot extract distinctive points when removing the sparsity loss.
Grid Size and Volumetric Resolution The grid size 1/U controls the number of keypoints because L s enforces the model to predict a single local maxima per grid of size (1/U ) 3 . Fig. 7-b shows Table 5, indicating that U = 6 gives the best results. From Table 6, we can see that higher resolution improves performance. However, the performance drops when it reaches the resolution of 80. The potential reason is as such: the number of queries in a single grid increases when the resolution becomes higher, as mentioned in 3.2. In this case, finer details make the input to cosine similarity too long and contain spurious values.
Optimization
Step and Learning Rate Fig. 7-c shows the importance of optimization (see Alg. 1) for refining keypoint coordinates on the ModelNet40 dataset. It is noted that too many optimization steps will not bring more gains but increase the computational overhead. In this paper, we set the number of update steps to 10. The learning rate for optimization is also key to the final result. When the learning rate is set to 0.1, 0.01, 0.001 and 0.0001, the relative repeatability (%) on ModelNet40 dataset with the same experimental settings as Table 6 are 0.002, 0.622, 0.854 and 0.826, respectively. In addition, the comparison of computation cost of baselines and ours can be found in the Appendix D.
Conclusion and Discussion
We propose SNAKE, a method for 3D keypoint detection based on implicit neural representations. Extensive evaluations show our keypoints are semantically consistent, repeatable, robust to downsample, and generalizable to unseen scenarios. Limitations. The optimization for keypoint extraction during inference requires considerable computational cost and time, which may not be applicable for use in scenarios that require real-time keypoint detection. Negative Social Impact. The industry may use the method for pose estimation in autonomous robots. Since our method is not perfect, it may lead to wrong decision making and potential human injury.
A Network Architecture
Following [26], our implementation is a compilation of PointNet++ [27], 3D UNet [6], positional encoder and implicit surface occupancy decoder. The architecture of the implicit keypoint decoder is designed to be the same as the surface occupancy decoder. The dimensions of the feature embedding Z and Z are both set to 32, i.e., C 1 = C 2 = 32. And each point from a query set is also encoded into a 32-dimensional feature vector. More details can be found in the code we provide.
B Implementation Details B.1 Training SNAKE is implemented in PyTorch [25] using the Adam [14] optimizer with a mini-batch size of b on 4 NVIDIA A100 GPUs for el epochs. We use a learning rate of 10 −4 for the first ef epochs, which is dropped ten times for the remainder. As discussed in Sec. 3.2 (repeatability loss), we perform random rigid transformation T on the input P to generate a second view input T P . Then, we use some data augmentation on T P to increase data diversity by downsampling with a random rate between 0 and 4, and Gaussian noise. Training hyper-parameters on four datasets are provided in Table 7.
In our formulation, occupied points are those on the input surface, and the others are considered all unoccupied, including the points inside the surface. Therefore, we can only use input point clouds to learn the surface occupancy model. Specifically, we randomly sample the positives from the input point cloud. The negatives are randomly sampled in the unit 3D space. Although some of the negatives are indeed on the surface of the object, their number is so limited compared to the whole query sets that they do not affect the training.
B.2 Testing
For the SMPL dataset, the correspondence between the paired point clouds can be generated by SMPL vertex index. Since the keypoint SNAKE generates may not be in the input point cloud (we enforce the keypoint scatter on the underlying surface of the input), we take the point closest to the generated keypoint in the input as the final keypoint. We use the same strategy on the 3DMatch dataset when performing geometric registration because D3feat [1] predicts descriptors for each point in the input. The testing hyper-parameters are shown in Table 7.
C Results
C.1 Additional comparison with UKPGAN on keypoint repeatability
Due to the absence of pretrained model on the ModelNet40 and 3DMatch dataset, we do not report the keypoint repeatability of UKPGAN [40] on the main paper. We have tried to train UKPGAN (official implementation) on the ModelNet40 and 3DMatch datasets from scratch but observed divergence under default hyper-parameters. The training always reports NaN losses in early epochs. This instability also implies limitations in implementing the idea of joint reconstruction and keypoint detection with GAN-based methods. As such, we provide a new experiment to compare their repeatability on the KeypointNet dataset, on which the UKPGAN provided a pre-trained model. Table 2 in the main paper show that SNAKE achieves significant gains over UKPGAN in most cases. Interestingly, when the inputs are disturbed, the performance of UKPGAN increases rather than decreases. Via visualizing the results in Fig. 8, we find that when the input point clouds are disturbed, the keypoints predicted by UKPGAN are clustered in a small area, which improves the repeatability of keypoints but fails to cover the input uniformly. This illustrates that the GAN-based method adopted by UKPGAN to control the keypoint sparsity is not robust to input point cloud disturbance. The keypoints of ours still remain meaningful under the drastic changes of inputs.
Tabke 1 and
Ours
C.2 Quantitative Results
The specific numerical results on semantic consistency and repeatability are summarized in Table 9-15, which correspond to Figure 5 in the main paper. We present the mean and standard deviation of our results over 6 models trained under different random seeds.
C.3 Qualitative Visualization of Saliency Field and Keypoints
We show more qualitative results on keypoint semantic consistency between intra-class instances from rigid objects plane, guitar, motorcycle, and deformable human shapes in Figure 10-13. Owing to entangling shape reconstruction and keypoint detection, SNAKE can extract aligned representation for intra-class instances. As shown in Figure 14-19, we provide more visualizations of keypoints under some disturbances on object-level (ModelNet40) and scene-level (Redwood) datasets. It can be seen that SNAKE can generate more consistent keypoints than other methods under significant variations of inputs. We also show the detected keypoints of the same object/scene from different views to demonstrate the repeatability of keypoint in Figure 20-22.
C.4 Qualitative Visualization of Surface Occupancy Field and Shape Reconstruction
As shown in Figure 9, we show visualizations of the occupancy field and shape reconstruction on the ModelNet40 dataset. These five samples are taken from the unseen test set. As shown by the second row, only points on the input surface have a high occupancy value, and the other points (inside or outside of the surface) have a near-zero occupancy value. Under our definition, two surfaces can be obtained through the marching cube, and we only show the outer surface.
D Computation Cost
As shown in Table 8, we report the time taken to generate keypoints of hand-crafted detector ISS, deep-learning (DL) based methods USIP [16], UKPGAN [40] and ours. ISS [46] is implemented by Open3d [48] and deployed on an AMD EPYC 7742 64-Core CPU. DL-based methods are deployed on an NVIDIA GeForce RTX 3090 GPU. USIP requires the lowest computational time to generate keypoints, while UKPGAN requires the highest cost since it takes much time to compute smoothed density values. The inference time of our model is comparable to ISS when we do not refine the keypoint by optimization (J=0), and the repeatability is still as high as around 81% when the input point number is 4096. The time increases with the increasing number of optimization iterations J.
As discussed before, when J becomes larger (below 15), the performance of keypoint gets better. It suggests that there is a trade-off between keypoint performance and inference speed of our method. The GPU memory cost (MB) for USIP, UKPGAN, and SNAKE during a single batch inference is 3747, 10727, and 2785, which illustrates that SNAKE requires the lowest GPU memory cost to generate keypoints.
E Illustrations on the Assets We Used and Released
The license of assets we used is as follows: All existing datasets and codes we used in this paper are allowed for research and do not contain personally identifiable information or offensive content. Note that SMPL only has human shapes without the identity information of the person, such as the face or body texture. Our code is released under the MIT license. Table 15: Relative repeatability (%) when the input is randomly downsampled by some rates on the Redwood dataset. This table corresponds to Figure 5-(g) in the main paper. Figure 9: Visualization for surface occupancy field and surface reconstruction of test instances (unseen) from ModelNet40 dataset. The second row shows the middle slice of the surface occupancy field of these objects. The third row shows the projected surface occupancy field on the same slice by taking the maximum value. The fourth row shows the outer surface reconstructed by applying marching cubes on the surface occupancy field, using a threshold of 0.4.
Figure 1 :
1A comparison between existing 3D keypoint detection formulations and our newly proposed one. (a) USIP-like methods directly predict keypoint coordinates from input point clouds P . (b)
Figure 2 :
2Framework:
Figure 3 :
3Comparison with human annotations on KeypointNet[41] dataset.
Figure 4 :Figure 5 :
45Semantic consistency of keypoints on rigid and deformable objects. Our keypoints are more evenly scattered on the underlying surface of objects, more symmetrical, and more semantically consistent under significant shape variations when compared to other methods. The saliency field projected slice shows that SNAKE decodes well-aligned saliency values for keypoints in different instances but with similar semantics, such as the wingtip of the airplane and the leg of the human. Here, small saliency is shown in bright red and gets darker with a larger value. Quantitative results on four datasets. Keypoint semantic consistency (a)(e) on KeypointNet and SMPL. Relative repeatability for two-view point clouds with different distance threshold (b), downsample rate (c), Gaussian noise N (0, σ noise ) (d) on ModelNet40. The results of (f)(g)(h) are tested on Redwood with the same settings in (b)(c)(d). The specific numerical results can be found in the Appendix C.2.
Figure 8 :
8Keypoints of the KeypointNet data under some input disturbances.
(a) MIT License for KeypointNet dataset. (b) Software Copyright License for non-commercial scientific research purposes on SMPL-Model. (c) GPL-3.0 License for ModelNet40, 3DMatch, Redwood dataset, and USIP. (d) Microsoft research license for 3DMatch registration benchmark.
Figure 10 :Figure 11 :Figure 12 :
101112Keypoint semantic consistency comparison on the plane. Keypoint semantic consistency comparison on the guitar. Keypoint semantic consistency comparison on the motorcycle.
Figure 13 :Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 :Figure 19 :Figure 20 :Figure 21 :Figure 22 :
13141516171819202122Keypoint semantic consistency comparison on the human shape. Keypoints of the chair under some input disturbances. Keypoints of the desk under some input disturbances. Keypoints of the flower under some input disturbances. Keypoints of the indoor scene (1) under some input disturbances. Keypoints of the indoor scene (2) under some input disturbances. Keypoints of the indoor scene (3) under some input disturbances. Keypoints repeatability comparison when the input is not corrupted. Note that in the Redwood dataset (right panel), two-view point clouds are partially overlapped. Keypoints repeatability comparison when the input is 8x down sampled. Note that in the Redwood dataset (right panel), two-view point clouds are partially overlapped. Keypoints repeatability comparison when the input is added Gaussion noises (std=0.06). Note that in the Redwood dataset (right panel), two-view point clouds are partially overlapped.
as the evaluation metric. Given two point clouds captured from different viewpoints, a keypoint in the first point cloud is repeatable if itsOurs
USIP
Shape
Recon.
ISS
Original
8x Downsample
0.06 std. Noise
Original
8x Downsample
0.06 std. (m) Noise
Table 1 :
1Relative repeatability (%) with differ-
ent distance thresholds on the KeypointNet
dataset.
0.03 0.05 0.07 0.09 0.10
UKPGAN 0.199 0.454 0.661 0.81 0.864
Ours
0.643 0.806 0.892 0.936 0.948
Table 2 :
2Relative repeatability (%) when input
point clouds are disturbed ( =0.03). Here, ori.
means the original input.
ori. d.r.=4 d.r.=8 σ=0.02 σ=0.03
UKPGAN 0.199 0.570 0.427 0.608
0.558
Ours
0.643 0.594 0.525 0.626
0.536
Table 3 :
3Registration result on 3DMatch. We combine the off-the-shelf descriptor D3Feat[1] and different keypoint detectors to perform two-view point cloud registration.Feature Matching Recall (%)
Registration Recall (%)
Inlier Ratio (%)
Detector
Descriptor 2500 1000 500 250 100 2500 1000 500 250 100 2500 1000 500 250 100
D3Feat
D3Feat
95.6 94.5 94.3 93.3 90.6 84.4 84.9 82.5 79.3 67.2 40.6 42.7 44.1 45.0 45.6
Random
D3Feat
95.1 94.5 92.8 90.0 81.2 83.0 80.0 77.0 65.5 38.8 38.6 33.6 28.9 23.6 17.3
ISS
D3Feat
95.2 94.4 93.4 90.1 81.0 83.5 79.2 76.0 64.3 37.2 38.2 33.5 28.8 23.9 17.4
SIFT
D3Feat
94.9 94.0 93.0 91.2 81.3 84.0 79.9 76.1 60.9 38.6 38.4 33.6 28.8 23.3 17.4
UKPGAN
D3Feat
94.7 94.2 93.5 92.6 85.9 82.8 81.4 77.1 69.7 47.4 38.8 35.5 34.0 33.1 27.7
Ours
D3Feat
95.5 95.0 94.7 92.9 89.5 85.1 83.7 81.2 74.6 50.9 41.3 39.0 37.0 33.5 30.0
Table 4 :
4Ablations for the designs of loss function. occ. = occupancy, sur. = surface, rep. = repeatability, spa. = sparsity and rr. = relative repeatability.rr. (%) on [39] mIoU (%) on [41]
Threshold
0.04 0.05 0.06 0.08 0.09 0.1
w/o occ. & sur. 0.92 0.94 0.95 0.22 0.25 0.28
w/o sur.
0.28 0.36 0.42 0.31 0.35 0.39
w/o rep.
0.22 0.28 0.34 0.30 0.35 0.39
w/o spa.
0
0
0
0
0
0
w/ all
0.85 0.89 0.90 0.30 0.37 0.42
Table 5 :
5Impact of different local grid size used in the L o and L s on ModelNet40.U
4
6
8
10
rr. (%) ( =0.04) 0.79 0.85 0.79 0.77
Table 6: Impact of different global volumetric
resolution on ModelNet40.
H(= W = D) 32 48 64 80
rr. (%) ( =0.04) 0.62 0.79 0.85 0.78
Table 7 :
7Training and testing hyper-parameters. Sem.=Semantic consistency evaluation,
Table 8 :
8Average time (s) taken to compute keypoints from input point clouds on ModelNet40 dataset.
The hyper-parameters of ours can be found in the Table 7. Decimals in parentheses in italics are
relative repeatability (%). Here, the experiment setting is the same as in Sec. 4.2.
Input Point #
ISS
USIP
UKPGAN
Ours
J=0
J=5
J=10
2048
0.07 (0.088) 0.006 (0.748)
14.41
0.08 (0.795) 0.50 (0.835) 0.81 (0.851)
4096
0.11 (0.096) 0.007 (0.799)
36.80
0.09 (0.811) 0.50 (0.850) 0.83 (0.864)
Table 9 :
9mIoU (%) with different geodesic distance thresholds on the KeypointNet dataset. This table corresponds toFigure 5-(a) in the main paper.0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Random
0.005
0.010
0.017
0.020
0.023
0.026
0.028
0.032
0.036
0.042
0.049
ISS
0.008
0.012
0.024
0.040
0.060
0.088
0.121
0.160
0.198
0.242
0.286
SIFT3D
0.005
0.010
0.015
0.022
0.043
0.065
0.089
0.120
0.160
0.189
0.221
Harris3D
0.005
0.010
0.014
0.023
0.040
0.060
0.084
0.110
0.150
0.180
0.216
USIP
0.003
0.006
0.013
0.024
0.045
0.078
0.116
0.160
0.212
0.264
0.314
UKPGAN
0.005
0.009
0.021
0.036
0.059
0.084
0.114
0.147
0.179
0.207
0.238
Ours
0.006±0.000 0.012±0.000 0.025±0.001 0.039±0.001 0.058±0.001 0.091±0.002 0.144±0.005 0.214±0.005 0.291±0.005 0.361±0.002 0.412±0.002
Table 10 :
10mIoU (%) with different Euclidean distance thresholds on SMPL mesh. This table corresponds toFigure 5-(e) in the main paper.0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Random
0.008
0.011
0.015
0.021
0.038
0.056
0.075
0.103
0.136
0.161
0.195
ISS
0.078
0.095
0.101
0.113
0.129
0.148
0.174
0.206
0.231
0.258
0.293
SIFT3D
0.009
0.011
0.016
0.026
0.043
0.064
0.084
0.108
0.146
0.183
0.213
Harris3D
0.012
0.013
0.016
0.021
0.032
0.047
0.065
0.097
0.129
0.159
0.187
USIP
0.037
0.043
0.051
0.081
0.129
0.198
0.278
0.338
0.390
0.440
0.492
UKPGAN
0.036
0.041
0.059
0.085
0.126
0.171
0.235
0.302
0.369
0.424
0.476
Ours
0.063±0.018 0.079±0.019 0.094±0.023 0.128±0.028 0.182±0.036 0.255±0.041 0.355±0.041 0.457±0.046 0.557±0.043 0.639±0.037 0.704±0.036
Table 11 :
11Relative repeatability (%) with different distance thresholds on the ModelNet40 dataset. This table corresponds toFigure 5-(b) in the main paper.0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Random
0.056
0.094
0.14
0.191
0.249
0.308
0.368
0.429
ISS
0.058
0.096
0.14
0.192
0.247
0.306
0.367
0.427
SIFT3D
0.055
0.092
0.138
0.191
0.249
0.308
0.369
0.429
Harris3D
0.056
0.096
0.147
0.21
0.277
0.347
0.415
0.48
USIP
0.771
0.799
0.815
0.827
0.836
0.844
0.851
0.857
Ours
0.763±0.011 0.864±0.009 0.897±0.007 0.910±0.005 0.917±0.005 0.923±0.005 0.927±0.005 0.930±0.005
Table 12 :
12Relative repeatability (%) when the input is randomly downsampled by some rates on the ModelNet40 dataset. This table corresponds toFigure 5-(c) in the main paper.1
2
4
8
16
Random
0.094
0.093
0.093
0.091
0.092
ISS
0.096
0.088
0.088
0.083
0.076
SIFT3D
0.092
0.089
0.087
0.082
0.075
Harris3D
0.096
0.093
0.093
0.093
0.092
USIP
0.799
0.748
0.685
0.554
0.321
Ours
0.864±0.009 0.851±0.009 0.820±0.008 0.730±0.009 0.528±0.012
Table 13 :
13Relative repeatability (%) when the input is disturbed by Gaussian noise N (0, σ) on the ModelNet40 dataset. This table corresponds to Figure 5-(d) in the main paper.0.00
0.02
0.04
0.06
0.08
0.10
0.12
Random
0.094
0.062
0.038
0.027
0.021
0.016
0.014
ISS
0.096
0.061
0.037
0.025
0.02
0.016
0.015
SIFT3D
0.092
0.06
0.036
0.025
0.019
0.016
0.014
Harris3D
0.096
0.063
0.038
0.029
0.02
0.015
0.015
USIP
0.799
0.872
0.844
0.746
0.558
0.341
0.192
Ours
0.864±0.009 0.869±0.008 0.841±0.015 0.766±0.013 0.619±0.041 0.464±0.049 0.354±0.045
Table 14 :
14Relative repeatability (%) with the different distance thresholds (m) on the Redwood dataset. This table corresponds to Figure 5-(f) in the main paper. 205±0.005 0.246±0.007 0.286±0.008 0.323±0.008 0.359±0.009 0.393±0.010 0.425±0.010 0.454±0.0090.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
Random
0.09
0.126
0.163
0.204
0.246
0.287
0.326
0.362
ISS
0.087
0.119
0.156
0.191
0.228
0.264
0.301
0.336
SIFT3D
0.088
0.123
0.168
0.21
0.254
0.297
0.33
0.367
Harris3D
0.079
0.109
0.14
0.175
0.209
0.243
0.278
0.31
USIP
0.255
0.285
0.314
0.342
0.368
0.392
0.417
0.439
Ours
0.
Table 16 :
16Relative repeatability (%) when the input is disturbed by Gaussian noise N (0, σ) on the Redwood dataset. This table corresponds toFigure 5-(h) in the main paper.0.00
0.02
0.04
0.06
0.08
0.10
Random
0.287
0.289
0.275
0.252
0.23
0.21
ISS
0.264
0.26
0.268
0.259
0.25
0.214
SIFT3D
0.297
0.289
0.27
0.261
0.241
0.217
Harris3D
0.243
0.239
0.225
0.206
0.193
0.178
USIP
0.392
0.386
0.375
0.341
0.317
0.295
Ours
0.393±0.010 0.392±0.008 0.381±0.009 0.359±0.009 0.318±0.007 0.256±0.013
AcknowledgmentsThis research is jointly supported by following projects: the National Science and Technology Major Project of the Ministry of Science and Technology of China (No.2018AAA0102900); the Key Field R&D Program of Guangdong Province (No.2021B0101410002); Sino-German Collaborative Research Project Crossmodal Learning (NSFC 61621136008/DFG SFB/TRR169); the National Natural Science Foundation of China (No.62006137); Beijing Outstanding Young Scientist Program (No.BJJWZYJH012019100020098). We would like to thank Pengfei Li for discussions about implicit field learning. We would also like to thank the anonymous reviewers for their insightful comments.
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"Vivek F Farias \nOperations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n\n",
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"Tianyi Peng \nOperations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n\n",
"Andrew Zheng \nOperations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n\n"
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"Operations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n",
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"Operations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n",
"Operations Research Center\nTepper School of Business\nDepartment of Aeronautics and Astronautics\nOperations Research Center\nMassachusetts Institute of Technology\nCarnegie Mellon University\nMassachusetts Institute of Technology\nMassachusetts Institute of Technology\n"
] | [] | We consider experiments in dynamical systems where interventions on some experimental units impact other units through a limiting constraint (such as a limited inventory). Despite outsize practical importance, the best estimators for this 'Markovian' interference problem are largely heuristic in nature, and their bias is not well understood. We formalize the problem of inference in such experiments as one of policy evaluation. Off-policy estimators, while unbiased, apparently incur a large penalty in variance relative to state-of-the-art heuristics. We introduce an on-policy estimator: the Differences-In-Q's (DQ) estimator. We show that the DQ estimator can in general have exponentially smaller variance than off-policy evaluation. At the same time, its bias is second order in the impact of the intervention. This yields a striking bias-variance tradeoff so that the DQ estimator effectively dominates state-of-the-art alternatives. From a theoretical perspective, we introduce three separate novel techniques that are of independent interest in the theory of Reinforcement Learning (RL). Our empirical evaluation includes a set of experiments on a city-scale ride-hailing simulator.1 arXiv:2206.02371v2 [cs.LG] 9 Jun 2022 third policy (simple randomization). This immediately suggests framing the problem as one of Off-Policy Evaluation, and borrowing one of many existing unbiased estimators, e.g.[62,61,42,27,34,35].This tack appears to be promising, e.g.[55], but we observe that the resulting variance is necessarily large (Theorem 3).Our Contributions: Against the above backdrop, we propose a novel on-policy treatment-effect estimator, which we dub the 'Differences-In-Q's' (DQ) estimator, for experiments with Markovian interference. In a nutshell, we characterize our contribution as follows:The DQ estimator has provably negligible bias relative to the treatment effect. Its variance can, in general be exponentially smaller than that of an efficent off-policy estimator. In both stylized and large-scale real-world models, it dominates state-of-the-art alternatives.We next describe these relative merits in greater detail:1. Second-order Bias: We show (Theorem 1) that when the impact of an intervention on transition probabilities is O(δ), the bias of the DQ estimator is O(δ 2 ). The DQ estimator thus | 10.48550/arxiv.2206.02371 | [
"https://arxiv.org/pdf/2206.02371v2.pdf"
] | 249,395,257 | 2206.02371 | c3bf25311fb01a3da9874cde1e1cb9d9a57b84f0 |
Markovian Interference in Experiments
Vivek F Farias
Operations Research Center
Tepper School of Business
Department of Aeronautics and Astronautics
Operations Research Center
Massachusetts Institute of Technology
Carnegie Mellon University
Massachusetts Institute of Technology
Massachusetts Institute of Technology
Andrew A Li
Operations Research Center
Tepper School of Business
Department of Aeronautics and Astronautics
Operations Research Center
Massachusetts Institute of Technology
Carnegie Mellon University
Massachusetts Institute of Technology
Massachusetts Institute of Technology
Tianyi Peng
Operations Research Center
Tepper School of Business
Department of Aeronautics and Astronautics
Operations Research Center
Massachusetts Institute of Technology
Carnegie Mellon University
Massachusetts Institute of Technology
Massachusetts Institute of Technology
Andrew Zheng
Operations Research Center
Tepper School of Business
Department of Aeronautics and Astronautics
Operations Research Center
Massachusetts Institute of Technology
Carnegie Mellon University
Massachusetts Institute of Technology
Massachusetts Institute of Technology
Markovian Interference in Experiments
We consider experiments in dynamical systems where interventions on some experimental units impact other units through a limiting constraint (such as a limited inventory). Despite outsize practical importance, the best estimators for this 'Markovian' interference problem are largely heuristic in nature, and their bias is not well understood. We formalize the problem of inference in such experiments as one of policy evaluation. Off-policy estimators, while unbiased, apparently incur a large penalty in variance relative to state-of-the-art heuristics. We introduce an on-policy estimator: the Differences-In-Q's (DQ) estimator. We show that the DQ estimator can in general have exponentially smaller variance than off-policy evaluation. At the same time, its bias is second order in the impact of the intervention. This yields a striking bias-variance tradeoff so that the DQ estimator effectively dominates state-of-the-art alternatives. From a theoretical perspective, we introduce three separate novel techniques that are of independent interest in the theory of Reinforcement Learning (RL). Our empirical evaluation includes a set of experiments on a city-scale ride-hailing simulator.1 arXiv:2206.02371v2 [cs.LG] 9 Jun 2022 third policy (simple randomization). This immediately suggests framing the problem as one of Off-Policy Evaluation, and borrowing one of many existing unbiased estimators, e.g.[62,61,42,27,34,35].This tack appears to be promising, e.g.[55], but we observe that the resulting variance is necessarily large (Theorem 3).Our Contributions: Against the above backdrop, we propose a novel on-policy treatment-effect estimator, which we dub the 'Differences-In-Q's' (DQ) estimator, for experiments with Markovian interference. In a nutshell, we characterize our contribution as follows:The DQ estimator has provably negligible bias relative to the treatment effect. Its variance can, in general be exponentially smaller than that of an efficent off-policy estimator. In both stylized and large-scale real-world models, it dominates state-of-the-art alternatives.We next describe these relative merits in greater detail:1. Second-order Bias: We show (Theorem 1) that when the impact of an intervention on transition probabilities is O(δ), the bias of the DQ estimator is O(δ 2 ). The DQ estimator thus
Introduction
Experimentation is a broadly-deployed learning tool in online commerce that is simple to execute, in principle: apply the treatment in question at random (e.g. an A/B test), and 'naively' infer the average effect of the treatment by differencing the average outcomes under treatment and control.
About a decade ago, Blake and Coey [8] pointed out a challenge in such experimentation on Ebay:
"Consider the example of testing a new search engine ranking algorithm which steers test buyers towards a particular class of items for sale. If test users buy up those items, the supply available to the control users declines."
This violation of the so-called Stable Unit Treatment Value Assumption (SUTVA) [13] has been viewed as problematic in online platforms as early as Reiley's seminal 'Magic on the Internet' work [43]. Blake and Coey [8] were simply pointing out that the resulting inferential biases were large, which is particularly problematic since treatment effects in this context are typically tiny.
The interference problem above is germane to experimentation on commerce platforms where interventions on a given experimental unit impact other units, since all units share a common inventory of 'demand' or 'supply' depending on context.
Despite the ubiquity of such interference, a practical solution is far from settled. An ongoing line of work addresses the problem via experimental design, assigning treatments carefully to mitigate the bias of 'naively'-derived estimators. In the best cases such designs provably reduce bias by exploiting certain application specific structures, but often it is unclear whether the problem at hand affords such structure (a case in point being the search-engine example above, as will be apparent later). As such, experimentation on online platforms still largely relies on simple randomization, i.e. A/B tests. Motivated by this fact, we focus instead on designing effective estimators assuming simple randomization. We demonstrate a novel estimator which, thanks to an effective bias-variance tradeoff, is a compelling alternative to both alternative state-of-the-art estimators as well as bespoke experimental designs when they apply.
Markovian Interference and Existing Approaches: We study a generic experimentation problem within a system represented as a Markov Decision Process (MDP), where treatment corresponds to an action which may interfere with state transitions. This form of interference, which we refer to as Markovian, naturally subsumes the platform examples above, as recently noted by others either implicitly [50] or explicitly [29,55]. In that example, a user arrives at each time step, the platform chooses an action (whether to treat the user), and the user's purchase decision alters the system state (inventory levels).
Our goal is to estimate the Average Treatment Effect (ATE), defined as the difference in steadystate reward with and without applying the treatment. In light of the above discussion, we assume that experimentation is done under simple randomization (i.e. A/B testing). Now without design as a lever, there are perhaps two existing families of estimators:
1. Naive: We will explicitly define the Naive estimator in the next section, but the strategy amounts to simply ignoring the presence of interference. This is by and large what is done in practice. Of course it may suffer from high bias (we show this momentarily in Section 1.1), but it serves as more than just a strawman. In particular, bias is only one side of the estimation coin, and with respect to the other side, namely variance, the Naive estimator is effectively the best possible.
Off-Policy Evaluation (OPE):
Another approach comes from viewing our problem as one of policy evaluation in reinforcement learning (RL). Succinctly, it can be viewed as estimating the average reward of two different policies (no treatment, or treatment) given observations from some leverages the one piece of structure we have relative to generic off-policy evaluation: treatment effects are typically small. Our analysis introduces a novel Taylor-like expansion of the ATE (Theorem 5) that in addition to the current setting, is of general interest in the theory of RL (for instance, in the context of Policy Optimization).
Variance:
We show (Theorem 2) that the DQ estimator is asymptotically normal, and provide a non-trivial, explicit characterization of its variance. By comparison, we show (Theorem 4) that this variance can, in general, be exponentially (in the size of the state space) smaller than the variance of any unbiased off-policy estimator. Our analysis introduces two new techniques. First, we prove what we dub an 'Entrywise Non-expansive Lemma', that we believe is crucial to elucidating the variance reduction afforded by on-policy methods. Second, we introduce a novel linearization trick which dramatically simplifies the analysis of variance in RL via the delta method.
Summarizing the above points, we are the first (to our knowledge) to explicitly characterize the favorable bias-variance trade-off in using on-policy estimation to tackle off-policy evaluation. This new lens has broader implications for OPE and policy optimization in RL (e.g., this leads to a new approach with a provably lower bias than some widely used methods in policy optimization, see Section 6.3).
3. Practical Performance: Despite the technical novelty described above, we view this as our most important contribution. We conduct experiments in both a caricatured one-dimensional environment proposed by others [29], as well as a city-scale simulator of a ride-sharing platform.
We show that in both settings the DQ estimator has MSE that is substantially lower than (a) naive, and several state-of-the-art off-policy estimators, and even (b) estimators given access to incumbent state-of-the-art experimental designs.
An Illustrative Example
It will be useful at this point to consider a simple example which highlights (a) the model of interference that we address, (b) the shortcomings of existing approaches to inference under such interference, and (c) our own approach to the problem. Importantly, all results presented in this simple example will extend to general MDPs by virtue of our analysis in Sections 3 and 4.
Consider the continuous-time Markov chain depicted in Fig. 1; this is simply an M/M/N/N queue (or the 'Erlang B' model). The state space, ranging from 0 to N , can be thought of as the quantity of some resource (e.g. rental homes of a similar type and geography) currently 'occupied'.
Customers arrive according to a Poisson process with rate λ, and independently with probability p, occupy a resource if available, for an exponentially-distributed duration with mean 1/µ. In spite of its simplicity, this model is closely related to one previously studied by [29] in the context of interference in commerce platforms. Now consider a treatment whose effect is to increase the probability p by some (unknown) quantity δ ≥ 0. Our goal is to measure the effect of the treatment on the steady-state rate of occupation (i.e. the steady-state rate of rightward transitions). We wish to estimate the treatment effect from a simple A/B test; i.e., an experiment that randomly applies (or does not apply) the treatment to each arriving customer. We now describe the various candidate estimators under this experimental design.
Existing Approach 1 -Naive: Given the observed trajectory during this experiment, the 'naive'
approach measures the empirical rates at which customers with and without treatment occupy resources, and takes the difference -effectively ignoring interference. While this is largely what is done in practice, unfortunately the resulting estimator is biased. Specifically, its expected value overestimates the true treatment effect, loosely because it ignores the fact that an increase in p, while increasing the immediate likelihood of occupation, has the secondary effect of decreasing the availability of resources, and thus preventing new occupations in the future. This is interference. Figure 2: The discrete Markov chain analogous to the continuous-time chain depicted in Fig. 1, for the case N = 1. Arrows indicate transition probabilities, rather than rates. Without loss of generality, the parameters are normalized so that λ + µ = 1.
0 1 pλ (1 − p)λ + µ µ λ
To be concrete, consider the simplest case: N = 1. Fig. 2 depicts the equivalent discrete Markov chain, where we have assumed (without loss of generality) that λ + µ = 1. A new occupation occurs whenever the chain transitions from state 0 to state 1. We are interested in the rate of such transitions, which can be worked out to be pλµ/(pλ + µ). The additive increase in this term when p is replaced with p + δ, is the so-called average treatment effect (ATE) that we are after. For this example, it suffices to know that the ATE is Ω(δ).
The chain we actually observe, e.g. resulting from an A/B test, applies the treatment with 1/2 probability at each time period, affecting the transition probabilities when in state 0. Given a single trajectory {s t } of length T from this chain, the Naive estimator then iŝ
ATE NV = 1 |T 1 | t∈T 1 I {st=0,s t+1 =1} − 1 |T 0 | t∈T 0 I {st=0,s t+1 =1} ,
where T 1 and T 0 are, respectively, the sets of time periods in which the treatment was and was not applied. An explicit calculation then shows
lim TÂ TE NV − ATE ≈ pλ µ ATE,(1)
where the approximation (≈) hides terms of size O(δ 2 ). In words, unless the system is extremely unoccupied (µ pλ), the Naive estimator has bias that is on the order of the treatment effect. It is worth noting that the variance of this Naive estimator is effectively O(1) -i.e. it is as small as we can hope for.
Existing Approach 2 -Off-Policy Evaluation: We may view the problem at hand as one of
Off-Policy Evaluation in reinforcement learning. To do this, we associate an MDP with our chain.
The actions in this MDP correspond to treating or not treating an arrival at a given state in the chain. The reward associated with this action is 1 if the subsequent transition is to the right and 0 otherwise. The ATE then corresponds to the difference in average steady-state rewards between the two policies which always select, respectively, the treatment and non-treatment actions.
The task of estimating the ATE is now trivially viewed as one of OPE. This in turn, immediately suggests a whole host of existing OPE estimators that yield unbiased estimates of the ATE, e.g. [27,34,35]. This natural approach appears to be promising, e.g. [55], but outside of secondary issues (e.g. discounted vs. average reward), the primary issue is that being unbiased appears to come at a price: variance. Specifically, one may show that any unbiased OPE estimator has variance that grows exponentially with the number of states in our chain, as e Ω(N ) .
This sets up two extremes of a bias-variance tradeoff in our simple example: the Naive estimator has O(1) variance, but its bias is on the order of the treatment effect itself. Any unbiased OPE estimator on the other hand will have variance that scales like e Ω(N ) .
Our example ρ min = e −Ω(N ) . In Theorem 3 we prove a lower bound on the variance of any unbiased estimator in the context of general MDPs, which is exponentially larger -scaling as Ω(1/ρ min ).
These two bounds show that the variance reduction relative to OPE is generic (Theorem 4).
In summary, this example illustrates precisely the bias-variance trade-offs embodied by each estimator in Table 1. In particular, the DQ estimator has bias second-order in the estimand, with variance exponentially smaller than any unbiased OPE estimator -capturing a particularly advantageous spot in the bias-variance curve. These results hold in generality for a large class of problems, which we formalize in the next section.
Estimator Bias Variance
Naive In this example ρ min = e −Ω(N ) , but in general ρ min can be up to 1/N .
Ω(δ) O(1) Off-Policy Evaluation 0 e Ω(N ) Differences-In-Q's (DQ) O(δ 2 ) O(N )
Aside -Alternative Experimental Designs: Whereas our focus is on estimation assuming simplerandomization, a more sophisticated two-sided randomization (TSR) design has also been studied for this specific system in [29]. In their scheme, both customers and resources are randomized independently into treatment and control, and the intervention is applied only if both the customer and the resource are treated. We provide empirical comparisons against this approach in Section 5, which show that DQ outperforms TSR in typical supply / demand regimes, despite a simpler design.
Related Literature:
The largest portion of work in interference is in experimental design, with the design levers ranging from stopping times in A/B tests [37,28,69,30], to any form of more-sophisticated 'clustering' of units [12,21,24,15,46,65,67,18,19], to clustering specifically when interference is represented by a network [44,66,52,2,7,48,73], to the proportion of units treated [26,60,4], to the timing of treatment [56,9,22], and beyond [3,36,63,44,11,6,25,52,20]. As alluded to earlier, these sophisticated designs can be powerful, but cost, user experience, and other implementation concerns restrict their application in practice [38,39].
We view this paper as orthogonal to this literature, but will eventually compare against a recent state-of-the-art design, so-called two-sided randomization [29,5], that is specific to the context of two-sided marketplaces (e.g. the one we simulate).
As stated earlier, the problem we study is one of off-policy evaluation (OPE) [49,58]. The fundamental challenge in OPE is high variance, which can be attributed to the nature of the algorithmic tools used, e.g. sampling procedures [62,61,42]. Recent work on 'doubly-robust' estimators [27,34,35] has improved on variance (incidentally, our estimator is loosely tied to these, as we discuss in Section 6), but again we will show, via a formal lower bound, that unbiased estimators as a whole have prohibitively large variance. Finally, our motivation is close in spirit to a recent paper [55], which applies OPE directly in Markovian interference settings; we make a direct experimental comparison in Section 5.
In the policy optimization literature, 'trust-region' methods [53] and conservative policy iteration [32] use a related on-policy estimation approach to bound policy improvement. Relative to the existing literature, we develop an on-policy surrogate with provably lower bias than extant proposals; see Section 6.3. Furthermore, the explicit application of on-policy estimation in the context of OPE, and in particular the striking bias-variance tradeoff this enables, are novel to this paper.
Model
Having discussed each estimator in a specific example, we now formalize the general inference problem that we tackle, casting it in the language of MDPs. Vis-à-vis the existing literature, this lens allows us to reason about the problem using a large, well-established toolkit, and makes obvious the fact that OPE provides unbiased estimation of the ATE. We then present what we call the 'Naive' estimator (alluded to in the introduction). This is the lowest-variance estimator one can hope for in this setting, but it can have significant bias, as we see in Eq. (1).
We begin by defining an MDP with state space S. We denote by s t ∈ S the state of the MDP at time t ∈ N. Every state is associated with a set of available actions A which govern the transition probabilities between states via the (unknown) function p : S × A × S → [0, 1]. We assume that A = {0, 1} irrespective of state; for descriptive purposes, we will associate the '1' action with the use of a prospective intervention, so that '0' is associated with not employing the intervention. We denote by r(s, a) the reward earned in state s having employed action a. A policy π : S → A maps states to random actions. We define the average reward λ π , under any (ergodic, unichain) policy π, according to:
λ π = lim T →∞ 1 T T t=1
r(s t , π(s t )).
There are three policies we define explicitly:
The Incumbent Policy π 0 : This policy never uses the intervention, so that π 0 (s) = 0 for all s. This is 'business as usual'. Denote the associated transition matrix as P 0 (i.e. the entries of P 0 are exactly p(·, 0, ·))
The Intervention Policy π 1 : This policy always uses the intervention, so that π 1 (s) = 1 for all s. This reflects the system, should the intervention under consideration be 'rolled out'. Denote the associated transition matrix as P 1 .
The Experimentation Policy π p : This policy corresponds to the experiment design. Simple randomization would select π(s) = 1 with some fixed probability p, say 1/2, independently at every period. This corresponds to the sort of search engine experiment alluded to in the introduction.
The transition matrix associated with this design is then P 1/2 = 1 2 P 0 + 1 2 P 1 . The Inference Problem: We are given a single sequence of T states, actions, and rewards, observed under the experimentation policy π p (recall that cost and constraints [38,39] prohibit us from running π 0 or π 1 separately until convergence). We observe the sequence {(s t , a t , r(s t , a t )) : t = 1, . . . , T }, wherein a t π p (s t ). Our goal is to estimate the average treatment effect (ATE): ATE λ π 1 − λ π 0 .
The Naive Estimator and Bias. A natural approach to estimating the ATE is to use simple randomization (i.e. P 1/2 ) and the Naive estimator, which we define in the language of MDPs as:
ATE NV = 1 |T 1 | t∈T 1 r(s t , a t ) − 1 |T 0 | t∈T 0 r(s t , a t ), where T 1 = {t : a t = 1} and T 0 = {t : a t = 0}.
In the context of the example of Section 1.1, this corresponds to simply taking the difference between the probability of renting a resource among test users (T 1 ), and control users (T 0 ). What goes wrong is simply that the two empirical averages above, that seek to estimate λ π 1 and λ π 0 respectively, employ the wrong measure over states. As we saw, this is sufficient to introduce bias that is on the order of the treatment effect being estimated.
The Differences-In-Q's Estimator
We are now prepared to introduce our estimator for inference in the presence of Markovian interference. Before defining our estimator, which we will see is only slightly more complicated than the Naive estimator, we recall a few useful objectis in average-reward MDPs. Denote the average cost of a policy π by λ π . The V -function of a policy π, V π , characterizes the "rewardto-go" V π (s) := E ∞ t=0 r(s t , a t ) − λ π s 0 = s . It is also known that (V π , λ π ) is the fixed point of the Bellman operator T π with T π (V π , λ π ) = V π . Here T π : R |S| × R → R |S| is given by T π (V, λ) = r π − λ1 + P π V where r π : S → R is defined according to r π (s) = E [r(s, π(s))].
Finally, the Q-function associated with π, denoted Q π : S × A → R, is defined according to Q π (s, a) := E ∞ t=0 r(s t , a t ) − λ π s 0 = s, a 0 = a . Put simply, the Q-function measures the 'excess' reward obtained starting from s with the action a relative to the average reward under π.
An Idealized First Step
In motivating our estimator, let us begin with the following idealization of the Naive estimator, where we denote by ρ 1/2 the steady state distribution under the randomization policy π 1/2 : E ρ 1/2 Â TE NV = s ρ 1/2 (s) [r(s, 1) − r(s, 0)] . It is not hard to see that in the example of Section 1.1, we continue to have |E ρ 1/2 [ÂTE NV ] − ATE| ≈ pλ µ ATE, i.e. this idealization of the Naive estimator continues to have bias on the order of the treatment effect. Consider then, the following alternative:
E ρ 1/2 Â TE DQ = s ρ 1/2 (s) Q π 1/2 (s, 1) − Q π 1/2 (s, 0) , where the term E ρ 1/2 [ÂTE DQ ]
can for now just be thought of as an idealized constant (ÂTE DQ is defined soon in (2)).
Compared to E ρ 1/2 [ÂTE NV ], we see that E ρ 1/2 [ÂTE DQ
] takes a remarkably similar form, except that as opposed to an average over differences in rewards, we compute an average of differences in Q-function values. The idea is that doing so will hopefully compensate for the shift in distribution induced by π 1/2 , as it does in the example of Section 1.1.
Is the dramatic mitigation of bias we see in the example generic? If the experimentation policy mixes fast, our first set of results essentially answers this question in the affirmative. In particular, we make the following mixing time assumption:
Assumption 1 (Mixing time). There exist constants C and λ such that for all s ∈ S, d TV (P k 1/2 (s, ·), ρ 1/2 ) ≤ Cλ k where d TV (·, ·) denotes total variation distance.
We then have that the second order bias we saw in Section 1.1 is, in fact, generic:
Theorem 1 (Bias of DQ). Assume that for any state s ∈ S, d TV (p(s, 1, ·), p(s, 0, ·)) ≤ δ. Then,
ATE − E ρ 1/2 Â TE DQ ≤ C 1 1 − λ 2 r max · δ 2
where r max := max s,a |r(s, a)| and C is a constant depending (polynomially) on log(C).
The Differences-In-Q's Estimator
Motivated by the development in the previous subsection, the Differences-In-Q's (DQ) estimator we propose to use is simplŷ
ATE DQ = 1 |T 1 | t∈T 1Q π 1/2 (s t , a t ) − 1 |T 0 | t∈T 0Q π 1/2 (s t , a t ),(2)
where we take an empirical average over the state trajectory produced under the randomization policy, andQ π 1/2 is an estimator of the Q-function. For concreteness, we obtainQ π 1/2 by solving min V ,λ s∈S t,st=s
r(s t , a t ) −λ +V (s t+1 ) −V (s t ) 2 .(3)
Our main result characterizes the variance and asymptotic normality ofÂTE DQ :
Theorem 2 (Variance and Asymptotic Normality of DQ). The DQ estimator is asymptotically normal
so that √ T Â TE DQ − E ρ 1/2 Â TE DQ d → N (0, σ 2 DQ ), with limiting standard deviation σ DQ ≤ C 1 1 − λ 5/2 log 1 ρ min r max .
where ρ min := min s∈S ρ 1/2 (s) and C is a constant depending (polynomially) on log(C).
The fact that σ DQ in Theorem 2 only depends on 1/ρ min logarithmically is somewhat surprising.
In fact, a coarse analysis will lead to σ D = Ω 1 ρ min , which shows no advantage compared to the unbiased OPE estimators (which we will see momentarily). The key enabler for this striking result is a novel lemma that exploits an entry-wise bound for controlling the variance, even at states that are rarely visited (we dub this the "Entry-wise Non-expansive Lemma"; see Lemma 3). The lemma admits a simple form and may have broader implications for analyzing variance in OPE estimators (see Discussions in Section 6). In addition, our asymptotic normality analysis borrows the delta-method framework used in the context of on-policy LSTD [41], but with a novel linearization that dramatically simplifies the analysis. See Section 3.4 for more details.
One Extreme of the Bias-Variance Tradeoff: We may heuristically think of the Naive estimator as representing one extreme of the bias-variance tradeoff among reasonable estimators. For the sake of comparison, by the Markov Chain CLT, the Naive estimator is also asymptotically normal with standard deviation Θ(r max /(1 − λ) 1/2 ). This rate is efficient for the estimation of the mean of a Markov chain [23]. On the other hand, while the Naive estimator is effectively useless for the problem at hand given its bias is in general Θ(δ), that of the DQ estimator is O(δ 2 ).
Proof of Theorem 1
The proof of Theorem 1 is a simple proof built on a perturbation formula for stationary distributions of Markov chains. We in fact construct a novel Taylor series representation of the ATE parameterized by δ that controls the perturbation around P 1/2 , which yields the Naive estimator as the zeroth-order truncation of the series; and the idealized DQ estimator as the natural first-order correction. Theorem 1 then proceeds by bounding the remainder. This strategy additionally allows us to generalize the DQ estimator to arbitrarily high-order bias corrections, by computing Q-functions iteratively. Here we present the proof (with some details omitted for simplicity).
We first define few pieces of useful notation. Let ρ 0 ∈ R |S| , ρ 1/2 ∈ R |S| , ρ 1 ∈ R |S| be the vectors of the stationary distributions of P 0 , P 1/2 , P 1 accordingly. Let r 0 ∈ R |S| , r 1/2 ∈ R |S| , r 1 ∈ R |S| be the reward vectors associated with policies π 0 , π 1/2 , π 1 , i.e., r a (s) = r(s, a) and r 1/2 = 1 2 r 0 + 1 2 r 1 . To begin, we parameterize P 0 := P 1/2 − δA and P 1 := P 1/2 + δA by δ with fixed P 1/2 and some
fixed matrix A ∈ R |S|×|S| with A 1,∞ ≤ 1 ( A 1,∞ = max i j |A ij |) 2 .
Then, ρ 0 and ρ 1 can also be viewed as a function of δ. Also recall ATE = ρ 1 r 1 − ρ 0 r 0 . Our goal is to represent ATE as a function of δ and then study the Taylor expansion of such a function. To do so, we use the following known perturbation formula of Markov chains.
Lemma 1 (Stationary Distribution Perturbation, Theorem 4.1 [45]). Suppose P ∈ R n×n and P ∈ R n×n are transitions matrices of two finite-state aperiodic and irreducible Markov Chains and ρ ∈ R n , ρ ∈ R n are the stationary distributions accordingly.
Then ρ = ρ + ρ (P − P )(I − P ) # where (I − P ) # is the group inverse of I − P given by (I − P ) # = (I − P + 1ρ ) −1 − 1ρ .
Let us apply Lemma 1 to ρ 1 r 1 based on the perturbation between ρ 1/2 and ρ 1 .
ρ 1 r 1 = ρ 1/2 r 1 + ρ 1 (P 1 − P 1/2 )(I − P 1/2 ) # r 1 = ρ 1/2 r 1 + δ · ρ 1 A(I − P 1/2 ) # r 1(4)
Note that we can apply Lemma 1 again to the ρ 1 in the RHS of Eq. (4) and then repeat this process,
ρ 1 r 1 = K k=0 δ k · ρ 1/2 A(I − P 1/2 ) # k r 1 + δ K+1 · ρ 1 A(I − P 1/2 ) # K+1 r 1(5)
for any K = 0, 1, 2, . . . . Essentially Eq. (5) provides the K-th order Taylor expansion for ρ 1 r 1 with an explicit remainder. Furthermore, we can bound the remainder by
ρ 1 A(I − P 1/2 ) # K+1 r 1 (i) ≤ ρ 1 1 A 1,∞ I − P # 1/2 1,∞ K+1 r 1 max (ii) ≤ I − P # 1/2 K+1 1,∞ r max (iii) ≤ 2 ln(C) + 1 1 − λ K+1 r max
Here in (i) we use that for any vector a, b and matrix B, we have |a b| ≤ a 1 b max and a B 1 ≤ a 1 B 1,∞ . In (ii) we use that ρ 1 1 = 1, A 1,∞ ≤ 1. In (iii), we use the following lemma implied by the mixing time assumption and the series expansion of (I − P ) # .
Lemma 2. Suppose for any s ∈ S, d TV (P k 1/2 (s, ·), ρ 1/2 ) ≤ Cλ k . Then (I − P 1/2 ) # 1,∞ ≤ 2 ln(C)+1 1−λ .
Appplying a similar process to ρ 0 r 0 , we obtain the Taylor expansion for the ATE.
ATE = K k=0 δ k · ρ 1/2 A(I − P 1/2 ) # k r 1 − ρ 1/2 (−A)(I − P 1/2 ) # k r 0 + δ K+1 · a K(6)
where |a K | ≤ 2 2 ln(C)+1
1−λ K+1 r max .
It is easy to see that the Naive estimator ρ 1/2 (r 1 − r 0 ) corresponds to the zeroth-order truncation. In fact, the DQ estimator, i.e., E ρ 1/2 Â TE DQ , exactly matches the first-order truncation. To see this, by the definition of E ρ 1/2 Â TE DQ and Q-functions,
E ρ 1/2 Â TE DQ = s ρ 1/2 (s) Q π 1/2 (s, 1) − Q π 1/2 (s, 0) = s ρ 1/2 (s) r 1 (s) + s V 1/2 (s )P 1 (s, s ) − r 0 (s) − s V 1/2 (s )P 0 (s, s ) = ρ 1/2 r 1 − r 0 + (P 1 − P 0 )V 1/2
where V 1/2 is the induced vector of the V -function of policy π 1/2 . By the well-known fact that V 1/2 = (I − P 1/2 ) # r 1/2 induced by the Bellman equation, we then have
E ρ 1/2 Â TE DQ = ρ 1/2 r 1 − r 0 + (P 1 − P 0 )(I − P 1/2 ) # r 1/2 = ρ 1/2 r 1 − ρ 1/2 r 0 + δρ 1/2 A(I − P 1/2 ) # (r 1 + r 0 ).
Then indeed E ρ 1/2 Â TE DQ is the first-order Taylor truncation. Together, this completes the proof.
Generalization to Higher-Order Bias Correction. In fact, the K-th order Taylor expansion of ATE allows us to design estimators that can correct higher-order bias, based on computing difference-in-Q functions iteratively. See details in Section 6.1.
Proof Sketch of Theorem 2
We aim to use the Markov chain CLT ( [31]) to show asymptotic normality of our estimator.
The Markov chain CLT states that for a Markov chain X 1 , X 2 , . . . , and a bounded function u with domain on the state space, there exists Σ u such that Delta method. Unfortunately, the estimatorÂTE DQ can not be directly written as an empirical average of some function u. To address this issue, we use the the delta method (traced back to [16], see Lemma 5). In particular, we writeÂTE DQ = f (u T ) as a function of a random vector u T given by u T := 1 T T t=1 u(X t ). Under some minor conditions, the delta method states that
√ T 1 T T t=1 u(X t ) − u * d → N (0, Σ u ) where u * is√ T (f (u T ) − f (u * )) d → N (0, σ 2 f ) where σ 2 f := ∇f (u * ) Σ u ∇f (u * ) and ∇f (u * )
is the gradient of f evaluating at the point u * . This forms the basis for proving Theorem 2.
Linearization. To simplify the analysis for σ f , instead of computing Σ u explicitly, we "linearize" the function f by definingf (X t ) := ∇f (u * ) (u(X t ) − u * ) and the delta method in fact implies (see
Lemma 6) √ T 1 T T t=1f (X t ) d → N (0, σ 2 f ), i.e.
, the linearized f converges with the same limiting variance as the original f. Therefore, we can focus onf for analyzing σ f .
Bounding σ f with Entry-wise Non-expansive Lemma.
To bound σ f , we will invoke Lemma 4, which states that σ f ≤ √ 2 2 ln(C)+1 1−λf max wheref max := max s |f (s)|. Then the problem reduces to boundingf max , which will be controlled by the following key lemma.
Lemma 3 (Entry-wise non-expansive lemma). Let W : R |S| → R |S| be a map denoted by W (ρ) :=
(I −P 1/2 ) # (P 1 −P 0 ) ρ. Let c := 4 ln(C)+ln(1/ρ min )+1 1−λ . Then, for any s ∈ S, 1 c W (ρ 1/2 )(s) ≤ ρ 1/2 (s).
The Price of Being Unbiased
Thus far, we have seen that the DQ estimator provides a dramatic mitigation in bias (Theorem 1) at a relatively modest price in variance (Theorem 2). This suggests another question: could we hope to construct an unbiased estimator that has low variance (i.e. comparable to either the Naive or DQ estimators). We will see that the short answer is: no.
The Variance of an Optimal Unbiased Estimator
As noted earlier, a plethora of Off-policy evaluation (OPE) algorithms might be used to provide an unbiased estimate of the ATE. Rather than consider a particular OPE algorithm, here we produce a Cramér-rao lower bound on the variance of any unbiased OPE algorithm. While such a bound is obviously of independent interest (since OPE is a far more general problem than what we seek to accomplish in this paper), we will primarily be interested in comparing this lower bound to the variance of the DQ estimator from Theorem 2.
Theorem 3 (Variance Lower Bound for Unbiased Estimators). Assume we are given a dataset {(s t , a t , r(s t , a t )) : t = 0, . . . , T } generated under the experimentation policy π 1/2 , with s 0 distributed according to ρ 1/2 . Then for any unbiased estimatorτ of ATE, we have that
T · Var(τ ) ≥ 2 s ρ 1 (s) 2 ρ 1/2 (s) s p(s, 1, s )(V π 1 (s ) − V π 1 (s) + r(s, 1) − λ π 1 ) 2 + 2 s ρ 0 (s) 2 ρ 1/2 (s) s p(s, 0, s )(V π 0 (s ) − V π 0 (s) + r(s, 0) − λ π 0 ) 2 σ 2 off .
It is worth remarking that this lower bound is tight: in the appendix we show that an LSTD(0)type OPE algorithm achieves this lower bound. While this is of independent interest vis-à-vis average cost OPE, we turn next to our ostensible goal here -evaluating the 'price' of unbiasedness.
We can do so simply by comparing the variance of the DQ estimator with the lower bound above.
In fact, we are able to exhibit a class of one-dimensional Markov chains (in essence the model in Section 1.1) for which we have:
Theorem 4 (Price of Unbiasedness). For any 0 < δ ≤ 1 5 , there exists a class of MDPs parameterized by n ∈ N, where n is the number of states, such that
σ DQ σ off = O n c n , for some constant c > 1. Furthermore, |(ATE − E[ÂTE DQ ])/ATE| ≤ δ.
Another Extreme of the Bias-Variance Tradeoff: Theorems 2, 3, and 4 together reveal the opposite extreme of the bias-variance tradeoff. Specifically, if we insisted on an unbiased estimator for our problem (of which there are many, thanks to our framing of the problem as one of OPE), we would pay a large price in terms of variance. In particular Theorem 4 illustrates that this price can grow exponentially in the size of the state space. This jibes with our empirical evaluation in both caricatured and large-scale MDPs in Section 5.
Taken together our results reveal that the DQ estimator accomplishes a striking bias-variance tradeoff: it has substantially smaller variance than any unbiased estimator (in fact, comparable to the Naive estimator), all while ensuring bias that is second order in the impact of the intervention.
Experiments
This section will empirically investigate the DQ estimator and a number of alternatives in two settings: the simple example of Section 1.1, originally proposed by [29]; and more realistically, a city-scale simulator of a ride-hailing platform similar to what large ride-hailing operators use in production. The alternatives we consider include: 1) the Naive estimator; 2) TSRI-1 and TSRI-2, the "two-sided randomization" (TSR) designs/estimators from [29]; and 3) a variety of OPE estimators.
For the OPE estimators, we note that off-policy average reward estimation has only recently been addressed in [68,72], and we implement their specific estimators which we simply denote as TD and GTD respectively. We also implement an extension to an LSTD type estimator proposed in [55].
A Simple Example
We first study all of our estimators in the example of Section 1.1, a simple setting that does not call for any sort of value function approximation. Our goal now is to understand the relative merits of practical implementations of these estimators, in terms of their bias and variance.
To recap, this MDP is a stylized model of a rental marketplace, consisting of a 1-D Markov chain on N = 5000 states parameterized by a 'customer arrival' rate λ and a 'rental duration' rate µ.
At a given state n (so that n units of inventory are in the system), the probability that an arriving customer rents a unit is impacted by the intervention. As such if the intervention increases the probability of a customer renting, this reduces the inventory availability for customers that arrive later. Our MDP and experimental setup exactly replicates that of [29], with N = 5000, λ = 1, µ = 1.
We run all estimators over 100 separate trajectories of length t = 10 4 N of the above MDP initialized in its stationary distribution. Figure 3 summarizes the results of this experiment. Beginning with the left panel, which reports estimated quantities at t = 10 4 N , we immediately see:
TSR improves on Naive: The actual ATE in the experiment is 1.5%. Whereas it has the lowest variance of the estimators here, the Naive estimator has among the highest bias. The two TSR estimators reduce this bias substantially at a modest increase in variance. It is worth noting, as a sanity check, that these results precisely recreate those reported in [29].
OPE estimators are high variance: The OPE estimators have the highest variance of those considered here. The TD estimator has the lower variance but this is simply because it is implicitly regularized. Run long enough, both estimators will recover the treatment effect.
DQ shows a compelling bias-variance tradeoff: In contrast, the DQ estimator has the lowest bias at t = 10 4 N and its variance is comparable to the TSR estimators (It is worth noting that run long enough, the DQ estimator had a bias of ∼ −5 × 10 −7 ).
Conclusions hold across experimental budgets: Turning our attention briefly to the right chart in Figure 3, We note that specialized designs such as TSR can still be valuable in specific settings: when λ µ, for example, TSR is nearly unbiased (see [29]), and can outperform DQ; see the appendix for such a study.
A Large-Scale Ridesharing Simulator
We next turn our attention to a city-scale ridesharing simulator similar to those used in production at large ride-hailing services. We will consider the problem of experimenting with changes to dispatching rules. Experimenting with these changes naturally creates Markovian interference by impacting the downstream supply/ positioning of drivers. Relative to the earlier toy example, the corresponding MDP here has an intractably large state-space, necessitating value function approximation for the DQ and OPE estimators.
The Simulator: Ridesharing admits a natural MDP; see e.g. [50]. The Experiment: We experiment with dispatch policies. Specifically, we consider assigning a request to an idle driver or a 'pool' driver, i.e. a driver who already has riders in their car. A dispatch algorithm might prefer the former, but only if the cost of the resulting trip is at most α% higher than the cost of assigning to a pool driver. We consider three experiments, each of which changes α from a baseline of 0 to one of three distinct values: 30%, 50% or 70%, with ATEs of 0.5%, -0.9%, and -4.6% respectively. As we noted earlier, we would expect significant interference in this experiment (or indeed any experiment that experiments with pricing or dispatch) since an intervention changes the availability / position of drivers for subsequent requests. shared a common linear approximation architecture with basis functions that count the number of drivers at every occupancy level. We note that this approximation introduces its own bias which is not addressed by our theory. We immediately see:
Strong Impact of Interference: As we might expect, interference has a significant impact here as witnessed by the large bias in the Naive estimator.
Incumbent estimators do not improve on Naive: None of the incumbent estimators improve on Naive in this hard problem. This is also the case for the TSR designs, which in this large scale setting surprisingly appear to have significant variance. The OPE estimators have lower variance due to the regularization caused by value function approximation.
DQ works: In all three experiments, the bias in DQ (although in a relative sense higher than in the toy model) is substantially smaller than the alternatives, and also smaller than the ATE. This is evident in the left panel in Figure 4. Notice that in the rightmost experiment (ATE = 0.5), DQ is the only estimator to learn that the ATE is positive. Like in the toy model, the right panel shows that these results are robust over experimentation budgets.
Discussion: Bias-Variance Tradeoffs, Policy Optimization
To summarize, we have shown that the DQ estimator achieves a surprising bias-variance tradeoff by applying on-policy estimation to the Markovian interference problem, and more generally to OPE.
Here we draw further connections between the Naive, DQ, and OPE estimators and provide methods to realize other points on the bias-variance curve. Furthermore, we draw surprising connections between our estimator and trust-region methods in policy optimization, and show that DQ can serve as a drop-in replacement for these policy optimization surrogates with provably lower bias.
A k th -order Bias Correction
As alluded to in Section 3, we can view the DQ estimator as a first-order correction to the Naive estimator, based on a Taylor series expansion of the ATE. This immediately motivates a k th -order correction, with the goal of obtaining estimators with bias O(δ k+1 ) for arbitrary k.
This correction turns out to have a suprising and intuitive form 3 . In short, to obtain the k th order correction term for k odd (the correction for k even is 0), we simply compute the DQ estimator, but replace rewards in the MDP with the (k − 1) th order Difference-in-Q functions -effectively a Difference-in-Qs-of-Difference-in-Qs.
Precisely, for some reward function f : S → R, we can define an auxiliary MDP with the same transition probabilities, but rewards f (s) at each state. Let Q(s, a; f ) be the corresponding Q function, under policy π 1/2 . We now define the k th order Q-function to be Q (k) (s, a) = Q(s, a; f (k−1) ),
where the rewards are defined as f (k−1) (s) = 1 2 Q (k−1) (s, 1) − Q (k−1) (s, 0) ; in other words, the previous-order Difference-in-Q functions. We take as a base case f (0) (s) = r(s). Finally, we can define the K th -order Difference-in-Qs estimate of the ATE to be the sum of all lower-order correction terms:ÂTE
(K) DQ = k odd,k≤K E ρ 1/2 Q (k) (s, 1) − Q (k) (s, 0) .
In principle this approach enables off-policy evaluation with arbitarily low bias -entirely via estimation of on-policy quantities. One can verify that ATE
(1) DQ is the expected value of the DQ estimator. We now generalize Theorem 1 to provide a bias bound for the k th order correction 4 :
Theorem 5. For any K = 0, 1, 2, . . . , we have ATE − ATE
(K) DQ ≤ C 1 1−λ K+1 δ K+1 r max , where
C is a constant depending (polynomially) on log(C).
Interpolating from OPE to Naive via Regularization
Here, we view the DQ estimator again as an intermediate point on the bias-variance curve between Naive and OPE. This time, however, we interpolate between these extremes by regularizing certain key nuisance parameters in estimating the ATE.
An OPE meta-estimator. First, we situate the DQ estimator in the context of existing OPE techniques. Consider the following exact identity for the ATE: ATE = E ρ 1/2 [ζ(s)(Q π 1/2 (s, 1) − Q π 1/2 (s, 0))] where ζ(s) = 1 2 ρ 1 (s)+ρ 0 (s) ρ 1/2 (s)
is the likelihood ratio of the stationary distributions. A variety of OPE estimators -including doubly-robust ( [34,64]) and primal-dual ( [14,59]) estimators -in fact estimate ATE explicitly by plugging in estimatesζ,Q π 1/2 of the likelihood ratio and value functions (referred to as the "doubly-robust meta-estimator" in [34]):
ATE DR = 1 |T 1 | t∈T 1ζ (s t )Q π 1/2 (s t , 1) − 1 |T 0 | t∈T 0ζ (s t )Q π 1/2 (s t , 0)(7)
Explicit regularization. In estimatingζ(s), one can directly penalize its deviation from one, where increasing the penalty interpolates from OPE to DQ. Given that estimation ofζ(s) is the key difference between DQ and unbiased OPE -and therefore the source of the massive variance gap (Theorems 2 and 3) -we would expect this to be a particularly powerful approach to OPE, and indeed similar penalties have produced strong empirical performance [47]. Similarly, one can directly penalize the deviation ofV π 1/2 from zero, as in regularized variants of LSTD (see e.g. [40]). As we increase the regularization penalty onζ(s), we interpolate from OPE to DQ; additionally increasing the regularization penalty onV π 1/2 then interpolates from DQ to Naive. Approaches combining both forms of regularization have been explored in [70].
Function approximation. More generally, one can restrictζ(s) andV π 1/2 to lie in particular function classes, with one extreme being any mapping S → R, and the other extreme being the constant functionsV π 1/2 (s) = c orζ(s) = 1. As one example, when the state space is massive we may approximate it using state aggregation. At the extreme, aggregating all states into a single aggregate state implies that the value function (or likelihood ratio) must be a constant. As the aggregation forζ(s) goes from fine to coarse, we interpolate between OPE and DQ; increasing the coarseness of V π 1/2 (s) then interpolates between DQ and Naive.
DQ as a Policy Optimization Objective
Surrogate objectives in trust-region methods DQ also has a suprising relationship to trustregion methods [53,54,33]. At each iteration, these methods essentially solve an offline policy optimization problem: using data collected under some "behavioral"policy π b , they evaluate (and subsequently optimize) a candidate policy π. The policy evaluation step is exactly an OPE problem, and they construct a surrogate objective based on an identity sometimes referred to as Dynkin's identity [17]:
λ(π) = E π b [r(s, a)] + E s∼ρπ b ,a∼π ρπ(s) ρπ b (s) (Q π b (s, a) − V π b (s)) , where ρ π b , V π b , Q π b
is the stationary distribution, V -function, and Q-function of the policy π b respectively and ρ π is the stationary distribution of π. The referenced trust-region methods then effectively take the likelihood ratio ρ π (s)/ρ π b (s) to be identically one, yielding the (idealized) surrogate objective:
λ TR (π) = E π b [r(s, a)] + E s∼ρπ b ,a∼π [Q π b (s, a) − V π b (s)].
A DQ-based surrogate As it turns out, DQ derives from a very similar identity -with a small but critical difference which allows DQ to obtain even lower bias. Applying the peturbation bound in Lemma 1 to λ(π), we obtain a slightly different identity for λ(π): λ(π) = E π b [r π (s)] + E s∼ρπ b ,a∼π ρπ(s) ρπ b (s) (Q π b (s, a; r π ) − V π b (s; r π )) where r π (s) = E a∼π [r(s, a)] is the expected reward under π, and recall that Q π b (·, ·; r π ), V π b (·, ·; r π ) are the value functions for an auxiliary MDP with the same transition probabilities, but taking r π to be the reward function. The biased form of this estimator, which forms the basis of the DQ estimator 5 , is:λ DQ (π) = E π b [r π (s)] + E s∼ρπ b ,a∼π [Q π b (s, a; r π ) − V π b (s; r π )] which is preciselyλ TR (π), but computed on an MDP with rewards r π .
Lower-order bias This striking resemblance between the surrogateesλ DQ (π) andλ TR (π) naturally raises the question of how they compare. As it turns out, the simple act of replacing the rewards with r π inλ DQ has significant consequences in terms of bias:
Theorem 6 (Bias of the DQ surrogate). Suppose it holds that d TV (p(s, a, ·), p(s, a , ·)) ≤ δ for all s, a, a , and that d TV (π(s, ·), π (s, ·)) ≤ δ for all s. Then, the biases ofλ TR (π) 6 andλ DQ (π) satisfy
|λ TR (π) − λ π | = O(δ(δ ) 2 ) |λ DQ (π) − λ π | = O((δδ ) 2 )
This characterization is sharp, in that there exist non-pathological examples where the exact bias ofλ DQ is a factor δ smaller than that ofλ TR . Crucially, this means that even if the distance between policies has no non-vacuous upper bound (i.e. δ = 2), as long as the resulting transition functions are similar, then the bias ofλ DQ will be small, whereas the bias ofλ TR can be of the order of δ. This immediately suggests that optimizingλ DQ with respect to π should allow for both larger and more accurate policy improvement steps.
Conclusion
We propose a novel estimator, the DQ estimator, to solve the interference problem in experiments with simple randomized designs. The DQ estimator achieves second-order bias in estimating the average treatment effect, while its variance can be exponentially smaller than that of any unbiased estimator. We conducted a large scale ride-hailing experiment that demonstrated the superior performance of the DQ estimator over state-of-the-art approaches. The striking and rigorous bias-variance trade-offs induced by the DQ estimator and its generalizations provide a new lens for general off-policy evaluation and policy optimization in reinforcement learning.
[70] M. Yang
Appendix A Notation
For a vector a ∈ R n , we use a 1 = n i=1 |a i | and a ∞ = max n i=1 |a i |. For a matrix M ∈ R n×m , we use M 1,∞ = max 1≤i≤n m j=1 |a ij | to represent the maximal row-wise l 1 -norms. We use 1
to represent the vectors with all ones. We use A # to represent the group inverse of A. For an irreducible and aperiodic Markov chain with associated transition matrix P and the stationary distribution ρ, there is (I − P ) # = (I − P + 1ρ ) −1 − 1ρ .
B Analysis of the Example
To begin, let us derive the ATE. Under policy π 0 , the transition matrix is
P 0 = (1 − p)λ + µ pλ µ λ
and the stationary distribution is ρ 0 = [ µ µ+λp , λp µ+λp ] accordingly. Similarly, one can verify under policy π 1 , the transition matrix is
P 1 = (1 − p − δ)λ + µ (p + δ)λ µ λATE = r 1 ρ 1 − r 0 ρ 0 = µλ(p + δ) µ + λ(p + δ) − µλp µ + λp = δµ 2 λ (µ + λ(p + δ))(µ + λp) .
Consider the transition matrix for π 1/2 ,
P = (1 − p − δ/2)λ + µ (p + δ/2)λ µ λ .
Then one can verify that the stationary distribution ρ 1/2 is
ρ 1/2 = µ µ + λ(p + δ/2) , λ(p + δ/2) µ + λ(p + δ/2) .
The naive estimator is
E[ÂTE NV ] = δλµ µ + λ(p + δ/2) .
Next, we consider the computation of E ρ 1/2 [ÂTE DQ ], which can be written as
E ρ 1/2 [ÂTE DQ ] = ρ 1/2 (Q 1 − Q 0 )
where Q a is the Q-value vector for the policy π 1/2 under the action a. Furthermore, consider the following Bellman equation for Q-value function:
Q(s, a) = r(s, a) − λ 1/2 + s ,a P a (s, s ) 1 2 Q(s , a ).
One can verify that one solution of the above equations is
Q(0, 0) = µλp µ + λp , Q(0, 1) = 0 Q(1, 0) = µλ(p + δ) µ + λp , Q(1, 1) = 0
Therefore,
E ρ 1/2 [ÂTE DQ ] = µ µ + λ(p + δ/2) (Q(0, 1) − Q(0, 0)) = µ µ + λ(p + δ/2) µλδ µ + λp .
For the bias induced by the DQ estimator, we have
ATE − E ρ 1/2 [ÂTE DQ ] = δµ 2 λ (µ + λ(p + δ))(µ + λp) − µ µ + λ(p + δ/2) µλδ µ + λp = δ µ 2 λ µ + λp 1 µ + λ(p + δ) − 1 µ + λ(p + δ/2) ≈ δ 2 λ (µ + λp) ATE
where ≈ ignore the terms O(δ 3 ). This completes the analysis.
C Proof of Theorem 2 C.1 Entry-wise Non-Expansive Lemma
To begin, we present a lemma that is the key enabler of establishing the striking variance improvement of the DQ estimator over the un-biased estimator. The lemma simply states This is to say, the mapping 1 c W does not expand ρ 1/2 in terms of entry-wise values. To see the necessity of this lemma and gain some intuition, consider a special case where P 0 , P 1 , P 1/2 , ρ 0 , ρ 1 , ρ 1/2 are all known, while only the rewards are unknown and can only be sampled under the distribution ρ 1/2 . For simplicity, assume r 0 = r 1 = r and the sample for r t = r(s t ) + t is i.i.d from ρ 1/2 with some exogenous noise t ∼ N (0, 1). Let us denote the empirical average estimator for r ber. By
CLT, we have √ T (r − r) d → N (0, D −1 )
where D is a diagonal matrix with entries D s,s = ρ 1/2 (s). This limiting variance captures the intuition that, for the state that is rarely visited, the variance forr(s) − r(s) can blow up. In fact, consider an un-biased estimator (ρ 1 − ρ 0 )r for the ATE, (ρ 1 − ρ 0 )r, (this is the un-biased estimator that achieves the optimal variance), we have
√ T (ρ 1 − ρ 0 )r − ATE d → N (0, σ 2 0 ) where σ 2 0 := (ρ 1 − ρ 0 )D −1 (ρ 1 − ρ 0 ) = s (ρ 1 (s) − ρ 0 (s)) 2 ρ 1/2 (s) .
Note that there is no guarantee for the likelihood ratio ρ 0 (s) ρ(s) and ρ 1 (s) ρ(s) and in general σ 2 0 = Ω 1 ρ min and this is the price to pay for the un-biased off-policy evaluation.
On the other hand, one can verify that the DQ estimator is simply
ATE D = ρ 1/2 (P 0 − P 1 ) (I − P 1/2 ) #r .
This leads to the limiting variance of ATE DQ :
√ T ATE D − E ρ 1/2 [ATE D ] d → N (0, σ 2 1 ) where σ 2 1 := ρ 1/2 (P 0 − P 1 ) (I − P 1/2 ) # D −1 (ρ 1/2 (P 0 − P 1 ) (I − P 1/2 ) # ) = ρ 1/2 (P 0 − P 1 ) (I − P 1/2 ) # D −1/2 2 .
By the definition of W mapping, we then have
σ 2 1 = s W (ρ 1/2 )(s) 1 ρ 1/2 (s) 1/2 2 (i) ≤ s 1 c 2 ρ 1/2 (s) = 1 c 2 .
where (i) is due to Lemma 3. Then σ 1 is in the order of log(1/ρ min ). In fact, without Lemma 3, a loose analysis will provide σ 2 1 = Ω(1/ρ min ), which is in the same order of σ 2 0 , that shows no advantage of using DQ estimator. Essentially Lemma 3 characterizes the explicit superiority of evaluating on-policy quantities over off-policy quantities. We believe this novel lemma is of independent interest for the field of OPE. The proof is postponed to the end of the section.
C.2 Outline of the Proof
In this section, we present the outline of the proof for Theorem 2. We aim to use Markov chain CLT ( [31]) to provide the asymptotic normality of our estimator. Note that Markov chain CLT states that for a Markov chain X 1 , X 2 , . . . , and a bounded function u with the domain on the state space, there exists Σ u such that
√ T 1 T T t=1 u(X t ) − u * d → N (0, Σ u )
where u * is the expected value of u under the stationary distribution of the Markov chain.
Delta method. Unfortunately, the estimatorÂTE DQ can not be directly written as an empirical average of some function u. To address this issue, we use "delta method" (traced back to [16], see Lemma 5). In particular, we writeÂTE DQ = f (u T ) as a function of a random vector u T given by
u T := 1 T T t=1 u(X t )
. Under some minor conditions, "delta method" states that
√ T (f (u T ) − f (u * )) d → N (0, σ 2 f )
where σ 2 f := ∇f (u * ) Σ u ∇f (u * ) and ∇f (u * ) is the gradient of f evaluating at the point u * . This forms the basis of proving Theorem 2.
Linearization. To simplify the analysis for σ f , instead of computing Σ u explicitly, we "linearize" the function f by definingf (X t ) := ∇f (u * ) (u(X t ) − u * ) and the delta method in fact implies (see
Lemma 6) √ T 1 T T t=1f (X t ) d → N (0, σ 2 f ),
i.e., the linearized f converges with the same limiting variance as the original f. Therefore, we can focus onf for analyzing σ f .
Bounding σ f with Entry-wise Non-expansive Lemma.
To bound σ f , we will invoke Lemma 4, which states that
σ f ≤ √ 2 2 ln(C) + 1 1 − λf max
wheref max := max s |f (s)|. Then the problem boils down to boundf max , which will be controlled by Lemma 3.
Next, we present the proof in full details.
C.3 Delta Method and Linearization
To begin, consider the Markov chain X t = (s t , a t , s t+1 ). For a ∈ {0, 1}, denote F (a) , h (a) by
F (a) (X t ) := 2E st E s t+1 · 1(a t = a) (8) h (a) (X t ) := 2r(s t , a t ) · E st · 1(a t = a)(9)
where E s is a vector with all entries zero except that the s-th entry is one. Let F
T := 1 T T t=1 F (a) (X t ) h (a) T = 1 T T t=1 h (a) (X t ).
We aim to writeÂTE DQ := f (F (0)
T , F(1)T , h (0) T , h(1)
T ) as a function of F (0)
T , F(1)T , h (0) T , h(1)
T for applying delta method. To do so, let D One can verify thatV
= (D (0) T + D (1) T − F (0) T − F (1) F ) # (h (0) T + h (1) T )
gives the estimation of V -function in Eq. (3). Further, one can verify that with a plugging-in estimator for Q, the DQ estimator is given bŷ
ATE DQ = f (F (0) T , F (1) T , h (0) T , h (1) T ) =: 1 (F (1) T − F (0) T )(D (0) T + D (1) T − F (0) T − F (1) F ) # (h (0) T + h (1) T ) + 1 (h (1) T − h (0) T ).
By Markov chain CLT, we have when T goes to infinity
F (0) T → F * 0 := DP 0 , F(1)T → F * 1 := DP 1 h (0) T → h * 0 := Dr 0 , h (1) T → h * 1 := Dr 1
where D is a diagonal matrix with entries D s,s = ρ 1/2 (s). Then by the delta method (see Lemma 5),
we have 7 √ T (f (F (0) T , F (1) T , h (0) T , h (1) T ) − f (F * 0 , F * 1 , h * 0 , h * 1 )) d → N (0, σ 2 f ) which is equivalent to √ T (ÂTE DQ − E ρ 1/2 [ÂTE DQ ]) d → N (0, σ 2 f ) since f (F * 0 , F * 1 , h * 0 , h * 1 ) = E ρ 1/2 [ÂTE DQ ].
To analyze σ f , we consider the "linearization" of f around
u * := (F * 0 , F * 1 , h * 0 , h * 1 ). In particular, let u(X t ) = (F (0) (X t ), F (1) (X t ), h (0) (X t ), h (1) (X t )). Let (λ, V )
be the average reward and the "true" V -function under the policy π 1/2 . One can verify that
√ T 1 T T t=1f (X t ) d → N (0, σ 2 f ).
Here σ 2 f is explicitly given by (by Markov Chain CLT)
σ 2 f :=(I − P 1/2 ) # s ,s 1 t(s 1 )
where t(s) = a,s f (s, a, s )P a (s, s )ρ 1/2 (s) 1 2 .
C.4 Bound σ f
Next, we aim to provide a bound for σ f . Note that that the mixing time of X t is the same as s t and by Lemma 4, we have Let z s := (1 D(P 1 − P 0 )(I − P 1/2 ) # D −1 )E s . By the definition off , we havẽ
σ f ≤ √ 2f max 2 ln(C) + 1 1 − λ wheref max = max s,f max ≤ 2(z max + 2)(V max + r max ) where z max := max s |z s |, V max := max s |V (s)|. For V max , we have V ∞ = (I − P 1/2 ) # r ∞ ≤ (I − P 1/2 ) 1,∞ r max ≤ 2 ln(C) + 1 1 − λ r max .
For z max , note that
z s = 1 D(P 1 − P 0 )(I − P 1/2 ) # D −1 )E s = ρ 1/2 (P 1 − P 0 )(I − P 1/2 ) # D −1 E s = ρ 1/2 (P 1 − P 0 )(I − P 1/2 ) # E s 1 ρ 1/2 (s) .
Then, we can invoke Lemma 3 to obtain that
z s = W (ρ 1/2 )(s) 1 ρ 1/2 (s) ≤ 4 ln(C) + ln (1/ρ min ) + 1 1 − λ .
Combining all together, we have
σ f ≤ C log 1 ρ min 1 1 − λ 5/2 r max
for some constant C that depends (polynomially) on log(C), which completes the proof of Theorem 2.
C.5 Proof of Lemma 3
The only thing remaining is the proof of Lemma 3.
Note that we have W (ρ 1/2 ) = (I − P 1/2 ) # (P 1 − P 0 ) ρ 1/2 . Let v := (P 1 − P 0 ) ρ 1/2 . We will show first (i) 1 2 v is entry-wise non-expansive; and then (ii) (I − P 1/2 ) v is entry-wise bounded. To begin, we claim that |(P 1 − P 0 )(s, s )| ≤ 2P 1/2 (s, s ) for any s and s . This is due to 2P 1/2 = P 0 + P 1 and for any a ≥ 0, b ≥ 0, we have |a − b| ≤ a + b.
Furthermore, note that ρ 1/2 P 1/2 = ρ 1/2 . Then for any s ,
|v(s )| = s ρ 1/2 (s)(P 1 − P 0 ) s,s ≤ s ρ 1/2 (s)|(P 1 − P 0 ) s,s | ≤ s ρ 1/2 (s)2P 1/2 (s, s ) ≤ 2ρ 1/2 (s ).
This is to say, v 2 is entry-wise bounded by ρ 1/2 . Furthermore, this bound continues to hold after any transformation for v under P 1/2 :
|(v P k 1/2 )(s )| = s v(s)P k 1/2 (s, s ) ≤ 2 s ρ 1/2 (s)P k 1/2 (s, s ) ≤ 2ρ 1/2 (s ). Next, consider v (I − P 1/2 ) # E s = ∞ k=0 v (P k 1/2 − 1ρ 1/2 )E s =: ∞ k=0 a k .
Note that |(v P k 1/2 )E s | ≤ 2ρ 1/2 (s). Further, |v 1ρ 1/2 E s | ≤ |v 1|ρ 1/2 (s) ≤ 2ρ 1/2 (s). Therefore, for any k, |a k | ≤ 4ρ 1/2 (s). We also have
|a k | ≤ v 1 P k − 1ρ 1,∞ E s max ≤ 2Cλ k .
Using the same trick in proving Lemma 2, we have
1 ρ 1/2 (s) ∞ k=0 |a k | ≤ ∞ k=0 min 4, 2Cλ k 1 ρ 1/2 (s) ≤ 2 log λ (C/ρ 1/2 (s))−1 k=0 2 + k=log λ (C/ρ 1/2 (s)) C ρ 1/2 (s) λ k = 4 ln(C/ρ 1/2 (s)) − ln(λ) + 2 1 − λ ≤ 4 ln(C/ρ 1/2 (s)) 1 − λ + 2 1 − λ ≤ 4 ln(C) + ln (1/ρ min ) + 1 1 − λ .
Then W (ρ 1/2 )(s) = | k a k | ≤ cρ 1/2 (s) with c := 4 ln(C)+ln(1/ρ min )+1
1−λ . This completes the proof.
D Proof of Theorem 3
The proof is based on multi-variate Cramér-Rao bound. To begin, we assume P 0 (s, s ) > 0, P 1 (s, s ) > 0 for all (s, s ). 8
Consider the parameters θ = (F 0 , F 1 ) which controls the transition matrices
P 0 (s, s ) = F 0 (s, s ) s F 0 (s, s ) , P 1 (s, s ) = F 1 (s, s ) s F 1 (s, s )
.
Given the observations X t = (s t , a t ), t = 0, 1, . . . , T under the policy π 1/2 . We can compute the log-likelihood l(X 1 , . . . , X T | θ) = s,a,s n s,a,s · ln(P a (s, s )) − T ln (2) where n s,a,s = t 1(s t = s, a t = a, s t+1 = s ). Then, the entry of the Fisher information matrix 8 The general case follows a similar proof and is omitted for simplicity.
with θ * = (P 0 , P 1 ) is given by
I k,m = −E X ∂l(X|θ * ) ∂θ k ∂θ m = −E X s,a,s n s,a,s P a (s, s ) · ∂P a (s, s ) ∂θ k ∂θ m + E X s,a,s n s,a,s P a (s, s ) 2 · ∂P a (s, s ) ∂θ k ∂P a (s, s ) ∂θ m = −T s,a,s 1 2 ρ 1/2 (s) · ∂P a (s, s ) ∂θ k ∂θ m + T s,a,s 1 2 ρ 1/2 (s) P a (s, s ) · ∂P a (s, s ) ∂θ k ∂P a (s, s ) ∂θ m = −T ∂1 ∂θ k ∂θ m + T s,a,s 1 2 ρ 1/2 (s) P a (s, s ) · ∂P a (s, s ) ∂θ k ∂P a (s, s ) ∂θ m = T s,a,s 1 2 ρ 1/2 (s) P a (s, s ) · ∂P a (s, s ) ∂θ k ∂P a (s, s ) ∂θ m . Consider θ k = F 0 (i, j), θ m = F 0 (i, l), we have 1 T I k,m = 1 2 ρ 1/2 (i) P 0 (i, j) 1(j = l) − 1 2 ρ 1/2 (i). For θ k = F 1 (i, j), θ m = F 1 (i, l), we have 1 T I k,m = 1 2 ρ 1/2 (i) P 1 (i, j) 1(j = l) − 1 2 ρ 1/2 (i).
Otherwise it is easy to see that I k,m = 0.
Next, consider an unbiased estimatorτ (X 1 , . . . , X T ) for ATE. We can write ATE = f (F 0 , F 1 ) as a function of F 0 and F 1 . Further, one can verify that
∂f (θ * ) ∂F 0 (i, j) = −ρ 0 (i)(V π 0 (j) − V π 0 (i) + r 0 (i) − λ π 0 ) ∂f (θ * ) ∂F 1 (i, j) = ρ 1 (i)(V π 1 (j) − V π 1 (i) + r 1 (i) − λ π 1 ).
Finally, we aim to use the multi-variate Cramér-rao bound. To do so, let v
(1) i be an vector with the j-th element being v
(1)
i (j) = ρ 1 (i)(V π 1 (j) − V π 1 (i) + r 1 (i) − λ π 1 ). Let I (1) i (j, l) = T 2 ρ 1/2 (i) P 1 (i, j) 1(j = l) − T 2 ρ 1/2 (i)
be a matrix. Similarly, define v bound for the singular Fisher information matrix [57], we have
T Var(τ ) ≥ i v (1) i (I (1) i ) −1 v (1) i + i v (0) i (I (0) i ) −1 v (0) i = 2 i ρ 0 (i) 2 ρ 1/2 (i) j P 0 (i, j)(V π 0 (j) − V π 0 (i) + r 0 (i) − λ π 0 ) 2 + 2 i ρ 1 (i) 2 ρ 1/2 (i) j P 1 (i, j)(V π 1 (j) − V π 1 (i) + r 1 (i) − λ π 1 ) 2
which completes the proof.
D.1 Unbiased Estimator that achieves the lower-bound
In this section, we construct an LSTD(0)-type OPE estimator that achieves the aforementioned Cramér-Rao lower bound. To do so, we solve the following least square optimization problems that are similar to Eq. (3),
(V 1 ,λ π 1 ) = arg min V ,λ s∈S t,st=s,at=1 r(s t , a t ) −λ +V (s t+1 ) −V (s t ) 2(10)(V 0 ,λ π 0 ) = arg min V ,λ s∈S t,st=s,at=0 r(s t , a t ) −λ +V (s t+1 ) −V (s t ) 2 .(11)
Then, the estimation for the average treatment effect is given by
τ off :=λ π 1 −λ π 0 .
To analyze the variance ofτ , we follow the similar analysis as in Theorem 2. To begin, one can verify thatλ π 0 − λ π 0 = ρ 0 − ρ 0 r 0 whereρ 0 is the empirical stationary distribution for the empirical transition matrixP 0 (ρ 1 andP 1 can be defined accordingly).
Next, by the perturbation bound ofρ 0 , we havê
ρ 0 − ρ 0 = ρ 0 (P 0 − P 0 )(I −P 0 ) # .
Hence,λ
0 − λ π 0 = (ρ 0 − ρ 0 )r 0 = ρ 0 (P 0 − P 0 )(I −P 0 ) # r 0 .
Note thatP 0 is a function of F T . Similarly, we can define
f 1 (F (1) T ) :=λ 1 − λ π 1 = ρ 1 (P 1 − P 1 )(I −P 1 ) # r 1
Then by Lemma 6, we have the asymptotic normality for τ off :
√ T (τ off − ATE) = √ T (f 1 (F (1) T ) − f 0 (F (0) T )) d → N (0, σ 2 off ).
In order to compute σ off by using Lemma 6, we will linearize f 1 − f 0 around (F * 0 , F * 1 ). To do so,
consider ∂f 0 (F 0 ) ∂(F 0 )(i, j) = ρ 0 ∂(P 0 − P 0 ) ∂F 0 (i, j) (I − P 0 ) −1 (r 0 − λ π 1) + ρ 0 (P 0 − P 0 ) ∂(I − P 0 ) −1 ∂(F 0 )(i, j) (r 0 − λ π 1) = ρ 0 ∂P 0 ∂F 0 (i, j) V 0 = k ρ 0 (i)V 0 (k) ∂P 0 (i, k) ∂F 0 (i, j) Note thatP (i, k) =F 0 (i, k)/ lF 0 (i, l). Therefore, ∂f 0 (F 0 ) ∂(F 0 )(i, j) = k ρ 0 (i)V 0 (k) ∂ F 0 (i,k) l F 0 (i,l) ∂F 0 (i, j) = k ρ 0 (i)V 0 (k) 1(j = k) l F 0 (i, l) − F 0 (i, k) ( l F 0 (i, l)) 2 = k ρ 0 (i)V 0 (k) 1(j = k)ρ(i) − ρ(i)P 0 (k|i) ρ(i) 2 = ρ 0 (i) ρ(i) V 0 (j) − ρ 0 (i) ρ(i) k P 0 (k|i)V 0 (k) = ρ 0 (i) ρ(i) (V 0 (j) − V 0 (i) + r 0 (i) − λ π 0 ). Hence, the linearization of f 0 is ij ∂f 0 (F 0 ) ∂(F 0 )(i, j) F 0 (s, s , a) ij − F 0 (i, j) = 2 · 1(a = 0) ρ 0 (s) ρ(s) (V 0 (s ) − V 0 (s) + r 0 (s) − λ π 0 ) − ij ρ 0 (i)(V 0 (j)P 0 (j|i) − V 0 (i) + r 0 (i) − λ π 0 ) = 2 · 1(a = 0) ρ 0 (s) ρ(s) (V 0 (s ) − V 0 (s) + r 0 (s) − λ π 0 ).
The similar linearization can be done for f 1 . Then the linearization of f 1 − f 0 is g((s, s , a)) = −2 · 1(a = 0) ρ 0 (s) ρ(s)
(V 0 (s ) − V 0 (s) + r 0 (s) − λ π 0 ) + 2 · 1(a = 1) ρ 1 (s) ρ(s) (V 1 (s ) − V 1 (s) + r 1 (s) − λ π 1 ).
Note that for any E[g(X k )|X 1 = (s, s , a)] = 0 for any (s, s , a) and k ≥ 2. Hence
σ 2 off = Var ρ (g) + 2 ∞ k=2 Cov ρ [g(X k )g(X 1 )] = Var ρ (g) = 2 s,s ρ 0 (s) 2 P 0 (s |s) ρ(s) (V 0 (s ) − V 0 (s) + r 0 (s) − λ π 0 ) 2 + 2 s,s ρ 1 (s) 2 P 1 (s |s) ρ(s) (V 1 (s ) − V 1 (s) + r 1 (s) − λ π 1 ) 2
which completes the proof.
E Proof of Theorem 4
We construct a birth-death Markov chain with n states. Let P ∈ R n×n be a transition matrix
where P (s, s + 1) = 1 4 − δ, P (s, s − 1) = 1 4 and P (s, s) = 1/2 + δ (exception at two ends with P (0, 0) = 3/4 + δ and P (n − 1, n − 1) = 3/4).
Let the stationary distribution of P be ρ. Then ρ(s) = c (1 − 4δ) s for 0 ≤ s ≤ n − 1 and c := 1 s (1−4δ) s is a constant. By [10], we have the spectral gap of the chain is in the order of γ = O(1/n). Furthermore, the mixing time of the chain is bounded by
P k − 1ρ 1,∞ ≤ 1 ρ min (1 − γ) k (I − P ) # 1,∞ ≤ log 1 ρ min O(n) = O(n 2 ).
Following the same proof in Theorem 2, we have the on-policy variance is bounded by
σ on = O(n 6 ).
On the other hand, consider the node k where n s=k ρ(s) ≤ c δ/n 2 and n s=k−1 ρ(s) > c δ/n 2 for some sufficient small constant c . Let P 1 be the same as P except ∀s ≥ k P 1 (s, s + 1) = 1 4 P 1 (s, s) = 1 2 .
Let ρ 1 be the stationary distribution of P 1 . One can verify that ρ 1 (n) = O(1/n 2 ). We then construct r such that r(n, 1) = 1 and λ π 1 = 0. Then
σ off ≥ 2 ρ 1 (n) 2 ρ(n) 3 4
= Ω e cn n 2
for some constant c. Therefore,
σ on σ off = O n e c n
for some constant c . Next, consider the bias of DQ estimator. Suppose ATE = δ without loss (one can always achieve this by adding some constants to r). Let P 0 = 2 · P 1 − P and ρ 0 be the stationary distribution of P 0 . One can verify that
ρ 1 − ρ 1 = O(δ/n 2 ), ρ 0 − ρ 1 = O(δ/n 2 ).
Furthermore, following the proof in Theorem 1, we have
|(ATE − E[ÂTE DQ ])/ATE| ≤ ( ρ 1 − ρ 1 + ρ 0 − ρ 1 ) I − P # 1,∞ ≤ C · c δ 1 n 2 n 2 ≤ δ
for sufficient small constant c . This completes the proof.
F Technical Lemmas
Lemma 2. Suppose P ∈ R n×n is the transition matrix of a finite-state aperiodic and irreducible Markov Chain and ρ is the stationary distribution. Suppose there exists C and λ such that for any k = 0, 1, . . . P k − 1ρ 1,∞ ≤ Cλ k . Then
(I − P ) # 1,∞ ≤ 2 ln(C) + 1 1 − λ .
Proof. Note that
A = (I − P + 1ρ ) −1 − 1ρ = ∞ k=0 P k − 1ρ .
Then
A 1,∞ ≤ ∞ k=0 P k − 1ρ 1,∞ ≤ ∞ k=0 min 2, Cλ k ≤ log λ (1/C)−1 k=0 2 + ∞ k=log λ (1/C) Cλ k ≤ 2 log λ (1/C) + 1 1 − λ = 2 ln(C) − ln(λ) + 1 1 − λ (i) ≤ 2 ln(C) + 1 1 − λ where (i) is due to − ln(x) ≤ 1 − x for x > 0.
Lemma 4. For a finite-state aperiodic and irreducible Markov Chain X 1 , X 2 , . . . , X t . Let P be the transition matrix, ρ be the stationary distribution, and S be the state space. Suppose there exists C and λ such that for k = 0, 1, . . . , P k − 1ρ 1,∞ ≤ Cλ k . Then for any bounded function f : S → [a, b], there exists σ such that when T goes to infinity,
1 √ T T t=1 (f (X t ) − f * ) d → N (0, σ 2 )(12)
where f * = E ρ (f ) is the expected value of f under the stationary distribution and
σ ≤ √ 2(b − a) 2 ln(C) + 1 1 − λ .(13)
Proof. Note that Eq. (12) is simply due to the Markov chain CLT ( [31]). Let D be an diagonal matrix with entries D ii = ρ i . [31] further states that
σ 2 = Var ρ (f ) + 2 ∞ k=2 E ρ [(f (X 1 ) − f * )(f (X k ) − f * )] = (f − f * ) D(f − f * ) + 2 ∞ k=1 (f − f * ) DP k (f − f * ) = 2 ∞ k=0 (f − f * ) D(P k − 1ρ )(f − f * ) − (f − f * ) D(f − f * ) ≤ 2 ∞ k=0 (f − f * ) D(P k − 1ρ )(f − f * ) ≤ 2 (f − f * ) D 1 I − P # 1,∞ f − f * max (i) ≤ 2 f − f * 2 max 2 ln(C) + 1 1 − λ .
where (i) is due to Lemma 2. Therefore,
σ ≤ √ 2(b − a)
2 ln(C) + 1 1 − λ .
Lemma 5 (Theorem 6.2 [41]). Let U k be a sequence of random variables in R p converging in probability to u. Let a k be a deterministic non-negative sequence increasing to ∞. Let √ α k (U k − u) converge in distribution to N (0, Γ). Let f : R p → R q be a function twice differentiable in a neighborhood of u. Then, denoting the Jacobian of f at u by ∇f (u), we have 1. f (U k ) converges in probability to f (u).
2.
√ α k (f (U k ) − f (u)) converges in distribution to N (0, ∇f (u * )Γ∇f (u * ) ).
Lemma 6.
Consider an irreducible and aperiodic finite-state space Markov Chain X 1 , X 2 , . . . , X t . Let S be the state space and ρ be the stationary distribution. Let u : S → R p be a function with each component u i , 1 ≤ i ≤ p. Let u * = s∈S ρ(s)u(s) be the expected value of u under the stationary distribution ρ.
Let f : R p → R be a function twice differentiable in a neighbor of u * . Then, there exists σ ≥ 0 such that when T → ∞,
√ T f 1 T T i=1 u(X t ) − f (u * ) d → N (0, σ 2 ) √ T p i=1 (u i (X t ) − u * i ) · ∂f (u * ) ∂u i d → N (0, σ 2 )
Proof. To begin, note that by Markov Chain CLT (Corollary 5 [31]), we have
√ T 1 T T i=1 u(X t ) − u * d → N (0, Σ)
for some covariance matrix Σ ∈ R p×p . In particular,
Σ := E ρ [(u(X 1 ) − u * )(u(X 1 ) − u * ) ] + 2 ∞ k=2 E ρ [(u(X 1 ) − u * )(u(X k ) − u * ) ](14)
where E ρ denotes the expectation when the initial distribution of the Markov chain is ρ.
Then, since f is twice differentiable in a neighbor of u * , we can invoke Lemma 5 to get
√ T f 1 T T i=1 u(X t ) − f (u * ) d → N (0, σ 2 ) where σ 2 := ∇f (u * ) Σ∇f (u * ). Next, let F (X) := p i=1 (u i (X) − u * i ) · ∂f (u * ) ∂u i = (u(X) − u * ) ∇f (u * )
. Then using the fact 1 T T t=1 u(X t ) → u * and invoking Markov Chain CLT again, we have
√ T 1 T T t=1 F (X t ) d → N (0, σ 2 F ) where σ 2 F := E ρ [F (X 1 ) 2 ] + 2 ∞ k=2 E ρ [F (X 1 )F (X k )].
Expanding F (X) by (u(X) − u * ) ∇f (u * ), we have
σ 2 F = E ρ [((u(X 1 ) − u * ) ∇f (u * )) 2 ] + 2 ∞ k=2 E ρ [(u(X 1 ) − u * ) ∇f (u * )(u(X k ) − u * ) ∇f (u * )] = ∇f (u * ) E ρ [(u(X 1 ) − u * )(u(X 1 ) − u * ) ]∇f (u * ) + ∇f (u * ) ∞ k=2 E ρ [(u(X 1 ) − u * )(u(X k ) − u * ) ]∇f (u * ) (i) = ∇f (u * ) Σ∇f (u * ) = σ 2
where (i) uses Eq. (14). This implies that F (the linearization of f at the point u * ) will converge with the same limiting variance as f . We replicate exactly the environment of [29]. We model a rental marketplace with N = 5000 homogeneous listings. Customers arrive according to a Poisson process with rate N λ, decide whether to rent a listing (with rental probability controlled by the intervention), and if they do rent, they occupy a listing for an exponentially distributed time with mean 1 µ . Specifically, we define our MDP to be the discrete-time jump chain of this process, with events indexed by t and state s t ∈ {0, 1 . . . N } representing the current inventory of listings. At the t th event, the system chooses to apply control (a t = 0) or treatment (a t = 1). One of the following state transition and reward scenarios may then happen: 1. A previously occupied rental becomes available, i.e. s t+1 = s t + 1 and r t = 0; this occurs with probability (N −st)µ N µ+N λ 2. A customer arrives, with probability N λ N µ+N λ , and subsequently:
(a) Rents a listing, so s t+1 = s t − 1 and r t = 1; this occurs with probability stv(at) N +stv(at) where v(0) = 0.315 and v(1) = 0.3937 are the average utility under control and treatment, respectively.
(b) Does not rent a listing , so s t+1 = s t and r t = 0; this occurs with probability N N +stv(at) .
3. No state change occurs; i.e. s t+1 = s t and r t = 0.
[29] also describes a two-sided randomization scheme, where listings are also assigned to control or treatment, and the customer's purchase probability depends on both the customer's treatment assignment a t , as well as the number of control listings and the number of treatment listings. This corresponds to a more complicated MDP with a two-dimensional state s t = (s co t , s tr t ), where s co t corresponds to the number of available control listings, and s tr t the number of available treatment listings. The average utility of a control listing is v co (0) = v co (1) = v(0), while the average utility of a treatment listing is v tr (0) = v(0) and v tr (1) = v(1). We defer to [29] for further details of this scheme.
G.1.2 Implementation details
Here we list algorithms and hyperparameters tuned for this experiment. Hyperparameters were chosen to minimize MSE averaged over 10 held-out trajectories. As in [29], we also include a burn-in period of T 0 = 5N .
1. Naive. This has no hyperparameters.
2. TSRI. This has several hyperparameters, which affect both the experimental design (customer randomization probability p and listing randomization probability p L ), as well as the estimator (parameters k and β, as described in [29]). We set p, p L , β assuming λ, µ are known, exactly as prescribed in [29]. Specifically, we compute the values reported in Table 2 as:
p = 1 − e −λ/µ + 0.5e −λ/µ p L = 0.5 1 − e −λ/µ + e −λ/µ β = e −λ/µ
We report results for both k = 1 and k = 2.
3. DQ with LSTD, which we estimate using a slight modification of Equation (3). Specifically, we directly estimate the state-action value function Q instead of separately estimating the state value function V and P 1 , P 0 , and we add an L 2 regularization term. In short, we approximate and solve for a fixed point to the regularized least-squares problem:
Q = arg min Q Q − r − P Q + λ 2 2 + α Q 2 2
where Q ∈ R 2(N +1) is the vector of estimated Q(s, a) values, andP ∈ R 2(N +1)×2(N +1) is the state-action transition matrix. We use sample means in each state to construct plug-in estimates of r, P and λ.
4. Off-Policy with LSTD, which we note is novel in the literature. In Section D.1 we describe this algorithm, provide convergence guarantees, and show that this algorithm is efficient. This can be construed as a direct analog of [55]'s off-policy estimator, which applies LSTD in the discounted-reward setting. It has no hyperparameters.
5. Off-Policy with TD, where Q -functions and off-policy average rewards are calculated according to the Differential TD algorithm of [68]. This approach has two hyperparameters: the learning rate for the Q -function γ/ √ t, and the learning rate for the mean reward estimate βγ/ √ t.
For these experiments, we exclude the Off-Policy GTD variant described in [72] as their convergence guarantees do not apply to the tabular setting.
Algorithm Hyperparameters TSRI p = 0.816, p L = 0.683, k ∈ {1, 2}, β = 0.368 DQ (LSTD) α ∈ {0.01, 0.1, 1, 10, 100} Off-Policy (TD) β ∈ {0.2, 0.5}, γ ∈ {0.001, 0.01, 0.1, 1.} Table 2: Hyperparameters for the synthetic example of [29]. Parameter settings reported in the main text are in bold.
G.1.3 Additional results
We note that there are scenarios for which which specialized designs and estimators -specifically TSR, in this example -can provide a superior bias-variance tradeoff. [29] shows that the TSRI estimators become unbiased when λ µ. We ran the synthetic example setting λ = 10, µ = 1 (also mirroring results from [29]), and indeed for this setting for reasonable horizons TSR achieves lower RMSE. Recall, however, that TSR is ill-defined for settings where there is no natural notion of two-sided randomization (i.e. in any MDP without a notion of two sides), and its bias properties are clearly highly instance-specific and depend on knowledge of λ, µ. DQ still outperforms all alternatives besides TSR in this setting, and even in this extremely unbalanced setting bachieves a much lower asymptotic bias than TSR (-5e-3 vs 1e-2, as a proportion of the treatment effect magnitude). .
G.1.4 Computing environment
These experiments were performed on a personal desktop with a 24-core Intel Xeon X5670 CPU and 128 GB RAM. Total compute time per seed averaged less than two hours.
G.2 Ridesharing Simulator G.2.1 Environment
We implement a ridesharing simulator, with code available on Github.
1. Riders are generated based on trips resampled from the NYC Taxi Dataset [1] (specifically, from January 11, 2015), with a random willingness-to-pay per second distributed as LogNormal(log(0.01), 1.). The the rider's outside option is assumed to be the trip they actually took in the dataset, and the cost (i.e., negative utility) the rider incurs for this option is the fare recorded in the dataset, plus the trip time times the rider's WTP per second.
2. Drivers enter the system at pickup locations in the same dataset, but at a lower arrival rate (tuned to achieve a utilization of ∼ 70%). Drivers stay in the system for an exponential time with a mean of one hour, and stop serving new requests once they exit the system.
3. When a request enters the system, the pricing engine computes the cost to serve that request with an idle driver (where cost is based on recent per-mile and per-minute fare rates), and discounts this by 10%; this is the price offered to the rider. The pricing engine also offers the rider a worst-case time-to-destination (ETD) guarantee, which is 1.5 times the time to serve the request with an idle driver. The rider then chooses to accept or reject the offer, based on whether their worst-case utility for the trip exceeds the utility of the outside option. If the rider rejects the offer they exit the system. 4. If the rider accepts, the request is submitted to the dispatch engine. The dispatcher searches for the nearest idle driver and the 10 nearest pool drivers to the request. This list of candidates is filtered to those who can serve the request while satisfying the ETD guarantees of all riders.
The pool candidates are then further filtered to those whose cost to service the request is at most 1 1+αt times the cost of the idle driver, where α t = α co = 0 in control (a t = 0) and α t = α tr in treatment (a t = 1), where we vary α tr ∈ {0.3, 0.5, 0.7}. Finally, the minimum cost driver among this set is dispatched.
We can implement two-sided randomization in this market as follows. Each driver is also randomized into either treatment or control. The dispatcher then dispatches to the minimum cost driver among the following set:
• All idle drivers (i.e., drivers currently assigned no passengers).
• Control pool drivers, whose cost is at most 1 1+αco times the minimum cost idle driver.
• Treatment pool drivers, whose cost is at most 1 1+atαtr+(1−at)αco times the minimum cost idle driver.
G.2.2 Algorithms
We use the same approximation architecture for each algorithm, where Q(s, a) = θ φ(s, a) is a linear function of features φ : S × A → R d with coefficients θ. We take features φ(s t , a t ) to consist of the number of drivers in the system with each of 0, 1, 2, and 3 open seats remaining, as well as the price and cost of the current request.
The algorithms are then:
1. Naive, with no hyperparameters.
2. TSRI, again with hyperparameters p, p L , k, β. We set these based on the relative supply and demand characteristics of the simulator. Specifically, with analogy to the synthetic problem, the system averages around 600 drivers, with 3 passenger seats per driver, for a total of N ≈ 1800 available units of capacity. The arrival rate is 4 passengers per second, yielding λ ≈ 4/1800, while the average trip lasts 12 minutes, yielding µ ≈ 720. Ultimately we have λ/µ ≈ 1.6, and set the algorithm hyperparameters accordingly.
3. DQ with LSTD, with a single regularization hyperparameter α. Here we solve for θ by approximating and solving for a fixed point to the regularized least-squares problem:
θ = arg min θ Φθ − r − P Φθ + λ 2 2 + α θ 2 2
where Φ ∈ R |S|×|A| is the matrix of state-action feature representations.
4. Off-Policy with LSTD, where we solve simultaneously for θ 1 , λ 1 by solving for the unique fixed point of the projected Bellman equation Φ 1 Φ 1 θ 1 = Φ 1 (r 1 − 1λ 1 ) + Φ 1 P 1 Φ 1 θ 1 , where Φ 1 ∈ R |S| is the matrix of state-action features corresponding to action 1, and r 1 ∈ R |S| is the vector of rewards for action 1. We solve an analogous equation for θ 0 , λ 0 . This effectively extends the algorithm of Section [?] to the setting of linear function approximation. This has no hyperparameters.
5. Off-Policy with TD, where Q -functions and off-policy average rewards are calculated according to the extension of [68] to linear function approximation, as provided in [72]. This approach has two hyperparameters: the learning rate for the Q -function γ/ √ t, and the learning rate for the mean reward estimate βγ/ √ t.
6. Off-Policy with Gradient TD (GTD), as in [72]. This has the same hyperparameters β, γ as TD.
A single hyperparameter was selected for each algorithm across all treatment effect settings, based on a scalarization of MSE across all settings, and tuned on 10 held-out trajectories for each setting.
Algorithm Hyperparameters TSRI p = 0.9, p L = 0.6, k ∈ {1, 2}, β = 0.2 DQ (LSTD) α ∈ {0.01, 0.1, 1, 10, 100} Off-Policy (TD) β ∈ {0.2, 0.5}, γ ∈ {0.001, 0.01, 0.1, 1.} Off-Policy (GTD) β ∈ {0.2, 0.5}, γ ∈ {0.001, 0.01, 0.1, 1.} Table 3: Hyperparameters for the ridesharing setting. Parameter settings reported in the main text are in bold.
Figure 1 :
1A continuous-time Markov chain. Arrows indicate rates of transition between states.
the expected value of u under the stationary distribution of the Markov chain. See proof details in Appendix C.
Figure 3 :
3Toy-example from[29]. Left: Estimated ATE at time t/N = 10 4 across 100 trajectories. Dashed line indicates actual ATE. Diamonds indicate the asymptotic mean for each estimator. DQ shows compelling bias-variance tradeoff for this experimental budget. Right: Relative RMSE vs. Time; DQ dominates the alternatives at all timescales.
we show the relative RMSE (i.e. RMSE normalized by the treatment effect) of the various estimators considered here across all experimental budgets t. RMSE effectively scalarizes bias and variance and we see that on this scalarization the DQ estimator dominates the other estimators considered here over all choice of t.
The state at the time of a request corresponds to that of all drivers at that time: position, assigned routes, riders, and the pickup/dropoff location of the request. Actions correspond to driver assignments and pricing decisions. The reward for a request is the price paid by the rider, less cost incurred to service the request. Our simulator models Manhattan. Riders and drivers are generated according to real world data, based on [1]; this yields ∼ 300k requests and ∼ 7k unique drivers per real day. An arriving request is served a menu of options generated by a price engine. The rider chooses an option based on a choice model calibrated on taxi prices (for the outside option) and delay disutility. A dispatch engine assigns a driver to the rider; the engine chooses the driver who can serve the rider at minimal marginal cost, subject to the product's constraints. Finally drivers proceed along their assigned routes until the next request is received. The simulator implements pooling. Users can switch out demand and supply generation, pricing and dispatch algorithms, driver repositioning, and the choice model via a simple API. Other simulators exist in the literature[50, 71], but either lack an open-source implementation, or implement a subset of the functionality here.
Figure 4 :
4Ridesharing model Left:ÂTE at t = 3 × 10 5 over 50 trajectories. Dashed line indicates actual ATE. DQ has lowest bias, and is only estimator to estimate correct sign of the treatment at all effect sizes. Right: RMSE vs. Time; DQ dominates at all time scales.
Figure 4
4summarizes the results of the above experiments, wherein each estimator was run over 50 independent simulator trajectories, each over 3 × 10 5 requests. The DQ and OPE estimators
p+δ) , λ(p+δ) µ+λ(p+δ) ] . Let r 0 = [λp, 0] , r 1 = [λ(p + δ), 0] be the reward vector under actions 0 or 1. Then, the ATE is
Lemma 3 (
3Entry-wise non-expansive lemma). Let W : R |S| → R |S| be a map denoted by W (ρ) := (I − P 1/2 ) # (P 1 − P 0 ) ρ. Then, for any s ∈ S, 1 c W (ρ 1/2 )(s) ≤ ρ 1/2 (s)
R
|S| be the empirical average of the function F (a) and h (a)
T
(s, s ).
f
(s, a, s ) := ∇f (u * ) (u(s, a, s ) − u * )= (1 D(P 1 − P 0 )(I − P 1/2 ) # D −1 )E s (r(s, a) − λ + V (s ) − V (s)) + 2(1(a = 1) − 1(a = 0))(V (s ) +r(s, a)) − c where c := E ρ 1/2 [2(1(a = 1) − 1(a = 0))(V (s ) + r(s, a) − λ)]. By Lemma 6, we have
a,s |f (s, a, s )|. Then the problem boils down to boundf max .
. Then, by the multi-variate Cramér-rao
T
(P 0 (i, j) = F (0) T (i, j)/ k F (0) T (i, k), F (0) is defined inEq. (8)). Therefore, we can define f
Figure 5 :
5Simple example from[29], with λ = 10. Left: Estimated ATE at time t/N = 10 3 across 100 trajectories. Dashed line indicates actual ATE. Diamonds indicate the asymptotic mean for each estimator. Over this horizon, TSRI-1 and TSRI-2 exhibit small bias and variance, although asymptotically DQ still has lower bias.
Approach -The DQ Estimator: Continuing to keep in mind the MDP policy evaluation lens from above, observe that the Naive estimator effectively computes the average difference in instantaneous rewards, averaged over states visited under the policy corresponding to simple randomization. Our estimator makes one change to the Naive estimator: instead of computing the average difference in instantaneous rewards, we instead compute the average difference in Q-functions 1 . Intuitively, doing so allows us to partially account for the long-term effects of selecting the treatment over no-treatment at any given state, and consequently, we hope for a less biased estimate of the treatment effect.It turns out that the DQ estimator (denoted byÂTE DQ ) provides a dramatic reduction in bias.from which we find that it is O(δ 2 ). See Appendix B for details. This is second order relative to the ATE, and, of course, a marked improvement over the Naive estimator's bias. It turns out that this reduction in bias is generic: one of our primary contributions (Theorem 1) is to prove the DQ estimator's bias is O(δ 2 ) in general MDPs.Turning next to variance, we can show that the variance of the DQ estimator in our example is O(N ). In contrast, an optimal unbiased estimator has variance e Ω(N ) , so that the DQ estimator provides an exponential reduction in variance for a relatively small increase in bias. In fact, these relative merits are also generic. Specifically, in Theorem 2 we upper bound the variance of the DQ estimator for general MDPs. This upper bound scales as log(1/ρ min ), where ρ min is probability of the least-visited state under the stationary distribution, which can be as large as 1/N ; in the given 1 Q-functions are formally introduced in Section 3.Starting with bias, the DQ estimator's bias can be worked out explicitly here,
lim
TÂ
TE DQ − ATE ≈
δ
2
λ
(µ + λp)
ATE
Table 1 :
1The bias-variance tradeoff of different estimators. Bias is parameterized by the additive impact δ of the intervention on transition probabilities -note that the ATE itself can be Ω(δ). 'Variance' shows the limiting variance of each estimator on this example, as a function of the cardinality N of the state space. In full generality, variance is O(log(1/ρ min )) for DQ, and Ω(1/ρ min ) for OPE, where ρ min is the frequency of the least-visited state under the steady-state distribution.
This is always possible since dTV(p(s, 1, ·), p(s, 0, ·)) ≤ δ.
For now, we assume for simplicity that rewards are only a function of state. Similar results can be derived for the more general case where r is a function of both state and action, although the resulting formulas are more complex
The variance of such plug-in estimators can be bounded by iteratively applying Lemma 3, which is omitted for simplicity.
The DQ estimator in our original setting can actually be derived as eitherλTR(π1)−λTR(π0) orλDQ(π1)−λDQ(π0), but this results from a very suprising cancellation of terms in the subtraction; i.e.λTR(π) andλDQ(π) are individually different estimators with very different properties, as we will see; and the higher-order corrections must be derived fromλDQ.6 This result forλTR is in fact a slightly refined version of the key perturbation bound in[53].
The group inverse is continuous if we consider the set of matrices with rank |S| − 1 ([51]).
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3 Computing environment These experiments were performed on an internal cluster. Each run of the simulator took an average of four hours, allocating a single CPU and 8GB of RAM. G , G.2.3 Computing environment These experiments were performed on an internal cluster. Each run of the simulator took an average of four hours, allocating a single CPU and 8GB of RAM.
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"Thermally-driven Multilevel Non-volatile Memory with Monolayer MoS2 for Neuro-inspired Artificial Learning"
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"Sameer Kumar Mallik \nInstitute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India\n\nHomi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia\n",
"Roshan Padhan \nInstitute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India\n\nHomi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia\n",
"Mousam Charan Sahu \nInstitute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India\n\nHomi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia\n",
"Suman Roy \nInstitute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India\n\nHomi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia\n",
"Gopal K Pradhan \nDepartment of Physics\nSchool of Applied Sciences\nKIIT Deemed to be University\n751024BhubaneswarOdishaIndia\n",
"Prasana Kumar Sahoo \nMaterials Science Centre\nQuantum Materials and Device Research Laboratory\nIndian Institute of Technology Kharagpur\nKharagpurWest BengalIndia\n",
"Saroj Prasad Dash \nDepartment of Microtechnology and Nanoscience\nQuantum Device Physics Laboratory\nChalmers University of Technology\nGöteborgSweden\n",
"Satyaprakash Sahoo \nInstitute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India\n\nHomi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia\n"
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"Institute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India",
"Homi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia",
"Institute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India",
"Homi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia",
"Institute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India",
"Homi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia",
"Department of Physics\nSchool of Applied Sciences\nKIIT Deemed to be University\n751024BhubaneswarOdishaIndia",
"Materials Science Centre\nQuantum Materials and Device Research Laboratory\nIndian Institute of Technology Kharagpur\nKharagpurWest BengalIndia",
"Department of Microtechnology and Nanoscience\nQuantum Device Physics Laboratory\nChalmers University of Technology\nGöteborgSweden",
"Institute of Physics\nLaboratory for Low Dimensional Materials\nBhubaneswar-751005India",
"Homi Bhabha National Institute\nTraining School Complex, Anushakti Nagar\n400094MumbaiIndia"
] | [] | The demands of modern electronic components require advanced computing platforms for efficient information processing to realize in-memory operations with a high density of data storage capabilities towards developing alternatives to von Neumann architectures. Herein, we demonstrate the multifunctionality of monolayer MoS2 mem-transistors which can be used as a high-geared intrinsic transistor at room temperature; however, at a high temperature (>350 K), they exhibit synaptic multi-level memory operations. The temperature-dependent memory mechanism is governed by interfacial physics, which solely depends on the gate field modulated ion dynamics and charge transfer at the MoS2/dielectric interface. We have proposed a non-volatile memory application using a single FET device where thermal energy can be ventured to aid the memory functions with multi-level (3-bit) storage capabilities. Furthermore, our devices exhibit linear and symmetry in conductance weight updates when subjected to electrical potentiation and depression. This feature has enabled us to attain a high classification accuracy while training and testing the Modified National Institute of Standards and Technology datasets through artificial neural network simulation. This work paves the way for new avenues in 2D semiconductors toward reliable data processing and storage with highpacking density arrays for brain-inspired artificial learning. *Corresponding Author: [email protected] exorbitant processes are still demanded for making charge-trap based memory devices.Additionally, the size of the overall devices increases with multiple gate stacking and various 3 tailor-made designs. To achieve higher storage density per unit cell area, growing attention has spurred the creation of ultra-thin 2D materials based multibit memory devices.Recently, memtransistors (integration of transistors with memristive functionalities)based on 2D MoS2 have been proposed with various architectures showing controlled transport with digital switching ratios, 17,18 multibit optoelectronic memory, 19 neuromorphic computing applications etc. 20,21 Especially, the electrolyte gated transistors show better conductance modulation benefitted from its ion-gating mechanism over electrostatic charge trap phenomena, making them viable candidates for brain-inspired computation and logic-inmemory applications. 22,23 Despite the recent development of synaptic ion-gated transistors, multi-level memory based on ion-gating mechanisms is still lacking. However, liquid electrolytes practically limit the large-density integration of devices and high-temperature applications. Consequently, it is desirable to instigate new multi-level memories with eccentric functionalities that confer the possibility of robust device architecture with higher throughput. Furthermore, with the growing packing density of FET arrays on a single wafer, high performance integrated circuits (ICs) can reach an operating temperature as high as 500K, 24 making it essential to understand and exploit novel properties of 2D materials based devices at high temperatures. Thermally-driven memory is one of the applications where thermal energy can aid memory functions with multi-level storage capabilities.Our work demonstrates robust MoS2/SiO2/Si three-terminal devices with very different functionality and a novel mechanism that drives to a high geared intrinsic transistor at room temperature and exhibits a multi-bit memory cell above room temperature. The proposed thermally-driven programmable memory operations, i.e., READ, WRITE, ERASE, are found to be modulated by gate voltage biasing history. The step-like READ-RESET ratio and retention curves are obtained for several pulse voltage conditions. The multi-level states are investigated with varying pulsed width, gate voltage amplitudes, and drain/source voltages. We | null | [
"https://export.arxiv.org/pdf/2305.02259v1.pdf"
] | 258,461,191 | 2305.02259 | 01207c2b9c627a875de29421727abcc5d3f183a7 |
Thermally-driven Multilevel Non-volatile Memory with Monolayer MoS2 for Neuro-inspired Artificial Learning
Sameer Kumar Mallik
Institute of Physics
Laboratory for Low Dimensional Materials
Bhubaneswar-751005India
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
400094MumbaiIndia
Roshan Padhan
Institute of Physics
Laboratory for Low Dimensional Materials
Bhubaneswar-751005India
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
400094MumbaiIndia
Mousam Charan Sahu
Institute of Physics
Laboratory for Low Dimensional Materials
Bhubaneswar-751005India
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
400094MumbaiIndia
Suman Roy
Institute of Physics
Laboratory for Low Dimensional Materials
Bhubaneswar-751005India
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
400094MumbaiIndia
Gopal K Pradhan
Department of Physics
School of Applied Sciences
KIIT Deemed to be University
751024BhubaneswarOdishaIndia
Prasana Kumar Sahoo
Materials Science Centre
Quantum Materials and Device Research Laboratory
Indian Institute of Technology Kharagpur
KharagpurWest BengalIndia
Saroj Prasad Dash
Department of Microtechnology and Nanoscience
Quantum Device Physics Laboratory
Chalmers University of Technology
GöteborgSweden
Satyaprakash Sahoo
Institute of Physics
Laboratory for Low Dimensional Materials
Bhubaneswar-751005India
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
400094MumbaiIndia
Thermally-driven Multilevel Non-volatile Memory with Monolayer MoS2 for Neuro-inspired Artificial Learning
1 *Corresponding Author: [email protected] 2
The demands of modern electronic components require advanced computing platforms for efficient information processing to realize in-memory operations with a high density of data storage capabilities towards developing alternatives to von Neumann architectures. Herein, we demonstrate the multifunctionality of monolayer MoS2 mem-transistors which can be used as a high-geared intrinsic transistor at room temperature; however, at a high temperature (>350 K), they exhibit synaptic multi-level memory operations. The temperature-dependent memory mechanism is governed by interfacial physics, which solely depends on the gate field modulated ion dynamics and charge transfer at the MoS2/dielectric interface. We have proposed a non-volatile memory application using a single FET device where thermal energy can be ventured to aid the memory functions with multi-level (3-bit) storage capabilities. Furthermore, our devices exhibit linear and symmetry in conductance weight updates when subjected to electrical potentiation and depression. This feature has enabled us to attain a high classification accuracy while training and testing the Modified National Institute of Standards and Technology datasets through artificial neural network simulation. This work paves the way for new avenues in 2D semiconductors toward reliable data processing and storage with highpacking density arrays for brain-inspired artificial learning. *Corresponding Author: [email protected] exorbitant processes are still demanded for making charge-trap based memory devices.Additionally, the size of the overall devices increases with multiple gate stacking and various 3 tailor-made designs. To achieve higher storage density per unit cell area, growing attention has spurred the creation of ultra-thin 2D materials based multibit memory devices.Recently, memtransistors (integration of transistors with memristive functionalities)based on 2D MoS2 have been proposed with various architectures showing controlled transport with digital switching ratios, 17,18 multibit optoelectronic memory, 19 neuromorphic computing applications etc. 20,21 Especially, the electrolyte gated transistors show better conductance modulation benefitted from its ion-gating mechanism over electrostatic charge trap phenomena, making them viable candidates for brain-inspired computation and logic-inmemory applications. 22,23 Despite the recent development of synaptic ion-gated transistors, multi-level memory based on ion-gating mechanisms is still lacking. However, liquid electrolytes practically limit the large-density integration of devices and high-temperature applications. Consequently, it is desirable to instigate new multi-level memories with eccentric functionalities that confer the possibility of robust device architecture with higher throughput. Furthermore, with the growing packing density of FET arrays on a single wafer, high performance integrated circuits (ICs) can reach an operating temperature as high as 500K, 24 making it essential to understand and exploit novel properties of 2D materials based devices at high temperatures. Thermally-driven memory is one of the applications where thermal energy can aid memory functions with multi-level storage capabilities.Our work demonstrates robust MoS2/SiO2/Si three-terminal devices with very different functionality and a novel mechanism that drives to a high geared intrinsic transistor at room temperature and exhibits a multi-bit memory cell above room temperature. The proposed thermally-driven programmable memory operations, i.e., READ, WRITE, ERASE, are found to be modulated by gate voltage biasing history. The step-like READ-RESET ratio and retention curves are obtained for several pulse voltage conditions. The multi-level states are investigated with varying pulsed width, gate voltage amplitudes, and drain/source voltages. We
Introduction
In this modern era of electronic devices, ultrahigh data storage density is imperative for smarter computing platforms and vigorous information processing applications. Owing to the physical limits of silicon-based metal oxide field effect transistors (MOSFETs), the quest for alternative prospective materials has been the focus of intensive research to improve storage capacity. The discovery of atomically thin graphene has put forward direction toward significant active research in other two-dimensional (2D) materials over the last decade. 1 Molybdenum disulfide (MoS2), 2 being an in-demand member of the transition metal dichalcogenides (TMDCs) family, 3 has met broadband applications in electronics and optoelectronics manifesting its potency for future miniaturized electronic components. 4,5 Atomically thin MoS2 channel based FETs exhibit high carrier mobility ~200 cm 2 V -1 s -1 , and profound ON/OFF switching characteristics ~10 8 . 6 However, its large surface-to-volume ratio imposes an inevitable exposure of the atomically thin channel to the oxide trap/defect states, which draw hysteresis and threshold voltage instabilities hindering device performances. 7,8 Nevertheless, the large hysteresis produced during the dual sweep and pulsed gate operations has recently been exploited for nonvolatile (which include 2D flash memory, 9 magnetic random access memory, 10 resistive random access memory 11,12 ) and volatile (such as dynamic random access memory, 13 16 However, cumbersome and proposed the energy band diagrams that explain the single and multi-level memory effects in MoS2 FETs to validate the results. Furthermore, we demonstrate excellent linearity and symmetry upon electrical potentiation and depression. As a result, a high classification accuracy is achieved during training and testing of the Modified National Institute of Standards and Technology (MNIST) datasets in artificial neural network (ANN) simulation. These findings indicate the potential of in-memory applications in ion-driven MoS2-based transistors for building multilevel memory and synapse arrays, facilitating complex data processing tasks.
Results and Discussion
Monolayer (ML) MoS2 (triangular domains of size ~30 m) are directly synthesized on SiO2(285 nm)/Si substrate by using the salt-assisted chemical vapor deposition (CVD) technique (see the Methods section for more details). 25 Figure 1(a) displays the prominent Raman active modes i.e. E2g and A1g of as-grown MoS2. The frequency difference (∆ = 19 cm -1 ) between these phonon modes are characteristic signature of ML nature of our sample.
The Lorentz fit results provide the FWHM of E2g (~2.5 cm -1 ) and A1g (~5.2 cm -1 ) modes comparable to that of exfoliated MoS2 which depict high crystallinity of our CVD grown samples. 26 Figure 1(b) displays the optical micrograph of as-fabricated MoS2 devices with silver (Ag) as drain/source electrodes. The near equal work function of Ag and electron affinity of ML MoS2 allows the formation of a low Schottky barrier with negligible strain effect and superior carrier transport when Ag is used as metal contact to ML MoS2. 25,27 The channel lengths (Lch) of proposed devices are varied from 1 to 4 m with channel width fixed at 10 m.
Photoluminescence (PL) mapping of the MoS2 device shown in Fig. 1(c) provides intense and uniform luminescence, which illustrates exemplary optical properties of the ML MoS2 throughout the channel regions.
Initially, the room temperature basic transistor properties are obtained for Lch = 2 m in a high vacuum environment (10 -6 mbar) under dark conditions. Figure 1(d) demonstrates the linear output characteristics (i.e., drain current Id vs. drain-source voltage Vds) for the Vds sweep range from -100 to 100 mV at fixed back gate voltages (Vg = 0 to 40V in steps of 5V). Linear Id vs. Vds curves and Id values in few A ensure the Ohmic contact has been formed between Ag and MoS2, 28 which drives excellent carrier transport with low power consumption as per the previous literature. 25 The stable electrostatic gate control with n-type transistor behavior can be observed in Fig. 1 (e) during the single sweep transfer characteristics (Id vs. Vds) for Vg sweep range from -40 to 40V with fixed Vds values (25,50,75, and 100 mV). The semilogarithmic plot shows a high on/off current ratio (Ion/Ioff) of about 10 5 providing excellent transistor switching operations. The field effect mobility (µ !" ) and threshold voltage (Vth) is extracted from the slope of the linear region of the transfer curve for Vds = 100 mV, 25 which is shown in Fig. 1(f). In our case, such high carrier mobility of ~22 cm 2 V -1 s -1 is free from grain boundary and strain effects and therefore meets excellent agreement with previously reported results on unpassivated ML MoS2. 25,29 The transistor properties of other channel lengths are provided in the Supplementary Information (see Figure S1). The as-prepared backgated MoS2 devices with Ag as metal electrodes show superior transistor performance at room temperature and are further investigated for temperature-dependent behavior. The room temperature transfer characteristics in our case show robust transistor performance due to its minimal hysteresis behavior. A transfer curve at temperature 250K and 300K for Lch = 2 m is shown in Fig. 2 The lower panel of Fig. 2 31 However, the nature of the anti-clockwise hysteresis in our case is distinctly different from that of previous reports and requires an independent mechanism that can corroborate our forthcoming results. The fundamental idea that propels the origin of anticlockwise hysteresis in our case is as follows; in NaCl-assisted CVD growth of MoS2, the minute amount of mobile charges such as sodium ions (Na + ) exist in the bulk of SiO2 gate dielectric is inevitable during high temperature synthesis process. It is apparent that the mobile ions in gate oxide can be a plausible factor responsible for threshold voltage shift and could induce hysteretic transfer characteristics in FETs. 33 As shown in schematic of Fig The insets of each figure represent the magnified view of the WRITE process and its progression with temperature. At 300 K, READ provides initial drain current values, which increase suddenly as the pulse amplitude increases from 0 to 30V due to the accumulation of charge carriers into the MoS2 channel, as shown in Fig. 3 (a). However, during the WRITE process, while the voltage pulse is maintained at 30 V, the drain current drops quickly, followed by a slight decrease attributed to two types of charge trapping, i.e., fast and slow at the MoS2/SiO2 interface. This degradation of drain current due to positive gate stress is well consistent with previously reported results. 25 The final READ (R1) process provides nearly identical drain current values as that of initial READ (R0) such that the ratio between them,
i.e., R1/R0 remains near unity. The drain current reduction during WRITE operation is best fitted with two trap model by employing the following equation;
= % + * !(#!# $ ) & ' , + * !(#!# $ ) & ( ,(1)
where & , and ' are time constants for fast and slow trapping processes, respectively, A and B are initial current amplitudes, and % and % are adjustable offsets. At 350 K, the fleeting drain current during the WRITE operation shows a near similar behavior, except a slow increase of current values can be observed after an initial degradation of current, as shown in Fig. 3 (b). In this case, the R1/R0 ratio still provides near unity value. However, when the temperature exceeds 350 K, i.e., at 400 K, the drain current during the WRITE process reforms its behavior and is observed to increase sharply till the gate stress is applied. Interestingly, as shown in Fig. 3 (c), the R1/R0 ratio also increases to 1.5, providing an initial signature for possible memory applications of our MoS2 FETs. When the temperature is further increased to 450 K, a more pronounced increment in the drain current values is observed during the WRITE operation, as shown in Fig. 3 (d). In this case, the enhancement in R1/R0 ratio to 3.57 indicates thermal modulation of drain currents in the presence of gate bias. It is worth mentioning that the sharp increment of current during the WRITE process always follows a very short-lived degradation of drain currents (as indicated by an arrow in the inset), which uncovers two possible mechanisms that are taking place at high temperatures. The origin of the initial degradation of drain currents is previously discussed, which corresponds to the trapping of charge carriers at the MoS2/SiO2 interface. However, the trend is carefully analyzed to shed light on the mechanism of the increased drain current. We notice that the increasing trend of drain current closely resembles a diffusion-like current and is accordingly best fitted with the following current monitoring equation, 34
= % − ( ( )* ( + ( ) )(2)
where, % , and A are standard derived parameters, L is the channel length, and D is the diffusion coefficient. The temperature-dependent progression of underlying mechanisms is shown in
(a) (b) (c) (d) E f E c E v
Tunnel Barrier
Si ++ To explain the non-volatile memory states in MoS2 FETs originated by the thermallydriven accumulation of ions at the MoS2/SiO2 interface, we propose a hypothesized mechanism involving WRITE, READ, and ERASE schemes. interface. This is known as ERASE process, where the additional injected carriers are removed from the conduction channel at a high negative gate field, regressing the device to the original neutral state. The read process is further performed at Vg = 0V, giving rise to a low conductivity state, where the carriers mediated by ions don't contribute to the resultant drain current, as shown in Fig. 3 (h). Apart from single memory operation, which consists of a pair of (high and low) conductive read states, the proposed mechanism could adequately explain the multi-level memory effects arising in our MoS2 FETs. As illustrated in Fig. 3 The READ current rises progressively with increasing the pulse number, a phenomenon that represents the continual accumulation of carriers in MoS2 as prolonging the gate stress on the memory.
WRITE Operation
To further explore the pulse-controlled memory behaviour, the multi-level memory cycles are expanded to different programming pulsed width measurements, attributed to distinct conductance states based on the number of cycles. Figure S7 represents such memory cycles for pulse width 2, 3, and 4s, respectively. It is worth noticing that using the programmed memory operations with longer pulse widths yields higher conductance states, which reflects that n-bit memory can be achieved by varying the pulse time. Additionally, we witness an interesting result on data erasing, i.e., the multi-level storage information can be erased gradually by applying low negative voltages accompanying the complete RESET at a high negative bias (-70 V). Figure 4 reported READ windows of about 1.9 and 7.4 in few-layered MoS2 and ReS2 memory devices, respectively, for single-step memory operations. 35 Our results show unambiguously higher READ values in the case of ML MoS2 FETs. More importantly, our studies show multi-level memory states in our device can take this ratio up to ~220 and beyond by using longer pulse trains with varying pulse widths. It is also worth noting that combinational pulse waveforms can get desired READ window and store information in multiple storage cells. The retention curves of the observed memory behavior are obtained after 3 successive WRITE operations for various pulse width conditions, shown in Fig. 4 (d). The retention behavior shows that the data levels of different pulse conditions are well-discernable after 1500s. It may also be noted that our device shows much better retention capabilities with increasing pulse widths. Such memory devices can be used in electronic components that require low-scale information storage functionalities such as cache memory, disposable electronic tags, buffer memories, etc.
Apart from various pulse conditions, i.e., several pulses and pulse width, multi-level data storage can be obtained by changing the gate voltage amplitudes. In previous reports using floating gate memory configurations, optical response, and plasma treatment, the gate voltagedependent multi-bit generation is achieved in 2D materials and heterostructures. 19,[36][37][38][39] Here, we also find a strong dependence on the drain current levels as we increase the Vg from 10V
to 40V, as shown in Fig. 4 (e). The programmable memory operations are obtained for 5 successive pulses, which represent a similar progressive increase in READ states, i.e., the higher gate voltage amplitude corresponds to larger READ windows. This indicates gate voltage amplitude is crucial for obtaining stable and distinct memory states. We are also keen to observe the effects of drain-source voltage, i.e., Vds, on the memory operations in the MoS2
FETs, shown in Fig. 4 (f). All these above experiments strongly vow the versatile nature of our memory device, which can be tuned with different controllable parameters such as the number of pulses, pulse-width, pulse-amplitude, and drain-source voltage.
As discussed earlier, non-volatile memory (NVM) devices present a potential application as an in-memory computing element, 40 Figure 5(c), our proposed synaptic transistor shows remarkable linear and symmetrical conductance updates. Notably, the dynamics of potentiation and depression, specifically the dynamic range, linearity, and asymmetry, play a pivotal role in ensuring the precision of learning and recognition simulation. 46 The outstanding linearity in our case could provide a potential opportunity to achieve high classification accuracy in ANN training. We have attained a dynamic range of conductance ratio [GMax/GMin] equal to ~20, which may be deemed as the on/off ratio value for accomplishing superior performance in ANN tasks. We confidently ascribe the enhanced linearity of the conductance updates to the ion-driven charge transfer, storage, and release.
Additionally, we employ a synaptic model to replicate the LTP and LTD features demonstrated in turn, are linked to 10 output nodes, as depicted in Fig. 5 (d). These 10 output nodes correspond to the output classes of the MNIST dataset, which are digits ranging from "0" to "9". The NN training algorithm used the rectified linear unit activation function and cross entropy loss as the cost function with a back propagation method. The training session is repeated five times for 300 epochs, and the mean values for the training accuracy are plotted as a function of the training epochs, as shown in Fig. 5 (f). Our synaptic device achieves a maximum training accuracy of ~98% and eventually drops to a stable limit of ~95% for the pattern recognition task at the end of the training cycles, which strongly suggests the neuromorphic adaptation of our artificial synaptic transistors. Figure 5 (g) displays the relationship between the number of hidden nodes and accuracy at 200 epochs. As shown in the graph, an increase in hidden nodes leads to a higher recognition rate, with a maximum rate achieved at 512 nodes. These findings indicate the potential of our ion-driven MoS2-based transistors for building multilevel in-memory synapse arrays, facilitating complex data processing tasks.
Conclusions
In summary, we have demonstrated a novel functional three terminal device with saltassisted CVD-grown ML MoS2 as channel material which serves as a high geared intrinsic transistor at room temperature and becomes a multi-level memory cell at relatively high temperature. The memory behavior is attributed to the hysteresis collapse and switch from clockwise to anticlockwise, which further expands with increasing temperature. The reverse hysteresis is ascribed to the migration of active ions and accumulation at the MoS2/SiO2 interface due to electrostatic gating at a higher temperature of around 450 K. The proposed thermally-driven programmable memory operations, i.e., READ, WRITE, ERASE, are found to be modulated by gate voltage biasing history. The step-like READ-RESET ratio and retention curves are obtained for various pulse conditions, which stipulates vigorous data storage capabilities in thermally-driven memory cells. Furthermore, the multi-level memory states are investigated with varying pulsed width, gate voltage amplitudes, and drain-source voltages. We corroborate these results with the relative energy band diagrams to explain the single and multi-level memory effects in MoS2 FETs. Moreover, our device demonstrates excellent linearity and symmetry upon electrical potentiation and depression for hightemperature synaptic applications. As a result, a high classification accuracy of 95% is achieved during training and testing of the Modified National Institute of Standards and Technology (MNIST) datasets in artificial neural network (ANN) simulation. Our work serves as a precursor to novel pathways in the realm of 2D semiconductors, aimed towards achieving robust high temperature in-memory computing applications with enhanced memory capabilities for artificial cognitive development after the human brain.
Methods
Synthesis of monolayer MoS2:
The monolayer MoS2 is grown using a salt-assisted CVD synthesis as reported elsewhere. 25 However, some growth parameters are varied from the previous literature, not only to achieve large scale triangular domain growth with high crystallinity but also to diffuse mobile ions into the SiO2 gate dielectric. More specifically, the amount of the salt precursor, i.e., NaCl, is increased to promote the nucleation density, enhancing the lateral dimension of MoS2 domains. Additionally, the precursor (MoO3+NaCl)
height is increased so that the distance between the precursor and growth substrate is minimized to 3 mm to channelize the sulfur feeding during growth time.
Supporting Information
Supporting Information is available free of charge which contains additional results of device characterizations.
semi-floating gate transistor 14 ) memory applications. Considering the superior charge trap capacity of conventional Al2O3/HfO2/Al2O3 gate stack and few layer MoS2 as the channel, Zhang et. al. establish a large memory window (~20 V) with efficient modulation of the program and erase operation. 15 A stable two-terminal floating gate memory cell is realized by Vu et. al. on a vertically stacked MoS2/hBN/graphene heterostructure providing program/erase operations with retention >10 4 s.
Figure 1 .
1Structural characterization, design, and electrical properties of MoS2 transistor
(a), along with possible mechanisms illustrated in Fig. 2 (b) that justify the nature of the hysteresis loop. Low backward sweep current (BSC) indicates clockwise hysteresis in a dual sweep transfer curve, whereas high BSC reflects anti-clockwise hysteretic nature. The red (blue) curve signifies a forward (backward) sweep to realize the clockwise hysteresis nature at low and room temperature having hysteresis width less than 1V.As shown inFig. 2 (b), two cross-sectional schematic representations of the device portray the possible interfacial mechanism that involves charge-trapping processes at high negative and positive gate voltages. As illustrated in the schematic, at the beginning of the forward sweep, when a particular negative bias is applied to the gate terminal, an initial de-trapping process in the SiO2 releases additional electrons to the MoS2 channel, which causes a left shift of the transfer characteristics. As we reach a high positive gate bias, the capturing of electrons at the trapping sites of SiO2 dominates. As a consequence, during the backward sweep, the channel of MoS2 contains fewer electrons, which leads to low BSC.25
. 2(c), under the application of high negative bias at the gate terminal, the Na + ions move away from the channel-interface region, and a random distribution of ions can be realized in the gate dielectric during the forward sweep. However, under high positive bias, these ions drift towards the MoS2 channel, which eventually increases the drain current values during the backward sweep. We reproduce the above results on evolution of hysteresis by fabricating devices with different channel lengths (Lch = 1, 3, and, 4 µm) and the obtain results are shown inFig. S3. Even with different channel lengths the repeatability of our results are highly reproducible over the temperature range from 100 to 450 K indicating the high temperature anti-clockwise hysteresis behavior is not limited to a confined channel length. To support our Na + ion-driven mechanism, it is extremely important to compare these results with conventional (without NaCl) CVD grown MoS2 based transistors. For this, we have conducted the dual sweep transfer characteristics on such CVD grown pristine MoS2 samples to examine the role of Na + ions on device performances. It is worth mentioning that the hysteresis behavior of without NaCl CVD grown MoS2 based transistors show a consistent clockwise behavior throughout the temperature range from 250 to 450 K, as shown inFig. 2 (d) and supplementary Fig. S4. This behavior is attributed to the increased thermally activated trap centers at MoS2/SiO2 interface which trap/de-trap charge carriers depending on the voltage polarization. This comparison confirms our proposed mechanism of the Na + ion-driven high temperature anti-clockwise hysteresis in MoS2 transistors. In the subsequent sections, we shall expound the mechanism more lucidly and observe how the ions migrate inside SiO2, generating a diffusion-like drain current behavior in the conduction channel with temperature.
Figure 2 .
2Temperature-dependent hysteresis evolution and proposed mechanism. (a) Dual sweep transfer curves of the MoS2 device (Lch = 2 µm) are plotted against temperatures at 250 K, 300 K, and 450 K. The gate voltage sweep range is taken from -60V to 40V and again back to -60V with a sweep rate of 0.5 V/s; Vds is fixed at 100 mV. The forward (backward) sweep is represented by solid red (blue) lines indicating a clockwise hysteresis loop marked by arrows. Cross-sectional schematic representations of the device portray the possible interfacial mechanism that involves (b) charge-(de)trapping processes in clockwise hysteresis at high Vg<0) and positive gate voltages (Vg>0). (c) the movement of mobile ions in gate oxide under (f) high negative (Vg<0) and (g) positive gate voltages (Vg>0). (d) Dual sweep hysteresis behaviors on CVD grown MoS2 synthesized using conventional techniques (without NaCl). The transfer curves show the emergence of clockwise hysteresis with respect to temperature. As shown, the degree of hysteresis increases maintaining the same direction (clockwise) as we increase the temperature from 250 to 450 K. The transient drain current analysis provides an elementary route to understand subsequent intermediate mechanisms at different time scales that alter with temperature under positive bias at the gate terminal. Figure 3 (a-d) represents the waveform nature of the drain current by employing a single pulse gate voltage at four different temperature regions starting from 300 to 450 K. Each pulse waveform has a period of ~12s which consists of three components of equal intervals; an initial READ (R0) at Vg = 0 V, WRITE (W) at Vg = 30 V followed by a final READ (R1). A constant Vds of 300 mV is used for all the above operations.
Fig. S5 (
S5see supplementary Information), where the left Y axis represents the trap time constants (outcome of trap model), and the right Y axis represents the diffusion coefficient (outcome of current monitoring model). As the temperature rises, the two trap constants (fast and slow) initially provide finite values and then become negligible around 400 K. However, the diffusion coefficient has a finite value above 400 K, stipulating its dominant nature at high temperatures. The charge trapping process is more favourable at room temperature, quenched by the migration of ions near the MoS2/SiO2 interface at high temperature. Here, it is worth mentioning that the accumulation of ions enroute the memory capabilities in ML MoS2 FETs, which will be further discussed in the following sections.
Figure 3 .
3Transient drain current analysis and proposed energy band diagrams (a-d) represent the waveform nature of the drain current by employing a single pulse gate voltage at four different temperature regions starting from 300K to 450K. Each pulse waveform has a period of ~12s which consists of three components of equal intervals; one is at Vg = 30 V, known as WRITE (W) operation, and two are at Vg = 0V, known as READ (R0 and R1) operations. The drain-source voltage is kept fixed at 300 mV. The insets of each figure represent the magnified view of the WRITE process and its progression with temperature. The
Figure 3 (
3e-h) demonstrates the energy band diagrams for different gate voltage polarities to acknowledge basic n-type transistor functions and extrapolation of programmable conductance states for both single and multi-level memory capabilities. As shown in Fig. 3 (e), when a high positive voltage (Vg > 0) is applied to the gate terminal of the MoS2 FETs, the Fermi-level shifts up, and accumulation of carriers (shown in orange) in the conduction band of MoS2 essentially switch on the transistor offering a n-type FET behavior. Additionally, as the device is characterized at elevated temperature, the thermally induced ions drift and migrate towards MoS2/SiO2 interface due to high positive gate bias. Many accumulated ions persuade high carrier densities at the interface during positive gate stress. As a result, lowered tunneling barrier favors injecting additional charge carriers (shown in blue) into the conduction band of MoS2. It should be noted that this injection process continues till the positive gate voltage is maintained, which constantly populates the 14 conduction electrons in the MoS2 channel. This behavior is consistent with the observed results inFig. 3 (d), where the drain current progressively increases during the pulsed gate voltage maintained at 30V. This is denoted by the WRITE process, where conductance states can be achieved by allowing continuous migration and accumulation of ions near the SiO2/MoS2 interface, increasing the carrier densities in the MoS2 channel over a progressive period by applying a positive gate voltage pulse.Figure 3(f) demonstrates the band diagram for the high conductivity read state after immediate withdrawal of the gate field (Vg = 0V). The Fermi level is lowered as the transistor goes to the off state at zero gate bias. However, the injected carriers mediated by the ions have a low probability of tunneling back through the barrier. As a result, despite the transistor being in the off state at zero gate bias, a finite density of carriers (shown in blue) is retained in the conduction channel of the FET, giving rise to a high conductivity state after the writing process. When a high negative bias is applied to the gate terminal (Vg < 0), the transistor remains in the off state, and the Fermi level is further lowered. However, as shown inFig. 3 (g), the accumulated ions diffuse and migrate away from the MoS2/SiO2
(i), successive WRITE operations lower the barrier allowing the charge carriers to tunnel into the conduction band of MoS2, independently increasing the carrier densities and lifting the Fermi level at each pulse cycle. Continuing this process generates higher conductance states where each state corresponds to pulse cycle biasing history. This way, we could observe multi-level memory effects in our MoS2 FETs.The development of a novel thermally-driven and ion-mediated non-volatile multi-level storage capacity can further be realized owing to the anti-clockwise hysteresis nature of our MoS2 FETs at high temperatures.Figure S6(see supplementary Information) shows dual sweep transfer characteristics with varying hysteresis widths for 10 successive cycles at four different temperatures. In each cycle, the gate voltage is swept from -10 to 40 V and back to -10 V. With repeating cycles, the hysteresis curves initially provide overlapping current loops at 300K. However, the current loops shift upwards at elevated temperatures with each successive cycle, particularly at 475K. This illustrates multiple conductance states can be achieved corresponding to several dual sweep cycles, which spurred the attainment of multilevel memory generation of our MoS2 FETs.Figure 4 (a)shows multi-level programmed state operations driven by gate pulse trains having pulse width 1s. Based on our single-state memory function at 450 K, a positive gate pulse train (10 cycles) with an amplitude of 30V is provided for multiple writing processes, as shown in the lower panel (blue waveforms) ofFig. 4 (a). The reading of distinct conductance states is performed at Vg = 0 V after each writing process establishing a multi-level storage unit in the case of ML MoS2 FET. Note that a complete RESET of the device is necessary (with Vg = -70 V) before the initial writing process to recover from any higher conductance state (blue bar). The drain-source voltage is kept fixed at 300 mV at every step of the programmable memory operations. The upper panel (red waveforms) shows the fleeting drain currents during RESET, WRITE, and READ programs. The green gradient bars (a guide to eyes) represent the individual READ processes after periodic WRITE operations, which assimilate increased distinct conductance states with several pulse cycles.
(b) represents a multi-level ERASE operation with programmable erase pulse train of pulse width 4s where the ERASE and READ processes are performed at -40 and 0 V, respectively (see the voltage waveform in the lower panel). Similar to the multi-level WRITE operation, a progressive reduction of the READ current with successive ERASE operations indicates continual decumulation of charge carriers in the conduction channel. This means any desired lower conductance state can be achieved from a higher conductance state by performing ERASE operations with n-cycle pulse trains. The multi-level WRITE and ERASE operations in our device show exceptional pulse control of charge injection and release, providing a complete yet simplest way to achieve large-scale data storage capabilities in MoS2 FETs. The increment and decrement of current updates with the number of pulses provide a potential platform to achieve high temperature in-memory neuromorphic computing for online learning, discussed subsequently.
Figure 4 .
4Dynamic responses of multi-level ionotronic memory (a) Multi-level programmed state operations at 450K driven by gate pulse trains with pulse width 1s. A Positive gate pulse train (10 cycles) with an amplitude of 30V is provided for multiple writing processes, as shown in the lower panel (blue waveforms). The upper panel (red waveforms) shows the fleeting drain currents during RESET, WRITE, and READ programs. The green gradient bars represent the individual READ processes after periodic WRITE operations, which assimilate to increased distinct conductance states with several pulse cycles. (b) Multi-level ERASE operation with a programmable pulse train of pulse width 4s where the ERASE and READ processes are performed at -40V and 0V, respectively (see the voltage waveform in the lower panel). (c) READ-to-RESET ratios for 10 successive pulses of widths 1s, 2s, 3s, and 4s. (d) The retention curves of the observed memory behavior after 3 successive WRITE operations for 1500 s with various pulse width conditions. Multi-level programmed state operations at 450 K driven by gate pulse trains (e) with gate pulse amplitude varied from 10 to 40V (f) with drain-source voltage for 0.1, 0.2, and 0.3V. The thermally-driven multi-level memory functions discussed above can generate large set of data storage possibilities by controlling the pulse trains of different pulse widths with nsuccessive cycles. The pulse-dependent distinct READ states are shown in Fig. S8 (see supplementary information). The READ-to-RESET ratio, a desirable feature of a memory device, is calculated for all the pulse-controlled READ states shown in Figure 4 (c). For each pulse width, the ratio values are plotted in logarithmic scale for 10 successive pulses represented by colored half-filled circles spanned over a large range from 1 to 220. The ratio value 1 is represented by the OFF-state in all pulse widths. The ON/OFF ratio for the first pulse varies from 3 to 16 as the pulse width varies from 1s to 4s. Interestingly, the pulse width with longer time has higher and distinct READ window values showing strong dependence on various pulse conditions for each corresponding cycle. An attempt has been made to realize the ON/OFF ratios in thermally-aided mem-transistors in 2D materials; for example, Goyal et. al.
which can be accomplished via the multilevel conductance response of an NVM, providing the capability to store analog synaptic weights of an Artificial Neural Network (ANN) on-chip.41 ANNs draw inspiration from the neural connectivity of the human brain, although they do not align with any particular biological learning paradigm. Notably, in-memory computing is capable of executing matrixvector multiplication (MVM) operations, which are the primary computations utilized in the domain of Artificial Intelligence (AI).42 According to this approach, a crossbar array of NVM devices can undertake the MVM operation by encoding the input vector as an analog voltage and the weight matrix as analog conductance values stored in the memory devices. It may be worth mentioning that the main focus of most of the neuromorphic computing literature is based on conductance weight updates at room temperature for online learning. 20,22,43 As theFigure. 5 Artificial synaptic learning of MoS2 based ionotronic devices (a) Demonstration of biological synapse operation between pre and post-synaptic neurons generating EPSC. (b) illustration of our proposed MoS2 based artificial synapse in accordance with biological response. (c) Long-term plasticity characteristics of the proposed device emulating long term potentiation (LTP) and depression (LTD) with 20 consecutive identical pulses of (40V, 1s), and (-40V, 4s), respectively. The Vds of 0.1V is applied to read the change in EPSC. (d) The schematic diagram of a three-layer artificial neural network with 784 input nodes, 512 hidden nodes, and 10 output nodes for the classification of MNIST 28×28 pixels. (e) Extraction of nonlinearity factors for LTP and LTD, (f) Training accuracy evolution (%) vs. the number of epochs for the software-based ideal numeric case and the proposed ion-driven synaptic device during NN training. (g) Training accuracy evolution (%) vs. the number of hidden nodes ranging from 100 to 600. A biological synapse in the human cerebrum transmits neurological impulses through electrical and chemical signals between pre and post-synaptic neurons. Neurotransmitters released from the axon of the presynaptic neuron attach to neuro-receptors at the dendrite of the postsynaptic neuron, generating an excitatory postsynaptic current (EPSC), as illustrated inFig. 5 (a). The magnitude of EPSC depends on the synaptic weight, which can be modified by the presynaptic spike. This is critical for memory and cognitive processing.21 The proposed ionotronic device based on ML MoS2 can function as an artificial synaptic transistor, emulating the biological synapse, as presented inFigure 5(b). MoS2 plays the role of the synapse, where the back-gate voltage is considered as the presynaptic stimulation, and the drain/source electrode acts as the postsynaptic terminal responsible for accumulating EPSC.Figure 5 (c) depicts the change in channel conductance with the number of repetitive presynaptic stimulation pulses. To imitate long-term synaptic plasticity in our device, identical potentiation (consisting of voltage pulses with amplitude of 40V and width of 1s) and depression (consisting of voltage pulses with amplitude of -40V and width of 4s) pulses are consecutively implemented to the gate terminal for 20 times as a presynaptic spike. To perceive the change in channel conductance, a bias (Vds) of 0.1V is applied. As shown in
ω and the maximum (GMax) and minimum (GMin) conductance determine the nonlinearity coefficients ( ) for potentiation ( 2 ) and depression ( 3 ), which are 0.64 and 0.41, respectively. These parameters play a crucial role in the off-chip training procedure for pattern recognition in neural network training. 48 To verify the image classification performance of our synaptic device, we simulate an ANN using the experimentally determined long-term plasticity characteristics to facilitate supervised learning of large image handwritten digits (28×28 pixels) from the Modified National Institute of Standards and Technology (MNIST) dataset. 49 An open-source software (Pytorch) is utilized for implementing the simulations. 50 The training images are divided into batches of 32, with each batch image linearized to a 784×1 input matrix of pixel intensities normalized to [0, 1]. A total of 784 input nodes were then linked to 512 hidden nodes, which,
Device Fabrication :
:The monolayer MoS2 based mem-transistors are fabricated by using a photolithography (Heidelberg μPG 101) system. Initially, the salt-assisted CVD grown MoS2 samples on SiO2/Si substrate are coated with a positive photoresist (ma-p-1205) by a spin coater (SUSS Microtech) and then baked at 80 °C for 1 min. Under the inspection of a highresolution microscope, the contact patterns of several channel lengths are exposed on the monolayer MoS2 flakes with a 405 nm laser in photolithography. The exposed patterns are developed with an alkaline solution (1:4, NaOH:DI water) for 1 min. The sample is then mounted on a thermal evaporation chamber for the deposition of the silver (Ag) electrode, followed by the dissolution of residual resists in acetone for 10 mins, known as lift-off process.The heavily p-doped silicon functioned as the gate electrode, and 285 nm SiO2 functioned as the gate dielectric. Before taking all the electrical measurements for this work, the devices areannealed at 200 °C for 20 hours in a high vacuum (~10 -6 mbar) condition. Electrical Characterization: The fabricated device is mounted on a cryogenic four-probe station (Lake Shore) to probe the top and bottom electrodes. The chamber is maintained at a high vacuum of ~10 -6 mbar during all the measurements. For electrical characterization, a semiconductor parameter analyzer system (Keithley 4200A-SCS) is used. A high-speed pulse generator module 4220-PGU and measurement unit 4225-PMU integrated within Keithley 4200 system are used for electrical pulse characterization. In order to prevent photo-excitation of charge carriers, all experiments are performed in dark conditions.Raman spectroscopy and PL mapping: The Raman spectra are collected with a confocal micro-Raman spectrometer (Renishaw Invia) using a laser excitation wavelength of 532 nm in a backscattering configuration employing a 100 Å~ (NA = 0.8) objective. The laser power on the sample was kept low to avoid local heating. The laser exposure time on the sample is kept fixed for 10s with 2 accumulations. However, for PL mapping, the exposure time is reduced to 1s to avoid local heating during the spectral acquisition.
( a )
aThe Raman spectroscopic signature of ML MoS2 show E2g and A1g modes. The Lorentz fit results (solid red lines) provide the high crystallinity of CVD grown ML MoS2 sample. (b) and the current on/off ratio ~ 10 5 . (f) The field-effect mobility ( #$ ) and threshold voltage (Vth) is extracted from the slope of the linear region of the transfer curve. The hysteresis behavior over the temperature range from 100 to 450 K with gate voltage forward sweep from -60 to 40 V and backward sweep from 40 to -60 V for different channel lengths are plotted in Fig. S2. The indiscernible dual sweep curves from 100 to 400 K stipulateOptical image of the MoS2 device having equal channel widths (10 m) and length varies from
1 m to 4 m, Ag is being used as source/drain electrodes (shown in false color). (c)
Photoluminescence mapping confirms the uniform optical grade quality of the MoS2 sample
with robust device fabrication. (d) The output characteristics (Id vs Vds) for fixed gate voltages
(Vg ranging from 0V to 40 V) show the Ohmic contact nature of the as fabricated device. (e)
The transfer characteristics (Id vs Vg) is displayed in both linear and semi-logarithmic scale
for fixed drain-source voltages (25 mV to 100 mV) showing stable electrostatic gate control
The temperature-dependent dual sweep transfer characteristics are conducted to
perceive the evolution of hysteresis in as-fabricated devices under high vacuum (~10 -6 mbar).
weak hysteretic behavior manifesting lessened gate stress effects, unvaried threshold voltages
during gate operations, 8 and hence elicit excellent transistor performance. Nevertheless,
noticeable hysteresis is observed above 400 K, which inflates further at elevated temperatures.
According to previous reports, the origin of hysteresis in MoS2 based FETs can have various
source factors at room temperature, such as adsorption of oxygen/water molecules onto MoS2
channel surface at high bias in ambient pressure, 7 charge trapping/detrapping at MoS2-dieletric
interface, etc. 30 However, at high temperatures, the charge trap density increases due to the
thermal activation of deep trap states in the dielectric, allowing a faster trapping process which
essentially causes outspread hysteresis. 30 A recent report by He et. al. explains that a charge
injection from the back gate to dielectric that drives unique hysteretic crossover characteristics
at relatively higher temperatures. 31 Kaushik et. al. describe independent mechanisms occurring
at room and high temperatures in the case of FETs based on multilayered exfoliated MoS2, that
switches the hysteresis loop from clockwise to anticlockwise. 32
field of neuromorphic computing is growing rapidly, it is crucial to shed light on its high temperature applications. However, a few attempts have been made to understand high temperature neuromorphic learning aspects in integrated synaptic devices.44,45 In this regard, by tuning the presynaptic voltage pulses, we demonstrate near linear synaptic weight updates for high temperature neuro-inspired online learning behavior in our MoS2 based memtransistor, and the results are discussed in the following section.
Conflict of InterestThe authors declare no competing financial interest.Data Availability StatementThe data that support the findings of this study are available in the supplementary material of this article and from the corresponding author upon reasonable request.KeywordsHigh temperature transport; Reverse hysteresis; Monolayer MoS2 transistors; Multi-level Nonvolatile memory; Neuromorphic Computing
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| [] |
[
"GEODESIC AND CONFORMALLY REEB VECTOR FIELDS ON FLAT 3-MANIFOLDS",
"GEODESIC AND CONFORMALLY REEB VECTOR FIELDS ON FLAT 3-MANIFOLDS"
] | [
"Tilman Becker "
] | [] | [] | A unit vector field on a Riemannian manifold M is called geodesic if all of its integral curves are geodesics. We show, in the case of M being a complete flat 3-manifold not equal to E 3 , that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field X on a closed orientable complete flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to X that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of X. Finally, similar results for non-closed flat 3-manifolds are discussed. | 10.2139/ssrn.4195760 | [
"https://export.arxiv.org/pdf/2207.03274v2.pdf"
] | 250,334,683 | 2207.03274 | b10ebd2f73c9749bf98f6646454288375dd7908e |
GEODESIC AND CONFORMALLY REEB VECTOR FIELDS ON FLAT 3-MANIFOLDS
24 May 2023
Tilman Becker
GEODESIC AND CONFORMALLY REEB VECTOR FIELDS ON FLAT 3-MANIFOLDS
24 May 2023
A unit vector field on a Riemannian manifold M is called geodesic if all of its integral curves are geodesics. We show, in the case of M being a complete flat 3-manifold not equal to E 3 , that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field X on a closed orientable complete flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to X that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of X. Finally, similar results for non-closed flat 3-manifolds are discussed.
Introduction
Let M be a manifold and X a nowhere vanishing vector field on M . Then X is called geodesible if there exists a Riemannian metric g on M such that X is of unit length and all of its integral curves are geodesics with respect to g. If the metric g is fixed, X is called geodesic. Analogously, a one-dimensional foliation is called geodesible (resp. geodesic) if there is a geodesible (resp. geodesic) vector field spanning it. Note that, by definition, a geodesible foliation is always orientable.
The class of geodesible foliations or vector fields is a large and interesting one. Examples include:
(1) Killing vector fields of unit length, (2) vector fields that admit a global closed (hyper-)surface of section, (3) Reeb vector fields of contact forms or stable Hamiltonian structures.
For a discussion of these examples and geodesible foliations in general, see [6]. Of particular interest are geodesic foliations of Riemannian manifolds of constant sectional curvature. The study of such foliations was initiated by Gluck and Warner [8] in the 1980s, who described the possible ways the 3-sphere can be fibered by great circles. More than two decades later, Salvai [14] and Harrison [10,12] have given similar characterizations of line fibrations of E 3 (that is, R 3 with the Euclidean metric). More generally, one may consider geodesic foliations of arbitrary flat 3manifolds. The following is the first result of the present paper. Theorem 1. Let M be a complete flat 3-manifold not equal to E 3 . Then any onedimensional oriented geodesic foliation of M is tangent to a two-dimensional totally geodesic foliation.
Remark. (i) Note that we do not assume M to be oriented or closed. Furthermore, the statement is false for geodesic foliations of E 3 . In fact, there exist fibrations of E 3 by pairwise non-parallel oriented lines (so-called skew fibrations), see [10].
(ii) As pointed out by the referee, Theorem 1 may also be proved using Theorem 1 (d) and a slight variation of Lemma 17 in [12].
We will prove Theorem 1 in Section 2 using the fact that every complete flat 3-manifold that is not equal to E 3 can be written as the quotient E 3 /Γ, where Γ is a subgroup of the isometry group of E 3 that contains either a translation or a screw motion. We then look at the lifted geodesic foliation of E 3 (i.e., a fibration by oriented lines), which must be invariant under the action of Γ. Using some elementary geometric arguments, we will see that this forces the line fibration to be tangent to a fibration by affine planes, which is equivalent to the statement in the theorem.
There is also an interesting relation between geodesic foliations and contact structures. Recall that a contact structure on a (2n + 1)-dimensional manifold M is a maximally non-integrable hyperplane field ξ ⊂ T M . That is, if we write ξ locally as the kernel of a 1-form α, then the contact condition α ∧ (dα) n = 0 must hold everywhere. Any such α is called a contact form defining the contact structure ξ. Now given a geodesic foliation, one can consider the hyperplane field given by the orthogonal complement of the corresponding fiber at each point. If this hyperplane field defines a contact structure, one says that the contact structure is induced by the geodesic foliation. Perhaps the most basic example is the Hopf fibration of the standard 3-sphere, which is a great circle fibration inducing the standard contact structure. Gluck [7] has recently shown that any great circle fibration of the 3-sphere induces a contact structure, and that any such contact structure is diffeomorphic to the standard one. However, this does not hold in dimensions ≥ 5, see [9]. The situation in the Euclidean case is a little more restrictive, as was shown by Harrison [11,12]: A geodesic vector field X spanning a line fibration of E 3 induces a contact structure if and only if rank ∇X ≥ 1, where ∇ is the Levi-Civita connection. Similar to the case of S 3 , it has also been shown that any of these contact structures is diffeomorphic to the standard one, see [11] and [1].
To every contact form α there is associated a specific vector field, called the Reeb vector field of α (denoted by R α ). It is the unique vector field spanning the one-dimensional kernel of dα, normalized so that α(R α ) = 1. If a geodesic vector field X induces a contact structure, there is always a corresponding contact form whose Reeb vector field is given by X, namely, α = i X g, where g is the underlying Riemannian metric. Now say we are given a Riemannian 3-manifold M of constant sectional curvature equal to 1. Then M is the quotient of S 3 by the action of some finite subgroup Γ < Isom(S 3 ) of isometries. If X is a geodesic vector field of M , we can lift it to a vector field on S 3 spanning a great circle fibration. By Gluck's result, the lifted vector field induces a contact structure. Since orthogonal complements are preserved under the action of Γ, this implies that X, too, induces a contact structure. In particular, X is the Reeb vector field of some contact form. Similarly, using Harrison's result, a geodesic vector field X on a flat 3-manifold induces a contact structure if and only if rank ∇X ≥ 1, and X is also Reeb in this case. However, consider for example the constant geodesic vector field ∂ z on E 3 . This clearly does not induce a contact structure (the orthogonal complement being a constant plane field), but it is the Reeb vector field of a contact form, namely the standard one given by dz + x dy. That is, unlike in the case of positive constant curvature, the class of geodesic Reeb vector fields on flat 3-manifolds does not coincide with the class of geodesic vector fields that induce contact structures. The natural question then is: What is a (necessary and sufficient) criterion for a geodesic vector field on a flat 3-manifold to be conformally Reeb? Here, a vector field X is said to be conformally Reeb if there is a contact form α and a positive function λ such that X = λ R α (see also [13]). This question is also motivated by Example (3) above: Every Reeb vector field is geodesible, but the converse is not true. Indeed, consider for example the manifold S 2 × S 1 and the geodesible vector field X = ∂ ϕ , where ϕ is the angular coordinate of the S 1 -factor. This vector field cannot be Reeb (not even up to rescaling): If α were a contact form whose Reeb vector field is parallel to X, then dα would restrict to an exact area form on S 2 × {point}, which is not possible due to Stokes' theorem. Generally, vector fields that admit closed global surfaces of section are always geodesible (Example (2) above) but never Reeb. The question can then be seen as a special case of the more general question of whether or not a given geodesible vector field is conformally Reeb. We give an answer to the above question for geodesic vector fields on closed orientable complete flat 3-manifolds. Recall that by the classical Bieberbach theorems [2,3] (see also [16]), any such 3-manifold M can be written, up to affine diffeomorphism (that is, a diffeomorphism preserving the Levi-Civita connection), as the quotient M = T 3 /Γ, where Γ < Isom(T 3 ) is a finite subgroup of isometries acting freely on T 3 , and T 3 is the standard flat 3-torus. Then, a geodesic vector field X on M can be lifted to a geodesic vector field X T on T 3 . We obtain the following result.
Theorem 2. Let X be a geodesic vector field on a closed orientable complete flat 3-manifold M . Then X is conformally Reeb for a contact form α if and only if there is a geodesic vector field Y on M inducing a contact structure ξ such that X is everywhere transverse to ξ.
In this case, writing M as M = T 3 /Γ with Γ < Isom(T 3 ), there is a fibration ζ : T 3 → S 1 whose fibers are totally geodesic 2-tori such that the lifted vector fields X T and Y T are tangent to the fibers of ζ. Furthermore, the lifted contact structures ker α T and ξ T on T 3 are both diffeomorphic to
ker sin vol X |Γ| A ζ E 1 + cos vol X |Γ| A ζ E 2 ,
where • E 1 and E 2 are 1-forms dual to a global orthonormal parallel frame (E 1 , E 2 ) spanning the fibers of ζ,
• A := ζ −1 (a) E 1 ∧ E 2 is the (Euclidean) area of a typical fiber, • vol X is the volume of X.
See section 3 for the definition of the volume of a geodesible vector field. Corollary 3. Let X be a geodesic vector field on a closed flat 3-manifold. If X is conformally Reeb for a contact form α, then the lifted contact structure kerα on R 3 is diffeomorphic to the standard contact structure ker (dz + x dy).
Theorem 2, and subsequently Corollary 3, will be proven in section 4 using the characterization in Theorem 1.
Remark. Both the "if"-and the "only if"-part of the first statement of Theorem 2 are false in general for non-closed manifolds, see section 5.
Acknowledgements. I want to thank my advisor Hansjörg Geiges, as well as Murat Saglam, for many helpful discussions regarding this work. I would also like to thank the referee for the very detailed report, and for pointing out the alternative proof of Theorem 1.
This work is part of a project of the SFB/TRR 191 'Symplectic Structures in Geometry, Algebra and Dynamics', funded by the DFG (Projektnummer 281071066 -TRR 191).
Proof of Theorem 1
Let M be a complete flat 3-manifold not equal to E 3 . Then M can be identified with E 3 /Γ, where Γ < Isom(E 3 ) is a non-trivial discrete subgroup of isometries acting freely. Now let F be a (one-dimensional, oriented) geodesic foliation of M , spanned by a unit vector field X. Then F lifts to a geodesic foliationF of E 3 , spanned by the lifted vector field ∼ X. Here, we viewF = {ℓ} just as a set of lines. For a point p ∈ E 3 , denote by ℓ p ∈F the fiber through p. Note thatF must be invariant under the action of Γ, that is, ℓ γ(p) = γ(ℓ p ) for every γ ∈ Γ and p ∈ E 3 . Now it clearly suffices to prove the statement for the lifted foliationF , since the covering map π : E 3 → M is locally isometric. That is, we have to show that the fibrationF of E 3 by oriented lines is tangent to a fibration by affine planes. A line fibration of this type is also called one-parameter, cf. [12]. To do so, let us take a closer look at the group Γ < Isom(E 3 ). It is well known that every isometry of E 3 (also called Euclidean motion) is given by the composition of a reflection in a plane or rotation about some axis, and some (perhaps trivial) translation. Then one can easily see that any fixed-point free Euclidean motion must be one of the following three:
• a translation;
• a screw motion, i.e. rotation about some axis followed by translation in the direction of this axis; • a glide reflection, i.e. reflection in some plane followed by translation parallel to this plane. Note that applying a glide reflection twice yields a (pure) translation again. Hence, we may assume that the group Γ contains a non-trivial translation or screw motion. We will treat these two cases separately.
First case (Γ contains a translation): Assume that there is some
T v ∈ Γ, where T v is the translation by some vector v ∈ R 3 . If ∼ X is constant, there is nothing to prove. Otherwise, there is a point p 0 ∈ E 3 such that ℓ p0 does not point in the direction of v. Let P ⊂ E 3 be the affine plane through p 0 spanned by v and the cross-product ∼ X(p 0 ) × v.
Then P is transverse to ℓ p0 , so we can consider the projection π : E 3 → P onto P in the direction of ℓ p0 . Define a vector field Y on P by Y (p) := dπ p ( ∼ X(p)), and denote by ℓ Y p the line in P spanned by Y (p). Note that ℓ Y p is just given by the projected line π(ℓ p ). The Z-action on E 3 generated by the translation T v restricts to a Z-action on P , and Y is invariant under this action. Now partition P as P = A ⊔ B, where A = {Y = 0} and B = {Y = 0}. Note that B is precisely the set of points p ∈ P for which ℓ p is parallel to ℓ p0 . Therefore, we may assume that A = ∅, for otherwise, ∼ X is constant and therefore trivially one-parameter. Also, if Y (p) = 0 at some point p ∈ P , then ℓ Y p must be disjoint from B. Indeed, if there were a point q ∈ B ∩ ℓ Y p , then the fiber ℓ p would intersect ℓ q transversely, which is of course not possible. Now we consider two cases. First, assume that there is a point q ∈ A for which Y (q) is parallel to v. Then Y must be parallel to Y (q) on the whole line ℓ Y q . Indeed, if that were not the case, then the set of lines {ℓ Y p : p ∈ ℓ Y q } would fill out a cone that intersects p 0 + Z v, see Figure 1. In particular, there would be some line ℓ Y p intersecting a point in B, which is not possible, as we have seen above. For the same reason, Y must be non-vanishing on ℓ Y q (in fact, we have that Y (p) = Y (q) for all p ∈ ℓ Y q ). It follows that the plane spanned by ℓ Y q and ∼ X(q) is fibered by pairwise parallel lines in F . The same holds for every parallel translate of that plane, and we conclude thatF is one-parameter. Figure 1. The set of lines spanned by Y contains the grey cylinder, which intersects the set of points {p 0 + Z v}.
p0 v q Y (q) ℓ Y q
Thus, we may assume that Y is nowhere parallel to v. Let Q ⊂ E 3 be the affine plane through p 0 spanned by v and ∼ X(p 0 ). Then Q contains infinitely many fibers of F parallel to ℓ p0 , namely, the fibers over points in p 0 + Z v. Note that these points correspond to points in B ⊂ P . If Q ′ is any other affine plane parallel to Q, then there must be fibers contained in Q ′ as well. To see this, denote by U Q and U Q ′ the set of points in Q and Q ′ , respectively, where ∼ X is transverse to Q (resp. Q ′ ). Then, the flow of ∼ X maps U Q diffeomorphically to U Q ′ . But since there is a Z-family of fibers tangent to Q, we see that U Q is either empty or disconnected, so the same must be true for U Q ′ . In particular, U Q ′ = Q ′ , so that there must be fibers in F tangent to Q ′ . All of these fibers must be parallel to ℓ p0 , for otherwise Y is parallel to v (and non-zero) somewhere, and we are in the first case again. Furthermore, the translates of these fibers by integer multiples of v are again fibers of F contained in Q ′ . But then every disc of radius > |v| in P must intersect B in at least one point. Now by an argument similar to the one in the first case, we see that for every point q ∈ A and p ∈ ℓ Y q , we have that Y (p) = Y (q), and we conclude thatF is one-parameter.
Second case (Γ contains a screw motion): Assume that Γ contains a screw motion γ, where γ is given by some rotation followed by translation by some vector v ∈ R 3 . We may assume that the angle of rotation is an irrational multiple of 2π, for otherwise, applying γ some number of k times yields a (pure) translation, and we are in the first case again.
Denote by P the plane through the origin orthogonal to v, and for t ∈ R let P t := P + tv, the parallel translate of P by the vector tv. Consider the fiber ℓ 0 through the origin, and let ℓ t := ℓ tv . We need the following additional lemma.
Lemma 4. Either ℓ t ⊂ P t for all t, or ℓ 0 is parallel to v.
Proof. The statement is equivalent to saying that if ℓ t is transverse to P t for some t, then ℓ t is parallel to v. Therefore, for the sake of contradiction, let us assume that there is some t ∈ R such that ℓ := ℓ t is transverse to P t (and hence transverse to P ) and not parallel to v. For simplicity assume that t = 0. Let π : E 3 → P be the orthogonal projection onto P . Then ℓ projects to a line π(ℓ) ⊂ P . Now, consider the projected lines π(ℓ t ) for t ∈ R. If π(ℓ t ) = π(ℓ) for all t ∈ R, then the lines ℓ t must be pairwise parallel, thus the plane Q spanned by v and ℓ is fibered by (parallel) lines. Then γ must preserve Q in order for the fibrationF to be preserved, which is only possible if γ is trivial, a contradiction. Hence, we may assume that there is some t 0 ∈ R such that π(ℓ t0 ) = π(ℓ). We may further assume (without loss of generality) that t 0 < 0 and that every ℓ t , for t ∈ [t 0 , 0], intersects P transversely (by choosing t 0 close enough to 0). Now let
N := t∈[t0,0] ℓ t ⊂ E 3 .
Then the projection π(N ) ⊂ P contains the cone K ⊂ P given by the convex hull of π(ℓ) and π(ℓ t0 ) (see Figure 2). Let θ be the angle between π(ℓ) and π(ℓ t0 ). Since the angle of rotation of γ is irrational, there is some k ∈ N such that the projection of γ k (ℓ) ∈ F onto P is a line obtained by rotating −π(ℓ) towards the interior of K by an angle of less then θ. In other words, π(γ k (ℓ)) ⊂ Int K. From this we deduce that γ k (ℓ) intersects N . However, since k > 0 > t 0 , we see that γ k (ℓ) N , hence γ k (ℓ) intersects some line in N transversely, a contradiction.
Proof of Theorem 1 (cont.) Using Lemma 4, we now have to consider two cases. The first is that ℓ t ⊂ P t for all t ∈ R. Let us show that, under this assumption, every fiber ofF must be contained in one of the planes P t (in particular, the fibration will be one-parameter). To see this, note first that each of the oriented lines ℓ t divides P t into two open, oriented half-planes ℓ + t and ℓ − t , where ∂ℓ + t = ℓ t and ∂ℓ − t = −ℓ t (that is, ℓ t with the opposite orientation). Here, the orientations of ℓ + t and ℓ − t come from a consistently chosen orientation of the P t . Now assume that there is a point p ∈ P such that ℓ p intersects P (and hence every P t ) transversely. We may assume that ℓ p is not parallel to v, so that the orthogonal projection π(ℓ p ) ⊂ P of ℓ p is a line again. Furthermore, we may assume that π(ℓ p ) intersects ℓ 0 transversely (if that is not the case, simply replace P by an appropriate P t , and ℓ 0 by ℓ t , for some t ∈ R). Now, without loss of generality, let us assume that p ∈ ℓ + 0 . Then the point p t given by the intersection of ℓ p with P t must be contained in ℓ + t for every t ∈ R, for otherwise, the line ℓ p intersects one of the ℓ t transversely. But since π(ℓ p ) intersects ℓ 0 transversely, there is some T > 0 such that π(p t ) ∈ ℓ − 0 for all t > T . Again, since the angle of rotation of γ is irrational, we
N ℓ ℓt 0 γ k (ℓ) Rv θ K P π(ℓ) π(ℓt 0 ) π(γ k (ℓ)) Figure 2. The line γ k (ℓ) intersects N transversely.
can approximate ℓ 0 arbitrarily well by π(γ k (ℓ 0 )) = π(ℓ k ) for large enough k ∈ N,
hence we can approximate ℓ − 0 by π(γ k (ℓ 0 )) − . In particular, there is some k ≥ T such that π(p k ) ∈ ℓ − 0 ∩ π(ℓ k ) − . But then p k ∈ ℓ − k , a contradiction.
The other case is that ℓ 0 is parallel to v (and then, in particular, ℓ t = ℓ 0 for all t). We will show that in this case, every fiber must be parallel to v, and so the fibration is trivially one-parameter. Arguing again by contradiction, we assume that there are fibers not parallel to v. Choose a small closed disc D ⊂ P such that (i) for every p ∈ D, the fiber ℓ p is transverse to D;
(ii) for every p ∈ ∂D, the fiber ℓ p is not parallel to v. Such a disc can be found as follows. First, take a disc D = D r (0) (the closed disc about 0 of radius r > 0) that satisfies (i). Now if (ii) does not hold, then there is some p 0 ∈ ∂D such that ℓ p0 is parallel to v. By applying γ successively (once again using the fact that its rotational angle is irrational) we find that for a dense subset of ∂D, the corresponding fibers must be parallel to v. Then by continuity, this must hold for every fiber through points in ∂D. But then the set of all fibers through ∂D form a straight cylinder parallel to v, and thus every fiber inside that cylinder must be parallel to v as well. In other words, the fibration is constant over D. But since the fibration is assumed to be globally non-constant, we find a larger disc, again called D, so that (i) is still satisfied and the fibration is not constant over D. Then D has to satisfy (ii) as well. Now let Σ := {ℓ p : p ∈ ∂D} be the surface consisting of all fibers through points in ∂D. Let Σ t := Σ ∩ P t , with P t = P + tv as before, and let π(Σ t ) be its projection to P . We shall prove that there is some T > 0 such that for all t ∈ R with t > T or t < −T we have that
(1) D ⊂ Int π(Σ t ),
where Int π(Σ t ) denotes the interior of π(Σ t ), that is, the connected component of P t \ π(Σ t ) bounded by π(Σ t ) with compact closure. Indeed, π(Σ t ) is obtained from π(Σ 0 ) = ∂D by flowing in the direction of the projected lines π(ℓ p ), p ∈ ∂D. Denote this flow by Φ τ . For T large enough and t > T or t < −T , the set Φ t (∂D) lies outside of D, that is, Φ t (∂D) ⊂ P \ D. The fact that we can write D instead of Int D here is because the ℓ p project to lines and not points, due to property (ii) above; hence no point on ∂D is fixed under the flow Φ. Since none of the projected lines point to the origin (due to Lemma 4), the origin stays in the interior while applying the flow, from which (1) follows. This is illustrated in Figure 3. Now let k > T and consider the surfaceΣ := γ k (Σ). Then, sinceF is invariant under the action of Γ, we see thatΣ, too, is a union of fibers ofF . Hence, either Σ andΣ are disjoint, or they intersect in a set of common fibers. In particular, the intersection Σ ∩Σ is either empty or there is a non-empty intersection in every t-level, that is, Σ t ∩Σ t = ∅ for every t. On the other hand, from (1) we deduce that π(Σ k ) = ∂D ⊂ Int π(Σ k ), henceΣ k ⊂ Int Σ k . Similarly, one shows that Σ 0 ⊂Σ 0 , see Figure 4 below. But this means that Σ ∩Σ = ∅ while Σ k ∩Σ k = ∅, a contradiction.
The volume of a geodesible vector field
In this section, we introduce the notion of volume for a geodesible vector field, following [5]. Let X be a non-vanishing vector field on a manifold M . By a result of Wadsley [15], X is geodesible if and only if there exists a 1-form α such that α(X) = 1 and i X dα = 0. This 1-form is called connection form (sometimes also characteristic form) for X. Note that, in particular, this characterization implies the geodesibility of Reeb vector fields. Now, assume that M is closed and dim M = 3. Define the volume of X as
vol X := M α ∧ dα.
This does not depend on the specific choice of the connection form α, as follows from the following identity for arbitrary 1-forms α and β:
(2) α ∧ dα − β ∧ dβ = (α − β) ∧ (dα + dβ) + d(α ∧ β).
Namely, if α and β are connection 1-forms for X, then (α − β) ∧ (dα + dβ) = 0, thus, (2) implies that Figure 4. The surfaces Σ andΣ intersect transversely.
M α ∧ dα − M β ∧ dβ = M d(α ∧ β) = 0, Σ Σ P k P0
where the last equality follows from the assumption of M being closed. Similarly, one can define the volume of geodesible vector fields on higher-dimensional manifolds. I refer the reader to [5] for more details, also regarding the computation of vol X . Now, let p : M → N be a k-fold covering and X a geodesible vector field on N . Let α be a connection form for X. Then the 1-form p * α is clearly a connection form for the lifted vector field Y (in particular, Y is geodesible). The volumes of X and Y are related as
(3) vol Y = M p * (α ∧ dα) = k N α ∧ dα = k vol X .
Formula (2) can also be used to derive a slight generalization of Proposition 2.1 in [5], see Lemma 5 below. Before formulating the statement, let us introduce some notation that will also be used throughout the remainder of this paper. Given two vector fields X and Y on a manifold M , we write X ∼ Y if there is a positive function λ ∈ C ∞ (M ) such that Y = λX.
Lemma 5. Let α 0 and α 1 be two contact forms on a closed 3-manifold M such that R α0 ∼ R α1 . Then, the contact structures ker α 0 and ker α 1 are diffeomorphic.
Proof. Note that it suffices to prove that α 0 ∧ dα 0 and α 1 ∧ dα 1 define the same orientation of M . Indeed, if this is the case, the 2-forms dα 0 and dα 1 restrict to nondegenerate forms defining the same orientation on any hyperplane field transverse to R α0 ∼ R α1 . Then, the 1-form α t := (1 − t) α 0 + t α 1 is contact for every t ∈ [0, 1], and the statement follows from Gray stability (see [4,Thm. 2
.2.2]).
Therefore, for the sake of contradiction, assume that the orientations induced by α 0 and α 1 are opposite. Since R α0 ∼ R α1 , the 2-forms dα 0 and dα 1 must be multiples of each other, so we may write dα 1 = µ dα 0 where µ is a function M → R >0 . Also, set λ := α 1 (R α0 ) ∈ C ∞ (M, R >0 ). It follows that
α 1 ∧ dα 1 = λµ α 0 ∧ dα 0 , and (α 0 − α 1 ) ∧ (dα 0 + dα 1 ) = (1 − λ)(1 + µ) α 0 ∧ dα 0 . Then, identity (2) implies that M (1 − λµ) α 0 ∧ dα 0 = M α 0 ∧ dα 0 − M α 1 ∧ dα 1 = M (α 0 − α 1 ) ∧ (dα 0 + dα 1 ) = M (1 − λ)(1 + µ) α 0 ∧ dα 0 .
But then M (µ−λ) α 0 ∧dα 0 must vanish, which is impossible since µ−λ is assumed to be negative everywhere. Hence, we arrive at a contradiction.
Proof of Theorem 2
Let M be a closed orientable complete flat 3-manifold. As discussed in the introduction, M can be written (up to affine diffeomorphism) as M = T 3 /Γ, where Γ < Isom(T 3 ) is a finite subgroup of isometries of T 3 acting freely and orientationpreservingly. Here, T 3 is the standard flat 3-torus, that is,
T 3 = E 3 /(2π Z 3 )
, where the Z 3 -action is generated by translations in the standard coordinate directions. By the following proposition, it is in fact enough to consider M = T 3 . Proposition 6. Let X be a geodesic vector field on M = T 3 /Γ and X T its lift to T 3 . Then X is conformally Reeb if and only if X T is conformally Reeb.
Proof. Assume first that X is conformally Reeb. That is, there is a contact form α on M such that X ∼ R α . Let π : T 3 → M be the natural projection. Then p * α is again a contact form, and clearly R p * α ∼ X T .
Conversely, assume that X T ∼ Rα for some contact formα on T 3 . Since |Γ| < ∞, we can average under the action of Γ to get a 1-form
α := 1 |Γ| γ∈Γ γ * α .
Then α is again a contact form, since γ * X T = X T and dγ maps the hyperplane field X ⊥ T orientation-preservingly to itself, for every γ ∈ Γ. Here we are using the fact that in dimension 3, a 1-form β with β(X T ) > 0 is contact if and only if dβ is non-vanishing on any hyperplane field transverse to X T . It also follows that R α = Rα ∼ X T . Now since γ * α = α for every γ ∈ Γ, α descends to a contact form on M , whose Reeb vector field is a multiple of X.
We may now, for the remainder of the section, assume that M = T 3 . We may further assume that the geodesic vector field X is not constant, for otherwise, there is an embedded 2-torus transverse to X and so by Stokes' theorem, X cannot be (conformally) Reeb. Let ∼ X be the lift of the geodesic vector field X to E 3 . By Theorem 1, ∼ X is tangent to a fibration P of affine planes. Now choose a parallel orthonormal frame (E 1 , E 2 , E 3 ) of E 3 such that E 1 and E 2 span the fibers of P. This frame descends to an orthonormal frame of T 3 , which we call (E 1 , E 2 , E 3 ) again. Then E 1 and E 2 span the leaves of the totally geodesic foliation P T of T 3 covered by P. Let us see that P T is in fact a T 2 -fibration over S 1 . First note that the leaves are embedded copies of T 2 . Indeed, each leaf P T ∈ P T is covered by a plane P ∈ P, hence P T is either a 2-torus, or a dense immersed cylinder S 1 × R, or a dense immersed copy of R 2 . But since X is constant on each leaf, the existence of dense leaves would force X T to be globally constant, which we already ruled out.
Next, define a map ζ : T 3 → S 1 as follows. Fix a 2-torus P T ∈ P T covered by a plane P ∈ P. Then, the orbit of P under the action of 2π Z 3 is a discrete set of equidistant planes. This follows from the fact that P T is not dense in T 3 , and that the Z 3 -action on E 3 commutes with the action of R 3 by translations. For the same reason, the minimal distance t 0 of two such planes does not depend on the specific fiber P T of P T . Then, for q ∈ T 3 , we may define ζ(q) as
ζ(q) := 2π t q t 0 mod 2π ∈ S 1 = R/2πZ,
where t q > 0 is the smallest number such that Φ tq (q) (where Φ is the flow of E 3 ) lies in the fiber P T . This defines a fibration of T 3 whose fibers are the elements of P T . Now we can write X as
(4) X = sin θ(ζ) E 1 + cos θ(ζ) E 2
for some function θ : S 1 → S 1 . Using the identification S 1 = R/2πZ, we may think of θ (or any other function S 1 → S 1 ) as a function R → R, such that θ(t+2π)−θ(t) ∈ 2π Z for all t ∈ R. In fact, we have that θ(t+2π)−θ(t) = 2π deg θ, where deg θ is the degree of θ as a map of S 1 . By θ ′ we mean the usual derivative of θ when viewed as a function defined on R.
We will first prove the following proposition.
Proposition 7. Let X be a geodesic vector field on T 3 . Then the following are equivalent.
(i) X induces a contact structure.
(ii) deg θ = 0 and for any a, b ∈ R, a < b we have that
θ(b) − θ(a) > −π, if deg θ > 0, and θ(b) − θ(a) < π, if deg θ < 0. (iii) The set B := ϕ ∈ C ∞ (S 1 , S 1 ) : ϕ ′ = 0, |ϕ − θ| < π 2 is non-empty (here, |.| is the Euclidean norm (modulo 2π)).
Proof. We first show that (iii) implies (ii). So assume that (iii) holds, and choose some ϕ ∈ B. Note that deg θ = deg ϕ = 0. If deg θ > 0, then ϕ ′ must be positive everywhere. Then, for a < b,
θ(a) − π 2 < ϕ(a) < ϕ(b) < θ(b) + π 2 ,
which implies that θ(b) − θ(a) > −π. A similar argument applies for the case of deg θ being negative. Conversely, if (ii) holds, we need to show that B = ∅. We will do so by constructing some ϕ ∈ B explicitly. First, perhaps after applying a C 0 -small perturbation, we may assume that θ has finitely many local minima and maxima, respectively, and no other critical points. That is, there is a subdivision 0 < a 1 < b 1 < . . . < a n < b n < 2π, such that θ has a local maximum at every a k and a local minimum at every b k . Define intervals I k by
I k := [θ(a k ) − π/2, θ(b k ) + π/2].
Note that it follows from (ii) that I k does indeed define an interval with non-empty interior. Moreover, we have that (5) max I k > min I l for every k = 1, . . . , n and every l = 1, . . . , k. Now we want to find some numbers
c 1 ≤ c 2 ≤ . . . ≤ c n ,
such that c k ∈ I k for every k. Given such numbers, we can define a function φ with the following properties:
• |φ − θ| < π/2 everywhere; • φ is constantly equal to c k on [a k , b k ];
• φ is non-decreasing. This is illustrated in Figure 5. This construction is possible since θ is strictly increasing on (b k , a k+1 ). Then φ can be approximated by a function in B. Therefore, all we are left to do is find numbers c k as above. This is best done reversely, starting with c n . Set c n := max I n . The remaining c k are defined inductively as
c k := min{c k+1 , max I k } ≤ c k+1 .
Note that c k ∈ I k since c k is given by the maximum of some I l , l ≥ k, so that c k > min I k by (5). This concludes the proof of the equivalence of (ii) and (iii). Figure 5. Construction of a function φ that can be approximated by a function in B.
2π deg θ c k I k π 2 − π 2 a k b k 2π θ φ
Now, let us see how (iii) implies (i). Given ϕ ∈ B as in (iii), consider the 1-form
α = sin ϕ(ζ) E 1 + cos ϕ(ζ) E 2 , where (E 1 , E 2 , E 2 ) is the dual frame to (E 1 , E 2 , E 3 ). A simple calculation shows that dα = ϕ ′ (ζ) cos ϕ(ζ) E 3 ∧ E 1 − ϕ ′ (ζ) sin ϕ(ζ) E 3 ∧ E 2 , hence α ∧ dα = ϕ ′ (ζ) E 1 ∧ E 2 ∧ E 3 = 0,
so α is a contact form. Its Reeb vector field is given by
R α = sin ϕ(ζ) E 1 + cos ϕ(ζ) E 2 .
We claim that, for a suitably chosen ϕ ∈ B, there are functions f, g ∈ C ∞ (S 1 , R >0 ), such that
(6) f (ζ) X = R (1/g(ζ))α . Generally, for h ∈ C ∞ (T 3 , R >0 ), we have that R (1/h)α = h R α + Y ,
where Y is the unique vector field satisfying α(Y ) = 0 and
(7) i Y dα = dh(R α ) α − dh. Now, if h = g • ζ for some function g ∈ C ∞ (S 1 , R >0 ), then dh(R α ) = dg • dζ(R α ) = 0, so (7) translates to i Y dα = −dh = −dg • dζ = −g ′ (ζ) E 3 ,
where we again think of g as a 2π-periodic function R → R, with g ′ being its usual derivative. Then, to solve equation (6), we need to find functions f and g such that
Y := f (ζ) X − g(ζ) R α satisfies (8) 0 = α(Y ) = f (ζ) α(X) − g(ζ),
as well as
(9) i Y dα = −g ′ (ζ) E 3 .
Now (8) is equivalent to g = f cos(ϕ − θ), which is positive iff f is positive, since |ϕ − θ| < π/2. Then (9) translates to
f ϕ ′ sin(ϕ − θ) E 3 = −(f ′ cos(ϕ − θ) − f (ϕ ′ − θ ′ ) sin(ϕ − θ)) E 3 ,
where we refrained from writing ζ in the arguments for simplicity. This, in turn, reduces to
f ′ cos(ϕ − θ) + f θ ′ sin(ϕ − θ) = 0.
This differential equation is being solved by
f (x) := exp − x 0 tan(ϕ(t) − θ(t)) θ ′ (t)dt > 0.
However, for a generic choice of ϕ, f is not 2π-periodic, hence it does not define a function on S 1 . Note that f is 2π-periodic if and only if
I(ϕ) := 2π 0 tan(ϕ(t) − θ(t)) θ ′ (t)dt
vanishes. Therefore, we need to show that the function I : B → R has a zero. First observe that since 2π 0 tan(ϕ(t) − θ(t))(ϕ ′ (t) − θ ′ (t)) dt = x0 x0 tan(u) du = 0, we have that
I(ϕ) = 2π 0 tan(ϕ(t) − θ(t)) ϕ ′ (t) dt.
Now, it is easy to see that B is convex. Hence, it suffices to find functions ϕ + , ϕ − ∈ B such that I(ϕ + ) > 0 and I(ϕ − ) < 0. For then we can simply interpolate between ϕ + and ϕ − to find a zero of I. To achieve this, one can adjust the construction of ϕ in the proof of (ii) ⇒ (iii) so that ϕ < θ wherever ϕ is not constant (in fact, the function ϕ drawn in Figure 5 has this property). By approximating this function with a function in B, we obtain a function ϕ − ∈ B with I(ϕ − ) < 0. The function ϕ + is constructed similarly.
To finish the proof, we show that (i) implies (ii). Assume that X ∼ R α for some contact form α of T 3 . Suppose, for the sake of contradiction, that (ii) does not hold. Assume for the moment that deg θ > 0. Then (ii) being false means that there are a, b ∈ [0, 2π] with a < b such that θ(b) − θ(a) = −π, as well as c, d ∈ [0, 2π] with b < c < d such that θ(c) = θ(b) and θ(d) = θ(a) (since deg θ > 0). Furthermore, we may choose a, b and c, d so that (10) θ
(x) ∈ [θ(b), θ(a)] for all x ∈ [a, b] ∪ [c, d].
Now choose a point p ∈ E 3 that projects to a point in ζ −1 (a) ⊂ T 3 and let P be the affine plane in E 3 through p spanned by E 3 and ∼ X(p). Assume for the moment that P covers a 2-torus in T 3 , which we call Σ. Consider the two subsets
Σ 1 := Σ ∩ {a ≤ ζ ≤ b}, Σ 2 := Σ ∩ {c ≤ ζ ≤ d}.
Both Σ 1 and Σ 2 are diffeomorphic to cylinders, and their boundaries are integral curves of X. Choose any orientation for Σ and orient Σ 1 and Σ 2 accordingly as submanifolds of Σ. Denote the (oriented) boundary curves of Σ 1 and Σ 2 by
∂Σ 1 = γ a ⊔ γ b , ∂Σ 2 = γ c ⊔ γ d .
We may choose the orientation of Σ so that γ a and γ b are negatively tangent to X, whereas γ c and γ d are positively tangent, see Figure 6. It follows that
Σ1 dα = γa α + γ b α < 0, and Σ2 dα = γc α + γ d α > 0.
However, it follows from (10) that X (and then also R α ) is positively transverse to the interiors of both Σ 1 and Σ 2 . Then, since Σ 1 and Σ 2 are oriented consistently, Σ1 dα and Σ2 dα must have the same sign, and we arrive at a contradiction.
We are left to deal with the case of P covering some dense infinite cylinder in T 3 (instead of a 2-torus). Parametrize P using coordinates s and t, so that ∂ s is identified with ∼ X(p). Consider subsets of the form
P s0 := P ∩ {−s 0 ≤ s ≤ s o } ⊂ P
for some s 0 > 0. Then P s0 covers a cylinder in T 3 , which we call Σ = Σ s0 . Defining Σ 1 = Σ s0 1 as before, we now have
∂Σ 1 = ∂ v Σ 1 ∪ ∂ h Σ 1 , where ∂ v Σ 1 = γ a ⊔ γ b and ∂ h Σ 1 = ∂Σ 1 ∩ ∂Σ.
In other words, ∂ v Σ 1 and ∂ h Σ 1 are the "vertical" and "horizontal" part of ∂Σ 1 , respectively. Note that, since α is non-zero on the vertical boundary components, we have that
(11) ∂v Σ t 0 1 α > ∂v Σ s 0 1 α for t 0 > s 0 . Now let C := ∂v Σ s 0 1 α
for some s 0 , and choose t 0 > s 0 large enough so that
∂ h Σ t 0 1 α < C.
This can be done due to the fact that P covers a dense cylinder in T 3 . Then (11) implies that
sgn ∂Σ t 0 1 α = sgn ∂v Σ t 0 1 α ,
and the same may be assumed for Σ t0 2 . Then, using the same reasoning as in the first case, we arrive at a contradiction again. Figure 6. Σ 1 and Σ 2 .
Σ1 Σ2 Σ a b c d ζ X X γa γ b γc γ d
The case deg θ < 0 is analogous. Now we are still left to show that deg θ is indeed non-zero. Note that if deg θ = 0 and the image of θ is contained in an open interval of length at most π, then there is an embedded 2-torus transverse to X, so that X cannot be conformally Reeb. Therefore, we may again assume that there are a < b < c < d with θ(b) − θ(a) = ∓π, θ(c) = θ(b) and θ(d) = θ(a), and we arrive at a contradiction using the same argument as before.
Proof of Theorem 2. Assume first that there is a geodesic vector field Y on M inducing a contact structure ξ such that X is everywhere transverse to ξ. In other words, X and Y are nowhere orthogonal. Then the same is true for the lifted vector fields X T and Y T on T 3 . In particular, both X T and Y T are non-constant, so by the discussion prior to Proposition 7, there are fibrations T X and T Y of T 3 by 2-tori tangent to X T and Y T , respectively. These two fibrations must coincide: Indeed, if this were not the case, we could consider a loop in some T ∈ T X that is transverse to X T and also transverse to T Y . Along this loop, X T is constant, whereas Y T must make at least one complete turn (since Y T induces a contact structure), hence X T and Y T are orthogonal somewhere, a contradiction. Therefore, writing X T as X T = sin θ(ζ) E 1 + cos θ(ζ) E 2 as in (4), we find that Y T is of the form (12) Y T = sin ϕ(ζ) E 1 + cos ϕ(ζ) E 2 for some function ϕ : S 1 → S 1 with ϕ ′ = 0. We also have that |ϕ − θ| < π/2 everywhere. Thus, it follows from Proposition 7 that X is conformally Reeb. Conversely, assume that X is conformally Reeb, that is, X ∼ R α for some contact form α on M = T 3 /Γ. Write the lifted vector field on T 3 again as X T = sin θ(ζ) E 1 + cos θ(ζ) E 2 . Then, by Proposition 7, there is a function ϕ : S 1 → S 1 such that ϕ ′ = 0 and |ϕ − θ| < π/2 everywhere. In particular, the geodesic vector field Y T := sin ϕ(ζ) E 1 + cos ϕ(ζ) E 2 induces a contact structure and is nowhere orthogonal to X T . If M = T 3 , then Y = Y T and we are done. So suppose that M is not equal to T 3 , that is, M = T 3 /Γ, where Γ is a non-trivial subgroup of Isom(T 3 ). We want to adjust the construction of Y T (resp. ϕ) so that it is invariant under the action of Γ, and therefore descends to a geodesic vector field Y on M . First note that every element of Γ must be a screw motion of finite order in Γ, since glide reflections are not orientation-preserving. If γ ∈ Γ is such a screw motion, then γ must preserve the fibration T of 2-tori defined by ζ. Indeed, if there were some T ∈ T such that γ(T ) / ∈ T , then γ(T ) would intersect every fiber of T transversely. Now since X T is constant along each fiber of T and also constant along γ(T ) (since γ * X T = X T ), it would follow that X T is globally constant. In particular, X T cannot be Reeb, a contradiction. But this means that the axis of rotation of γ (and consequently its translational part) must be orthogonal to T . In other words, the translation vector of γ is a multiple of E 3 . Now choose γ 0 ∈ Γ so that the absolute value of its translational part is minimal among all elements of Γ. Then γ 0 generates Γ (in particular, Γ is cyclic). Write γ 0 as γ 0 = T λ E3 • R φ , where R φ is rotation about the axis spanned by E 3 of angle φ, and T λ E3 is translation by the vector λ E 3 for some real number λ. Then, it suffices to choose ϕ so that
ϕ(t + λ) = ϕ(t) + φ for all t, for then ϕ • ζ • γ 0 = ϕ • ζ + φ which implies that (γ 0 ) * Y T = Y T .
To find an appropriate ϕ, we can construct ϕ first on the interval [0, λ] as in the proof of Proposition 7, and then extend it to the whole real line via ϕ(t+λ) := ϕ(t)+φ. Of course one has to be a little careful regarding smoothness of ϕ, but this can be arranged easily. The vector field Y T we end up with is invariant under Γ, hence it descends to a vector field Y .
To prove the second statement of the theorem, note that in Proposition 7 it is actually shown that if X T is conformally Reeb for some contact form α T , then it is also conformally Reeb for a multiple of the contact form α ϕ = sin ϕ(ζ) E 1 + cos ϕ(ζ) E 2 whose kernel defines the contact structure ξ T . Then, by Lemma 5, ker α T and ξ T are diffeomorphic. Now set n := 2π deg ϕ = 2π deg θ = θ(2π) − θ(0).
Then α ϕ pulls back to α n := sin(n ζ) E 1 + cos(n ζ) E 2 via the diffeomorphism
T 3 −→ T 3 , p −→ (Φ ϕ −1 (n ζ(p)) • Φ −ζ(p) )(p),
where Φ denotes again the flow of E 3 . On the other hand,
|Γ| vol X = vol XT = T 3 α T ∧ dα T = T 3 θ ′ (ζ) E 1 ∧ E 2 ∧ E 3 = θ(2π) ζ −1 (2π) E 1 ∧ E 2 − θ(0) ζ −1 (0) E 1 ∧ E 2 = n A,
where the first equation follows from (3). Hence, n = |Γ| vol X /A. .
Proof of Corollary 3.
Choose global coordinates (x, y, z) for R 3 such that the frame (E 1 , E 2 , E 3 ) on T 3 is covered by the coordinate frame (∂ x , ∂ y , ∂ z ). Then, by Theorem 2, kerα is diffeomorphic to the kernel of α n = sin(nz) dx + cos(nz) dy, which pulls back to α st = dz + x dy via the diffeomorphism (x, y, z) −→ z sin(ny) − x cos(ny) n , z cos(ny) + x sin(ny) n , y .
Open flat 3-manifolds
We start with the following general result.
Proposition 8. Let M be an orientable 3-manifold with H 2 dR (M ) = 0, and X a non-vanishing vector field on M whose flow induces a free, proper R-action. Then X is the Reeb vector field of a contact form.
Proof. Since X induces a free and proper R-action that is also orientation-preserving, the orbit space B = M/R is an orientable 2-dimensional manifold, and the projection p : M → B defines a principal line bundle, which is necessarily trivial. That is, we can identify M with B×R, where the R-fibers correspond to the integral curves of X. Now B is a deformation retract of M , so we have that H 2 dR (B) = H 2 dR (M ) = 0. Hence, there is an exact area form ω = dβ on B. Let t denote the coordinate of the R-fibers of M = B × R. Then the 1-form α := dt + p * β is contact, and R α = ∂ t = X. Corollary 9. Let X be an aperiodic geodesic vector field on E 3 or R 2 × S 1 . Then X is the Reeb vector field of a contact form.
Remark. Of course, in the case of E 3 , every geodesic vector field is aperiodic; hence, Corollary 9 implies that every geodesic vector field on E 3 is Reeb.
The following two examples are to show that Theorem 2 is not true in general for non-closed manifolds.
Example 10. (i) Let M be equal to S 1 × R 2 or T 2 × R with coordinates (x, y, z) and consider the geodesic vector field X = sin θ(z) ∂ x + cos θ(z) ∂ y ,
where θ : R → R is a smooth function defined as follows. Set θ(0) = θ(2π) = 0, θ(π) = −π, and θ(z) ≈ −z, 0 ≤ z ≤ π, z − 2π, π ≤ z ≤ 2π, where the approximation is C 0 -close. Then extend θ to a 2π-periodic function defined on R. Since θ(π) − θ(0) = −π, the condition of Proposition 7 (or Theorem 2) is not satisfied. However, X is still conformally Reeb. To see this, consider the 1-form β = F (z) dx + y sin θ(z) dz, where Then dβ = cos θ(z) dz ∧ dx + sin θ(z) dy ∧ dz is non-degenerate on the plane field η spanned by ∂ z and cos θ(z) ∂ x − sin θ(z) ∂ y , and i X dβ = 0. Furthermore,
β(X) = F (x) sin θ(x) ≈ sin x(1 − sin x) ≥ 0, if 0 ≤ x ≤ π, sin x(sin x − 1) ≥ 0, if π < x ≤ 2π.
That is, β(X) ≥ −ε for some arbitrarily small ε > 0. Now choose ε so that 1 + 2 ε θ ′ > 0 everywhere, and consider the 1-form α := β + 2 ε α θ ,
where α θ = sin θ(z) dx + cos θ(z) dy. Then dα = (1 + 2 ε θ ′ ) >0 dβ is again non-degenerate on η, and α(X) = β(X) + 2 ε ≥ ε > 0. Therefore, as X is transverse to η and i X dα = 0, it follows that α is a contact form with Reeb vector field R α = (1/α(X)) X.
(ii) Let M = T 2 × R with coordinates (x, y, z) and choose a diffeomorphism ϕ : R ∼ = − → (−π/4, π/4). Define geodesic vector fields X and Y on M by X = ∂ y + ∂ z , Y = sin ϕ(z) ∂ x + cos ϕ(z) ∂ y .
Then Y induces a contact structure and X, Y = cos ϕ(z) > 0. But X is transverse to the 2-torus {z = 0}, hence X cannot be conformally Reeb.
Figure 3 .
3D is contained in the interior of π(Σ t ).
F
(
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Weyertal 86-90, 50931 Köln, Germany Email address: [email protected]. Mathematisches Institut, deUniversität zu KölnMathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany Email address: [email protected]
| [] |
[
"Minimality of B-free systems in number fields",
"Minimality of B-free systems in number fields"
] | [
"Aurelia Dymek ",
"Stanisław Kasjan ",
"Joanna Kułaga-Przymus "
] | [] | [] | Let K be a finite extension of Q and O K be its ring of integers. Let B be a primitive collection of ideals in O K . We show that any B-free system is essentially minimal. Moreoever, the B-free system is minimal if and only if the characteristic function of B-free numbers is a Toeplitz sequence. Equivalently, there are no ideal d and no infinite pairwise coprime collection of ideals C such that dC ⊆ B. Moreover, we find a periodic structure in the Toeplitz case. Last but not least, we describe the restrictions on the cosets of ideals contained in unions of ideals.Contents the corresponding set of multiplies, by M B . These sets were studied intensively from the number theoretic viewpoint (see[15], and [9] for more references). Our approach is dynamical and it is the continuation of the line of research initiated by Sarnak. Let η ∈ {0, 1} Z stand for the characteristic function of F B and let X η be the orbit closure of η with respect to the left shift σ on {0, 1} Z . The resulting subshift (X η , σ) is called a B-free system. In 2010 Sarnak suggested to study the so-called square-free system, i.e. the B-free system for B being the set of squares of all primes. The underlying motivation was to understand the random nature of the arithmetic Möbius function µ -note that µ 2 is the characteristic function of the set of square-free integers.As each set bZ is a subgroup of the additive group Z and also an ideal of the ring Z, there are two natural ways of generalizing the notion of a B-free system to higher dimensions. Namely, as the counterpart of the set of multiples M B we choose • countable unions of sublattices of Z m , • countable unions of ideals in the rings of integers O K of algebraic number fields K. Similarly to [7], we consider a collection B of non-zero ideals in O K . Put F B := O K \ b∈B b and let X η be the orbit closure ofRecall that:(A) Each B-free subshift is essentially minimal, i.e. (X η , σ) has a unique minimal subset (which is the orbit closure of a Toeplitz sequence ([9, Theorem A])).(B) The B-free subshift (X η , σ) is minimal if and only if it is a Toeplitz system ([9, Corollary 1.4]).Then, in [17] the following result related to (B) was shown:(B') The minimality of (X η , σ) implies that η itself is a Toeplitz sequence, under the extra assumption that B is taut.Finally, Keller [18] showed the following:(A') The unique minimal subset of (X η , σ) is also a B-free system corresponding to a set B * , which is a certain modification of B. | 10.3934/dcds.2023056 | [
"https://arxiv.org/pdf/2207.05396v1.pdf"
] | 250,450,902 | 2207.05396 | 4bd1b6ab5f13667d23e4f03e3d9ae5142debed29 |
Minimality of B-free systems in number fields
Jul 2022
Aurelia Dymek
Stanisław Kasjan
Joanna Kułaga-Przymus
Minimality of B-free systems in number fields
Jul 2022
Let K be a finite extension of Q and O K be its ring of integers. Let B be a primitive collection of ideals in O K . We show that any B-free system is essentially minimal. Moreoever, the B-free system is minimal if and only if the characteristic function of B-free numbers is a Toeplitz sequence. Equivalently, there are no ideal d and no infinite pairwise coprime collection of ideals C such that dC ⊆ B. Moreover, we find a periodic structure in the Toeplitz case. Last but not least, we describe the restrictions on the cosets of ideals contained in unions of ideals.Contents the corresponding set of multiplies, by M B . These sets were studied intensively from the number theoretic viewpoint (see[15], and [9] for more references). Our approach is dynamical and it is the continuation of the line of research initiated by Sarnak. Let η ∈ {0, 1} Z stand for the characteristic function of F B and let X η be the orbit closure of η with respect to the left shift σ on {0, 1} Z . The resulting subshift (X η , σ) is called a B-free system. In 2010 Sarnak suggested to study the so-called square-free system, i.e. the B-free system for B being the set of squares of all primes. The underlying motivation was to understand the random nature of the arithmetic Möbius function µ -note that µ 2 is the characteristic function of the set of square-free integers.As each set bZ is a subgroup of the additive group Z and also an ideal of the ring Z, there are two natural ways of generalizing the notion of a B-free system to higher dimensions. Namely, as the counterpart of the set of multiples M B we choose • countable unions of sublattices of Z m , • countable unions of ideals in the rings of integers O K of algebraic number fields K. Similarly to [7], we consider a collection B of non-zero ideals in O K . Put F B := O K \ b∈B b and let X η be the orbit closure ofRecall that:(A) Each B-free subshift is essentially minimal, i.e. (X η , σ) has a unique minimal subset (which is the orbit closure of a Toeplitz sequence ([9, Theorem A])).(B) The B-free subshift (X η , σ) is minimal if and only if it is a Toeplitz system ([9, Corollary 1.4]).Then, in [17] the following result related to (B) was shown:(B') The minimality of (X η , σ) implies that η itself is a Toeplitz sequence, under the extra assumption that B is taut.Finally, Keller [18] showed the following:(A') The unique minimal subset of (X η , σ) is also a B-free system corresponding to a set B * , which is a certain modification of B.
Background and main results
Given a set B of natural numbers, we say that an integer n is B-free, if no number in B divides n. The set of B-free integers is denoted by F B and its complement, which is D = d ⊆ O K : d is a non-zero ideal 1 such that dC ⊆ B for some infinite pairwise coprime collection C of ideals in
O K } ,(1)B * = (B ∪ D) prim and η * = ½ F B * .
Theorem A (see [18,Corollary 5] for O K = Z). Let B be a primitive collection of ideals in O K . Then η * is an O K -Toeplitz array and X η * is a unique minimal subset of X η . Moreover, η * ≤ η. Notice that even in the one-dimensional case this is a strenghtening of the earlier results. Namely, our methods allow us to skip the technical (as it turns out) assumption that B is taut in order to deduce that η is a Toeplitz sequence whenever (X η , σ) is minimal. The proofs substantially differ from the earlier ones, even in the one-dimensional case.
The proof of Theorem A relies on the following result, which, in our opinion, catches the essence of what is happening in the Toeplitz case.
Theorem C. Let B be primitive. Then the following are equivalent: Let us now discuss the remaining important ingredients in more details. Clearly, what prevents η from being Toeplitz, is the presence of non-periodic positions. We provide a description of the set of such positions.
Proposition D. We have
(2) O K \ s⊂O K Per(η, s) = M D ∩ F B = O K \ i≥1 Per(η, s i ),
where D is given by (1), (s i ) i≥1 is an ideal filtration of O K and summation is over all nonzero ideals s ⊆ O K .
"As a bonus", Proposition D allows us to find a periodic structure of η in the Toeplitz case.
Theorem E. Let B be a primitive collection of ideals in O K . Suppose S 1 ⊂ S 2 ⊂ · · · ր B is a saturated filtration of B by finite collections of ideals in O K . Assume that η is an O K -Toeplitz array. Then ( b∈S i b) i≥1 is a periodic structure of η.
The last key ingredient of this paper is a result of arithmetic nature, on cosets of ideals ("arithmetic progressions") contained in the set of multiples in case when η is Toeplitz.
Theorem F (see Proposition 3.14 and the preceeding comments). Let B be a primitive collection of ideals in O K such that D = ∅. Let r ∈ O K and let a ⊂ O K be an ideal. If r + a ⊆ M B , then for some b ∈ B gcd((r), a) = (r) + a ⊆ b.
Structure of the paper The relation between the results stated above and the location of their proofs in the paper is as follows (the arrow from "result 1" to "result 2" means that "result 1" is used in the proof of "result 2"): In Section 2 we introduce and recall the basic objects from the following areas: number fields and ideals, B-free integers on number fields, Toeplitz sequences. A part of the subsection on Toeplitz sequences related to the notion of essential periods is new and may be of an independent interest. Then, in Section 3, we discuss one dimensional B-free systems. Even though the results included there are a special case of their multidimensional counterparts covered in Section 4, we decided to keep it this way, as some readers may be interested in one-dimensional case only. Moreover, the multidimensional objects are much more technical and Section 4 may be quite hard to digest otherwise. The structure of Section 3 and Section 4 is similar, see the diagram above and the diagram in the beginning of Section 3. Finally, in Section 5, we discuss the setting of sets of multiples corresponding to lattices and provide a counterexample to a lattice version of Theorem B.
Main objects
Number fields and ideals
Let K be an algebraic number field with degree d = [K : Q], with the integer ring O K . As in every Dedekind domain, all proper non-zero ideals in O K factor (uniquely, up to the order) into a product of prime ideals. We will denote ideals in O K by a, b, c . . . We have
a + b = {a + b : a ∈ a, b ∈ b}, ab = {a 1 b 1 + · · · + a k b k : a i ∈ a, b i ∈ b, 1 ≤ i ≤ k}.
It is natural to set gcd(a, b) := a + b and lcm(a, b) := a ∩ b
and speak of the greatest common divisor and the least common multiple, respectively. Note that the words "greatest" and "least" are a bit misleading here: gcd(a, b) is the smallest ideal containing both, a and b, while lcm(a, b) is the largest ideal contained both in a and b. Proper ideals a, b are said to be coprime whenever a + b = O K . Equivalently, a, b do not share factors: there are no non-trivial ideals a ′ , b ′ , c such that a = ca ′ and b = cb ′ . If a and b are coprime then lcm(a, b) = a ∩ b = ab.
The algebraic norm of an ideal c = {0} is defined as N (c) :
= |O K /c| = [O K : c].
Finally, recall that there is a natural isomorphism from O K to a lattice in R d , called the Minkowski embedding (see e.g. Chapter I, §5 in [24]). We refer the reader to [1,24] for more background information on algebraic number theory.
B-free integers in number fields
Let B be a collection of non-zero ideals in O K . Definition 2.1. We say that (i) ideal c is B-free, whenever c ⊆ b for all b ∈ B (equivalently, c cannot be written as a product of b with another ideal);
(ii) integer g ∈ O K is B-free if the principal ideal (g) := gO K is B-free.
We denote the set of B-free integers in O K by F B and its complement by M B .
Since for any ideal b ⊆ O K and g ∈ O K we have g ∈ b ⇐⇒ (g) ⊆ b, it follows immediately that
F B = O K \ b∈B b.
The characteristic function of F B will be denoted by η ∈ {0, 1} O K :
η(g) = 1 if g is B-free, 0 otherwise.
For a finite subset S ⊂ B define
ℓ S = b∈S b = lcm(S) and C S = {b + ℓ S : b ∈ B} = {gcd(b, lcm(S)) : b ∈ B}.
(cf. (13) below). We say that B is primitive whenever for any
b, b ′ ∈ B if b ⊂ b ′ then b = b ′ .
For any B, there exists a unique primitive subset B prim ⊂ B such that M B prim = M B :
B prim = B \ {b ∈ B : there exists b ′ ∈ B such that b b ′ }.
Without loss of generality we can therefore assume that B is primitive. Then S is a proper subset of C S . Let S ⊆ S ′ ⊂ B. Then
S ⊆ S ′ ⊆ C S ′ ⊆ M C S so that M S ⊆ M S ′ ⊆ M C S ′ ⊆ M C S . A finite set S ⊂ B is saturated, if C S ∩B = S.
In other words, the only elements of B that divide lcm(S) are precisely the elements of S. Given any S ⊂ B, the set S sat := C S ∩ B is saturated. Moreover, if S 1 ⊂ S 2 ⊂ · · · ր B is a filtration, then, for some subsequence (n k ), S sat n 1 ⊂ S sat n 2 · · · ⊂ր B is a saturated filtration of B, i.e. a filtration consisting of saturated sets (see [8], Section 3 for the details in dimension one).
Dynamical system outputting B-free integers Given an abelian group G and a finite alphabet A, there is a natural action on A G by G by commuting translations:
S g ((x g ′ ) h∈G ) = (x g ′ +g ) g ′ ∈G for g, g ′ ∈ G. In particular, on {0, 1} O K , we have S g ((x g ′ ) g ′ ∈O K ) = (x g ′ +g ) g ′ ∈O K , g ∈ O K .
To introduce a metric on {0,
1} O K , recall that (F n ) n≥1 ⊆ O K is a Følner sequence if for any g ∈ O K , lim n→∞ |(F n + g) ∩ F n | |F n | = 1.
If n≥1 F n = O K and F n ⊆ F n+1 for any n ≥ 1, then (F n ) n≥1 is called nested [11,12].
Fix a nested Følner sequence (F n ) n≥1 ⊆ O K (we can assume F 1 = {1}) and consider the metric given by the formula
(3) d(x, y) = 1, if x 1 = y 1 , 2 − max{n≥1; xg=yg for any g∈Fn} otherwise.
The above metric induces the product topology on {0,
1} O K . If X ⊆ {0, 1} O K is closed and (S g ) g∈O K -invariant, we say that X is a subshift. We will denote by X η ⊆ {0, 1} O K the smallest subshift containing η. A subshift X is called minimal if for any x ∈ X its orbit {S g x : g ∈ O K } is dense in X. We say that x ∈ {0, 1} O K is minimal, if its orbit
closure is a minimal subshift. Let us recall one more notion, in a sense complementary to the minimality in the case of subshifts. We say that a pair (x, y) ∈ X × X is proximal, if (4) lim inf g→∞ d(S g x, S g y) = 0.
By g n → ∞ as n → ∞ we mean that for any finite A ⊆ O K , there exists n 0 ≥ 1 such that g n ∈ A for any n > n 0 . If any pair of points in X is proximal, then subshift is called proximal. Usually the opposite notion to proximality is distality (a subshift X is called distal if (4) is not satisfied for any x = y ∈ X). But for any subshift X its distality implies its equicontinuity, so the finiteness of X, see [10, Theorem 2].
Consider
H B = b∈B O K /b
with the coordinatewise addition. It is the product of finite groups O K /b and the Haar measure P B on H B is the product of the corresponding counting measures. Moreover, there is a natural O K -action on H B by translations:
T g ((h b ) b∈B ) = (h b + g) b∈B , g ∈ O K . Let (5) H = {T g (0) : g ∈ O K },
where 0 is the neutral element of H B and let P be the Haar measure on H.
Let ϕ : H → {0, 1} O K be defined as ϕ(h)(g) = 1, if h b + g ≡ 0 mod b for each b ∈ B, 0, otherwise, where h = (h b ) b∈B . Notice that ϕ(0) = η = ½ F B . Moreover, ϕ = (½ C • T g ) g∈O K , where C = {g ∈ H : h b ≡ 0 mod b for each b ∈ B}.
In other words, ϕ is the coding of orbits of points under (T g ) g∈O K with respect to the partition {C, G \ C} of G. Finally, let ν η := ϕ * (P) be the pushforward of P under ϕ. We will call ν η the Mirsky measure.
Remark 2.2. All above objects are definined for B-free numbers by putting O K = Z and b = bZ for any b ∈ B ⊂ N. Remark 2.3. In the case of {p k : p ∈ P}-free numbers, in particular in the square-free case, ν η was considered by Mirsky [21,22] (cf. also [23]) who studied the frequencies of blocks on η.
Toeplitz sequences
Dimension one
Let A be a finite alphabet. We say that x = (x n ) n∈Z ∈ A Z is a Toeplitz sequence [16], whenever for any n ∈ Z there exists s n ∈ N such that x n+k·sn = x n for every k ∈ Z. In other words, any symbol appears on x with some period. The orbit closure of any Toeplitz sequence is called a Toeplitz subshift. Given x ∈ A Z , let
Per(x, s, a) := {n ∈ Z : x| n+sZ = a}, where a ∈ A, s ∈ N and
Per(x, s) = a∈A Per(x, s, a).
It is easy to see that x ∈ A Z is a Toeplitz sequence if and only if there exists a sequence (p n ) n≥1 such that p n | p n+1 and Z = n≥1
Per(x, p n ). 2 2 Indeed, one can simply take any p 1 such that 0 ∈ Per(x, p 1 ) and then proceed inductively. Suppose that we have chosen p 1 | p 2 | . . . | p 2n+1 such that [−n, n] ⊂ 1≤k≤2n+1 Per(x, p k ). There exist p 2n+2 , p 2n+3 such that −n − 1 ∈ Per(x, p 2n+2 ), n + 1 ∈ Per(x, p 2n+3 ). Replacing p 2n+2 with lcm(p 2n+1 , p 2n+2 ) and then p 2n+3 with lcm(p 2n+2 , p 2n+3 ), we obtain p 2n+1 | p 2n+2 | p 2n+3 and [−n − 1, n + 1] ⊂ 1≤k≤2n+3 Per(x, p k ).
Finally, let ? be a fixed symbol that is not in A, called a hole. For any s ≥ 1 and a Toeplitz subshift X we define the skeleton map at scale s by M s : X → (A∪?) Z , where (M s (y)) n = y n , n ∈ Per(y, s), ?, otherwise for any y ∈ X. Moreover, if y ∈ X x , where x is Toeplitz then
(6) M s (y) = σ j M s (x) = M s (σ j x)
for some 0 ≤ j ≤ s − 1. Indeed, since y ∈ X x , it follows immediately that Per(y, s) ⊃ Per(x, s) − j for some 1 ≤ j ≤ s − 1. By the minimality of X x , we have x ∈ X y and we can reverse the roles of x and y to obtain Per(y, s) = Per(x, s) − j. Moreover, (6) also holds.
Essential periods Downarowicz, in his survey [5], defines essential periods for a given
Toeplitz sequence x ∈ A Z in the following way: Clearly, (7) is equivalent to
(8) Per(x, s) = Per(x, q) =⇒ q ≥ s.
Downarowicz gives a reference to a paper by Williams [25]. However, she formulates the definition of an essential period differently. We claim that the two above notions mean the same.
Proposition 2.6. Definitions 2.4 and 2.5 are equivalent. Moreover, in formula (8), condition q ≥ s can be replaced by s | q.
This can be proven directly (and it is not difficult), however, let us first prove the following result which will be useful when we pass to the multidimensional setting. Proof. It is immediate that (a) and (b) are equivalent (to go from (a) to (b) we just apply (a) repeatedly and to go from (b) to (a) we just use that Per(x, s, a) − q is contained in Per(x, s, a) − qZ and both sides of (a) are unions of the same number of arithmetic progressions of the same difference s). The equivalence of (d) and (e) follows from the fact that Per(x, s) = a∈A Per(x, s, a), Per(x, q) = a ′ ∈A Per(x, q, a ′ ), where
Per(x, s, a) ∩ Per(x, q, a ′ ) = ∅ whenever a = a ′ .
If this intersection is nonempty then clearly a = x n = a ′ .
We prove now that (b) implies (c). Assume (b) and take n ∈ Per(x, s, a). We have n + sZ ⊂ Per(x, s, a) ⊂ Per(x, q, a). Therefore, for each n ∈ Per(x, s, a), we have x| n+sZ+qZ ≡ a. However, this means that n ∈ Per(x, gcd(s, q), a) since sZ+qZ = gcd(s, q)Z. This gives Per(x, s, a) ⊂ Per(x, gcd(s, q), a) for each a ∈ A. The opposite inclusion always takes place as gcd(s, q) | s. Notice also that (d) follows from (c) immediately as gcd(s, q) | q.
It remains to show that (d) implies (b). Take n ∈ Per(x, s, a). Then n + sZ ⊂ Per(x, s, a) ⊂ Per(x, q, a). Therefore, n+sZ+qZ ⊂ Per(x, q, a). In particular, x| n+qZ+sZ ≡ a, which yields n + qZ ⊂ Per(x, s, a). This completes the proof. Remark 2.8. As an immediate consequence, we obtain that condition (9) from Definition 2.5 is equivalent to (10) Per(x, s) ⊂ Per(x, q) =⇒ s | q.
Remark 2.9. We claim that condition (7) from Definition 2.4 is equivalent to (11) Per(x, s) = Per(x, q) =⇒ s | q.
In fact, (8), (10) and (11) are all equivalent. Indeed, (10) implies (11) and (11) implies (8).
It remains to show that (8) implies (10). Suppose that Per(x, s) ⊂ Per(x, q) and that (8) holds. It follows by Proposition 2.7 ((e) =⇒ (c)) that Per(x, s) = Per(x, gcd(s, q)), so we can apply (8) to conclude that gcd(s, q) ≥ s. The latter condition is however equivalent to s | q.
Proof of Proposition 2.6. We combine Remark 2.8 and Remark 2.9. To complete the proof, it remains to use the implication (e) =⇒ (c) from Proposition 2.7. Indeed, if Per(x, s) ⊂ Per(x, q) then Per(x, s) = Per(x, gcd(s, q)). Therefore, if (11) holds for s then s | gcd(s, q), which is equivalent to s = gcd(s, q), which, in turn, is the same as s | q.
Remark 2.10. Notice that s | q is equivalent to qZ ⊂ sZ. Moreover, for any k ∈ N, kZ ⊂ Z is a subgroup and an ideal of Z.
Periodic structure A periodic structure [25] of a Toeplitz sequence x ∈ A Z is any sequence (p k ) k∈N of essential periods such that p k | p k+1 for each k ∈ N and k∈N Per(x, p k ) = Z. Every Toeplitz sequence has a periodic structure.
Higher dimension (abelian discrete, finitely generated groups)
Let G be a discrete, finitely generated group and let Γ ⊆ G be a subgroup of finite index. We will be mostly interested in case G = Z d and therefore we restrict ourselves to abelian groups. For
x = (x g ) g∈G ∈ A G consider
Per(x, Γ, a) = {g ∈ G; x g+γ = a for any γ ∈ Γ}, a ∈ A,
Per(x, Γ) = a∈A
Per(x, Γ, a).
If Per(x, Γ) = ∅, then we say that Γ is a group of periods of x. We say that x is a G-Toeplitz array, if for all g ∈ G there exists a subgroup Γ ⊆ G of finite index such that g ∈ Per(x, Γ). The orbit closure of a G-Toeplitz array endowed with multidimensional shifts (S g ) g∈G is called a G-Toeplitz system. (1) x is a G-Toeplitz array,
(2) there exists a sequence (Γ n ) n≥1 of groups with finite indices such that Γ n+1 ⊂ Γ n for any n ≥ 1 and G = n≥1 Per(x, Γ n ).
If G = Z d then conditions (1) and (2) from the above theorem are equivalent to the following one:
(3) there exists a sequence (p n ) n≥1 ⊂ N such that p n | p n+1 and G = n≥1 Per(x, p n Z d ).
Indeed, it suffices to see that for any subgroup Γ n ⊂ Z d of finite index, there exists p n ∈ N with Γ n ⊃ p n Z d and proceed inductively, as in footnote 2 (for a detailed proof, see [2,Proposition 14]).
Remark 2.12. Take G = O K (as in Section 2.2). Since there is a group isomorphism between the additive structures of O K and Z d and it preserves the index of a subgroup, it follows immediately that for an O K -Toeplitz array x ∈ A O K , there is a corresponding Z d -Toeplitz array y ∈ A Z d . By condition (3) above, there exist a sequence of groups of periods (p n Z d ) n∈N for y such that p n | p n+1 and n≥1 Per(y, p n Z d ) = Z d . Any such group of periods p n Z d for y corresponds to a group of periods for x which is a principal ideal.
As in dimension one, let ? be a fixed symbol that is not in A, called a hole. For any subgroup Γ ⊆ G of finite index and a G-Toeplitz system X, we define the skeleton map at scale Γ by M Γ : X → (A∪?) G , where (M Γ (y)) n = y n , n ∈ Per(y, Γ), ?, otherwise for any y ∈ X. Moreover, if y ∈ X x , where x is a G-Toeplitz array then
(12) M Γ (y) = σ j M Γ (x) = M Γ (σ j x)
for some j ∈ G. The proof of (12) is the same as the proof of (6) in the one-dimensional case.
Essential periods There are two definitions of essential periods in this setting present in the literature. The goal of this section is to show that they are equivalent. Cortez, in her paper [2], defines the notion for G = Z d in the following way.
Definition 2.13 ([2, Definition 15]). A group Z ⊂ Z d of periods of x ∈ A Z d is called a group generated by essential periods of x if Per(x, Z) ⊆ Per(x, Z ′ ) implies that Z ′ ⊆ Z.
Cortez and Petite [3] define that notion for more general G-Toeplitz arrays differently (in their paper G is a discrete finitely generated group). We recall it in the abelian setting here.
Definition 2.14 ([3, Definition 4]). A syndetic group
Γ ⊂ G is called an essential group of periods of x ∈ A G if Per(x, Γ, a) ⊆ Per(S g x, Γ, a) for every a ∈ A implies that g ∈ Γ.
Remark 2.15. Notice that since Per(x, Γ, a) is a finite union of sets in the form Γ + g, g ∈ G, and for any a ∈ A one has Per(S g x, Γ, a) = Per(x, Γ, a) − g, the inclusion in Definition 2.14 implies Per(x, Γ, a) = Per(x, Γ, a) − g. Hence it is clear that for G = Z the two definitions are equivalent: x ∈ A Z , s ≥ 1 is an essential period if and only if sZ is an essential group of periods.
Remark 2.16. Cortez and Petite remark that any essential group of periods of x ∈ A Z d is a group generated by essential periods of x ([3, Remark 2]). In fact, Definition 2.13 can be easily extended to a general finitely generated group G. Lemma 6 in [3] shows that any essential group of periods of x ∈ A G is a group generated by essential periods of x. We will show that also the opposite implication is true, see Corollary 2.18 below. In order to do this, we will use a multidimensional version of Proposition 2.7. Proposition 2.17. Fix g ∈ G and let Γ ⊂ G be a subgroup. Let Γ ′ = Γ, g . The following conditions are equivalent:
(a) Per(x, Γ, a) = Per(x, Γ, a) − g for all a ∈ A, (b) Per(x, Γ, a) = Per(x, Γ, a) − Γ ′ for all a ∈ A, (c) Per(x, Γ, a) = Per(x, Γ ′ , a) for all a ∈ A, (d) Per(x, Γ, a) ⊂ Per(x, Γ ′ , a) for all a ∈ A, (e) Per(x, Γ) ⊂ Per(x, Γ ′ ).
Proof. Since Γ, g = {aγ + bg : a, b ∈ Z, γ ∈ Γ}, to show that (a) implies (b), we use Γ-invariance of Per(x, Γ, a) and just apply (a) repeatedly.
To go from (b) to (a) we just use that Per(x, Γ, a) − g ⊆ Per(x, Γ, a) − Γ ′ and both sides of (a) are unions of the same number of sets in the form Γ + g ′ , g ′ ∈ G.
The equivalence of (d) and (e) follows from the fact that Per(x, Γ) = a∈A Per(x, Γ, a),
Per(x, Γ ′ ) = a ′ ∈A Per(x, Γ ′ , a ′ ), where Per(x, Γ, a) ∩ Per(x, Γ ′ , a ′ ) = ∅ whenever a = a ′ (if this intersection is nonempty then clearly a = x n = a ′ ).
We prove now that (b) implies (c). Assume (b) and take γ ∈ Per(x, Γ, a). Then
x| γ+Γ ′ ≡ a. So γ ∈ Per(x, Γ ′ , a). This gives Per(x, Γ, a) ⊂ Per(x, Γ ′ , a) for each a ∈ A.
The opposite inclusion always takes place as Γ ⊂ Γ ′ . Notice also that (d) follows from (c) immediately.
It remains to show that (d) implies (a). Take γ ∈ Per(x, Γ, a).
Then by (d), γ − g ∈ γ + Γ ′ ⊂ Per(x, Γ ′ , a). In particular, x| γ−g+Γ ′ ≡ a, which yields x| γ−g+Γ ≡ a, so Per(x, Γ, a) − g ⊂ Per(x, Γ, a).
Since both sets are unions of the same number of sets in the form Γ + g ′ , g ′ ∈ G, they are equal. This completes the proof. For any x ∈ A G , the family of corresponding groups generated by essential periods is the same as the family of corresponding essential groups of periods.
Proof. Suppose that Γ is a group generated by essential periods of x and Per(x, Γ, a) ⊂ Per(x, Γ, a) − g for every a ∈ A and for some g ∈ G. Let Γ ′ = Γ, g . By Proposition 2.17 (a)⇒(e), Per(x, Γ) ⊂ Per(x, Γ ′ ). Since Γ is a group generated by essential periods of x,
Γ ′ ⊂ Γ. So g ∈ Γ.
Hence Γ is an essential group of periods.
Suppose now that Γ is an essential group of periods of x and Per(x, Γ) ⊂ Per(x, Γ ′ ) for some Γ ′ group of periods of x. Let g ∈ Γ ′ , a ∈ A and γ ∈ Per(x, Γ, a). Then x| γ+Γ+g ≡ a, so γ ∈ Per(x, Γ, a) − g. Thus, Per(x, Γ, a) ⊂ Per(x, Γ, a) − g. Since Γ is an essential group of periods of x, g ∈ Γ. Therefore Γ ′ ⊂ Γ, whence Γ is a group generated by essential periods of x.
Recall also that essential groups of periods exist. In fact, the next lemma tells us that we have even more.
Lemma 2.19 ([3, Lemma 7]). Let x ∈ A G . If Γ ⊆ G is a group of periods of x, then there exists K ⊆ G an essential group of periods of x such that Per(x, Γ) ⊆ Per(x, K).
Periodic structure and maximal equicontinuous factor Theorem 2.20 ([3, Corollary 6]). Let x ∈ A G be a G-Toeplitz array. Then there exists a sequence (Γ n ) n≥1 of essential groups of periods of x such that
Γ n+1 ⊆ Γ n and n≥1
Per (x, Γ n ) = G.
Sequence (Γ n ) from the theorem above is called a periodic structure of x. It can be used to describe the maximal equicontinuous factor of a G-Toeplitz system.
Theorem 2.21 ([3, Proposition 7])
. Let x ∈ A G be a G-Toeplitz array. If (Γ n ) n≥1 is a periodic structure of x, then ← − G = lim ← −n G/Γ n with addition of (e G , e G , . . .) is the maximal equicontinuous factor of (X x , (S g ) g∈G ).
We will also need the following related technical result (used in [3] to prove Proposition 7 therein). . Let x ∈ A G be a G-Toeplitz array and y ∈ X x . If Γ y ⊆ G is an essential group of periods of y, then C Γy (y) := x ′ ∈ X x : Per x ′ , Γ y , a = Per(y, Γ y , a) for all a ∈ A = {S g y; g ∈ Γ y } and S w (C Γy (y)) w∈G/Γy is a closed partition of X x .
3 Minimality and periodic structure of B-free systems
Background and main results
Recall that any B-free system is essentially minimal, i.e. it has exactly one minimal subset. In fact, we have the following. Recall the notion of densities for subsets of integers. Given A ⊂ Z, let δ(A) stand for the logarithmic density and d(A) stand for the natural density of A, provided that these densities exist:
δ(A) = lim N →∞ 1 log N 1≤n≤N 1 A (n) n and d(A) = lim N →∞ 1 N 1≤n≤N 1 A (n).
Recall that for the sets of multiples the logaritmic density always exists, by the result of Davenport and Erdös [4]. We say that B is Behrend, whenever δ(M B ) = 1 (which is equivalent to d(M B ) = 1). Recall also that a set B is called taut when
δ(M B ) > δ(M B\{b} ) for any b ∈ B.
It follows by [15,Corollary 0.19] that B is taut precisely if it is primitive and dC ⊆ B for any d ∈ N and any Behrend set C ⊂ N.
Recall that B is primitive if b ∤ b ′ for any b = b ′ ∈ B.
Kasjan, Keller and Lemańczyk described the minimality of B-free systems in [17]. Before we formulate their result, let us define some auxiliary objects. Given B, let (1) the window W is topologically regular, i.e. int(W ) = W ,
C S := {gcd(b, lcm(S)) : b ∈ B} for a finite subset S ⊂ B,(13)C ∞ := n ∈ N; ∀ finite S⊂B ∃ finite S ′ ; S⊆S ′ ⊂B n ∈ C S ′ \ S ′ .(2) F B = finite S⊂B F C S , (3) C ∞ = ∅,(4)
there are no d ∈ N and no infinite pairwise coprime set C ⊆ N\{1} such that dC ⊆ B,
(5) η = ϕ(0) is a Toeplitz sequence different from . . . 0.00 . . .,(6)
0 ∈ C ϕ and ϕ(0) = . . . 0.00 . . ., where C ϕ is the set of continuity points of ϕ,
(7) η ∈ M and η = . . . 0.00 . . ., where M = C ϕ , (8) X η is minimal, i.e. X η = M , and card (X η ) > 1, (9) the dynamics on (X η , σ) is a minimal almost 1-1 extension of H, R ∆(1) , the rotation ∆(1) on H.
The relations between the above conditions are as follows:
B is taut ⇐ (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4) ⇐⇒ (5) ⇐⇒ (6) ⇓ (9) ⇓ (7) ⇐⇒ (8) . Moreover, if ∆(Z) ∩ W = W (in particular if B is taut), then (1)−(9) are all equivalent.
Keller in [18, Lemma 1] showed that
(14) C ∞ = D := {d ∈ N : dC ⊂ B for some infinite pairwise coprime set C}.
Moreover, he gave the following more detailed description of the unique minimal subset of B-free systems than that in Theorem 3.1.
B * := (B \ M C∞ ) ∪ C prim ∞ and η * = ½ F B * .
Then η * ≤ η and it is a Toeplitz sequence. Moreover, X η * is a unique minimal subset of X η .
Remark 3.4. We have B * = (B ∪ D) prim . Indeed, we need to show that (B \ M D ) ∪ D prim = (B ∪ D) prim . By [18, Lemma 3 a)], B * = (B \ M D ) ∪ D prim is primitive. Therefore, also B \ M D is primitive (and disjoint with D prim ). Hence, (B ∪ D) prim = ((B \ M D ) ∪ D) prim = (B \ M D ) prim ∪ D prim = (B \ M D ) ∪ D prim .
Remark 3.5. Notice that by the above remark, we may define B * also for B that are not primitive, just by setting B * := (B ∪ D) prim (or even skip the "prim" in the supscipt in the above formula if we do not need that the resulting set B * is primitive). • Let B = 2P. Then B is primitive and |supp η mod 4| = 2, i.e. η ∈ Y . So, in this case we have ∆(Z) ∩ W = W .
• Choose b ∈ N, 0 < r < b and take B such that b ∈ B and (r + bN) prim ⊂ B. Then, clearly, |supp η mod b| ≤ b − 2, which gives η ∈ Y and, again, ∆(Z) ∩ W = W .
One of our main goals was to give an alternative proof of Theorem 3.3 adaptable to the multidimensional case. Our proof is based on the study of a periodic structure for B-free systems. Before we comment on the tools and the related results (often interesting on their own), let us first present the full one-dimensional version of Theorem B.
(a) (X η , σ) is minimal, (b) η is a Toeplitz sequence different from . . . 0.00 . . ., (c) D = ∅, (d) X η ⊆ Y , where Y = {x ∈ {0, 1} Z ; | supp x mod b| = b − 1 for any b ∈ B}.
As an immediate consequence, we obtain the following corollary.
Corollary 3.8. Let B ⊆ {2, 3, . . .} be primitive. Conditions (1) to (9) from Theorem 3.2 are all equivalent to (10) X η ⊆ Y , where Y = {x ∈ {0, 1} Z ; | supp x mod b| = b − 1 for any b ∈ B}.
The proof of both, Theorem 3.3 and Theorem 3.7, rely on the following result that is, in fact, a part of Theorem 3.3 which we think that deserves to be formulated separately, with more details (see also Remark 3.22 below). Recall that conditions (b) and (d) from Theorem 3.7 is related to the tautness of B. More precisely, we have the following two lemmas. Remark 3.12. It is not so hard to see that Y may not be closed, so even if η ∈ Y then X η ⊆ Y is not automatic. Indeed, let B be the set of squares of all primes. Then X η is hereditary and it is equal to the set of admissible sequences. Then η ∈ Y , so for any b ∈ B, there exists m such that that |supp(η) ∩ [−m, m] mod b| = b − 1. Moreover, each block that appears on η is of positive Mirsky measure, so, in particular, it appears on η infinitely often. This implies that |supp(η · ½ Z\[−n,n] ) mod b| = b − 1 for any n ≥ 1. In other words, η · ½ Z\[−n,n] ∈ Y . Clearly, η · ½ Z\[−n,n] converges to the all-zero sequence as n tends to infinity, which means that Y is not closed.
In fact, it follows from Theorem 3.7 that Y η := Y ∩X η is closed and non-empty precisely if η is Toeplitz. Indeed, if Y η is closed and non-empty then necessarily η ∈ Y η , which yields Y η = X η and this is equivalent to X η ⊂ Y , so, by Theorem 3.7, η is Toeplitz. To obtain the implication in the other direction, notice that if η is Toeplitz then X η = Y η and it is immediate that Y η is closed and non-empty.
Thus, if we are thinking of "translating" Theorem 3.7 to the multidimensional case, we may need a proof of the implication (b) =⇒ (c) that does not use the notion of tautness, which seems to lack a reasonable multidimensional analogue. If we look deeper in these proofs, we meet even more obstructions. Lemma 3.11 is a consequence of the following result.
4]). Assume that B ⊂ N is taut, d ∈ N, 1 ≤ r ≤ d. If r + dZ ⊆ M B , then there exists b ∈ B such that b | gcd(r, d).
Even if we "forget" about the problems related to the notion of tautness, the main tool in the proofs of Proposition 4.31 in [9] and Proposition 2.4 in [8] is the Dirichlet's theorem on primes in arithmetic progressions -it has no multidimensional counterpart that could be useful for us. However, if we replace the tautness of B by the stronger assumption that η is a Toeplitz sequence, new tools become available and it turns out that this weaker result stated below is the right (and sufficient) way to see things in higher dimension.
Theorem 3.14 (Theorem F in dimension one). Assume that η is a Toeplitz sequence,
d ∈ N, 1 ≤ r ≤ d. If r + dZ ⊆ M B ,
then there exists b ∈ B such that b | gcd(r, d).
In fact, the above theorem has the following consequence (cf. Lemma 3.11). Clearly, what prevents η from being a Toeplitz sequence is the presence of the nonperiodic positions. We give the following description of the set of such positions on η.
Proposition 3.16 (Proposition D in dimension one). Let (S i ) i≥1 be a saturated filtration of a primitive set B by finite sets. Then
(15) Z \ i≥1 Per(η, lcm(S i )) = M D ∩ F B = Z \ s⊂N Per(η, s),
where D is as in (14).
The above proposition also allows us to find a periodic structure for η in the Toeplitz case.
Theorem 3.17 (Theorem E in dimension one). Let B ⊂ N be primitive. Suppose S 1 ⊂ S 2 ⊂ · · · ր B is a saturated filtration of B by finite sets. Assume that η is a Toeplitz sequence. Then (lcm(S i )) i≥1 is a periodic structure of η.
Here is a table including the above results and their multidimensional counterparts formulated in Section 1.
We complete this section with a map of Section 3.2 which includes the proofs of our dimension one results.
Proofs
Arithmetic progressions in M B : proof of Theorem 3.14
The proof of Theorem 3.14 relies on two lemmas, whose proofs will be given in a moment. for some infinite pairwise coprime set C. Since η is a Toeplitz sequence, it follows from Theorem 3.2 ((5)⇔(4)) that D = ∅. Therefore, by Lemma 3.19, we have gcd(r, d) ∈ M B , which completes the proof.
Proofs of lemmas
Proof of Lemma 3.18. It suffices to show that if gcd(r, d) = 1 then r + dZ contains an infinite pairwise coprime set. Notice that for any finite subset of primes P , there exists k ∈ Z such that (16) gcd(r + kd, p∈P p) = 1 (take, for example, k = p∈P,p∤r p or k = 1 if {p ∈ P, p ∤ r} is empty). Suppose now that r + dZ does not contain an infinite pairwise coprime subset. Then there is a pairwise coprime set K of the maximal cardinality contained in r + dZ. Clearly, any element of r +dZ shares a factor with some element of this set. It remains to notice that (16) does not hold for P being the set of all prime factors of elements of K to obtain a contradition.
Proof of Lemma 3.19. Let C = {c i } i≥1 . Since dC ⊆ M B , for any i ≥ 1 there exist b i ∈ B and k i ∈ N such that
dc i = b i k i .
Suppose first that there exists b ∈ B such that b i = b for infinitely many i ≥ 1.
Then since c i , i ≥ 1, are pairwise coprime, b | d.
Suppose now that each b ∈ B appears among b i 's at most a finite number of times. In particular, passing to a subsequence if necessary, we may assume that b i 's are pairwise different (the corresponding c i 's still form an infinite pairwise coprime set). Since d has only finitely many factors, we can assume that gcd(d, k i ) is constant. Then
b i = d gcd(d, k i ) · c i · gcd(d, k i ) k i .
Since c i 's are pairwise coprime and c i ·gcd(d,k i )
k i | c i , it follows immediately that also c i ·gcd(d,k i ) k i (i ≥ 1)
are pairwise coprime. This completes the proof. The structure of this section is fairly simple, so we just proceed with the proofs.
Non
Proof of Proposition 3.16. It suffices to show that
(17) Z \ i≥1 Per(η, lcm(S i )) ⊆ M D ∩ F B ⊆ Z \ s∈N Per(η, s).
Clearly, Z \ s∈N Per(η, s) ⊆ Z \ i≥1 Per(η, lcm(S i )) and therefore the inclusions must be, in fact, equalities.
First inclusion. Take n ∈ i≥1 Per(η, lcm(S i )). Then n ∈ F B . Moreover, for any k ≥ 1 there exists t ∈ Z such that n + t lcm(S k ) ∈ M B , so there exists b k such that b k | n + t lcm(S k ).
Notice that b k / ∈ S k (if b k ∈ S k then b k | n which contradicts n ∈ F B ). Since (S k ) k≥1 is saturated, gcd(b k , lcm(S k )) < b k . Notice that gcd(b k , lcm(S k )) | n. Since n has only finitely many divisors, there exists d ∈ N and a subsequence (k ℓ ) ⊆ N such that gcd(b k ℓ , lcm(S k ℓ )) = d for some d ∈ N (and d | n).
Let ℓ 2 > ℓ 1 := 1 be such that
(18) b k ℓ 1 ∈ S ℓ k 2 (in particular, b k ℓ 1 | lcm(S ℓ k 2 )).
By the choice of (k ℓ ),
we have d | b k ℓ 1 , b k ℓ 2 , so d | gcd(b k ℓ 1 , b k ℓ 2 ).
On the other hand, using (18), we obtain
gcd(b k ℓ 1 , b k ℓ 2 ) | gcd(lcm(S k ℓ 2 ), b k ℓ 2 ) = d. Thus, gcd(b k ℓ 1 , b k ℓ 2 ) = d.
Suppose that we have constructed 1 = ℓ 1 < ℓ 2 < · · · < ℓ m such that
(19) gcd(b k ℓ i , b k ℓ i ′ ) = d for any 1 ≤ i < i ′ ≤ m.
There exists ℓ m+1 > ℓ m such that
(20) b k ℓ 1 , b k ℓ 2 , . . . , b k ℓm ∈ S k ℓ m+1 .
It follows by the choice of (k ℓ ) that d | b k ℓ i for 1 ≤ i ≤ m + 1. This gives
d | gcd(b k ℓ i , b k ℓ i ′ ) for any 1 ≤ i < i ′ ≤ m + 1.
On the other hand, using (20), we obtain gcd
(b k ℓ i , b k ℓ m+1 ) | gcd(lcm(S k ℓ m+1 ), b k ℓ m+1 ) = d. Thus, gcd(b k ℓ i , b k ℓ i ′ ) = d for any 1 ≤ i < i ′ ≤ m + 1
and the above inductive procedure therefore yields a sequence 1 = ℓ 1 < ℓ 2 < . . . such that
gcd(b k ℓ i , b k ℓ i ′ ) = d for any i = i; Let c i := b k i d .
It remains to notice that then C = {c i : i ≥ 1} is pairwise coprime and we have dC ⊆ B. Since d | n, it follows that n ∈ M D .
Second inclusion. Take n ∈ M D ∩F B . There exists d and an infinite pairwise coprime set C such that d | n and dC ⊆ B.
Suppose that n ∈ Per(η, s) for some s ∈ N. In other words, we have n + sZ ⊆ F B . Since d | n, it follows that
(21) dZ ∩ (n + sZ) = n + lcm(s, d)Z ⊆ n + sZ ⊆ F B .
Take c ∈ C coprime to s and d. Then Clearly, s i | s i+1 , so it remains to show that s i is an essential period of η. Let s < s i . Then there exists b ∈ S i such that
(22) b ∤ s. We have b ∈ Per(η, s i ). Suppose that b ∈ Per(η, s), i.e. b + sZ ⊆ M B
By Theorem 3.14 (for d = s, r = b), we get that for some
b ′ ∈ B, b ′ | gcd(b, s) | b.
Since B is primitive, b ′ = b, so b | s. But this contradicts (22), which completes the proof. Example 3.21. The first example of 0-1 nonperiodic Toeplitz sequence comes from Garcia and Hedlund [13]. The sequence is given by the following rule: in each odd-numbered step we fill every second available position with 0, in each even-numbered step we fill every second available position with 1. The corresponding periodic structure is (2 m ) m≥1 . On the other hand, it follows by Theorem 3.17 that (2 m ) m≥1 cannot be a periodic structure of any Toeplitz sequence given by some η * (corresponding to B * ). Since any two Toeplitz sequences from the same Toeplitz subshift have the same periodic structure, it follows that the Garcia-Hedlund sequence cannot be an element of a B-free subshift. Indeed, by Theorem 3.3, it would be then an element of the corresponding B * -free subshift which yields a contradiction.
Minimality: proof of Theorems 3.3 and 3.7
We will prove first Theorem 3.9, necessary for the proof of Theorem 3.3. Then we make some comments on set B * and compare it to the taut set B ′ from [9] giving the same Mirsky measure as B. Then we pass to the proof of Theorem 3.3. Finally, we prove Theorem 3.7, using both, Theorem 3.9 and Theorem 3.3.
Proof of Theorem 3.9. We will first show (a) ⇐⇒ (b). Clearly, if D = ∅ then B = B * . For the other direction, notice that
B = B * = (B \ M D ) ∪ D prim implies D prim ⊂ B.
However, by the definition of D, for any d ∈ D, some non-trivial multiple of d is a member of B, as dC ⊂ B where C is inifinite and pairwise coprime. If D = ∅ this contradicts the primitivity of B. Let us show that D * = ∅ (i.e. η * is Toeplitz, so (a) =⇒ (c)). Suppose that there is some d ∈ N and some infinite pairwise coprime set C such that
dC ⊆ B * = (B \ M D ) ∪ D prim .
Then, without loss of generality (taking a smaller but still infinite and pairwise coprime set C), one of the following holds:
(A) dC ⊆ B \ M D , (B) dC ⊆ D prim .
If (A) holds then dC ⊂ B, so d ∈ D and we obtain
dC ⊂ B \ M D ⊂ B \ dZ,
which yields a contradiction. Suppose now that (B) holds. Let C = {c 1 , c 2 , . . . } and let A i for i ≥ 1 be infinite, pairwise coprime and such that dc i A i ⊂ B. We can choose a i ∈ A i , i ≥ 1 so that {c i a i : i ≥ 1} is infinite and pairwise coprime (we just use that each A i is infinite and pairwise coprime). This gives again d ∈ D. This contradicts (B) (by the primitivity of D prim ). We conclude that indeed D * = ∅. Remark 3.22. In [9], a certain procedure was described to modify B to B ′ , so that:
• B ′ is taut, • η ′ ≤ η (where η ′ = 1 F B ′ ), • ν η = ν η ′ .
The idea was very similar to the one used to produce B * . We recall it here.
• Suppose that B is not taut, let c 1 be the smallest natural number such that there exists a Behrend set A 1 such that
c 1 A 1 ⊂ B. Replace B with B \ c 1 Z ∪ {c 1 }.
• If now B is taut, we stop. If not, we take the smallest natural number c 2 such that there exists a Behrend set A 2 such that c 2 A 2 ⊂ B and replace the original B with B \ (c 1 Z ∪ c 2 Z) ∪ {c 1 , c 2 }.
Either the above procedure ends after a finite number of steps or we arrive at the modified B of the form
B ′ = B \ i≥1 c i Z ∪ {c i : i ≥ 1}, where A i , i ≥ 1 are Behrend sets such that c i A i ⊆ B \ i−1 j=1 c j Z∪{c 1 , c 2 , .
. . , c i−1 } . It was shown in Lemma 4.10 in [9] that such B ′ fulfills the requirements listed in the beginning of this remark. In particular, this means that one could equivalently define B ′ as follows:
(23) B ′′ := (B \ M C ) ∪ C prim = (B ∪ C) prim ,
where C = {c ∈ N : there exists a Behrend set A such that cA ⊂ B}.
Indeed, we need to show that C prim = {c i : i ≥ 1} (the proof of the second equality above goes along the same lines as in Remark 3.4). The inclusion C prim ⊃ {c i : i ≥ 1} follows directly by the construction of {c i : i ≥ 1}. Suppose that we do not have an equality, i.e. there exists c ∈ C prim such that c = c i for all i ≥ 1. Let n ∈ N be such that c n−1 < c < c n .
Then cA ⊂ B \ n−1 i=1 c i Z for any Behrend set A but A c := { b c : b ∈ B, c | b} is Behrend. Notice that A c = A c ∩ b c : b ∈ B \ n−1 i=1 c i Z, c | b ∪ n−1 i=1 A c ∩ b c : b ∈ B ∩ c i Z, c | b .
By the above, the first set in the above sum is not Behrend. Recall that by [15, Corollary 0.14] a finite union of sets is Behrend, provided that at least one of them is Behrend. It follows immediately that there exists
1 ≤ i ≤ n − 1 such that A c ∩ b c : b ∈ c i Z, c | b is Behrend. However, A c ∩ b c : b ∈ B ∩ c i Z, c | b ⊂ c i gcd(c, c i ) Z.
Therefore,
1 = d M Ac∩{ b c :b∈c i Z,c|b} ≤ d c i gcd(c, c i ) Z = gcd(c, c i ) c i .
Hence gcd(c, c i ) = c i which contradicts c ∈ C prim ⊃ {c i : i ≥ 1}. Now, we claim that we have the following inclusions:
X η * ⊂ X η ′ ⊂ X η .
Indeed, for the second inclusion, see (27) in [19]. The first inclusion follows by the fact that X η * is the unique minimal subset of X η . So we must have X η * = X η * ∩ X η ′ and thus X η * ⊂ X η ′ . This also implies that (B ′ ) * = B * . Indeed, X η * is not only the unique minimal subset of X η but also of X η ′ . Therefore, we have X η * = X (η ′ ) * . Both, B * and (B ′ ) * are taut and (B ′ ) * = B * follows immediately by Corollary 4.36 in [9]. To sum up the above discussion, let us make the following observation. Up to applying "prim" at the end,
• the procedure that outputs B * means replacing the rescaled copies of infinite pairwise coprime subsets of B with the set of scales (cf. Remark 3.4),
• the procedure that outputs B ′ means replacing the rescaled copies of Behrend subsets of B with the set of scales (cf. (23)).
Recall now that each Behrend set contains an infinite pairwise coprime subset as the corresponding B-free subshift is proximal (see [9,Theorem 3.7]). This tells us immediately
that C ⊂ D so M B ⊆ M B ′ ⊆ M B * .
Proof of Theorem 3.3. Since, by Remark 3.4, B * = (B ∪ D) prim , we have η * ≤ η. Moreover, η * is Toeplitz, by Theorem 3.9. Now, we will prove that η * ∈ X η , i.e. that for any N ≥ 1 the block η * | [−N,N ] appears on η. Fix N ≥ 1 and let i ≥ 1 be sufficiently large to ensure that Let c 1 ∈ C 1 be coprime to L (such a number exists since L has finitely many factors and C 1 is infinite pairwise coprime). Then gcd(L, d 1 c 1 ) = d 1 | I 1 and therefore we can find k 1 ∈ Z such that
(24) M D ∩ F B ∩ [−N, N ] = (Z \ Per(η, lcm(S i ))) ∩ [−N, N ].
Recall that by
Lk 1 + I 1 ≡ 0 mod d 1 c 1 .
Let c 2 ∈ C 2 be coprime to L and c 1 (again, such a number exists because L and c 1 have finitely many factors and C 2 is infinite pairwise coprime). Since d 2 | L and L, c 1 and c 2 are pairwise coprime, we have gcd(Lc 1 , d 2 c 2 ) = d 2 | Lk 1 + I 2 . Therefore, we can find k 2 ∈ Z such that Lc 1 k 2 + Lk 1 + I 2 ≡ 0 mod d 2 c 2 .
Notice that since d 1 | L, we have
Lc 1 k 2 + Lk 1 + I 1 ≡ 0 mod d 1 c 1 .
Let c 3 ∈ C 3 be coprime to L, c 1 and c 2 . Since d 3 | L and L, c 1 , c 2 and c 3 are pairwise coprime, we have lcm(Lc 2 c 1 , d 3 c 3 ) = d 3 | Lc 1 k 2 + Lk 1 + I 3 . Therefore, we can find k 3 ∈ Z such that
Lc 2 c 1 k 3 + Lc 1 k 2 + Lk 1 + I 3 ≡ 0 mod d 3 c 3 .
Notice that since d ℓ c ℓ | Lc 2 c 1 for ℓ = 1, 2, we have
Lc 1 c 2 k 3 + Lc 1 k 2 + Lk 1 + I ℓ ≡ 0 mod d ℓ c ℓ for ℓ = 1, 2.
By repeating the above arguments, we obtain k j ∈ Z for j = 1, 2, . . . , t N such that
Lc t N−1 . . . c 1 k t N + Lc t N−2 . . . c 1 k t N−1 + . . . + Lc 1 k 2 + Lk 1 + I j ≡ 0 mod d j c j .(25)
for j = 1, 2, . . . , t N . Put
M N := Lc t N−1 . . . c 1 k t N + Lc t N−2 . . . c 1 k t N−1 + . . . + Lc 1 k 2 + Lk 1 .
We will show that and n + M N ∈ Per(η, lcm(S i )). It follows by Proposition 3.16 that n, n + M N ∈ M D ∩ F B (the latter set is the set of all non-periodic positions on η). In other words,
(26) η n+M N = η n = η * n for n ∈ Per(η, lcm(S i )) ∩ [−N, N ].n, n + M n ∈ F D ∪ M B = (F D ∩ F B ) ∪ M B .
Recall that by Remark 3.4, we have
B * = (B ∪ D) prim , so M B * = M B ∪ M D . Equiva- lently, F B * = F B ∩ F D . Therefore, • if n ∈ F D ∩ F B then n ∈ F B * , • if n ∈ M B then n ∈ M B * .
Hence, (26) indeed holds. Now, we will show that
(28) η n+M N = η * n for n ∈ (Z \ Per(η, lcm(S i ))) ∩ [−N, N ].
Since d j c j ∈ B and d j c j | I j + M N , it follows by (25) that
(29) η I j +M N = 0 for all 1 ≤ j ≤ t N .
Moreover, it follows by I j ∈ M D ⊆ M B * (recall again Remark 3.4), that we also have
η * I j = 0 for all 1 ≤ j ≤ t N .
Therefore, (28) indeed holds. Combining (26) and (28), we conclude that
(30) η| [−N,N ]+M N = η * | [−N,N ] .
Notice that in the above arguments we used only that It follows from (a) and from Theorem 3.3 that X η * = X η . Suppose that η * < η, i.e. there is n ∈ Z such that 0 = η * n < η n = 1. Let (p k ) k∈N be a periodic structure of η * and let k ∈ N be such that η * | n+p k Z = 0. It follows from (6) that
(31) |Per(η * , p k , 0) ∩ [0, p k − 1]| = |Per(η, p k , 0) ∩ [0, p k − 1]|
as η ∈ X η * . On the other hand, since η * ≤ η, it follows immediately that Per(η * , p k , 0) ⊃ Per(η, p k , 0).
It remains to notice that n ∈ Per(η * , p k , 0) \ Per(η, p k , 0), so
|Per(η * , p k , 0) ∩ [0, p k − 1]| > |Per(η, p k , 0) ∩ [0, p k − 1]|,
which contradicts (31). This yields η = η * , i.e. η itself is a Toeplitz sequence ((b) holds).
(b) ⇒ (d) Suppose η is a Toeplitz sequence. For any b ∈ B and 0 ≤ s b ≤ b let Y b ≥s b = {x ∈ {0, 1} Z ; |supp x mod b| ≤ b − s b } and Y b s b = {x ∈ {0, 1} Z ; |supp x mod b| = b − s b }. Then Y b
≥s b is closed and σ-invariant. By the minimality of (X η , σ), it follows that for any
0 ≤ s b ≤ b, b ∈ B, we have X η ∩ Y b ≥s b = X η or X η ∩ Y b ≥s b = ∅.
By Corollary 3.15, η ∈ Y , so for s b ≥ 2 we have
X η ∩ Y b ≥s b = ∅.
Since for any b ∈ B we have
X η = X η ∩ 1≤s b ≤b Y b s b , it follows immediately that X η = X η ∩ Y . (d) ⇒ (b) Suppose X η ⊆ Y . By Theorem 3.3, we have X η * ⊆ X η , so, in particular, η * ∈ X η is an element of Y . Since η * is a Toeplitz sequence, it follows by Corollary 3.15 that η * ∈ Y * , where Y * = {x ∈ {0, 1} Z ; |supp x mod b * | = b * − 1 for any b * ∈ B * }. Suppose there exists b ∈ B \ B * . Let b * ∈ B be such that b * | b. Then |supp η * mod b * | = b * − 1
and we conclude that
|supp η * mod b| = b b * |supp η * mod b * | = b b * (b * − 1) < b − 1.
Thus, η * ∈ Y , which we know that is not true. It follows that B = B * and therefore η = η * is a Toeplitz sequence.
Proof of Corollary 3.15. Suppose that η is a Toeplitz sequence and r + bZ ⊆ M B for some b ∈ B and 1 ≤ r ≤ b. Then by Theorem 3.14 (for Lemmas needed for the proof of Theorem 4.1 are used also in the proof of Theorem F. We decided to keep both results to make the arguments easier to digest. The auxiliary lemmas that we present now are valid in any Dedekind ring R but we phrase them here for R = O K . The proofs of the lemmas are contained in the next subsection. Let a 1 , . . . , a n , b O K be pairwise coprime (proper) ideals and x 1 , . . . ,
d = b), there is b ′ ∈ B such that b ′ | gcd(r, b). Since B is primitive, b = b ′ . Hence r = b. So η ∈ Y .x n ∈ O K . Then b ⊆ n i=1 (x i + a i ).
Given an ideal a and y ∈ O K , we set v a (y) := max{t : y ∈ a t }. Lemma 4.3. Let r, g ∈ O K and (r, g) = p α 1 1 · . . . · p αm m , where p i , 1 ≤ i ≤ m, are distinct prime ideals ordered in such a way that for some 0 ≤ m ′ ≤ m we have
(33) ν p i (g) = ν p i (r) ⇐⇒ i > m ′ .
Then for
x ∈ p m ′ +1 · . . . · p m \ m ′ i=1 p i , we have (r + xg) = (r, g)q 1 . . . q n ,
where q i are prime ideals and q i = p j for all 1 ≤ i ≤ n and all 1 ≤ j ≤ m.
We will need the following generalization of Lemma 4.3.
Lemma 4.4. Let r, g ∈ O K . Then there exists a sequence x 1 , x 2 , . . . ∈ O K such that
(r + x i g) = (r, g)q (i) 1 . . . q (i) m i , where q (i)
k are prime ideals and q
(i) k = q (j) l for i = j, 1 ≤ k ≤ m i , 1 ≤ l ≤ m j and q (i) k ∤ (r, g) for i ≥ 1.
In particular, for i = j, we have (r + x i g, r + x j g) = (r, g).
Note that Lemma 4.4 will play the same role in the proof of Theorem 4.1 as Lemma 3.18 in the proof of Theorem 3.14. To see that Lemma 4.4 is indeed a multidimensional version of Lemma 3.18 notice that we can rephrase Lemma 3.18. Proof of Theorem 4.1. Let x 1 , x 2 , . . . ∈ O K be the sequence from Lemma 4.4. For i ≥ 1,
let b i ∈ B be such that r + x i g ∈ b i .
By Lemma 4.4, we have
(r, g)q (i) 1 . . . q (i) m i = (r + x i g) ⊆ b i .
We claim that there exists b ∈ B such that b i = b for infinitely many i ≥ 1. Indeed, if this is not the case, then passing to a subsequence, we can assume that all b i are distinct. Moreover, since (r, g) has finitely many factors, passing to a further subsequence if necessary, we have that gcd(b i , (r, g)) = d for some ideal d.
Now, b i = gcd(b i , (r, g))c ′ i , where c i | q (i) 1 . . . q (i) m i . It follows that d ∈ D,
which is a contradiction. Consider the prime factorization of b:
b = q ′ 1 · . . . · q ′ n
and take i such that
(34) {q (i) 1 , . . . , q (i) m i } ∩ {q ′ 1 , . . . , q ′ n } = ∅.
We claim that this yields (r, a) ⊆ b. Indeed,
r + x i g ∈ b ⇐⇒ b | (r + x i g) = (r, g)q (i) 1 · . . . · q (i) m i ,
which, in view of (34), implies that b | (r, g 1 , . . . , g t ), i.e. (r, a) ⊆ b.
To prove Theorem F, we will need the following extension of Lemma 4.4. such that (r + x i,1 g 1 + . . . + x i,t g t ) = (r, g 1 , . . . , g t )q
(i)
1 . . . q (i) m i , where q (i)
j are prime ideals and q
(i) k = q (j) l for i = j, 1 ≤ k ≤ m i , 1 ≤ l ≤ m j and q (i) k ∤ (r, a 1 , . . . , a t ) for i ≥ 1.
Proof of Theorem F. Let g 1 , . . . , g t be a set of generators 3 of a. Let x 1 , x 2 , . . . be as in Lemma 4.6. Since B does not contain a rescaled copy of an infinite pairwise coprime set, we have
r + x i,1 g 1 + · · · + x i,t g t ∈ b
for infinitely many i's for some b ∈ B. Consider the prime factorization of b:
b = q ′ 1 · . . . · q ′ n
and take i such that
(35) {q (i) 1 , . . . , q (i) m i } ∩ {q ′ 1 , . . . , q ′ n } = ∅.
We claim that this yields (r, a) ⊆ b. Indeed,
r + x i,1 g 1 + · · · + x i,t g t ∈ b ⇐⇒ b | (r + x i,1 g 1 + · · · + x i,t g t ) = (r, g 1 , . . . , g t )q
(i) 1 · . . . · q (i) m i ,
which, in view of (35), implies that b | (r, g 1 , . . . , g t ), i.e. (r, a) ⊆ b. 3 It is known that every ideal in a Dedekind ring is generated by 2 elements, so we could assume that t ≤ 2. This, however, does not simplify the proofs substantially.
Proofs of lemmas
Proof of Lemma 4.2. This is a simple consequence of the Chinese Remainder Theorem for rings. Indeed, for any choice of y 1 , . . . , y n ∈ O K , one can find x ∈ O K such that
x ∈ b and x − y i ∈ a i for 1 ≤ i ≤ n. Now, it suffices to apply the above to any y 1 , . . . , y n such that
y i − x i ∈ a i to conclude that b ∩ n i=1 (x i + a i ) c = ∅.
Proof of Lemma 4.3. For every 1 ≤ i ≤ m, we will prove that v p i (r + xg) = α i for each
1 ≤ i ≤ m. Consider first i > m ′ . We have v p i (r) = v p i (g) = α i and v p i (x) = 1.
It follows that, as
v p i (r) = α i < v p i (xg), v p i (r + xg) = α i . Now, consider 1 ≤ i ≤ m ′ . We have v p i (x) = 0, v p i (r) ≥ α i , v p i (g) ≥ α i
and precisely one of the following:
(36) either v p i (r) > α i or v p i (g) > α i . It follows that v p i (r) = v p i (xg), thus v p i (r + xg) = min(v p i (r), v p i (xg)) = α i . This yields v p i (r + xg) = α i for each 1 ≤ i ≤ m.
Proof of Lemma 4.4. Let p 1 , . . . , p m , α 1 , . . . , α m and m ′ be as in the assumptions of Lemma 4.3 and let
x 1 ∈ p m ′ +1 · . . . · p m \ ( m ′ i=1 p i ).
Then
(r + x 1 g) = p α 1 1 · . . . · p αm m q (1) 1 · . . . · q (1) m 1 and q(1)
k ∈ {p 1 , . . . , p m } are prime ideals. Suppose that we have chosen x 1 , . . . , x i−1 and take
x i ∈ p m ′ +1 · . . . · p m \ m ′ j=1 p j ∪ i−1 j=1 m j l=1 (q (j) l + x j )
(Lemma 4.2 guarantees that such x i exists). It follows by Lemma 4.3 that
(r + x i g) = p α 1 1 · . . . · p αm m · q (k) 1 · . . . · q (k) m k , where q (i) k ∈ {p 1 , . . . , p m }. It remains to show that q (i) k = q (j) ℓ whenever j = i. Suppose otherwise, then (x j − x k )g ∈ q (j) ℓ Now, x j − x i / ∈ q (j) ℓ , as x i ∈ q (j)
ℓ + x j . Thus, g ∈ q (j) ℓ and, consequently,
r = r + x i g − x i g ∈ q (j)
ℓ . This is impossible, as q (j) ℓ = p i for all i. The remaining statement follows easily.
Proof of Lemma 4.6. The proof is inductive. For t = 1 the assertion becomes the one of Lemma 4.4. Now, take t > 1. There exists a sequence
x ′ 1 = (x ′ 1,1 , . . . , x ′ 1,t−1 ), x ′ 2 = (x ′ 2,1 , . . . , x ′ 2,t−1 ), · · · ∈ O t−1 K such that (r + x ′ i,1 g 1 + · · · + x ′ i,t−1 g t−1 ) = (r, g 1 , . . . , g t−1 ) · s (i) 1 · . . . · s (i) m ′ i , where s (i)
k are prime ideals such that s
(i) k = s (j) l , whenever i = j. Fix x ′ = (x ′ 1 , . . . , x ′ t−1 ) such that (r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 ) = (r, g 1 , . . . , g t−1 )s 1 · . . . · s m ′
and g t ∈ m ′ j=1 s j . By Lemma 4.4 (applied to r ′ = r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 and g ′ = g t ), there exists a sequence (x i ) i≥1 ⊆ O K such that, for every i ≥ 1:
(37) (r ′ + x i g ′ ) = (r ′ , g ′ )c (i) 1 · . . . · c (i) l i , i.e. (r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 + x i g t ) = (r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 , g t )c (i) 1 · . . . · c (i) l i , where c (i) 1 , . . . , c (i)
l i are prime ideals that do not divide (r ′ , g ′ ) and
{c (i) 1 , . . . c (i) l i } ∩ {c (j) 1 , . . . c (j) l j } = ∅ whenever i = j.
Now,
(r ′ , g ′ ) = (r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 , g t ) = (r, g 1 , . . . , g t−1 )s 1 · . . . · s m ′ + (g t ).
Clearly, (r ′ , g ′ ) ⊆ (r, g 1 , . . . , g t ).
On the other hand, since g t / ∈ s 1 ∪ . . . ∪ s m ′ , it follows that 1 ∈ (g t ) + s 1 · . . . . · s m ′ . Thus (r, g 1 , . . . , g t−1 ) + (g t ) ⊆ (r, g 1 , . . . , g t−1 )s 1 · . . . · s m ′ + (g t ),
and we get (r ′ , g ′ ) = (r, g 1 , . . . , g t ).
This equality, combined with (37), yields
(r + x ′ 1 g 1 + · · · + x ′ t−1 g t−1 + x i g t ) = (r, g 1 , . . . , g t )c (i) 1 · . . . · c (i) l i .
Non-periodic positions and periodic structure
The first goal of this section is to give a description of non-periodic positions on η, i.e. to prove Proposition D.
We have
F B * = F B ∩ F D ,
where D is defined in (1). Equivalently, this equality can be rewritten as
(38) η * n = η n , if n ∈ M D , 0, if n ∈ M D ,
Recall that for any ideal s ⊆ O K places which are s-periodic are denoted by Per(η, s) = {g ∈ O K ; η g = η g+s for any s ∈ s}.
Let (S i ) i≥1 be a saturated filtration of B by finite collections of ideals. For i ≥ 1 put
s i := b∈S i b = lcm(S i ).
Lemma 4.7. Let g ∈ G and {e G } = H 1 , H 2 ⊆ G be subgroups. Then
(H 1 + g) ∩ H 2 = ∅ if and only if g ∈ H 1 + H 2 .(39)
The proof of the above lemma is straightforward, so we skip it. Proof of Proposition D. It suffices to show that
(40) O K \ i≥1 Per(η, s i ) ⊆ M D ∩ F B ⊆ O K \ s⊂O K Per(η, s). Clearly, O K \ s⊂O K Per(η, s) ⊆ O K \ i≥1
Per(η, s i ) and therefore the inclusions must be, in fact, equalities.
First inclusion. Take n ∈ i≥1 Per(η, s i ). Then n ∈ F B . Moreover, n + s k ⊆ F B for any k ≥ 1. In other words, (n + s k ) ∩ M B = ∅, so there exists b k such that n ∈ b k + s k = gcd(b k , s k ), so that
nO K ⊆ b k + s k = gcd(b k , s k ).
Notice that b k ∈ S k (if b k ∈ S k then s k ⊆ b k and n ∈ b k + s k ⊆ b k which contradicts n ∈ F B ). It follows by Lemma 4.8 that the index of b k +s k is divisible by the index of nO K , so by Lemma 4.9, there are only finitely many possibilities for b k + s k . In other words, there exists an ideal d and a subsequence (k ℓ ) ⊆ N such that gcd(b k ℓ , s k ℓ ) = b k ℓ + s k ℓ = d (and n ∈ d, as n ∈ b k + s k for each k ≥ 1). Let ℓ 2 > ℓ 1 := 1 be such that
(41) b k ℓ 1 ∈ S k ℓ 2 (in particular, s k ℓ 2 ⊆ b k ℓ 1 ).
By the choice of (k ℓ ),
we have b k ℓ 1 , b k ℓ 2 ⊆ d, so b k ℓ 1 + b k ℓ 2 ⊆ d.
On the other hand, using (41), we obtain
d = b k ℓ 2 + s k ℓ 2 = gcd(b k ℓ 2 , s k ℓ 2 ) ⊆ gcd(b k ℓ 2 , b k ℓ 1 ). Thus, gcd(b k ℓ 1 , b k ℓ 2 ) = b k ℓ 1 + b k ℓ 2 = d.
Suppose that we have constructed 1 = ℓ 1 < ℓ 2 < . . . < ℓ m such that
(42) gcd(b k ℓ i , b k ℓ i ′ ) = d for any 1 ≤ i < i ′ ≤ m.
There exists ℓ m+1 > ℓ m such that
(43) b k ℓ 1 , b k ℓ 2 , . . . , b k ℓm ∈ S k ℓ m+1 .
It follows by the choice of (k ℓ ) that b k ℓ i ⊆ d for 1 ≤ i ≤ m + 1. This gives
gcd(b k ℓ i , b k ℓ i ′ ) = b k ℓ i + b k ℓ i ′ ⊆ d for any 1 ≤ i < i ′ ≤ m + 1.
On the other hand, using (43), we obtain
d = gcd(b k ℓ m+1 , s k ℓ m+1 ) ⊆ gcd(b k ℓ i , b k ℓ m+1 ). Thus, gcd(b k ℓ i , b k ℓ i ′ ) = d for any 1 ≤ i < i ′ ≤ m + 1
and the above inductive procedure therefore yields a sequence 1 = ℓ 1 < ℓ 2 < . . . such that
gcd(b k ℓ i , b k ℓ i ′ ) = d for any i = i ′ .
Let c i , for i ≥ 1, be an ideal such that c i d = b k ℓ i . It remains to show that C := {c i : i ≥ 1} is pairwise coprime. However, we have
d = gcd(b k ℓ i , b k ℓ i ′ ) = b k ℓ i + b k ℓ i ′ = (c i + c i ′ )d
and after dividing the above by d, we conclude that c i + c i ′ = O K for i = i ′ . This tells us that n ∈ M D .
Second inclusion. Take n ∈ M D ∩ F B . There exists a non-zero ideal d and an infinite pairwise coprime collection C of ideals such that n ∈ d and dC ⊆ B.
Suppose that n ∈ Per(η, s) for some non-zero ideal s. In other words, we have n + s ⊆ F B . Since n ∈ d, it follows that
(44) d ∩ (n + s) = n + d ∩ s ⊆ F B .
Take c ∈ C coprime to s and d. Then Clearly, s i+1 ⊆ s i . Suppose that s is such that
Per(η, s i ) ⊆ Per(η, s). For any b ∈ S i , we have b ⊆ Per(η, s i ) ⊆ Per(η, s), whence b + s ⊆ M B .
Take r ∈ s. It follows by Theorem F (for a = b) that for some b ′ ∈ B we have
(r) + b ⊆ b ′ .
In particular, b ⊆ b ′ , which, by the primitivity of B implies b ′ = b. Thus,
s ⊆ b,
as r ∈ s was arbitrary. Now, since b ∈ S i was arbitrary, we obtain s ⊆ s i and we conclude that s is an essential group of periods.
Since M 2 is (T g ) g∈G -invariant, there is an open set W with M 2 ⊆ W ⊆ V such that T g W ⊆ V for any g ∈ F N . But the orbit of x 0 is dense in X; therefore there exists g ′ ∈ G with T g ′ x 0 ∈ W . By (45), there exist g ∈ G and f ∈ F N such that T g x 0 ∈ U n and
g ′ = g + f . So U n ∋ T g x 0 = T g ′ −f x 0 = T −f (T g ′ x 0 ) ∈ V,
contradicting U n ∩ V = ∅. By uniqueness of M ′′ and by above we have M ⊆ M ′ ⊆ M ′′ and the reduction of M ′ has no impact on (d), the equalities hold. To finish the proof note that x M is minimal, so
x M ∈ M = M ′ = M ′′ .{B i } i≥1 such that {g ∈ G; x 0 | B i = x 0 | B i +g } is syndetic for any i ≥ 1.
Proof. It follows immediately from the equivalence (a) and (e) from Proposition 4.10.
Main part
Proof of Theorem C. We will first show (i) ⇐⇒ (ii). Clearly, if D = ∅ then B = B * . For the other direction, notice that
B = B * = (B \ M D ) ∪ D prim implies D prim ⊂ B.
However, by the definition of D, for any d ∈ D, some (non-trivial) multiple of d is a member of B, as dC ⊂ B where C is inifinite (and pairwise coprime). If D = ∅ this contradicts the primitivity of B.
Let us show that D * = ∅ (i.e. η * is Toeplitz, so (i) =⇒ (iii)). Suppose that there is some d and some infinite pairwise coprime set C such that
dC ⊆ B * = (B \ M D ) ∪ D prim .
Then, without loss of generality (taking a smaller but still infinite and pairwise coprime set C), one of the following holds:
(A) dC ⊆ B \ M D , (B) dC ⊆ D prim .
If (A) holds then dC ⊂ B, so d ∈ D and we obtain
dC ⊂ B \ M D ⊂ B \ d,
which yields a contradiction. Suppose now that (B) holds. Let C = {c 1 , c 2 , . . . } and let A i for i ≥ 1 be infinite, pairwise coprime and such that dc i A i ⊂ B. We can choose a i ∈ A i , i ≥ 1 so that {c i a i : i ≥ 1} is infinite and pairwise coprime (we just use that each A i is infinite and pairwise coprime). This gives again d ∈ D. This contradicts (B) (by the primitivity of D prim ). We conclude that indeed D * = ∅.
Finally, we will show (iii) =⇒ (i). If η is Toeplitz then Proof of Theorem A. Since B * = (B ∪ D) prim , we have η * ≤ η. Moreover, η * is Toeplitz, by Theorem C. Now, we will prove that η * ∈ X η . Let (F k ) k≥1 be a nested Følner sequence. We will show that for any N ≥ 1, η * | F N appears on η. Fix N ≥ 1 and let i ≥ 1 be sufficiently large to ensure that
(46) M D ∩ F B ∩ F N = (O K \ Per(η, s i )) ∩ F N .
Recall that by Proposition D we have M D = O K \ i≥1 Per(η, s i )), and Per(η, s i ) grows, as i grows. Moreover, by Proposition D, M D ∩ F B ∩ F N are all non-periodic positions on η restricted to F N . Let t N be the cardinality of M D ∩ F B ∩ F N and denote the elements of this set by I 1 , . . . , I t N . It follows by the definition of D that for any 1 ≤ j ≤ t N , there exists a non-zero ideal d j and an infinite pairwise coprime set C j such that
I j ∈ d j and d j C j ⊆ B. Let L := lcm(s i , d 1 , . . . , d t N ).
Let c 1 ∈ C 1 be coprime to L (such an ideal exists since L has finitely many factors and C 1 is infinite pairwise coprime). Then I 1 ∈ d 1 = L + d 1 c 1 = gcd(L, d 1 c 1 ) and therefore we can find k 1 ∈ L such that k 1 + I 1 ∈ d 1 c 1 .
Let c 2 ∈ C 2 be coprime to L and c 1 (again, such an ideal exists because L and c 1 have finitely many factors and C 2 is infinite pairwise coprime). Since L ⊆ d 2 and L, c 1 and c 2 are coprime, we have
k 1 + I 2 ∈ L + d 2 = d 2 Moreover, d 2 = gcd(Lc 1 , d 2 c 2 ) = gcd(L ∩ c 1 , d 2 c 2 ) = L ∩ c 1 + d 2 c 2
Therefore, k 1 + I 2 ∈ L ∩ c 1 + d 2 c 2 and thus, we can find k 2 ∈ L ∩ c 1 such that
k 2 + k 1 + I 2 ∈ d 2 c 2 .
Notice that since k 2 ∈ L ∩ c 1 ⊆ d 1 c 1 , we have
k 2 + k 1 + I 1 ∈ d 1 c 1 .
Let c 3 ∈ C 3 be coprime to L, c 1 and c 2 . Since L ⊆ d 3 and L, c 1 , c 2 and c 3 are pairwise coprime, we have
k 2 + k 1 + I 3 ∈ L ∩ c 1 + L + d 3 = d 3 Moreover, d 3 = L ∩ c 1 ∩ c 2 + d 3 c 3 .
Therefore, k 2 + k 1 + I 3 ∈ L ∩ c 1 ∩ c 2 + d 3 c 3 and thus, we can find k 3 ∈ L ∩ c 1 ∩ c 2 such that
k 3 + k 2 + k 1 + I 3 ∈ d 3 c 3 .
Notice that since k 3 ∈ d ℓ c ℓ for ℓ = 1, 2, we have
k 3 + k 2 + k 1 + I ℓ ∈ d ℓ c ℓ for ℓ = 1, 2.
By repeating the above arguments, we obtain k j ∈ L∩ j−1 i=1 c i and c j ∈ C j , j = 1, 2, . . . , t N , such that
(47) k t N + . . . + k 2 + k 1 + I k ∈ d j c j for j = 1, 2, . . . t N . Put M N := k t N + . . . + k 2 + k 1 ∈ L.
We will show that (48) η n+M N = η n = η * n for n ∈ Per(η, s i ) ∩ F N .
Take n ∈ Per(η, s i ) ∩ F N . Since M N ∈ L ⊆ s i , it follows that (49) η n+M N = η n for n ∈ Per(η, s i ) ∩ F N and n + M N ∈ Per(η, s i ). It follows by Proposition D that n, n + M N ∈ M D ∩ F B (the latter set is the set of all non-periodic positions on η). In other words,
n, n + M N ∈ F D ∪ M B = (F D ∩ F B ) ∪ M B .
Recall that we have
B * = (B ∪ D) prim , so M B * = M B ∪ M D . Equivalently, F B * = F B ∩ F D . Therefore, • if n ∈ F D ∩ F B then n ∈ F B * ,
• if n ∈ M B then n ∈ M B * .
Hence, (48) indeed holds. Now, we will show that
(50) η n+M N = η * n for n ∈ (O K \ Per(η, s i )) ∩ F N .
Since d j c j ∈ B and I j + M N ∈ d j c k , it follows by (47) that
(51) η I j +M N = 0 for all 1 ≤ j ≤ t N .
Moreover, it follows by I j ∈ M D ⊆ M B * that we also have
η * I j = 0 for all 1 ≤ j ≤ t N .
Therefore, (50) indeed holds. Combining (48) and (50), we conclude that
(52) η| F N +M N = η * | F N .
Notice that in the above arguments we used only that M N ∈ L and M N + I j ∈ d j c j (to obtain (49) and (51), respectively). Thus, by (47),
η| F N +M N = η| F N +M N +s for any s ∈ L ∩ s N j=1 c j .
By Lemma 4.8, L ∩ s N j=1 c j has finite index, so {s ∈ O K ; η| F N +M N = η| F N +M N +s } is syndetic. Moreover, since (F k ) k≥1 is increasing, so blocks (η| F N +M N ) N ≥1 have different lengths, so they are pairwise different. By Corollary 4.11, X η has a unique minimal subset. Indeed, since n + M N ∈ M D ∩ F B , we can show that η n+M N = η * n+M N for n ∈ Per(η, s i ) ∩ F N , using the same arguments as were used in the proof of Theorem A to show η n = η * n for n ∈ Per(η, s i ) ∩ F N so the same lines as we showed in the proof of Theorem A η n = η * n for n ∈ Per(η, s i ) ∩ F N , one can show η n+M N = η * n+M N for n ∈ Per(η, s i ) ∩ F N . Since I j ∈ d j ⊆ M D and I j + M N ∈ L ⊆ d j ⊆ M D , so η * I j = η * I j +M N for all j ∈ {1, . . . , s N }. By (53), S M N η| F N = S M N η * | F N , so d(S M N η, S M N η * ) ≤ 1 2 N . Hence (η, η * ) is proximal. To prove Theorem B, we will also need the following lemma. Proof of Theorem B. Obviously, (ii)⇒(i).
(i)⇒(ii). It follows from (i) and from Theorem A that X η * = X η . Suppose that η * < η, i.e. there is n ∈ O K such that 0 = η * n < η n = 1. Let p be such that η * | n+p = 0. Notice that both, Per(η, p, 0) and Per(η * , p, 0) are invariant under translations by the elements of p, so it is natural to look at them as subsets of the finite quotient group O K /p. It follows from (12) that (54) | Per(η * , p, 0) mod p| = | Per(η, p, 0) mod p| as η ∈ X η * . On the other hand, since η * ≤ η, it follows immediately that Per(η * , p, 0) ⊇ Per(η, p, 0).
It remains to notice that n ∈ Per(η * , p, 0) \ Per(η, p, 0), which contradicts (54). This yields η = η * , i.e. (ii) holds.
(ii) ⇒ (iv) Suppose η is an O K -Toeplitz array. For any b ∈ B and 0 ≤ s b ≤ N (b) let
Y b ≥s b = {x ∈ {0, 1} O K ; |supp x mod b| ≤ N (b) − s b } and Y b s b = {x ∈ {0, 1} O K ; |supp x mod b| = N (b) − s b }. Then Y b
≥s b is closed and (S g ) g∈O K -invariant. By the minimality of (X η , (S g ) g∈O K ), it follows that for any 0 ≤ s b ≤ N (b), b ∈ B, we have
X η ∩ Y b ≥s b = X η or X η ∩ Y b ≥s b = ∅. By Lemma 4.13, we have η ∈ Y , so for s b ≥ 2 we have X η ∩ Y b ≥s b = ∅. Because for any b ∈ B X η = X η ∩ 1≤s b ≤N (b) Y b s b ,
it follows immediately that X η = X η ∩ Y .
(iv) ⇒ (ii) Suppose X η ⊆ Y . By Theorem A, we have X η * ⊆ X η , so, in particular, η * ∈ X η is an element of Y . Since η * is an O K -Toeplitz array, it follows by Lemma 4.13 that η * ∈ Y * , where Y * = {x ∈ {0, 1} O K ; |supp x mod b * | = N (b * ) − 1 for any b * ∈ B * }.
Suppose there exists b ∈ B \ B * . Let b * ∈ B be such that b ⊆ b * . Then
|supp η * mod b * | = N (b * ) − 1
and we conclude that
|supp η * mod b| = N (b) N (b * ) |supp η * mod b * | = N (b) N (b * ) (N (b * ) − 1) < N (b) − 1.
Thus, η * ∈ Y , which we know that is not true. It follows that B = B * and therefore η = η * is an O K -Toeplitz array.
Proof of Lemma 4.13. Assume that η is an O K -Toeplitz array. Suppose η ∈ Y . Then for some b ∈ B and some r ∈ b r + b ⊆ M B .
By Theorem F, there exists b ′ ∈ B such that
(r) + b ⊆ b ′ .
Then by the primitivity of B, b = (r) + b = b ′ , which contradicts r ∈ b.
Counterexample to Theorem B for lattices
We mentioned in the introduction that as the counterpart of the set of multiples M B one can choose countable unions of sublattices of Z m . In this case the analogous result to Theorem B does not hold. Another example in [7] shows that the arithmetic characterization of proxmiality valid in the setting of ideals in integer rings also fails to hold in the lattice setting. These examples can be treated as a reason to work with integer rings in number fields and ideals rather than with lattices. Let us state now two necessary results. Z × dZ.
The proof of the above lemma is straightforward, so we skip it. We claim that:
1) η = ½ Z 2 \ i≥1 Λ i is not a Z 2 -Toeplitz array, 2) B does not contain a scaled copy of infinite pairwise coprime collection of lattices {Λ ′ i } i≥1 , i.e. {(a, b)Λ ′ i } i≥1 B for any (a, b) ∈ Z 2 , such that a, b = 0, where we consider the coordinatewise multiplication in Z 2 .
Indeed, by Lemma 5.2 (for Λ = Λ i , i ≥ 1), for i ≥ 1
a i c i Z × 2 i c i Z ⊆ Λ i .
Since {a i c i } i≥1 is infinite and pairwise coprime, there are arbitrarly long blocks of consecutive zeros on η 1 := η (n,0) n∈Z . Since a i ≥ 2, i ≥ 1, η (1,0) = 1. Suppose that there is Λ = (c, d)Z + (0, f )Z such that (1, 0) + Λ ⊆ F B .
By Lemma 5.2, cf gcd(d, f ) Z × dZ ⊆ Λ.
Hence, for any k ∈ Z,
η 1 1 + cf gcd(d, f ) k = 1,
but this contradicts that there are arbitrarly long blocks of consecutive zeros on η 1 . Hence η is not a Z 2 -Toeplitz array. Suppose that for some infinite pairwise coprime collection of lattices {Λ ′ i } i≥1 and some (a, b) ∈ Z 2 we have
{(a, b)Λ ′ i } i≥1 ⊆ B.
Since (a, b)Λ ′ i ∈ B, there exists j i ≥ 1 such that
(55) (a, b)Λ ′ i = Λ j i . Let Λ ′ i = (d ′ i , e ′ i )Z + (0, f ′ i )Z, i ≥ 1. Since {Λ ′ i } i≥1
is a collection of pairwise coprime lattices, by Theorem 5.1 we can assume {d ′ i f ′ i } i≥1 is infinite and pairwise coprime. Let
r i = d ′ i f ′ i , i ≥ 1.
Since r i = [Z 2 : Λ ′ i ], we have r i Z 2 ⊆ Λ ′ i for any i ≥ 1, so
ar i Z × br i Z ⊆ Λ j i .
By Lemma 5.2, the maximal lattice contained in Λ j i equals
a j i c j i Z × 2 j i c j i Z.
Hence
ar i Z × br i Z ⊆ a j i c j i Z × 2 j i c j i Z, i ≥ 1.
Thus, 2 j i c j i | br i , i ≥ 1.
Since r i , i ≥ 1, are pairwise coprime, we can assume that all of them are odd. Hence
2 j i | b, but {Λ ′ i } i≥1 is infinite, so 2 j i → ∞ as i → ∞.
It follows that b = 0, but then (a, b)Λ ′ i is of infinite index in Z 2 , which contradicts (55), as elements of B have finite indices.
fields and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 B-free integers in number fields . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Toeplitz sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 Dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.2 Higher dimension (abelian discrete, finitely generated groups) . . . . 8 3 Minimality and periodic structure of B-free systems 11 3.1 Background and main results . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.1 Arithmetic progressions in M B : proof of Theorem 3.14 . . . . . . . 15 3.2.2 Non-periodic positions (proof of Proposition 3.16) and periodic structure (proof of Theorem 3.17) . . . . . . . . . . . . . . . . . . . . . . 16 3.2.3 Minimality: proof of Theorems 3.3 and 3.7 . . . . . . . . . . . . . . . 18 4 Minimality and periodic structure of B-free systems 22 4.1 Arithmetic progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Non-periodic positions and periodic structure . . . . . . . . . . . . . . . . . 26 4.3 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.2 Main part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Counterexample to Theorem B for lattices 34
Theorem B. Let B be a primitive collection of ideals in O K . The following are equivalent:(i) (X η , (S g ) g∈O K ) is minimal, (ii) η is an O K -Toeplitz array different from 0, where 0 g = 0 for any g ∈ O K , (iii) D = ∅, (iv) X η ⊆ Y , where Y = {x ∈ {0, 1} O K ; | supp x mod b| = N (b) − 1 for any b ∈ B} (N (b)stands for the cardinality of the quotient O K /b).
(i) B = B * , (ii) D = ∅,(iii) η is a Toeplitz sequence different from 0. Moreover, we have D * := {d : dC ⊂ B * for some infinite pairwise coprime set C} = ∅, i.e. B * = (B * ) * and η * is Toeplitz.
Definition 2. 4 ([ 5 ,
45Definition 7.3]). Number s ∈ N is said to be an essential period of x ∈ A Z if Per(x, s) = ∅ and(7)q < s =⇒ Per(x, s) = Per(x, q).
Definition 2. 5
5([25]). Number s ∈ N is said to be an essential period of x ∈ A Z if Per(x, s) = ∅ and(9) (Per(x, s, a) = Per(x, s, a) − q for all a ∈ A) =⇒ s | q.
Proposition 2. 7 .
7Fix x ∈ A Z and s, q ∈ N. The following conditions are equivalent: (a) Per(x, s, a) = Per(x, s, a) − q for all a ∈ A, (b) Per(x, s, a) = Per(x, s, a) − qZ for all a ∈ A,
( c )
cPer(x, s, a) = Per(x, gcd(s, q), a) for all a ∈ A,
( d )
dPer(x, s, a) ⊂ Per(x, q, a) for all a ∈ A,
( e )
ePer(x, s) ⊂ Per(x, q).
Theorem 2 .
211 ([3, Proposition 5], cf. (10) and Remark 2.10). Let x ∈ A G . The following are equivalent:
Corollary 2 .
218 (cf. Remark 2.16).
Theorem 3.1 ([9, Theorem A], "baby version" of Theorem A in dimension one). For any B ⊂ N, the subshift (X η , σ) has a unique minimal subset, denoted by M . Moreover, M is the orbit closure of a Toeplitz sequence.
Theorem 3. 2
2([17, Theorem B], almost complete version of Theorem B in dimension one). Suppose that B is primitive. Let W = {h ∈ H : h b = 0 for all b ∈ B}, where H is defined by (5) for O K = Z and b = bZ, b ∈ B, i.e. H = {(n, n, n, . . . ) : n ∈ Z} ⊂ b∈B Z/bZ. Consider the following list of properties:
Theorem 3. 3 ([ 18 ,
318Corollary 5, Lemma 3], Theorem A in dimension one). For any primitive B ⊂ N let
Theorem 3.3 allows one to get rid of the technical assumption ∆(Z) ∩ W = W in Theorem 3.2, see Theorem 3.7 and Corollary 3.8.
Example 3 . 6 .
36It was shown in[17, Lemma 3.6] that η ∈ Y necessary for ∆(Z) ∩ W = W .
Theorem 3. 7 (
7Theorem B in dimension one). Let B ⊆ {2, 3, . . .} be primitive. The following conditions are equivalent:
Theorem 3 . 9 (
39Theorem C in dimension one). Let B ⊂ {2, 3, . . . } be primitive. Then the following are equivalent: (a) B = B * , (b) D = ∅, (c) η is a Toeplitz sequence different from . . . 0.00 . . .. Moreover, we have D * := {d ∈ N : dC ⊂ B * for some infinite pairwise coprime set C} = ∅, i.e. B * = (B * ) * and η * is Toeplitz.
Lemma 3.10 ([6, Proposition 5.3] and [17, Lemma 3.6]). Let B ⊂ N be primitive. If η is a Toeplitz sequence then B is taut.
Lemma 3 .
311 ([9, Corollary 4.32]). If B ⊂ N is taut then η ∈ Y .
Corollary 3. 15 .
15If η is a Toeplitz sequence then η ∈ Y .
Lemma 3. 18 .
18Let d, r ∈ N. Then there exists an infinite pairwise coprime set C such that gcd(r, d)C ⊆ r + dZ.
Lemma 3. 19 .
19Suppose that B is such that D = ∅. If dC ⊆ M B for some d ∈ N and some infinite pairwise coprime set C then d ∈ M B . Proof of Theorem 3.14. Suppose that r + dZ ⊆ M B . Then, by Lemma 3.18, we have gcd(r, d)C ⊆ r + dZ ⊆ M B
-periodic positions (proof of Proposition 3.16) and periodic structure (proof of Theorem 3.17)
∅
= (n + lcm(s, d)Z) ∩ dcZ ⊆ (n + lcm(s, d)Z) ∩ M B as dc ∈ B. This, however, contradicts (21). Proof of Theorem 3.17. For i ≥ 1 put s i := lcm(S i ). It follows from the equivalence (5) ⇔ (4) in Theorem 3.2 that D = ∅ and thus M D ∩ F B = ∅. Together with Proposition 3.16 this yields i≥1 Per(η, s i ) = Z.
Example 3 . 20 .
320Let B = {2 i c i ; i ≥ 1}, where c i ≥ 2 for i ≥ 1.In Example 3.1 in[9] it is shown that Z = i≥1 Per(η, 2 i+1 c 1 . . . c i ). Let S k = {2 i c i : 1 ≤ i ≤ k}. By Theorem 3.17, we have Z = i≥1 Per(η, 2 i c 1 , . . . , c i ).
Finally, we will show (c) =⇒ (a). If η is Toeplitz then F B ∩ M D = ∅ since (by Proposition 3.16) it is the set of non-periodic positions on η. Therefore, M D ⊆ M B and it follows immediately (see Remark 3.4) that M B * = M B . This yields B = B * by the primitivity of B and B * .
Proposition 3.16, we have M D ∩ F B = Z \ i≥1 Per(η, lcm(S i )), and Per(η, lcm(S i )) grows, as i grows. Moreover, by Proposition 3.16, M D ∩ F B ∩ [−N, N ] are all non-periodic positions on η restricted to [−N, N ]. Let t N be the cardinality of M D ∩ F B ∩ [−N, N ] and denote the elements of this set by I 1 , . . . , I t N . It follows by the definition of D that for any 1 ≤ j ≤ t N , there exists d j ∈ N and infinite pairwise coprime set C j ⊂ N such that d j | I j and d j C j ⊂ B. Let L := lcm(lcm(S i ), d 1 , . . . , d t N ).
Take n ∈ Per(η, lcm(S i )) ∩ [−N, N ]. Since lcm(S i ) | L | M N , it follows that(27) η n+M N = η n for n ∈ Per(η, lcm(S i )) ∩ [−N, N ]
L
| M N and d j c j | M N + I j (to obtain (27) and (29), respectively). Thus, by (25), η| [−N,N ]+M N = η| [−N,N ]+M N +s for any s ∈ Lc 1 . . . c t N Z. In particular, {s ∈ Z; η| [−N,N ]+M N = η| [−N,N ]+M N +s } is syndetic. Moreover, the blocks (η| [−N,N ]+M N ) N ≥1 have different lengths, so they are pairwise different. By [9, Corollary 2.17] (see Corollary 4.11), X η has a unique minimal subset. Proof of Theorem 3.7. Obviously, (b) ⇒ (a). (a) ⇒ (b).
Remark 3. 23 .
23Arguments used in the proof (b)⇒(d) of Theorem 3.7 comes from Example 3.16 in[9].
Lemma 4 . 5 .
45Let d, r ∈ N Then there exists a sequence (n i ) i≥1 ⊆ Z and an infinite pairwise coprime set C = {c 1 , c 2 , . . . } such that r + dn i = gcd(r, d)c i for i ≥ 1.
Lemma 4. 6 .
6Let r, g 1 , . . . , g t ∈ O K and (r, g 1 , . . . , g t ) = p 1 . . . p m . Then there exists a sequence x 1 = (x 1,1 , . . . , x 1,t ), x 2 = (x 21 , . . . , x 2t ), . . . ∈ O t K
Lemma 4.8 (see i.e. [20, Chapter I, Proposition 2.2]). Let H 1 , H 2 ⊆ G be subgroups of finite index. Then H 1 ∩ H 2 is also a subgroup of finite index in G and [G : H 1 ∩ H 2 ] = [G : H 1 ] · [H 1 : H 1 ∩ H 2 ].
Lemma 4.9 ([14, 2.4. s. 128]). If G is a finitely generated group, then it contains finitely many (possibly 0) subgroups of a given index n.The proof of Proposition D go along the same lines as the proof of Proposition 3.16.
n
∈ d = d(s + c) = ds + dc = d ∩ s + dc. Therefore (by Lemma 4.7 for G = O K , g = n, H 1 = d ∩ s i and H 2 = dc j ) we obtain (n + d ∩ s) ∩ M B ⊇ (n + d ∩ s) ∩ dc = ∅, which contradicts (44). Proof of Theorem E. For i ≥ 1 put s i := b∈S i b = lcm(S i ). By Proposition D, we get i≥1 Per(η, s i ) = O K .
Corollary 4 .
411 ([9, Corollary 2.17] for G = Z). Let (X, (S g ) g∈G ) be a subshift with a transitive point x 0 . Then (X, (S g ) g∈G ) is essentially minimal if and only if there exists an inifite family of pairwise distinct blocks
F
B ∩ M D = ∅ since (by Proposition D) it is the set of non-periodic positions on η. Therefore, M D ⊆ M B and it follows immediately (cf. Remark 3.4) that M B * = M B . This yields B = B * by the primitivity of B and B * .
Corollary 4. 12 .
12The pair (η, η * ) is proximal.Proof. Let N ≥ 1 and let F N , M N , s i , I j , d j , L, s N be the same as in the proof of Theorem A. Then(53) η| F N +M N = η * | F N +M N .
If η is an O K -Toeplitz array, then η ∈ Y .
Theorem 5. 1 ([ 7 ,
17Proposition 3.17]). Let m ≥ 2 and {Λ i } i≥1 be an inifite and pairwise coprime collection of lattices in Z m . Then {[Z m : Λ i ]} i≥1 contains an infinite pairwise coprime subset. Lemma 5.2. Let Λ = (a, b)Z + (0, d)Z be a lattice in Z 2 . Then the maximal (with respect to the inclusion relation) lattice contained in Λ equals ad gcd(b, d)
Example 5. 3 .
3Let a i , c i ∈ N \ {1}, i ≥ 1 be such that {a i c i } i≥1is an infinite and pairwise coprime set of odd numbers, {a i } i≥1 and {c i } i≥1 are infinite. LetΛ i = (a i , 2 i )Z + (0, 2 i c i )Z, i ≥ 1 and B = {Λ i } i≥1 .
The ideal d can be equal to O K .
Minimality and periodic structure of B-free systems4.1 Arithmetic progressionsThe main goal of this section is to prove Theorem F. We will prove first its easier version, valid for principal ideals.
Proposition 4.10 ([9, Proposition 2.15] for G = Z). Let G be countable abelian group. Let (X, (T g ) g∈G ) be a topological dynamical system with a transitive point x 0 ∈ X. Then the following conditions are equivalent: (a) (X, (T g ) g∈G ) has a unique minimal subset M , (b) there exists a closed, (T g ) g∈G -invariant subset M ′ ⊆ X such that for any x ∈ M ′ and y ∈ X, there exists (g n ) n≥1 ⊆ G such that g n → ∞ and T gn y → x as n → ∞, (c) there exists x M ∈ X such that for any y ∈ X there exists (g n ) n≥1 ⊆ G such that g n → ∞ and T gn y → x M ,Furthermore, if any of the above hold, thenis closed, non-empty and (T g ) g∈G -invariant. We will show that y∈X ω(y) is minimal. Let x ∈ y∈X ω(y). Then since y∈X ω(y) is closed and (T g ) g∈G -invariant, so X x ⊆ y∈X ω(y). Obviously, we have ω(x) ⊆ X x and y∈X ω(y) ⊆ ω(x). So ω(x) = X x = y∈X ω(y). Hence y∈X ω(y) is minimal. By (a), we get M = y∈X ω(y). Notice that any set M ′ satisfying (b) is contained in the set y∈X ω(y), so it also satisfies the equality M ′ = M .If {h ∈ G; T h x 0 ∈ V } is not syndetic, then for any n ∈ N we have {h ∈ G; T h x 0 ∈ V } + F n = G. So for any n ∈ N there exists k n ∈ G such that T kn−f x 0 ∈ X \ V for any f ∈ F n . Let T kn ℓ x 0 → z and g ∈ G. Then g ∈ F n ℓ for sufficiently large ℓ ≥ 1. Since(d)⇒(e) Let x ∈ M ′′ . Take U n := {y ∈ X; D(x, y) < 1 n }. From (U n ) n≥1 we take n such that diam U n < ε/2. We may assume that U n ∩ M 2 = ∅. Let V ⊇ M 2 be open such that V ∩ U n = ∅. By syndeticity, we get there exists N ≥ 1 such that (45) {g ∈ G; T g x 0 ∈ U n } + F N = G.
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. 87-100 ToruńChopina. 12Aurelia Dymek Faculty of Mathematics and Computer Science, Nicolaus Copernicus UniversityPoland [email protected] Dymek Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland [email protected]
. 87-100 ToruńChopina. 12Stanisław Kasjan Faculty of Mathematics and Computer Science, Nicolaus Copernicus UniversityPoland [email protected]ław Kasjan Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland [email protected]
. 87-100 ToruńChopina. 12Joanna Kułaga-Przymus Faculty of Mathematics and Computer Science, Nicolaus Copernicus UniversityPoland [email protected] Kułaga-Przymus Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland [email protected]
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"Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention ⋆"
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"Ioannis Stefanou \nUMR 6183\nNantes Université\nÉcole Centrale Nantes\nCNRS\nF-44000NantesGeMFrance\n",
"Franck Plestan \nUMR 6004\nNantes Université\nÉcole Centrale Nantes\nCNRS\nLS2N, F-44000NantesFrance\n"
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"UMR 6183\nNantes Université\nÉcole Centrale Nantes\nCNRS\nF-44000NantesGeMFrance",
"UMR 6183\nNantes Université\nÉcole Centrale Nantes\nCNRS\nF-44000NantesGeMFrance",
"UMR 6004\nNantes Université\nÉcole Centrale Nantes\nCNRS\nLS2N, F-44000NantesFrance"
] | [] | Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. A characteristic example is nonautonomous and discontinuous systems due to presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists in designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). At last, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid's diffusion. | 10.48550/arxiv.2207.07181 | [
"https://export.arxiv.org/pdf/2207.07181v3.pdf"
] | 250,607,712 | 2207.07181 | 8540cb85cd9e58aa6679a23a9b6d8a6c79f9b67a |
Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention ⋆
May 2023
Diego Gutierrez-Oribio
UMR 6183
Nantes Université
École Centrale Nantes
CNRS
F-44000NantesGeMFrance
Ioannis Stefanou
UMR 6183
Nantes Université
École Centrale Nantes
CNRS
F-44000NantesGeMFrance
Franck Plestan
UMR 6004
Nantes Université
École Centrale Nantes
CNRS
LS2N, F-44000NantesFrance
Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention ⋆
May 2023Passivity-based controlNon-smooth and discontinuous problemsUnderactuated systemsEarthquake control
Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. A characteristic example is nonautonomous and discontinuous systems due to presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists in designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). At last, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid's diffusion.
Introduction
Passivity is an important property in dynamical systems because it gives a sense on the system energy balance [11,26]. Roughly speaking, a system is said to be passive if it cannot produce energy on its own, and can only dissipate the energy that is stored in it at any time. Friction is a dissipative mechanism that is ubiquitous in mechanical systems [2,29]. Although friction may be a desirable property (as in brakes application), it can also lead to limit cycles, undesired stick-slip motion and instabilities. This last phenomenon can be explained qualitatively due to the competition of stored elastic energy ⋆ This paper was not presented at any IFAC meeting. Corresponding author Ioannis Stefanou.
Email addresses: [email protected] (Diego Gutierrez-Oribio), [email protected] (Ioannis Stefanou), [email protected] (Franck Plestan). and its dissipation via friction. If this stored energy can not be balanced by the frictional dissipation, then, an instability will be triggered. This is the case when the frictional force decreases with slip or slip-rate and can be explained through the loss of passivity of the system. The prevention of such instabilities is the main objective in this work.
Due to its energy dissipation nature, friction has been compensated in mechanical systems using passivitybased controllers. The passivity-based control term was introduced in [32] and it has an important role in the control theory with applications to electric motors, power electronics, chemical processes and mechanical systems (see [11,25,32,39]). For the case of totally actuated mechanical systems, one can mention [13] where a LuGre (dynamic) model of friction is compensated with an observer, or [36], where the stabilization of a system with Coulomb friction is analysed using sliding-modes. For the case of a system having less control inputs than degrees of freedom (underactuated system), the Inter-connection and Damping Assignment Passivity-based Control (IDA-PBC) presented in [31] was used in [38], [12] and [17] (with an adaptive IDA-PBC) to stabilize systems with dynamic, but not set-valued, frictional models. Furthermore, IDA-PBC requires the solution of partial differential equations (PDEs) in the control, which is cumbersome and in some cases a solution might not exist.
Despite the attractiveness of passivity concepts, the classical passivity theorems (e.g., [26,Chapter 6]) do not include set-valued frictional systems (like the Coulomb friction), which is the focus of this work. Furthermore, the classical theorems do not include nonautonomous systems either. There exist some works dealing with the feedback interconnection of multivalued systems using convex analysis (see, e.g., [1,7,9] and a very recent monograph [8]), yet they do not take into account nonautonomous systems. For this purpose, in this work we extend the classical theorem of passivity related to the negative feedback connection between two passive systems, in such a way to cover the general class of underactuated frictional, nonautonomous systems with set-valued right-hand side (RHS). This is accomplished by using an invariance-like principle [16,23].
Based on this theoretical result, a passivity-based control design is considered to restore the passivity property of a nonautonomous with set-valued RHS underactuated mechanical system. First, stabilization to a domain of zero velocities is obtained, recovering the passivity property by properly designing the underactuated control input. Then, a regulation result over constant references is obtained by augmenting the system with integral action. Finally, the actuator dynamics is considered and the control is designed to preserve the regulation result. The designed underactuated control is implemented in an earthquake prevention scenario of a seismic fault. This is an important and challenging example of a frictional underactuated system, where the designed control has to be able to dissipate the stored energy slowly, controlling the fast dynamics of an earthquake through a slow diffusion process. Simulations are presented to show how the passivity-based control is able to follow a slow reference dissipating slowly the stored energy, avoiding in this manner, an earthquake-like behaviour.
The outline of this work is as follows. The notation and useful definitions and existing theorems are presented in Section 2. The passivity extension for the negative feedback connection between two nonautonomous discontinuous systems, the main theorem of this paper, is presented in Section 3. The frictional underactuated mechanical system description, the link between passivity and the considered friction law and the control objectives are given in Section 4. The structured design of the passivity-based control is detailed in Section 5. The presentation of the fault model and the numerical simulations are shown in Section 6. Finally, some concluding remarks are discussed in Section 7.
Preliminaries
Consider the n-dimensional space ℜ n with the Euclidean norm ||·||. Elements of ℜ n are interpreted as column vectors and (·) T denotes the vector transpose operator. The identity matrix of dimension n is denoted by I n or simply I, if the size can be trivially determined by the context. Let v ∈ ℜ n , be the function sign(·) : ℜ n → ℜ n×n , defined as sign
(v) = diag[sign(v 1 ), ..., sign(v n )], with sign(v i ) = 1 v i > 0 [−1, 1] v i = 0 −1 v i < 0 ,
for all i = 1, ..., n and the function |·| : ℜ n → ℜ n is de-
fined as |v| = [|v 1 | , ..., |v n |] T .
Consider the state model given bẏ
x = f 1 (x, u), y = h(x, u),(1)
where
f 1 : ℜ n × ℜ p → ℜ n is locally Lipschitz, h : ℜ n × ℜ p → ℜ p is continuous, f 1 (0, 0) = 0 and h(0, 0) = 0.fulfils u T y ≥V (x) = ∂V ∂x f 1 (x, u) ∀ (x, u) ∈ ℜ n × ℜ p .
An important passivity theorem concerns the negative feedback connection between two passive systems, H 1 and H 2 (Fig. 1).
Letẋ = f 2 (x, t),(2)
where f 2 : ℜ n × ℜ ≥0 → ℜ n is piecewise continuous 1 in a domain G ⊂ ℜ n × ℜ ≥0 . The above system is nonautonomous and has set-valued RHS (see [30] for more details on discontinuous systems and [4,33] for control robot manipulators with discontinuous RHS for time invariant and nonautonomous systems, respectively). In the following, the solutions of discontinuous systems like (2) are understood in the Filippov's sense [15].
Theorem 2 (Invariance-like principle) [16,23] Let D ⊂ ℜ n be a domain containing x = 0. Suppose there exists a constant M such that ||f 2 (x, t)|| ≤ M , for almost all (x, t) ∈ D × ℜ. Let V : D × ℜ ≥0 → ℜ be a locally Lipschitz-continuous positive definite function such that
W 1 (x) ≤ V (x, t) ≤ W 2 (x), V (x, t) = ∂V ∂t + ∂V ∂x f 2 (x, t) ≤ −W (x),for all t ≥ 0 and for all x ∈ D, where W 1 (x) > 0, W 2 (x) > 0 and W (x) ≥ 0 are continuous functions on D. Choose r > 0 such that B r = { x ∈ ℜ n | ||x|| ≤ r} ⊂ D and let ρ < min ||x||=r W 1 (x). Then, every bounded Filippov solutions of system (2), such that x(t 0 ) ∈ { x ∈ B r | W 2 (x) ≤ ρ} are bounded and satisfy W (x(t)) → 0 as t → ∞. Consequently, x(t) approaches E = { x ∈ D | W (x) = 0} as t → ∞.
Moreover, if all assumptions hold globally and W 1 (x) is radially unbounded, the statement is true for all x(t 0 ) ∈ ℜ n .
Passivity Extension for Nonautonomous Discontinuous Systems
The classical definition of passivity in Definition 1 does not consider directly systems in the form of (2) due to the time dependency and its set-valued RHS. For this purpose, we generalize Theorem 1 for systems H 1 and H 2 that can be either time-variant dynamical systems with discontinuous RHS or time-variant discontinuous memoryless functions. Such theorem is the central result of this work and it will be used on each control design step.
Theorem 3 Assume each element of the feedback interconnection of Fig. 1 is passive and satisfies
e T i y i ≥V i + ϕ i (x), ϕ i : ℜ n → ℜ ≥0 , i = 1, 2.
Let D ⊂ ℜ n be a domain containing x = 0 and consider the locally Lipschitz-continuous positive storage function
V (x, t) = V 1 (x, t) + V 2 (x, t), V : D × ℜ ≥0 → ℜ such that W 1 (x) ≤ V (x, t) ≤ W 2 (x)
, for all t ≥ 0 and for all x ∈ D, where W 1 (x) > 0 and W 2 (x) > 0 are continuous on D. Choose r > 0 such that B r = { x ∈ ℜ n | ||x|| ≤ r} ⊂ D and let ρ < min ||x||=r W 1 (x). Then, every bounded Filippov solutions of the closed-loop system shown in Fig. 1 with u 1 = u 2 = 0, i.e. a system of the form (2), such that
x(t 0 ) ∈ { x ∈ B r | W 2 (x) ≤ ρ} are bounded and satisfy W (x(t)) → 0 as t → ∞, with W (x) = ϕ 1 (x) + ϕ 2 (x). Consequently, x(t) approaches E = { x ∈ D | W (x) = 0} as t → ∞.
Moreover, if all assumptions hold globally and W 1 (x) is radially unbounded, then the statement is true for all x(t 0 ) ∈ ℜ n .
PROOF. Taking the function V (x, t) = V 1 (x, t) + V 2 (x, t) as storage function of the closed-loop system, its derivative w.r.t. time is written aṡ
V ≤ −ϕ 1 (x) − ϕ 2 (x) + e T 1 y 1 + e T 2 y 2 ≤ −W (x) + (u 1 − y 2 ) T y 1 + (u 2 + y1) T y 2 ≤ −W (x) + u T 1 y 1 + u T 2 y 2 ,
which results in a classical passivity result of the feedback interconnection. Furthermore, in the case of u 1 = u 2 = 0, the derivative reads asV (x, t) ≤ −W (x) and all assumptions of Theorem 2 are fulfilled. Then, one can obtain the domain W (x) = 0, which is the domain where the system trajectories will be driven.
Underactuated Mechanical System with Coulomb Friction
Consider an n-DOF underactuated mechanical system modelled asδ
= |v| , u = v, Mv = F or e (u, v) − F or r (δ, u, v, p or , t),(3)
where δ ∈ ℜ n , u ∈ ℜ n , v ∈ ℜ n , represent the vectors of frictional slips, displacements and velocities (slip-rates), respectively. The state δ(t) represents the accumulated slip and it can not take negative values. The term p or ∈ ℜ q is the vector of control inputs, where q < n, resulting in having more degrees of freedom (DOF) than control inputs. M ∈ ℜ n×n is the inertia matrix and the term F or e (u, v) ∈ ℜ n is the vector of applied forces, which are considered to be viscoelastic forces defined as
F or e (u, v) = −K or u − H or v,(4)
where K or ∈ ℜ n×n is the stiffness matrix and H or ∈ ℜ n×n is the viscosity matrix. The term
F or r (δ, u, v, p or , t) = [F or r1 (δ 1 , u 1 , v 1 , p or , t), ..., F or rn (δ n , u n , v n , p or , t)]
T is the friction force and is written as follows
F or r i (δi, ui, vi, p or , t) = F or i (δi, vi, p or , t) if vi = 0 F or e i (ui, 0) if vi = 0 and F or e i (ui, 0) < Fs i Fs i sign(F or e i (ui, 0)) if vi = 0 and F or e i (ui, 0) ≥ Fs i (5) where i ∈ [1, n], F or (δ, v, p or , t) = [F or 1 (δ 1 , v 1 , p or , t), ..., F or n (δ n , v n , p or , t)]
T is an arbitrary friction function, p or ∈ ℜ q is the vector of control inputs and F s = [F s1 , ..., F sn ] T is a vector of static friction coefficients. The static friction counteracts the applied forces below a certain level and, thus, it prevents slip.
Remark 3
The inertia matrix, M , is considered to be constant and only translational displacements are on play, i.e., no Coriolis/Centripetal forces are considered in this work.
It is assumed that (3) has an equilibrium point at t = t * ∈ ℜ ≥0 . This equilibrium point is defined as (δ * , u * , v * ) and is described by
δ * = δ(t * ), v * = 0, u * = u(t * ), F or e (u * , 0) = F or r (δ * , u * , 0, p * , t * ), where p * = p or (t * ) ∈ ℜ q is the vector input at the equi- librium point. It is assumed that system (3) is on the verge of slip, i.e., F or ei (u * i , 0) ≥ F si for all i ∈ [1, n] in (5). Therefore, we set F or r (δ * , u * , 0, p * , t * ) = F * s , where F * s ∈ ℜ q
is the vector of friction at the equilibrium point. This is the point at which the system will be controlled. We then shift the system to this equilibrium point as fol-
lows. Let x = [x 1 , x 2 , x 3 ] T with x 1 = δ − δ * , x 2 = u − u * , x 3 = v − v * and p = p or − p * . Then, we obtaiṅ x 1 = |x 3 | , x 2 = x 3 , x 3 = F e (x 2 , x 3 ) − M −1 F r (x 1 , x 2 , x 3 , p, t),(6)
where
Fe(x2, x3) = −Kx2 − Hx3, Fr(x1, x2, x3, p, t) = F or r (x1 + δ * , x2 + u * , x3, p + p * , t) − F * s ,(7)
with K = M −1 K or and H = M −1 H or are defined. Recalling that the slip δ(t) is nonnegative, the new state x 1 (t) is nonnegative as well. The set of equilibrium points of system (6) is defined as
Γ(t) = { x * ∈ ℜ 3n | x * 3 = 0, K or x * 2 = −F r (x * 1 , x * 2 , 0, p, t)} .
The shifted system has the same form with (3) except for the new term F * s in the friction term F r (x 1 , x 2 , x 3 , p, t). This term represents a destabilizing force due to viscoelasticity and the associated to it stored potential energy of the system, i.e., F or e (u * , 0) = −K or u * = F * s . From the energetic point of view, if this stored energy can not be counteracted by the friction, the system will move abruptly and a part of its stored energy will be suddenly released (instability behaviour). The prevention of such fast-slip behaviour is the main objective in this work.
Coulomb Friction, Actuation and Passivity
The term F or (δ, v, p or , t) in (5) is modelled as Coulomb friction [2,29,34] and can be defined point-wise, for i ∈ [1, n], as
F or i (δ i , v i , p or i , t) = sign(v i )µ i (δ i , |v i | , t)A i (σ ni − p or i ), where µ i (δ i , |v i | , t)
is the friction coefficient, and σ ni and p or i are the normal stress and pressure applied at the surface area, A i , respectively. Such friction law is a setvalued function due to the term sign(v i ). The term p or i could be seen as an input to modify the friction: when the pressure p or i increases, the friction F or i (δ i , v i , p or i , t) decreases, and vice versa. Nevertheless, in real applications is not feasible to change the pressure at every point, i.e., it is not possible to change the value of every p or i independently.
A way to relate the point-wise pressure p or i with the pressure input p or in system (3) is through a relation matrix 2 C p ∈ ℜ n×q , i.e., [p or 1 , ..., p or n ] T = C p p or . This allows to reduce the number of inputs of the system by paying the price of underactuation. Notice that the matrix C p has to be full rank and to have nonzero rows. These conditions are justified by the physics of the problem and they are related to the controllability of the system.
Therefore, the Coulomb friction is defined for the whole system as (8) where the term µ(δ, |v| , t) ∈ ℜ n×n is defined as
F or (δ, v, p or , t) = sign(v)µ(δ, |v| , t)A(σ n − C p p or ),µ(δ, |v| , t) = diag [µ 1 (δ 1 , |v 1 | , t), ..., µ n (δ n , |v n | , t)], where µ i (δ i , |v i | , t), with i ∈ [1, n], are friction coeffi- cients. A is the surface area of the frictional interface, defined as A = diag [A 1 , ..., A n ].
The effective stress σ n − C p p or is defined, with σ n ∈ ℜ n as a vector of normal stresses (σ n = [σ n1 , ..., σ nn ] T ), and the matrix C p ∈ ℜ n×q as the relation matrix, ruling how the control input, p or , influences the system.
A schematic plot of F or r (δ, u, v, p or , t) defined as (5),
(8), with p or = 0 is shown in Fig. 2. The function h or (δ, u, v, 0, t) = [0 1×n , F or r (δ, u, v, 0, t) T ] T , h or : ℜ n × ℜ n × ℜ n × ℜ q × ℜ ≥0 → ℜ 2n is passive belonging to the sector [0, ∞], with [δ T , v T ] T as input 3 .
According to (7), F * s translates F or r (δ, u, v, 0, t) in the new system (6) (see Fig. 2(c-d)). As a result, the passivity property of the output h(
x 1 , x 2 , x 3 , 0, t) = [0 n , F r (x 1 , x 2 , x 3 , 0, t)], h : ℜ n × ℜ n × ℜ n × ℜ q × ℜ ≥0 → ℜ 2n is lost.
The new shifted friction term (8) can be written as
F (x 1 , x 3 , p, t) = g(x 1 , x 3 , t) − b(x 1 , x 3 , t)C p p, g(x 1 , x 3 , t) = sign(x 3 )µ(x 1 + δ 0 , |x 3 | , t)Aσ ′ n − F * s , b(x 1 , x 3 , t) = sign(x 3 )µ(x 1 + δ 0 , |x 3 | , t)A,(9)
where σ ′ n = σ n − C p p 0 is a vector of constant values. If the control input p ∈ ℜ q is taken into account in (9), the original passivity property could be recovered in the shifted friction term and a stability result for system (6) could be obtained.
Control Objectives
The control objectives are stated as follows:
(1) To design the control p in (6), (7) and (9) such that the output
y 2 = [− |x 3 | T , F r (x 1 , x 2 , x 3 , p, t) T ] T to become passive.
(2) To design an integral action to the latter control law, obtaining a reference tracking over the output error
y t = C t (r 3 − x 3 ),(10)
where C t ∈ ℜ q×n is a matrix to be defined and r 3 ∈ ℜ n is a vector of constant velocity references. (3) Considering dynamics in the input p (actuator dy-
namics) asṗ = C h (p ∞ − p),(11)
where C h ∈ ℜ q×q , to design the new control input p ∞ ∈ ℜ q capable to reproduce the same results as the ones obtained in objectives 2 and 3.
The above mentioned control objectives correspond to three distinct design steps, whose role is explained as follows. The first step allows the friction to recover the lost passivity, whereas the second step allows to release the stored energy of the system slowly, by choosing a small velocity reference r 3 . The final step accounts for the dynamics of the actuator and allows the design of the real control input p ∞ .
A block diagram of the full passivity-based control design is shown in Fig. 3 and the closed-loop system is illustrated in Fig. 4. The description of every part of this design is explained in the following sections.
The control design will be performed under the next following minimal assumptions for system (6):
Assumption 1
The initial condition of system (6) will be the origin: Assumption 4 The friction coefficient satisfies min{µ(x 1 + δ 0 , |x 3 | , t)A} = µ min > 0. Furthermore, µ min is a known constant.
x 1 (0) = x 2 (0) = x 3 (0) = 0.Assumption 5 The function h = [0 1×n , F r (x 1 , x 2 , x 3 , 0, t) T ] T belongs to the sector [L Fr , ∞], with L Fr = 0 n×n 0 n×n −l δ sign(x 3 ) −l v I n , input [x T 1 , x T 3 ] T and
l δ , l v > 0 assumed to be known constants (see Fig. 2).
Assumption 6 Relation matrix C p in (9) have full rank, has nonzero rows and it is known.
Remark 4 Assumptions 2-4 are fulfilled commonly in mechanical systems. Furthermore, µ min always exist due to thermodynamics (energy conservation).
Remark 5 Assumption 5 is physically justified by empirical frictional laws that are always bounded (see [2,29,42])
(c) (d)
Stabilization of the Frictional System
When the system is in motion, i.e. x 3 = 0, the frictional term F r (x 1 , x 2 , x 3 , p, t) in (5), (7), turns into F (x 1 , x 3 , p, t) described by (9), which will be considered in the subsequent analysis.
The feedback interconnection between a mechanical system and a frictional term, i.e. system (6), will be analysed using Theorem 3. Such intercon-nection can be seen in Fig. 5 and is the same as in Fig. 1, where system H 1 is defined asẋ 1 = e 11 , The next Lemma will show the passivity property of the frictional term, F (x 1 , x 3 , p, t), when the control input, p, is now taking into consideration.
x 2 = x 3 ,ẋ 3 = F e (x 2 , x 3 ) + M −1 e 21 (e 11 = |x 3 | and e 21 = −F (x 1 , x 3 , p, t)), with u 1 = 0 2n , e 1 = [e T 11 , e T 21 ] T , and y 1 = [x T 1 , x T 3 ] T . The system H 2 is defined as the memoryless function y 2 = [− |x 3 | , F (x 1 , x 3 , p, t)] with u 2 = 0 2n , and e 2 = [x T 1 , x T 3 ] T .
Lemma 1
The passivity map e T 2 y 2 is passive, i.e., e T 2 y 2 ≥ 0, if the control input p is defined as
p(x 1 , x 3 ) = −λ δ C T p x 1 − λ v C T p |x 3 | ,(12)
with control gains, λ δ , λ v , satisfying
λ δ > l δ + 1 µ min , λ v > l v µ min .(13)
PROOF. According to the definition of a sector in [26,Chapter 6] and eqs. (7), (9), Assumption 5 leads to
x T 3 g(x 1 , x 3 , t) ≥ −l δ |x 3 | T x 1 − l v x T 3 x 3 .(14)
Therefore, the passivity map, e T 2 y 2 , reads
e T 2 y 2 = −x T 1 |x 3 | + x T 3 g(x 1 , x 3 , t) − x T 3 b(x 1 , x 3 , t)C p p ≥ −(l δ + 1) |x 3 | T x 1 − l v x T 3 x 3 − x T 3 sign(x 3 ) T µ(x 1 + δ 0 , |x 3 | , t)AC p p ≥ −(l δ + 1) |x 3 | T x 1 − l v x T 3 x 3 − |x 3 | T µ min C p p,
where the sector condition (14) and Assumption 4 for the term µ(x 1 + δ 0 , |x 3 | , t)A have been used.
Selecting the control input p as (12), where λ δ , λ v are constants to be designed, the passivity map becomes
e T 2 y 2 ≥ |x 3 | T µ min λ δ l δ + 1 C p C T p − I n×n x 1 + |x 3 | T µ min λ v l v C p C T p − I n×n |x 3 | .
Due to Assumption 6, the product C p C T p is nonnegative (i.e., all its elements are nonnegative) and the elements of the diagonal are greater or equal to one. Therefore, last expression results to be passive, i.e. e T 2 y 2 ≥ 0, if the controller gains are chosen as (13). .
Notice that the designed control input p in (12) injects passivity into the shifted friction term g(x 1 , x 3 , t). However, a strict passivity condition can not be obtained due to the underactuation nature of the system, i.e. C p C T p ≥ 0. Nevertheless, this is not a critical condition for the stability result stated in the next Theorem.
Theorem 4 Every bounded solution x(t) of system (6) approaches to the domain E = { x ∈ ℜ 3n | x 3 = 0} as t → ∞, if the control input p(x 1 , x 3 ) is defined as in (12) and (13).
PROOF. Consider the positive definite function
V (x) = 1 2 x T 1 x 1 + 1 2 x T 2 K or x 2 + 1 2 x T 3 M x 3 ,
and its time derivative along the trajectories of system (6) aṡ
V = x T 1 |x 3 | + 1 2 x T 3 K or x 2 + 1 2 x T 2 K or x 3 + 1 2 −Kx 2 − Hx 3 − M −1 F (x 1 , x 3 , p, t) T M x 3 + 1 2 x T 3 M −Kx 2 − Hx 3 − M −1 F (x 1 , x 3 , p, t) = x T 1 |x 3 | − x T 3 F (x 1 , x 3 , p, t) − x T 3 H or x 3 = e T 1 y 1 − x T 3 H or x 3 ,
which results to be passive due to Assumption 3. If the controller gains are chosen as in (13), the frictional term is passive and e T 2 y 2 ≥ 0. Consequently, using Theorem 3, every bounded solution x(t) of system (6) (the feedback interconnection between two passive systems with u 1 = u 2 = 0 2n ) converges to the domain E = { x ∈ ℜ 3n | x 3 = 0} as t → ∞.
Notice that the above mentioned domain is bounded, given the first two equations of system (6), i.e., x 1 (t) and x 2 (t) will become constant and, therefore, they are bounded.
The presented stability result is not as strong as the asymptotic (or exponential) stability of the system origin. Nevertheless, recalling the definition of system (6), it results in an increasing evolution of the state x 1 (t) and the impossibility of returning it to the origin once it has started to evolve. Therefore, the obtained stability result is the best that one can obtain for these kind of frictional systems.
Regulation via Integral Control
The stability result of the previous section prohibits the abrupt release of the stored energy of the system by immobilizing it (convergence of system trajectories to zero velocities). However, the energy is still trapped into the system, requiring its stabilization continuously. For this purpose, tracking will be performed in order to allow the system to follow a constant (small) reference, r 3 , and release the stored energy with small velocities. In other words, the small reference, r 3 , will bring the system to another state of lower energy. This will be achieved by interconnecting the underactuated mechanical system with an integral extension of the tracking error.
Considering the new integral terṁ
ξ = y t = C t (r 3 − x 3 ),(15)
where ξ ∈ ℜ q and y t is the error variable defined in (10). Following a classical integral design (see e.g. [26,Chapter 12]), let us define the regulation error variables as
x ie (t) = x i (t) − x i (∞), p e (t) = p(t) − p(∞), ξ e (t) = ξ(t) − ξ(∞),(16)
where i = 1, 2, 3 and x i (∞), p(∞), ξ(∞) are the steady state values of the states, the control input and the integral action, respectively.
The error dynamics is written aṡ
x 1e = |x 3 | − |x 3 (∞)| = |x 3e + x 3 (∞)| − |x 3 (∞)| ≤ |x 3e | , (17) x 2e = x 3e , (18) x 3e = F e (x 2e , x 3e ) − M −1 ∆F (x 1e , x 3e , p e , t), (19) ξ e = −C t x 3e ,(20)
due to the fact that r 3 = r 3 (∞), because r 3 is a vector of constant references. The new nonlinear function ∆F (x 1e , x 3e , p e , t) is defined as
∆F (x 1e , x 3e , p e , t) = F (x 1 , x 3 , p, t) − F (x 1 (∞), x 3 (∞), p(∞), t) = F (x 1e + x 1 (∞), x 3e + x 3 (∞), p e + p(∞), t) − F (x 1 (∞), x 3 (∞), p(∞), t),(21)
which has the same characteristics as the term (9). Consequently, the control input p e of the error dynamics in (17)- (20) can be designed as
p e (x 1e , x 3e , ξ e ) = −λ δ C T p x 1e − λ v C T p |x 3e | + λ ξ C T p sign(x 3e + x 3 (∞))C p ξ e ,(22)
where λ ξ ∈ ℜ >0 is a gain to be designed. The first two terms of the latter control are designed to stabilize the mechanical system, equivalent to the system as in Theorem 4, while the new term includes the integral action to perform the regulation.
The interconnection of the mechanical system and the integral action is shown in Fig. 6. Control (22) interconnects the two systems as in Fig. 1: system H 1 is defined as (20) with u 1 = 0 2n , e 1 = [−x T 1e , −x T 3e ] T , and y1 = [01×n, (λ ξ b(x1 e , x3 e , t)CpC T p sign(x3 e + x3(∞))Cpξe) T ] T , and system H 2 is defined as (17)- (19) with u 2 = 0 2n , e2 = [01×n, (λ ξ b(x1 e , x3 e , t)CpC T p sign(x3 e + x3(∞))Cpξe) T ] T , and The next Theorem for the regulation solution holds.
y 2 = [x T 1e , x T 3e ] T .
Theorem 5 Every bounded solution (x(t), ξ(t)) of the closed-loop system (6), (15) approaches to the domain
E t = { (x, ξ) ∈ ℜ 3n × ℜ q | C t (r 3 − x 3 ) = 0 q } as t → ∞ if the control input p(x 1 , x 3 , ξ) is defined as p(x 1 , x 3 , ξ) = −λ δ C T p x 1 − λ v C T p |x 3 − r 3 | − λ v C T p |r 3 | + λ ξ C T p sign(x 3 )C p ξ,(23)
satisfying the condition (13), λ ξ > 0, and
C t = (C T p C p ) −1 C T p .(24)
PROOF. From the previous stability result, we know that the system (6) is passive, i.e., e T 2 y 2 ≥V +x T 3 H or x 3 , if p is designed as (12), (13). Such result can be inherited to the equivalent system (17)- (19). Thus, now the passivity property must be studied in system (20).
Let us study first the passivity map of the output
L(C p ξ e ) = λ ξ b(x 1e , x 3e , t)C p C T p sign(x 3e + x 3 (∞))C p ξ e with input C p ξ e , resulting in ξ T e C T p L = ξ T e C T p λ ξ b(x1 e , x3 e , t)CpC T p sign(x3 e + x3(∞))Cpξe = ξ T e C T p λ ξ Aµ(x1 + δ0, |x3| , t)sign(x3 e + x3(∞))CpC T p × sign(x3 e + x3(∞))Cpξe ≥ ξ T e C T p λ ξ µminsign(x3 e + x3(∞))CpC T p × sign(x3 e + x3(∞))Cpξe ≥ 0,
where the definition of b(x 1 , x 3 , t) in (9) and the Assumption 4 were used. Clearly, this output is passive.
Defining the storage function V ξ = Cpξe 0 L(σ)dσ for the system (15). Such function is positive semidefinite due to the passive property of the output L(C p ξ e ) and its derivative reads aṡ
V ξ = L(C p ξ e ) T C pξe = −x T 3e (C t C p ) T L(C p ξ e ) = e T 1 y 1 ,
if C t is the left pseudoinverse matrix of C p , i.e., C t is defined as in (24). Consequently, matrix C t is full rank due to Assumption 6. The last expression shows how the integral system is passive. Therefore, the feedback connection between the two systems will be passive.
Consequently, using Theorem 3, every bounded solution (x e (t), ξ e (t)) of system (17)- (20) converges to the domain
E t = { (x e , ξ e ) ∈ ℜ 3n ×ℜ q | x T 3e H or x 3e = 0 n } as t → ∞.
In order to obtain the domain in the original states, the error x 3e must fulfil the equation C t x 3e = C t (r 3 − x 3 ) obtaining the domain E t described in Theorem 5.
Finally, the original control p results from the control (22) and the definition of errors (16)
p − p(∞) = −λ δ C T p x 1 − λ v C T p |x 3 − r 3 | + λ ξ C T p sign(x 3 )C p ξ + λ δ C T p x 1 (∞) − λ ξ C T p sign(x 3 )C p ξ(∞),
where one can obtain expression (23) by replacing the steady state control p(∞)
= −λ δ C T p x 1 (∞) − λ v C T p |x 3 (∞)| + λ ξ C T p sign(x 3 )C p ξ(∞).
Actuator Dynamics
So far, the designed control (23) is able to either drive the system (6) states to a given domain E = { x ∈ ℜ 3n | x 3 = 0} as t → ∞, if r 3 = λ ξ = 0, or to perform a tracking over a given velocity constant reference if r 3 = 0 and λ ξ > 0. If now an actuator dynamics like (11) is considered in the model, p ∞ is the new control input to be designed. For this purpose, consider the control (23) as nominal controlp, i.e.,
p(x 1 , x 3 , ξ) = −λ δ C T p x 1 − λ v C T p |x 3 − r 3 | − λ v C T p |r 3 | + λ ξ C T p sign(x 3 )C p ξ.(25)
Then, one can get the nominal controlp ∞ from (6), (10), (11), (15) and (23) as
p ∞ = C −1 hṗ +p, = −λ δ C T p x 1 − λ δ C −1 h C T p |x 3 | − λ v C T p |x 3 − r 3 | − λ v C −1 h C T p sign(x 3 − r 3 )(ẋ 3 −ṙ 3 ) − λ v C T p |r 3 | − λ v C −1 h C T p sign(r 3 )ṙ 3 + λ ξ C T p sign(x 3 )C p ξ + λ ξ C −1 h C T p sign(x 3 )C p C t (r 3 − x 3 ).(26)
The time derivative of sign(x 3 ) is equal to zero because we are studying the case when the system is in motion (x 3 = 0).
In order to obtain the control p ∞ able to reproduce the nominal control (26), let us define the next error variablesp
= p −p,p ∞ = p ∞ −p ∞ ,(27)
leading to the error dynamics from (6), (11), (15) and (26) aṡ
x 1 = |x 3 | ,(28)x 2 = x 3 ,(29)x 3 = F e (x 2 , x 3 ) − M −1 F (x 1 , x 3 ,p +p, t),(30)ξ = C t (r 3 − x 3 ),(31)p = C h (p ∞ −p).(32)
Such error system can be seen in Fig. 3 and can be explained as the interconnection of two systems as in Fig. 1: system H 1 is defined as (32) with u 1 = 0 2n , e 1 = [−x T 1 , −x T 3 ] T , and y 1 = [0 1×n , (b(x 1 , x 3 , t)C pp ) T ] T , and system H 2 is defined as (28)-(31) with u 2 = 0 2n ,
e 2 = [0 1×n , (b(x 1 , x 3 , t)C pp ) T ] T , and y 2 = [x T 1 , x T 3 ] T .
Theorem 6 Every bounded solution (x(t), ξ(t), p(t)) of the closed-loop system (6), (11) and (15) approaches to the domain
E p = { (x, ξ, p) ∈ ℜ 3n × ℜ q × ℜ q | C t (r 3 − x 3 ) = 0 q , p =p} as t → ∞ if the control input p ∞ (x 1 , x 3 , ξ) is defined as p ∞ =p ∞ +p ∞ ,p ∞ = −µ min C T p |x 3 | ,(33)
with the nominalp ∞ defined as (26), fulfilling the conditions (13), λ ξ > 0 and matrix C t for the integral action (15) defined as in (24).
PROOF. System (28)-(31) is passive with the nominal controlp as shown in the previous regulation analysis. Therefore, the condition e T 2 y 2 ≥V +V ξ + x T 3eH x 3e is fulfilled. Thus, the passivity property must be studied now in system (32).
Defining the positive definite storage function V p = 1 2p T C −1 hp for the system (32), its derivative reads aṡ
V p =ṗ T C −1 hp = (p ∞ −p) T C T h C −1 hp = −µ min x T 3 sign(x 3 )C pp −p Tp ≤ e T 1 y 1 −p Tp ,
resulting to be strictly passive and, consequently, the feedback connection between the two systems will be passive.
Finally, to get the domain in which the trajectories will converge, we use Theorem 3 to obtain E p = { (x, ξ, p) ∈ ℜ 3n × ℜ q × ℜ q | x T 3e H or x 3e +p Tp = 0 q } , where the only possibility for the latter domain to be valid is if it takes the form of the one given in Theorem 6.
Earthquake Control
Consider a seismic fault as shown in Fig. 7. In this academic example, the fault is just beneath the surface and its dimensions are A = 3 × 3 [km 2 ] (x-and z-directions, respectively). The effective normal stress σ ′ n acting on the fault interface is assumed to vary linearly due to the lateral earth pressure. We assume also that the fault is adequately oriented in the tectonic stress regime for slip to occur. In this numerical application, the fault area is discretized into n = N x × N z = 10 × 10 elements. The above physical system can be described mathematically using eqs. (5), (6), (7), and (9), where x 1 represents the slip, x 2 the displacement and x 3 the slip-rate (velocity). Several methods in the literature can be used in order to discretize the differential operator representing the underlying continuum elastodynamic problem of seismic slip (e.g., Finite Element Method, Finite Differences, Boundary Element Method, spectral methods, model reduction methods, among others [5], [6], [14] and [28]). In most cases, the resulting discretized equations will finally take the form of (6) and, consequently, the control theory presented in this work can be applied.
The actuator dynamics (11) is also considered, where the control input p ∞ represents the pressure at the peak of four wells injecting fluid to the fault (q = 4), as shown in Fig. 7. The form of eq. (11) corresponds to a finite difference approximation of the diffusion equation, a Partial Differential Equation (PDE). Extension to PDE control could also be explored [18,19,27], but this is out of the scope of the current work. The theorems developed in the previous section can be applied as the diffusion equation remains passive. Then, through the diffusion process according to equation (11), the pressure p affects the fault friction by modifying the effective normal stress σ ′ n according to Terzaghi's principle of effective stress [43]. The control configuration of the wells on the fault can be seen in Fig. 7, where their influence is defined by the definition of matrix C p in the friction term (9).
Furthermore, an even more realistic scenario will be studied where the full state x(t) is not available, but only a measured output of the system (6) as y m = C m x 3 , where y m ∈ ℜ and C m ∈ ℜ 1×n . This single output represents an average velocity over the points of the fault. Therefore, the designed pressure at the fault p = p(x) and, consequently, the designed pressure at the wells, p ∞ = p ∞ (x), have to be now a feedback of the estimated states, i.e.p = p(x) andp ∞ = p ∞ (x), respectively. The design of a high-gain observer for this purpose is shown in Appendix A.
Without a control input, system (6) is unstable, resulting in an earthquake as shown in Fig. 8 (notice the time scale in seconds). It is worth mentioning that very few works are devoted to the control of such systems. In particular, an LQR control was designed to stabilize and perform tracking of an earthquake modelled by a MIMO system in [40], whereas a double-scale asymptotic approach was employed to design a transfer function-based control in [41]. These first applications of control theory to this problem have shown that earthquakes could be controlled, at least from a mathematical point of view, but they have not accounted for underactuation, the discontinuous nature of friction and diffusion. Therefore, the presented theoretical development a more realistic treatment of the problem.
The objective in the sequel is to implement the designed control law (26) and (33) with the integral dynamics (15) and (24), to drive the system states to the domain
E p = { (x, ξ, p) ∈ ℜ 3n × ℜ q × ℜ q | C t (r 3 − x 3 ) = 0 q , p =p} as t → ∞.
If one chooses a small velocity reference r 3 , this will result in a slow-aseismic response of the system. The desired reference r 3 is a smooth function reading as r 3 =ṙ(t)I n , r(t) = d max s 3 (10 − 15s + 6s 2 ), (34) where s = t/t op , d max is the target displacement and t op is the operational time of the tracking strategy. The constant d max is the distance the fault slides dynamically in order to reach its sequent stable equilibrium point. For this case, we selected d max = 500 [mm] (approximately two times equal to the seismic slip developed when the system is not controlled) and t op = 360 [days]. The desired total time is considerably larger than the fast slip in the earthquake behaviour (≃ 15 [s]) in order to slowly release and dissipate the seismic energy. Shorter t op can be chosen as well (e.g., of the order of hours) but in this case, the pressure at the tips of the wells, p ∞ , would be very high due to slow dynamics of the diffusion process (see (11)). The characteristic time of the diffusion process (see (11)) depends on the hydraulic diffusivity parameter, which has been taken equal to C h = 2.88 × 10 −7 I (representing injection in a sandstone) and a distance of the injection point to the fault equal to 1.5 [km].
Remark 6
The presented analysis for the regulation result in Section 5.2 fits only for constant references. Nevertheless, the resulting error could be improved by choosing references with low time derivatives, approximating its behaviour to constant references, like (34). One can improve this result by adding more (passive) integrator terms to cover a wider range of references r 3 (t), as stated in the internal model principle (e.g., [20]).
In this numerical example, we consider the friction coefficient µ(x 1 + δ 0 , |x 3 | , t) in (9) of the form µ i (x 1i ) = µ res − ∆µ · e − x 1 i /dc , with ∆µ < 0. Such function is defined as a slip-weakening friction law [24] and it evolves from an initial value µ max (static friction coefficient), to a residual one µ res (kinetic friction coefficient) in a characteristic slip d c . Its values were chosen as µ res = 0.5 (Assumption 4), ∆µ = µ res − µ max = 0.1 and d c = 10 [m]. Other friction laws could be used as well (see [42]).
Numerical Results
In order to illustrate the performance of the proposed passivity-based control strategy, simulations have been made based on the shifted system described by (5), (6), (7), (9), and (11). Such simulations were performed using the Differential Equations package of Julia [35] and an initial condition x(0) = 0 3n . In particular the TRBDF2 algorithm was used with events for detecting the transition between stick to slip and satisfy (5).
The control (26) and (33) with the integral dynamics (15) and (24) The results are presented in Figs. 9-10. The states now follow successfully a slow reference, dissipating the stored energy aseismically (notice the time scale of days in Figs. 9-10 instead of seconds of the instability Fig. 8). The discontinuous-like behaviour shown in the velocity x 3 is due to the stick-slip motion over the fault, resulting over the fact that Coulomb friction is a set-valued function (see (5), (7), and (9)). Nevertheless, the designed control is able to drive the tracking error C t (r 3 − x 3 ) close to zero, using the estimated states from the highgain observer (errors shown in Fig. 10 left and middle plots). Finally, the control signal from the wells p ∞ and the pressure p applied to the fault are depicted in Fig. 10, which show reasonable amplitudes to be used in real actuators.
Conclusions
In this work, we extend the classic theorem for the negative feedback interconnection of passive systems to account for nonautonomous and set-valued (discontinuous) ODEs. This generalization is based on an invariance-like principle and it allows the synthesis of controllers for underactuated mechanical systems with Coulomb friction. Based on this generalization, stabilization of the states to a domain of zero velocities and tracking over constant references, while assuming actuation dynamics, are achieved. The designed control injects passivity to (unstable) frictional systems using less control inputs than degrees of freedom. It also need minimum information about the plant, i.e., the minimum bound of the friction coefficient, the belonging sector of the friction law and the coefficient of the actuator dynamics. This in contrast with the IDA-PBC where it is necessary to solve PDEs, or other existing more involved approaches. In order to test the derived control strategy, an earthquake prevention case study is considered. In particular, the unstable dynamic slip of a mature seismic fault is prevented by injecting fluid through four wells located far from the fault. Numerical simulations show the successful tracking of the system output over a reference, despite the presence of the slow dynamics due to diffusion process and uncertainties with respect to the Coulomb frictional rheology, the (visco-)elastodynamic properties of the system and diffusivity of the fluid pressure in the rock. The results were accomplished with minimum measurements and the control signals (pressures) were of acceptable amplitudes for the actuators (pumps). This results in a promising solution for earthquake prevention and control. Fig. 9. Controlled system: The system is tracked aseismically to a new (stable) equilibrium state. Note the difference on the time scale (days) with respect to the earthquake-like behaviour described in Fig. 8 (seconds). The same color code as Fig. 8 was used.
Remark 1
1Theorem 3 is an extension of the classical result of the interconnection between two passive systems[26, Chapter 6]. Such extension covers non autonomous dynamical systems and discontinuities in both the dynamical system and the memoryless function. Theorem 1 is recovered then when both systems, H 1 and H 2 , fulfil the necessary smoothness conditions (locally Lipschitz around the origin) requested in the classical result. Remark 2 Theorem 3 provides the domain in which the trajectories of the closed-loop system will converge to, in contrast to the stabilization of the origin obtained from the classical feedback interconnection of two passive systems[26, Chapter 6].
3
See[26, Chapter 6] for more details about sector definition in passivity.
Assumption 2
2The inertia matrix M is symmetric and positive definite, i.e., M = M T > 0 n×n . Assumption 3 Matrices K or , K, H or , H, C h are positive definite. Furthermore, K or , H or and C h are symmetric matrices.
Fig. 2 .
2(a)-(b): Schematic representation of a component of F or r (δ, u, v, 0, t), showing how it is passive with respect to the input [δ T , v T ] T . (c)-(d): Loss of passivity due to the addition of the loading term F * s resulting in the new term Fr(x1, x2, x3, 0, t).
Fig. 3 .
3Steps of the passivity-based design.
Fig. 4 .
4Closed loop system.5 Passivity-based Control Design of Underactuated Frictional Systems
Fig. 5 .
5Control design: Step 1.
Fig. 6 .
6Control design: Step 2.
Fig. 7 .
7Illustration of a mature seismic fault discretized in Nx × Nz elements with four injection wells (inputs).
were implemented in the simulations with λ δ = 40 [Pa/m], λ v = 346.4 [Pa · s/m] and λ ξ = 5 × 10 3 [Pa/m]. These gains were designed to satisfy (13) with µ min = Aµres /2 (Assumption 4) and l δ = 4∆µ/d c , l v = 0 (Assumption 5) due to the previously presented definition and parameters of the friction coefficient µ(x 1 + δ 0 , |x 3 | , t). The control uses the estimated states from the observer (A.1) with ǫ = 0.1 and an initial condition x(0) = 0 3n .
Fig. 10 .
10Integral (left) and observation (middle) errors. Control signals (right): Pressures developed on the seismic fault (blue), p, and control pressures applied on the wells (black), p∞. The delay is because of the slow dynamics of the actuator, due to diffusion.
The systems H 1 and H 2 can be either time-invariant dynamical systems or (possibly timevariant) memoryless functions.Theorem 1 [26, Chapter 6] If the system H 1 is passive with input e 1 and output y 1 , and the system H 2 is passive with input e 2 and output y 2 , then the negative feedback connection of H 1 and H 2 is passive with input u = [u 1 , u 2 ] T and output y = [y 1 , y 2 ] T .H1
H2
y1
y2
e1
e2
u1
u2
+
+
+
-
Fig. 1. Negative feedback connection.
Fig. 8. Earthquake-like behaviour showing fast slip dynamics (instability). Each curve represents an element of the discretized seismic fault and its color varies linearly with depth from red (depth z = 0 [km]) to black (depth z = 3 [km]).Slip -
x1 [mm]
0
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25
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[mm]
0
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A function is said to be piecewise continuous in a domain G if it is continuous in G up to a set of measure zero defined by points of discontinuity of the function.
A relation (logical, boolean or binary) matrix is a matrix with only entries of zeros or ones[37].
AcknowledgementsThe authors would like to acknowledge the support of the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement no. 757848 CoQuake).A High-gain Observer DesignA high-gain observer design will be derived (see[26,Chapter 14],[3]) to obtain the estimates statesx of system (6) aṡwhere λ 1 , λ 2 ∈ ℜ, L 1 , L 2 ∈ ℜ n×m are gains to be designed,F e (x 2 ,x 3 ) = −K 0x2 − H 0x3 and K 0 , H 0 , M 0 , F r (x 1 ,x 2 ,x 3 ,p, t) are the the nominal matrices of K, H, M and the nominal function of F r (x 1 , x 2 , x 3 ,p, t), respectively.Based on[3]and[26,Chapter 14], the estimation errorx = x −x can be proved to be ISS with respect to the uncertain term δ(x,, if the observer gains are designed as λ 1 = 1 ǫ , λ 2 = 1 ǫ 2 , with ǫ ≈ 0 and L chosen to make the matrixà = 0 n×n I n×n − L 1 C mIs it worth noticing that the separation principle for nonlinear systems (e.g.,[3]) consider systems with sufficiently smooth right-hand sides. Therefore, the analysis of the full closed loop-system (plant, control and highgain observer) with discontinuous RHS presented in this paper, remains as future work.
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| [] |
[
"Uncertainty Estimation in Machine Learning",
"Uncertainty Estimation in Machine Learning"
] | [
"Valentin Arkov \nAutomated Control and Management Dept\nUfa State Aviation Technical University Ufa\n0000-0002-7913-4778Russia ORCID\n"
] | [
"Automated Control and Management Dept\nUfa State Aviation Technical University Ufa\n0000-0002-7913-4778Russia ORCID"
] | [] | Most machine learning techniques are based upon statistical learning theory, often simplified for the sake of computing speed. This paper is focused on the uncertainty aspect of mathematical modeling in machine learning. Regression analysis is chosen to further investigate the evaluation aspect of uncertainty in model coefficients and, more importantly, in the output feature value predictions. A survey demonstrates major stages in the conventional least squares approach to the creation of the regression model, along with its uncertainty estimation. On the other hand, it is shown that in machine learning the model complexity and severe nonlinearity become serious obstacles to uncertainty evaluation. Furthermore, the process of machine model training demands high computing power, not available at the level of personal computers. This is why so-called pre-trained models are widely used in such areas of machine learning as natural language processing. The latest example of a pre-trained model is the Generative Pre-trained Transformer 3 with hundreds of billions of parameters and a half-terabyte training dataset. Similarly, mathematical models built from real data are growing in complexity which is accompanied by the growing amount of training data. However, when machine models and their predictions are used in decision-making, one needs to estimate uncertainty and evaluate accompanying risks. This problem could be resolved with non-parametric techniques at the expense of greater demand for computing power, which can be offered by modern supercomputers available, including those utilizing graphical and tensor processing units along with the conventional central processors. | 10.1109/smartindustrycon57312.2023.10110722 | [
"https://arxiv.org/pdf/2206.01749v1.pdf"
] | 249,395,456 | 2206.01749 | 4e78382834106ae4a43fb6fe67cb8b3b3a6e814a |
Uncertainty Estimation in Machine Learning
Valentin Arkov
Automated Control and Management Dept
Ufa State Aviation Technical University Ufa
0000-0002-7913-4778Russia ORCID
Uncertainty Estimation in Machine Learning
parameter uncertaintysupervised learningmodelingprediction methodsforecasting
Most machine learning techniques are based upon statistical learning theory, often simplified for the sake of computing speed. This paper is focused on the uncertainty aspect of mathematical modeling in machine learning. Regression analysis is chosen to further investigate the evaluation aspect of uncertainty in model coefficients and, more importantly, in the output feature value predictions. A survey demonstrates major stages in the conventional least squares approach to the creation of the regression model, along with its uncertainty estimation. On the other hand, it is shown that in machine learning the model complexity and severe nonlinearity become serious obstacles to uncertainty evaluation. Furthermore, the process of machine model training demands high computing power, not available at the level of personal computers. This is why so-called pre-trained models are widely used in such areas of machine learning as natural language processing. The latest example of a pre-trained model is the Generative Pre-trained Transformer 3 with hundreds of billions of parameters and a half-terabyte training dataset. Similarly, mathematical models built from real data are growing in complexity which is accompanied by the growing amount of training data. However, when machine models and their predictions are used in decision-making, one needs to estimate uncertainty and evaluate accompanying risks. This problem could be resolved with non-parametric techniques at the expense of greater demand for computing power, which can be offered by modern supercomputers available, including those utilizing graphical and tensor processing units along with the conventional central processors.
I. INTRODUCTION
In recent decades, machine learning techniques have become popular due to the huge amount of digital data available, enough computing power achieved, and the ease of use of free software libraries, such as Sci-Kit Learn for Python.
Machine learning is often defined as the branch of artificial intelligence dealing with extracting information from data. Computer algorithms can tune the model structure and parameters using sample training data. During the model training, the chosen quality criterion is optimized. The next step consists of model validation. This validation step requires another portion of the data, preferably not used in the model training. The validation might be done repeatedly with several data splits into training and testing sets. Therefore, it is usually called cross-validation. In this way, the model generalization ability is accounted for.
In many cases, machine learning techniques are inherited from well-established statistical routines, such as least squares (LS). Note that the LS procedure yields the model parameters estimates along with their uncertainty, typically in the form of the probability distribution. This distribution is characterized by the mean and variance. This analytical approach is based on the proposition that two parameters are sufficient in describing the distribution close to Gaussian, which is often the case for the LS method. This is then followed by the estimation of the confidence bands (intervals) given the confidence probability, derived from analytical techniques.
However, the existing ML tools are mostly focused on model training and validation, with much less attention paid to the analysis of statistical properties of the estimates obtained. Furthermore, cross-validation seems much less established, as compared to the model training process. A possible reason might be the limiter computing power available to the researchers. This is affirmed by that the LS criterion is still in wide use despite all its disadvantages. The LS technique was developed for manual computations, but it is still in use as the major quality criterion even in deep learning algorithms realized at the most powerful supercomputers.
In this paper, the uncertainty of the estimates obtained through model training is investigated using cross-validation techniques. The proposed approach is close to the simulation of uncertainty in metrology, rather than to the analytical ones used in statistical routines.
Data Science also includes statistical methods and techniques with applications in various areas such as business process automation, as described by Gonchar in [1].
II. QUADRATIC CRITERIA IN MACHINE LEARNING
Cross-validation appears to be a very compute-consuming technique. Thorough validation might take much greater computing power than the model training itself [2]. As a result, during cross-validation, only a very few splits are typically used. For example, Lakshminarayanan, Pritzel, and Blundell describe their experimentation with deep learning ensembles including uncertainty evaluation in [3]. In these experiments, multi-layered artificial neural networks were trained on standard ML data sets such as Boston housing. The authors mention that the chosen datasets were split into several train-test folds, with a maximum of 20 splits. The regression and classification accuracy and confidence are evaluated after the ML model has been trained. While some sort of uncertainty band for each metric is provided, the amount of cross-validation cannot be sufficient.
To emphasize the importance of evaluating uncertainty, the following four types of data analytics should be mentioned here. First, descriptive analytics corresponds to the branch of statistics dealing with various forms of distributions. Here we have a mere description of the past events, mostly presented in the form of means and variances. Second, diagnostic analytics indicates possible causes for past events.
Third, predictive analytics is the tool for making predictions or forecasting. This implementation of machine models started in statistical learning in the form of regression models, see for example [4]. The authors emphasize that the incorporation of a vast number of features not relevant to the object under investigation does not necessarily improve the prognostic ability of the machine model, such as regression.
Prognostic systems can also include the evaluation of uncertainty, as indicated by Saxena in [5]. In that paper, uncertainty is presented as non-Gaussian non-parametric distribution to be approximated with some analytical function.
Fourth, prescriptive analytics usually provides suggested decisions to leverage the predictive power of the machine model, see for example papers by Vater, Harscheidt, and Knoll [6]. This stage is close to the process of the manufacturing management operation, rather than to mere analysis. Note that decision-making should include risk evaluation and management, which requires the estimation of the uncertainty for the predictions utilized.
The least squares technique is employed in modeling of both technical and economic systems, as exemplified by Wang in [7]. Mathematical modeling of technical objects is referred to as systems identification, while economic modeling is typically performed in the framework of econometric methodology. Despite the different terminology, one can easily reveal close similarities in the model representation.
Regression analysis in both systems identification and econometrics yields a similar model representation of coefficients along with their standard deviations, as shown below. This is then followed by the construction of confidence bands (limits) and significance testing under the chosen significance level. Again, this form of modeling is performed under the assumption of the normal distribution of the regression results, which is the implied consequence of the least-squares calculations.
Note that, unlike economic systems, technical ones are often manufactured at mass-production plants. Therefore, their construction and parameters are standardized and regulated by the design and technology documentation. This makes it possible to build precise mathematical models that only need to be refined individually using experimental data. This approach is described in detail, for example, by El-Sayed in [8].
Data mining represents another approach to searching for hidden patterns and interconnections within large data sets, see for example Leskovec [9]. Originally, this methodology was called Exploratory Data Analysis, compare for example the books by Tukey [10] and Bruce [11]. The techniques used for data mining and exploratory analysis still include regression with least-squares estimators of various kinds.
III. MODEL UNCERTAINTY
Mathematical models obtained using machine learning methods represent more complex entities than conventional constructs. The number of model coefficients is stably increasing every year with more computing power available. For example, the Generative Pre-trained Transformer GPT-3 model based on deep learning uses hundreds of billions of parameters for natural language processing. This eliminates any attempt to analytically evaluate the possible uncertainty of each coefficient. GPT tools are often described in terms of possible application areas, with much less attention paid to their uncertainty, see for example a paper by Korngiebel and Mooney [12] discussing how the generative transformers are replacing live human interaction.
On the other hand, cross-validation techniques provide additional tools for numeric evaluation of uncertainty, while imposing growing requirements for an even greater amount of computation than for estimating the model itself.
The least-squares method for regression analysis is considered a well-established and researched approach, see for example works of Snedecor [13]. This type of analysis, generally supposes homogeneous variance of the random error, referred to as homoskedasticity in econometrics.
The fundamentals of the least squares for linear regression analysis are detailed in the literature on econometric methods, such as in books by Fomby, Johnson, and Hill [14]. This embraces necessary assumptions, and the Gauss-Markov theorem followed by a discussion of the concepts of convergence and consistency. The Gauss-Markov theorem reflects a generalized approach to regression analysis, as in the works by Kong [15], Lyche [16], Drygas [17], or Zimmerman [18].
Ideally, the estimates of the regression equation coefficients are accompanied by their standard deviations shown below the coefficients as follows:
= + (σ ) (σ )(1)
Typically, the modeling pipeline includes the following stages: The last stage is connected to testing the conditions described in the Gauss-Markov Theorem, thus ensuring the desired properties of the estimates, such as asymptotic consistency, normality, and efficiency. Consider a conceptual experiment described by Greene [19]. The proposed Monte Carlo study yields the nonparametric sampling distribution for the LS estimator. Slightly modify the experimental setup for our purposes as follows. The x feature is drawn from the uniformly distributed population of random numbers. The y feature is linearly coupled with x. We then add a normally distributed random disturbance e, see below. U(150; 200) (2) e ~ N(0; 10) This training set is generated 1000 times with a sample size of 100 observations. For every training set, the linear regression coefficients are estimated. Take the slope coefficient for further detailed examination. Using this "sample" of estimates for the model slope, their distribution is evaluated in the form of a histogram and a box-and-whiskers plot, see Figure 1. As the estimates' distribution is expected to be normal, the Gaussian curve is also created, with the same mean value and standard deviation as for the slope estimates sample. We can further expand the presentation of the simulation results. In Figure 2, the whole set of linear predictions is shown, along with the background filled with the training samples. One can observe that the prediction variance grows as the x factor approaches the boundary of the known values. As we generate 1000 samples followed by estimating a linear model, the duration of the whole computational experiment is 1000 times greater than that of obtaining a single LS estimate. Following the analysis of variance for the estimates (usually referred to as ANOVA), the confidence bands are built as prediction intervals based on the prediction variance [20]. Thus the prediction uncertainty is evaluated in an "almost analytical" way, which is not feasible for complex ML models.
y = x -100 + e x ~
Extrapolating this analytical logic to machine learning, one could expect some sort of uncertainty estimation for the machine model parameters and predictions, as shown in Figure 3. The estimate of every parameter could be accompanied by its deviation, leading to the variation of the machine predictions. In most cases, however, this analytical approach is not feasible for several reasons, including the model nonlinearity. Therefore, LS techniques are focused on linear equations with nonlinear terms.
Fig. 3. Analytical estimation of uncertainty in machine learning
After the analysis of this historical background, one can discover that the machine learning routines very often possess much less depth in presentation and understanding. One might suppose that the deep analysis stage is often missing because the machine model/algorithm learns exclusively from the dataset and incorporates no prior/expert knowledge [21].
IV. PROBLEM STATEMENT
The problem of estimating the uncertainty is formulated as follows. Given the training set of linearly correlated data, the uncertainty of the predicted output feature values is to be estimated in the form of the confidence interval, preferably non-sensitive to outliers. To demonstrate the proposed approach, a simplified problem statement is used, with the potential of further generalization to higher-order multidimensional modeling.
Consider the problem statement in more detail.
The source data for machine learning are presented in the ready-to-use tabular form, where the input and output features are organized in columns as (x, y). Suppose y is linearly correlated with x. The coefficients of the linear equation are unknown, with the presence of some additive random noise that is normally distributed.
An arbitrary machine learning model M(c) is to be trained on the dataset available, with several parameters/coefficients c. The model training is performed to minimize the quadratic criterion comparing the actual and predicted output feature values: y 2 → min. During training, the model structure and coefficients are chosen to generalize the data inter-relation through data train/test split in some proportion.
Given the input feature values within the known interval X, the output feature prediction Y is then obtained using the trained model with the estimated model parameters C.
Finally, the uncertainty of the output predictions is to be estimated in the form of some interval, corresponding to the confidence band and less sensitive to possible outliers and fat tails of the distribution of Y. For demonstration purposes, two types of ML models are considered, namely the linear regression model and the random forest regressor.
V. COMPUTATIONAL EXPERIMENT
Note that regression analysis exemplifies the use of numerical/non-parametric methods for data analysis, as described by Kiusalaas in [20]. The problem of estimating the uncertainty of regressor predictions is proposed to resolve with a non-parametric approach, similar to the Monte Carlo experiments described by Green in [19]. This approach is further extended to output value predictions as follows.
The training set is generated every time with a new random generator state, which is refreshed automatically with every function call. This is then followed by ML model training. For demonstration purposes, two extreme cases of ML models were chosen, namely linear regression and random forest regressor. Both models are based on the LS criterion in their training. The prediction factors are linearly organized with the linspace function to cover the interval of values in the training data.
During multiple runs of data generating and model training, predicted outputs are calculated for the same predictor factor values with every trained ML model. Eventually, the predicted output values are gathered into an array which is then processed to estimate the distribution of the predictions.
The general shape of the distribution is presented in Figure 4 as histograms for two predictions: y(150) and y(200). While the LS model coefficients and predictions have Gaussian distributions, the random forest regressor produces slightly different results. In order to indicate the difference, the ideal bell-shaped curves of the Gaussian probability density functions are also shown, with the actual values of the mean and standard deviation calculated for the array of the output predictions.
In this figure, box-and-whiskers plots are also shown for these two predicted output values. As one can easily see, both distributions demonstrate much longer tails with a lot of outliers outside the standard quartile deviation borders. The central part of both histograms is narrow as compared to the Gaussian curve. Figure 5 shows the total array of the training sets (grey markers) along with the regressor predictions. The central tendency corresponding to the regression line is obtained as the median of all output predictions. The lower and upper bands for the predictions are calculated as the quartile deviation from the median in the way used in box plots as follows.
The quartiles Q1, Q2, and Q3 are obtained for the predictions using interpolation where appropriate. The Inter-Quartile Range (IQR) indicated the distance between the third and first quartiles. The lower and upper limits are then obtained to approximate the "three sigma" confidence interval for the confidence probability of about 99.7%. In addition to the estimation of uncertainty, one can discover the smoothing effect of the utilization of quartile measures. Both median and limit bands look quite smooth, as opposite to the predictions obtained on a single sample, see Figure 6.
In both Figures 5 and 6, the model prediction follows the general pattern of the least-squares estimate which produces a slightly lesser slope of the prediction line. This means that the slope regression coefficient is less than the original value in the model used in the data generation. Note that similar behavior is observed in most machine learning models using LS criteria, usually followed by the model quality evaluation with the mean squared error (MSE).
Recent developments in interactive parallel computations offer more possibilities for machine learning applications on parallel apps like R or Matlab [22]. In particular, interactive machine learning becomes available for Python programs utilizing TensorFlow [23]. This is further added with the automatic machine learning (AutoML) tools, thus making data analysis and machine learning easy to use for wider numbers of researchers. A popular example for AutoML would be Auto-sklearn [24], with much attention paid to hyperparameter optimization. Note that at each step of model training, Auto-sklearn produces an ensemble of models. This is followed by estimating uncertainty in the form of the median error with 5 th and 95 th percentiles, which is shown as a demonstration by-product.
VI. CONCLUSIONS AND FUTURE WORK
In this work, the basics of regression analysis have been extrapolated to demonstrate the need for uncertainty estimation in machine learning, currently absent in modern software kits. The proposed non-parametric approach is based on cross-validation and might yield uncertainty estimates for various types of machine learning predictions.
While the analytical estimation of uncertainty is not feasible because of the substantial model nonlinearity, the non-parametric approach can offer an alternative method for dealing with predictions. Furthermore, standard deviation as a measure of uncertainty is justified for Gaussian disturbances, which is not always the case in the real world. Instead, boxand-whiskers plots and quantile measures for uncertainty present a much more flexible and promising tool.
Further developments in the proposed uncertainty estimation approach might embrace other ML techniques, such as classification, clustering, and dimensionality reduction. When dealing with the actual big data, one would need to use supercomputers, as the proposed non-parametric estimation of uncertainty requires much greater computing power. The available graphical processing units could offer the acceptable level of computation speedup to perform multiple runs of model training and cross-validation resulting in quality estimates.
Fig. 1 .
1Distribution of slope coefficient estimates
Fig. 2 .
2Linear predictions over a series of samples
Fig. 4 .Fig. 5 .
45Boxplots and histograms for random forest predictionsThe upper and lower limits/bands for the predicted values are built as follows. Uncertainty of random forest prediction shown as quartile deviation
Fig. 6 .
6Output predictions of random forest regressor for single training sample
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| [] |
[
"Deep Neural Networks for Rank-Consistent Ordinal Regression Based On Conditional Probabilities",
"Deep Neural Networks for Rank-Consistent Ordinal Regression Based On Conditional Probabilities"
] | [
"Xintong Shi \nDepartment of Statistics\nUniversity of Wisconsin-Madison\n\n",
"Wenzhi Cao \nDepartment of Statistics\nUniversity of Wisconsin-Madison\n\n",
"Sebastian Raschka \nDepartment of Statistics\nUniversity of Wisconsin-Madison\n\n\nLightning AI\n\n"
] | [
"Department of Statistics\nUniversity of Wisconsin-Madison\n",
"Department of Statistics\nUniversity of Wisconsin-Madison\n",
"Department of Statistics\nUniversity of Wisconsin-Madison\n",
"Lightning AI\n"
] | [] | In recent times, deep neural networks achieved outstanding predictive performance on various classification and pattern recognition tasks. However, many real-world prediction problems have ordinal response variables, and this ordering information is ignored by conventional classification losses such as the multicategory cross-entropy. Ordinal regression methods for deep neural networks address this. One such method is the CORAL method, which is based on an earlier binary label extension framework and achieves rank consistency among its output layer tasks by imposing a weight-sharing constraint. However, while earlier experiments showed that CORAL's rank consistency is beneficial for performance, it is limited by a weight-sharing constraint in a neural network's fully connected output layer, which may restrict the expressiveness and capacity of a network trained using CORAL. We propose a new method for rank-consistent ordinal regression without this limitation. Our rank-consistent ordinal regression framework (CORN) achieves rank consistency by a novel training scheme. This training scheme uses conditional training sets to obtain the unconditional rank probabilities through applying the chain rule for conditional probability distributions. Experiments on various datasets demonstrate the efficacy of the proposed method to utilize the ordinal target information, and the absence of the weight-sharing restriction improves the performance substantially compared to the CORAL reference approach. Additionally, the suggested CORN method is not tied to any specific architecture and can be utilized with any deep neural network classifier to train it for ordinal regression tasks. | null | [
"https://export.arxiv.org/pdf/2111.08851v5.pdf"
] | 244,270,488 | 2111.08851 | f122c244beab7b8436c9ab43697ecf6d423896dd |
Deep Neural Networks for Rank-Consistent Ordinal Regression Based On Conditional Probabilities
Xintong Shi
Department of Statistics
University of Wisconsin-Madison
Wenzhi Cao
Department of Statistics
University of Wisconsin-Madison
Sebastian Raschka
Department of Statistics
University of Wisconsin-Madison
Lightning AI
Deep Neural Networks for Rank-Consistent Ordinal Regression Based On Conditional Probabilities
In recent times, deep neural networks achieved outstanding predictive performance on various classification and pattern recognition tasks. However, many real-world prediction problems have ordinal response variables, and this ordering information is ignored by conventional classification losses such as the multicategory cross-entropy. Ordinal regression methods for deep neural networks address this. One such method is the CORAL method, which is based on an earlier binary label extension framework and achieves rank consistency among its output layer tasks by imposing a weight-sharing constraint. However, while earlier experiments showed that CORAL's rank consistency is beneficial for performance, it is limited by a weight-sharing constraint in a neural network's fully connected output layer, which may restrict the expressiveness and capacity of a network trained using CORAL. We propose a new method for rank-consistent ordinal regression without this limitation. Our rank-consistent ordinal regression framework (CORN) achieves rank consistency by a novel training scheme. This training scheme uses conditional training sets to obtain the unconditional rank probabilities through applying the chain rule for conditional probability distributions. Experiments on various datasets demonstrate the efficacy of the proposed method to utilize the ordinal target information, and the absence of the weight-sharing restriction improves the performance substantially compared to the CORAL reference approach. Additionally, the suggested CORN method is not tied to any specific architecture and can be utilized with any deep neural network classifier to train it for ordinal regression tasks.
Introduction
Many real-world prediction tasks involve ordinal target labels. Popular examples of such ordinal tasks are customer ratings (e.g., a product rating system from 1 to 5 stars) and medical diagnoses (e.g., disease severity labels such as none, mild, moderate, and severe). While we can apply conventional classification losses, such as the multi-category cross-entropy, to such problems, they are suboptimal since they ignore the intrinsic order among the ordinal targets. For example, for a patient with severe disease status, predicting none and moderate would incur the same loss even though the difference between none and severe is more significant than the difference between moderate and severe. Moreover, unlike in metric regression, we cannot quantify the distance between the ordinal ranks. For instance, the difference between a disease status of none and mild cannot be quantitatively compared to the difference between mild and moderate. Hence, ordinal regression (also called ordinal classification or ranking learning) can be considered as an intermediate problem between classification and regression.
Among the most common machine learning-based approaches to ordinal regression is Li and Lin's extended binary classification framework [9] that was adopted for deep neural networks by Niu et Figure 1: Illustration of the difference between rank-consistent and rank-inconsistent methods.
al. in 2016 [14]. In this work, we solve the rank inconsistency problem ( Fig. 1) of this ordinal regression framework without imposing constraints that could limit the expressiveness of the neural network and without substantially increasing the computational complexity.
The contributions of our paper are as follows:
A new rank-consistent ordinal regression framework, CORN (Conditional Ordinal
Regression for Neural Networks), based on the chain rule for conditional probability distributions;
2. Rank consistency guarantees without imposing the weight-sharing constraint used in the CORAL reference framework [1];
3. Experiments with different neural network architectures and datasets showing that CORN's removal of the weight-sharing constraint improves the predictive performance compared to the more restrictive reference framework.
2 Related Work
Ordinal Regression Based on Extended Binary Classification Subtasks
Ordinal regression is a classic problem in statistics, going back to early proportional hazards and proportional odds models [13]. To take advantage of well-studied and well-tuned binary classifiers, the machine learning field developed ordinal regression methods based on extending the rank prediction to multiple binary label classification subtasks [9]. This approach relies on three steps: (1) extending rank labels to binary vectors, (2) training binary classifiers on the extended labels, and (3) computing the predicted rank label from the binary classifiers. Modified versions of this approach have been proposed in connection with perceptrons [3] and support vector machines [22,17,2]. In 2007, Li and Lin presented a reduction framework unifying these extended binary classification approaches [9].
Addressing Rank Consistency in Neural Networks for Ordinal Regression
In 2016, Niu et al. adapted Li and Lin's extended binary classification framework to train deep neural networks for ordinal regression [14]; we refer to this method as OR-NN. Across different image datasets, OR-NN was able to outperform other reference methods. However, Niu et al. pointed out that ORD-NN suffers from rank inconsistencies among the binary tasks and that addressing this limitation might raise the training complexity substantially. Cao et al. [1] recently addressed this rank inconsistency limitation via the CORAL method. To avoid increasing the training complexity, CORAL achieves rank consistency by imposing a weight-sharing constraint in the last layer, such that the binary classifiers only differ in their bias units. However, while CORAL outperformed the OR-NN method across several face image datasets for age prediction, the weight-sharing constraint may impose a severe limitation in terms of functions that the neural network can approximate. In this paper, we investigate an alternative approach to guarantee rank consistency without increasing the training complexity and restricting the neural network's expressiveness and capacity.
Other Neural Network-Based Methods for Ordinal Regression
Several deep neural networks for ordinal regression do not build on the extended binary classification framework. These methods include Zhu et al.'s [25] convolutional ordinal regression forest for image data, which combines a convolutional neural network with differentiable decision trees. Diaz and Marathe [5] proposed a soft ordinal label representation obtained from a softmax layer, which can be used for scenarios where interclass distances are known. Another method that does not rely on the extended binary classification framework is Suarez et al.'s distance metric learning algorithm [24]. Petersen et al. [16] developed a method based on differentiable sorting networks based on pairwise swapping operations with relaxed sorting operations, which can be used for ranking where the relative ordering is known but the absolute target values are unknown. Liu et al. adapted pairwise ranking constraints from RankingSVM [7] to reformlate the multi-category loss as a constrained optimization problem for ordinal regression [10].
This paper focuses on addressing the rank inconsistency on OR-NN without imposing the weightsharing of CORAL [1], which is why an exhaustive study of the methods mentioned above is outside the scope of this paper. However, additional experiments and comparisons with SORD [5] and CNNPOR [10]
Proposed Method
This section describes the details of our CORN method, which addresses the rank inconsistency in Niu et al.'s OR-NN [14] without requiring CORAL's [1] weight-sharing constraint. In an ordinal regression context, we refer to y [i] as the rank, where y [i] ∈ Y = {r 1 , r 2 , ...r K } with rank order r K ≻ r K−1 ≻ . . . ≻ r 1 . The objective of an ordinal regression model is then to find a mapping h : X → Y that minimizes a loss function L(h).
Preliminaries
Let D = x [i] , y [i] N i=1
Motivation
With CORAL, Cao et al. [1] proposed a deep neural network for ordinal regression that addressed the rank inconsistency of Niu et al.'s OR-NN [14], and experiments showed that addressing rank consistency had a positive effect on predictive performance.
Both CORAL and OR-NN built on an extended binary classification framework [9], where the rank labels are recast into a set of binary tasks, such that y
[i] k ∈ {0, 1} indicates whether y [i] exceeds rank r k .
The label predictions are then obtained via h x [i] = r q , where q ∈ {1, 2, ..., K} is the rank index, which is computed as
q = 1 + K−1 k=1 1 f k x [i] > 0.5 .(1)
Here, f k x [i] ∈ [0, 1] is the probability prediction of the k-th binary classifier in the output layer, and 1{·} is an indicator function that returns 1 if the inner condition is true and 0 otherwise.
The CORAL method ensures that the {f k } K−1 k=1 predictions are rank-monotonic, that is,
f 1 x [i] ≥ f 2 x [i] ≥ · · · ≥ f K−1 x [i]
, which provides rank consistency to the ordinal regression model. While the rank label calculation via Eq. 1 does not strictly require consistency among the K − 1 task predictions, f k x [i] , it is intuitive to see why rank consistency can be theoretically beneficial and can lead to more interpretable results via the binary subtasks. While CORAL provides this rank consistency, CORAL's limitation is a weight-sharing constraint in the output layer. Consequently, all binary classification tasks use the same weight parameters and only differ in their bias units, which may limit the flexibility and expressiveness of an ordinal regression neural network based on CORAL. The proposed CORN model is a neural network for ordinal regression that guarantees rank consistency without any weight-sharing constraint in the output layer (Fig. 2). Instead, CORN uses a new training procedure with conditional training subsets that ensures rank consistency through applying the chain rule of probability. raining example aper
... ... ! ! ! " ! # " ! " " " $%! Hidden layers !(# ! ) # ! # $ & !,! & !,# & !,* ' ! = [& !,! & !,# … & !,* ] where % " ! = + ! ( " & + * " where % # ! = + ! ( # & + * # where % +%" ! = + ! ( +%" & + * +%" + ! , [&] = -. ! & + # , [&] = -. # & + +)! , [&] = -. +)! & Logistic sigmoid function Outputs where ) , "#$ ! = - . / [!] > 1 "#$ | / [!] > 1 "#' 2 = [3 ! 3 # … 3 * ] 1 = [. ! . # … . +)! ]
One example as input
Rank-consistent Ordinal Regression based on Conditional Probabilities
Given a training set
D = x [i] , y [i] N i=1
, CORN applies a label extension to the rank labels y [i] similar to CORAL, such that the resulting binary label y [i] k ∈ {0, 1} indicates whether y [i] exceeds rank r k . Similar to CORAL, CORN also uses K − 1 learning tasks associated with ranks r 1 , r 2 , ..., r K in the output layer as illustrated in Fig. 2.
However, in contrast to CORAL, CORN estimates a series of conditional probabilities using conditional training subsets (described in Section 3.4) such that the output of the k−th binary task f k x [i] represents the conditional probability 2
f k x [i] =P y [i] > r k | y [i] > r k−1 ,(2)
where the events are nested:
y [i] > r k ⊆ y [i] > r k−1 .
The transformed, unconditional probabilities can then be computed by applying the chain rule for probabilities to the model outputs:P
y [i] > r k = k j=1 f j x [i] .(3)
Since
∀j, 0 ≤ f j x [i] ≤ 1, we have P y [i] > r 1 ≥P y [i] > r 2 ≥ ... ≥P y [i] > r K−1 ,(4)
which guarantees rank consistency among the K − 1 binary tasks.
Conditional Training Subsets
Our model aims to estimate f 1 x [i] and the conditional probabilities
f 2 x [i] , ..., f K−1 x [i] . Estimating f 1 x [i]
is a classic binary classification task under the extended binary classification framework with the binary labels y
[i]
1 . To estimate the conditional probabilities such aŝ P y [i] > r 2 | y [i] > r 1 , we focus only on the subset of the training data where y [i] > r 1 . As a result, when we minimize the binary cross-entropy loss on these conditional subsets, for each binary task, the estimated output probability has a proper conditional probability interpretation 3 .
In order to model the conditional probabilities in Eq. 3, we construct conditional training subsets for training, which are used in the loss function (Section 3.5) that is minimized via backpropagation. The conditional training subsets are obtained from the original training set as follows:
S 1 : all x [i] , y [i] , for i ∈ {1, ..., N }, S 2 : (x [i] , y [i] ) | y [i] > r 1 , . . . S K−1 : (x [i] , y [i] ) | y [i] > r k−2 , where N = |S 1 | ≥ |S 2 | ≥ ... ≥ |S K−1 |, and |S k | denotes the size of S k .
Note that the labels y [i] are subject to the binary label extension as described in Section 3.3. Each conditional training subset S k is used for training the conditional probability predictionP
y [i] > r k | y [i] > r k−1 for k ≥ 2.
Additional theoretical justification for constructing the conditional training subsets is provided in the Supplementary Material in section Theoretical Analysis of Conditional Probability Estimation. Section 5.1 compares the predictive performance of the CORN method with and without training subsets.
Loss Function
Let f j x [i] denote the predicted value of the j-th node in the output layer of the network (Fig. 2), and let |S j | denote the size of the j-th conditional training set. To train a CORN neural network using backpropagation, we minimize the following loss function:
L(X, y) = − 1 K−1 j=1 |S j | K−1 j=1 |Sj | i=1 log f j (x [i] ) · 1 y [i] > r j + log 1 − f j x [i] · 1 y [i] ≤ r j ,(5)
We note that in
f j (x [i] ), x [i]
represents the i-th training example in S j . To simplify the notation, we omit an additional index j to distinguish between x [i] in different conditional training sets.
To improve the numerical stability of the loss gradients during training, we implement the following alternative formulation of the loss, where Z are the net inputs of the last layer (aka logits), as shown in Fig. 2, and log σ z
[i] = log f j x [i] : L(Z, y) = − 1 K−1 j=1 |S j | K−1 j=1 |Sj | i=1 log σ z [i] · 1 y [i] > r j + log σ z [i] − z [i] · 1 y [i] ≤ r j . (6)
A derivation showing that the two loss equations are equivalent and a PyTorch implementation are included in the Supplementary Material in the section Numerically Stable Loss Function. In addition, the Supplementary Material includes a visual illustration of the loss computation based on the conditional training subsets ( Figure S1) and a theoretical Generalization Bounds analysis.
Rank Prediction
To obtain the rank index q of the i-th training example, and any new data record during inference, we threshold the predicted probabilities corresponding to the K − 1 binary tasks and sum the binary labels as follows:
q [i] = 1 + K−1 j=1 1 P y [i] > r j > 0.5 , where the predicted rank is r q [i] .
Experiments
Datasets and Preprocessing
The MORPH-2 dataset 4 [19] contains 55,608 face images, which were processed as described in [1]: facial landmark detection [20] was used to compute the average eye location, which was then used by the EyepadAlign function in MLxtend v0.14 [18] to align the face images. The original MORPH-2 dataset contains age labels in the range of 16-70 years. In this study, we use a balanced version of the MORPH-2 dataset containing 20,625 face images with 33 evenly distributed age labels within the range of 16-48 years.
The Asian Face dataset (AFAD) 5 [14] contains 165,501 faces in the age range of 15-40 years. No additional preprocessing was applied to this dataset since the faces were already centered. In this study, we use a balanced version of the AFAD dataest with 13 age labels in the age range of 18-30 years.
The Image Aesthetic dataset (AES) 6 [21] used in this study contains 13,868 images, each with a list of beauty scores ranging from 1 to 5. To create ordinal regression labels, we replaced the beauty score list of each image with its average score rounded to the nearest integer in the range 1-5. Compared to the other image datasets MORPH-2 and AFAD, the size of the AES dataset was relatively small, and we did not attempt to create a class-balanced version of this dataset for this study.
The Fireman dataset (Fireman) 7 is a tabular dataset that contains 40,768 instances, 10 numeric features, and an ordinal response variable with 16 categories. We created a balanced version of this dataset consisting of 2,543 instances per class and 40,688 from the 16 ordinal classes in total.
Each dataset was randomly divided into 75% training data, 5% validation data, and 20% test data. We share the partitions for all datasets, along with all preprocessing code used in this paper, in the code repository (see Section 4.4).
Neural Network Architectures
Comparison with binary label extension frameworks for ordinal regression
For the main method comparisons to other binary extension frameworks for ordinal regression on the image datasets (MORPH-2 and AFAD,), we used ResNet34 [6] as the backbone architecture since it is an established architecture that is known to achieve good performance on a variety of image classification datasets. Besides the hyperparameter settings listed in Tables 1 and 2; we adopt all other settings from the ResNet34 paper.
For the tabular Fireman dataset, we used a simple multilayer perceptron architecture (MLP) with leaky ReLU [12] activation functions (negative slope 0.01). Since the MLP architectures were prone to overfitting, a dropout layer with drop probability 0.2 was added after the leaky ReLU activations in each hidden layer. In addition, we used the AdamW [11] optimizer with a weight decay rate of 0.2. The number of hidden layers (one or two) and the number of units per hidden layer were determined by hyperparameter tuning (see Section 4.3 for more details).
In this paper, we focus on comparing the performance of a neural network trained via the rankconsistent CORN approach to the two prominent binary extension-based ordinal regression frameworks for deep learning, the Niu et al. [14] OR-NN method (no rank consistency) and CORAL (rank consistency by using identical weight parameters for all nodes in the output layer). As a performance baseline, we implement neural network classifiers trained with standard multicategory cross-entropy loss as a baseline, which we refer to as CE-NN. While all methods (CE-NN, OR-NN, CORAL, and CORN) use different loss functions during training, it is worth emphasizing that they can share similar backbone architectures and only require small changes in the output layer. For instance, to implement a neural network for ordinal regression using the proposed CORN method, we replaced the network's output layer with the corresponding binary conditional probability task layer.
Training and Evaluation
The model evaluations and comparisons are based on the mean absolute error (MAE) and root mean squared error (RMSE), which are defined as follows:
MAE = 1 N N i=1 |y i − h (x i )| and RMSE = 1 N N i=1 (y i − h (x i )) 2 ,
where y [i] is the ground truth rank of the i-th test example and h(x [i] ) is the predicted rank, respectively.
Then, using the best hyperparameter setting for each method, we repeated the model training five times using different random seeds (0, 1, 2, 3, and 4) for the random weight initialization and dataset shuffling. We considered the exact same hyperparameter ranges for each method. (A detailed list of the hyperparameter configurations we considered is shown in Table 1.) Then, we selected the best hyperparameter configuration, using grid search, based on its validation set performance for each method before computing the test set performance. Note that both the hyperparameter configuration and the best training epoch were determined based on the validation set before computing the final model performance on the independent test set. The best hyperparameter values for each method are listed in Table 2.
The models were trained for 200 epochs using stochastic gradient descent via adaptive moment estimation [8] with the default decay rates and carefully checked for convergence such that training and validation MAE started to diverge and the validation MAE started to stagnate or decline. The complete training logs for all methods are provided in the code repository (Section 4.4).
Hardware and Software
All neural networks were implemented in PyTorch 1.8 [15]. The models were trained on NVIDIA GeForce RTX 2080Ti graphics cards on a private workstation as well as T4 graphics cards using the Grid.ai platform. We make all source code used for the experiments available 8 and provide a user-friendly implementation of CORN in the coral-pytorch Python package 9 .
Results and Discussion
To compare deep neural networks trained with our proposed CORN method to CORAL [1], Niu et al.'s OR-NN [14], and the baseline cross-entropy loss (CE-NN), we conducted a series of experiments on three image datasets and one tabular dataset. As detailed in Section 4.2, the experiments on the MORPH and AFAD image datasets were based on the ResNet34 architecture. We used a multilayer perceptron for the tabular Fireman dataset. As the main results in Table 3 show, CORN outperforms all other binary label extension methods for ordinal regression on the MORPH-2 and AFAD image datasets and is tied with OR-NN on the Fireman tabular dataset. We repeated the experiments with different random seeds for model weight initialization and data shuffling, which ensures that the results are not coincidental.
It is worth noting that even though CORAL's rank consistency was found to be beneficial for model performance [1], it performs noticeably worse than OR-NN on the balanced MORPH-2 and AFAD datasets. This might likely be due to CORAL's weight-sharing constraint in the output layer, which could affect the expressiveness of the neural networks and thus limit the complexity of what it can learn. In contrast the CORN method, which is also rank-consistent, performs better than OR-NN on MORPH-2 and AFAD. Here, the performances are relatively close, and the 16-category prediction task is relatively easy for a fully connected neural network regardless of the loss function.
Ablation Study
Given the superior performance of CORN across several datasets, we studied the importance of the training subsets. In this ablation study, created an alternative CORN method without training subsets subsets. Here, the conditional probability of the k−th binary task is computed as
f k x [i] =P y [i] > r k ,(7)
which is a modified version of Eq. 2. Note that this modification results in meaningless probability scores, however, the rank consistency via Eq. 4 is still guaranteed since the probability scores are still computed via Eq. 3, and each score cannot be greater than 1.
We shall note that the modified CORN method without training subsets sees at least as many training examples as the regular CORN method. This is because each task now has access to the full training batch rather than a subset.
As the results in Table 4 show, the subsets do not only play a crucial role for yielding meaningful and theoretically justified rank probability values in CORN but they also improve the predictive performance. Across all datasets, with the exception of MORPH-2, the neural network trained with the regular CORN method outperforms the alternative version without subsets.
Conclusions
In this paper, we developed the rank-consistent CORN framework for ordinal regression via conditional training datasets. We used CORN to train convolutional and fully connected neural architectures on ordinal response variables. Our experimental results showed that the CORN method improved the predictive performance compared to the rank-consistent reference framework CORAL.
While our experiments focused on image and tabular datasets, the generality of our CORN method allows it to be readily applied to other types of datasets to solve ordinal regression problems with various neural network structures.
Acknowledgements 8 Supplementary Material
Theoretical Analysis of Conditional Probability Estimation
Suppose we are interested in estimating a series of conditional probabilities
f 1 (x) = P (y > r 1 | x) , f 2 (x) = P (y > r 2 | y > r 1 , x) , ..., f K−1 (x) = P (y > r K−1 | > r k−2 , x) , with the observed dataset D = x [i] , y [i] N i=1 , where f k (x)
is the functional form of the neural network model outputs that depend on the neural network model weights. The likelihood of the model weights can be written as
L = K−1 j=1 |Sj | j=1 f j x [j] 1{y [i] >rj } · 1 − f j x [j] 1{y [i] ≤rj } .(8)
Hence, minimizing the loss function (Eq. 5) is equivalent to solving the maximum likelihood estimate of the functional form representations of the conditional probabilities. This is also the theoretical justification that we construct the conditional training sets in the data preparation for the CORN loss function. Without using the conditional set in the loss function, the estimated probabilities do not have a conditional probability maximum likelihood interpretation. After solving the maximum likelihood estimates of the conditional probabilities, it is natural to use the probability chain rule to find the unconditional probabilities of exceeding rank r k in Eq. 3 given each input x.
Generalization Bounds
Analogous to CORAL [1] and based on established generalization bounds for binary classification, Theorem 1 shows that the final rank prediction by CORN generalizes well when the binary classification tasks generalize well. Theorem 1 (reduction of generalization error). Let C be the cost matrix for the ordinal label predictions, where C y,y = 0 and C y,r k > 0 for k ̸ = y. P is the underlying distribution of (x, y), i.e., (x, y) ∼ P . Furthermore, let h(x) be the model output yielding the predicted rank r q ; that is,
h(x) = r q . Let y (k) = 1{y > r k }, andŷ (k) = 1{P (y > r k ) > 0.5} = 1{f 1 f 2 . . . f k > 0.5}
be the prediction of y (k) . Given the binary classification tasks {f k } K−1 k=1 , which we obtain from minimizing the loss in Eq. 5, and the rank-monotonicŷ k , we have
E (x,y)∼P C y,h(x) ≤ K−1 k=1 C y,r k − C y,r k+1 E (x,y)∼P 1{ŷ (k) ̸ = y (k) }.(9)
Proof. For any x ∈ X , by Eq. 4 we havê
y (1) ≥ŷ (2) ≥ . . . ≥ŷ (K−1) .
If h(x) = y, then C y,h(x) = 0.
If h(x) = r q ≺ y = r s , then q < s. We havê
y (1) =ŷ (2) = . . . =ŷ (q−1) = 1 andŷ (q) =ŷ (q+1) = . . . =ŷ (K−1) = 0.
Also, y (1) = y (2) = . . . = y (s−1) = 1 and y (s) = y (s+1) = . . . = y (K−1) = 0.
Thus, 1{ŷ (k) ̸ = y (k) } = 1 if and only if q ≤ k ≤ s − 1. Since C y,y = 0,
C y,h(x) = s−1 k=q (C y,r k − C y,r k+1 ) · 1{ŷ (k) ̸ = y (k) } ≤ s−1 k=q C y,r k − C y,r k+1 · 1{ŷ (k) ̸ = y (k) } ≤ K−1 k=1 C y,r k − C y,r k+1 · 1{ŷ (k) ̸ = y (k) }.
Similarly, if h(x) = r q ≻ y = r s , then q > s and
C y,h(x) = q−1 k=s (C y,r k+1 − C y,r k ) · 1{ŷ (k) ̸ = y (k) } ≤ K−1 k=1 C y,r k+1 − C y,r k · 1{ŷ (k) ̸ = y (k) }.
In any case, we have
C y,h(x) ≤ K−1 k=1 C y,r k − C y,r k+1 · 1{ŷ (k) ̸ = y (k) }.(10)
By taking the expectation on both sides with (x, y) ∼ P , we arrive at Eq. 9.
Comparison with Other Deep Learning Methods for Ordinal Regression
We compare CORN with two additional, recent ordinal regression methods that do not rely on the binary extension framework:
1. the convolutional neural network with pairwise regularization for ordinal regression (CN-NPOR) method by Liu, Long, and Goh [10]; 2. the soft ordinal vectors (SORD) method by Diaz and Marathe [5].
To facilitate a fair comparison, we adopted the exact same architecture and preprocessing steps from [5] and [10]. Similar to CNNPOR and SORD, we used a VGG16 [23] backbone pre-trained on ImageNet [4] where only the last layer (output layer) was re-initialized with random weights following. Also, following the preprocessing steps in CNNPOR and SORD, the training images in the AES dataset were resized to 256 × 256 pixels and randomly cropped to 224 × 224 as well as randomly flipped across the horizontal axis.
As these additional results on the AES dataset show, CORN also outperforms other recent ordinal regression methods for deep learning (CNNPOR [10] and SORD [5]) overall when trained with a VGG16 backbone that was pre-trained on ImageNet (Table S2). Table S1: Best hyperparameter settings for the AES datasets. For CNNPOR [10] and SORD [5] settings, please refer to the respective papers.
Additional Results on Text Datasets using Recurrent Neural Networks
This section describes additional results we obtained from comparing CORN to other methods on text datasets using recurrent neural networks (RNNs) with long short-term memory (LSTM) cells.
Both the 100K Coursera's courses reviews dataset 10 and TripAdvisor hotels reviews dataset 11 contain reviews with 5 rating labels ranging from 1 to 5 stars. We used balanced versions of these datasets to distribute the ratings evenly. The balanced Coursera dataset contains 11,852 reviews, and the TripAdvisor dataset contains 7,000 reviews. Each dataset was randomly divided into 75% training data, 5% validation data, and 20% test data. The dataset splits and preprocessing code can be found in the code repository (see Section 4.4 of the main manuscript).
For method comparisons on the text datasets, we use a standard RNN with one LSTM cell. Similar to the image datasets, we compare the performance of a neural network trained via the rank-consistent CORN approach to both Niu et al.'s OR-RNN method (no rank consistency) and CORAL (rank consistency by using identical weight parameters for all nodes in the output layer). We also implemented RNN classifiers trained with standard multicategory cross-entropy loss as a baseline, which we refer to as CE-RNN. All methods share similar backbone architectures and only require minor changes in the output layer. The training and evaluation steps are similar to those of the image datasets in the main manuscript. The RNN models were trained for 200 epochs using ADAM with default settings. The model with the best validation set performance was then chosen as the final model for evaluation on the test set. The training logs for all runs are available in the CORN GitHub repository (see Section 4.4). The learning rates considered for hyperparameter tuning were 1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3, and we considered batch sizes 16, 32, 64, 128, 256, 512.
As the results in Table S4 show, CORN outperforms all other methods on the two text datasets, TripAdvisor and Coursera, in terms of the test set MAE. All experiments were repeated for different random seeds to ensure that the results were not coincidental.
It is worth noticing that while the CORN method showed superior performance in terms of test MAE, the CORAL method performed better in test RMSE compared with all other methods. One possible explanation is that since RMSE penalizes large gaps more harshly than MAE, CORAL may behave slightly better on outliers while CORN may make fewer mistakes in total. However, both methods show reliable performance over the text datasets. Table TableS5 is a more detailed version of the results table shown in the main paper, listing the performance for each individual random seed.
Detailed Performance
Numerically Stable Loss Function
We can convert the CORN loss function,
L(X, y) = − 1 k−1 j=1 |S j | k−1 j=1 |Sj | i=1 log f j (x [i] ) · 1 y [i] > r j + log 1 − f j x [i] · 1 y [i] ≤ r j ,(11)
into an alternative version
L(Z, y) = − 1 k−1 j |S j | k−1 j=1 |Sj | i=1 log σ z [i] · 1 y [i] > r j + log σ z [i] − z [i] · 1 y [i] ≤ r j ,(12)
where Z are the net inputs of the last layer (aka logits) and log σ z
[i] = log f j x [i] , since log 1 − 1 1 + e −z = log 1 − e z 1 + e z = log 1 1 + e z = log e z 1 + e z − log(e z ) = log 1 1 + e −z − z = log (σ(z)) − z.
This allows us to use the logsigmoid(z) function that is implemented in deep learning libraries such as PyTorch as opposed to using log(1-sigmoid(z)); the former yields numerically more stable gradients during backpropagation. A PyTorch implementation of the CORN loss function is shown in Fig. S1.
One example as input
Net inputs of output layer Figure S3: Outline of a neural network architecture that can be trained using CORN. Compared to a regular classification network, the only architecture modification is that the output layer consists of k − 1 instead of k nodes, where k represents the number of unique ordinal labels in the dataset. The hidden layers represent the layers of an existing backbone architecture, such as a standard ResNet-34.
Figure 2 :
2& > = ( = 9 : ; & > = ! A 9 : ; & > = # | ; & > = ! ⋯ 9 : ; & > = ( | ; & > = ()! Outline of the neural network architecture used for CORN.
Figure S1 :
S1logits[train_examples, task_index] loss = -torch.sum(F.logsigmoid(pred)*train_labels + (F.logsigmoid(pred) -pred)*(1-train_labels)) CORN loss function implemented in PyTorch v. 1.8.
Figure S2 :
S2Visual explanation of how the CORN loss is computed using the conditional training subsets.ing example er (simplified)...
[&] > = +)! | ; [&] > = +)#
are included in the Supplementary Material in section Comparison with Other Deep Learning Methods for Ordinal Regression.
denote a dataset for supervised learning consisting of N training examples, where x [i] ∈ X denotes the inputs of the i-th training example and y [i] its corresponding class label.
Table 1 :
1Configurations for hyperparameter tuning.Backbone
Learning rates
Batch sizes
Layer sizes
ResNet34 5e-5, 1e-4, 2.5e-4, 5e-4, 1e-3, 5e-3 16, 32, 64, 128, 256, 512
NA
MLP
1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3
16, 32, 64, 128, 256, 512 Layer1: 50, 100, 200, 300
Layer2: 50, 100, 200, 300
Table 2 :
2Best hyperparameter settings for image and tabular datasets.Datasets
Backbone Methods Learning rates Batch sizes Number of layers Layer hidden units
Image Datasets ResNet34 CE-NN
5e-4
256
-
-
Image Datasets ResNet34 OR-NN
5e-4
256
-
-
Image Datasets ResNet34 CORAL
5e-4
256
-
-
Image Datasets ResNet34
CORN
5e-4
16
-
-
Fireman
MLP
CE-NN
5e-4
64
2
300 × 200
Fireman
MLP
OR-NN
5e-4
128
2
300 × 300
Fireman
MLP
CORAL
5e-4
64
2
300 × 200
Fireman
MLP
CORN
1e-3
128
2
300 × 300
An additional study using a VGG16 backbone pre-trained on ImageNet and comparisons with SORD and CNNPOR can be found in the Supplementary Material in section Comparison with Other Deep Learning Methods for Ordinal Regression. In addition, results on text datasets and recurrent neural networks are included in the Supplementary Material in section Additional Results on Text Datasets using Recurrent Neural Networks.
Table 3 :
3Prediction errors on the test sets (lower is better). Each cell represents the average (AVG) and standard deviation (SD) for 5 random seeds runs. Best results are highlighted in bold. A ResNet34 backbone was used for the MORPH-2 and AFAD image datasets. A multilayer perceptron backbone was used for the Fireman tabular dataset. The class labels in all datasets were balanced. The full table of all random seeds runs can be found in the Supplementary Materials(Table S5).Method
Metrics format
MORPH-2 (Balanced)
AFAD (Balanced)
Fireman
MAE
RMSE
MAE
RMSE
MAE
RMSE
CE-NN
AVG±SD
3.73 ± 0.12 5.04 ± 0.20 3.28 ± 0.04 4.19 ± 0.06 0.80 ± 0.01 1.14 ± 0.01
OR-NN [14]
AVG±SD
3.13 ± 0.09 4.23 ± 0.10 2.85 ± 0.03 3.48 ± 0.04 0.76 ± 0.01 1.08 ± 0.01
CORAL [1]
AVG±SD
2.99 ± 0.04 4.01 ± 0.03 2.99 ± 0.03 3.70 ± 0.07 0.82 ± 0.01 1.15 ± 0.01
CORN (ours) AVG±SD
2.98 ± 0.02 3.99 ± 0.05 2.81 ± 0.02 3.46 ± 0.02 0.76 ± 0.01 1.08 ± 0.01
We found that OR-NN and CORN have identical performances on the tabular Fireman dataset (Ta-
ble 3), outperforming both the CE-NN and CORAL in both test MAE and test RMSE.
Table 4 :
4MAE prediction errors on the test sets for the ResNet34 backbone. The class labels in all datasets were balanced. Best results are highlighted in bold.CORN
CORN
w/o subsets
MORPH-2 2.98 ± 0.02 2.93 ± 0.04
AFAD
2.81± 0.02
3.06 ± 0.02
AES
0.43 ± 0.01 0.68 ± 0.01
Fireman
0.76 ± 0.01 0.81 ± 0.01
Table S2 :
S2Prediction errors on the test sets for the VGG16 backbone pre-trained on ImageNet. Best results are highlighted in bold.MAE (lower is better)
CE-NN OR-NN [14] CORAL [1]
CORN
(ours)
CNNPOR [10] SORD [5]
Nature
0.29
0.29
0.28
0.29
0.29
0.27
Animals 0.28
0.25
0.30
0.26
0.32
0.31
Urban
0.26
0.27
0.27
0.25
0.33
0.28
People
0.29
0.28
0.29
0.26
0.32
0.31
Overall
0.28
0.27
0.29
0.27
0.32
0.29
Table S3 :
S3Prediction errors on the test sets (lower is better). A ResNet34 backbone was used for the
MORPH-2 and AFAD image datasets. A multilayer perceptron backbone was used for the AES and
Fireman tabular datasets. The class labels in all datasets were balanced. Best results are highlighted
in bold.
Method
Seed
MORPH-2
AFAD
Fireman
MAE
RMSE
MAE
RMSE
MAE
RMSE
CE-NN
0
3.81
5.19
3.31
4.27
0.80
1.14
1
3.60
4.8
3.28
4.19
0.80
1.14
2
3.61
4.84
3.32
4.22
0.79
1.13
3
3.85
5.21
3.24
4.15
0.80
1.16
4
3.80
5.14
3.24
4.13
0.80
1.15
AVG±SD 3.73 ± 0.12 5.04 ± 0.20 3.28 ± 0.04 4.19 ± 0.06 0.80 ± 0.01 1.14 ± 0.01
OR-NN
[14]
0
3.21
4.25
2.81
3.45
0.75
1.07
1
3.16
4.25
2.87
3.54
0.76
1.08
2
3.16
4.31
2.82
3.46
0.77
1.10
3
2.98
4.05
2.89
3.49
0.76
1.08
4
3.13
4.27
2.86
3.45
0.74
1.07
AVG±SD 3.13 ± 0.09 4.23 ± 0.10 2.85 ± 0.03 3.48 ± 0.04 0.76 ± 0.01 1.08 ± 0.01
CORAL
[1]
0
2.94
3.98
2.95
3.60
0.82
1.14
1
2.97
4.03
2.99
3.69
0.83
1.16
2
3.01
3.98
2.98
3.70
0.81
1.13
3
2.98
4.01
3.00
3.78
0.82
1.16
4
3.03
4.06
3.04
3.75
0.82
1.15
AVG±SD 2.99 ± 0.04 4.01 ± 0.03 2.99 ± 0.03 3.70 ± 0.07 0.82 ± 0.01 1.15 ± 0.01
CORN
(ours)
0
2.98
4
2.80
3.45
0.75
1.07
1
2.99
4.01
2.81
3.44
0.76
1.08
2
2.97
3.97
2.84
3.48
0.77
1.10
3
3.00
4.06
2.80
3.48
0.76
1.08
4
2.95
3.92
2.79
3.45
0.74
1.07
AVG±SD 2.98 ± 0.02 3.99 ± 0.05 2.81 ± 0.02 3.46 ± 0.02 0.76 ± 0.01 1.08 ± 0.01
Table S4 :
S4Prediction errors on the test sets for the RNN backbone (lower is better). The class labels for both the Coursera and TripAdvisor were balanced. Best results are highlighted in bold. AVG±SD 1.07 ± 0.05 1.50 ± 0.04 0.95 ± 0.12 1.36 ± 0.14Method
Seed
TripAdvisor
Coursera
MAE
RMSE
MAE
RMSE
CE-RNN
0
1.13
1.56
1.01
1.48
1
1.04
1.53
0.97
1.05
2
1.05
1.54
1.12
1.65
3
1.23
1.81
1.18
1.76
4
1.03
1.52
0.84
1.26
AVG±SD 1.10 ± 0.09 1.59 ± 0.12 1.02 ± 0.13 1.53 ± 0.19
OR-RNN
[14]
0
1.06
1.53
0.98
1.34
1
1.09
1.50
0.93
1.24
2
1.11
1.53
1.12
1.47
3
1.23
1.52
1.11
1.53
4
1.07
1.40
0.85
1.16
AVG±SD 1.11 ± 0.07 1.50 ± 0.06 1.00 ± 0.12 1.35 ± 0.15
CORAL
[1]
0
1.15
1.58
0.99
1.29
1
1.14
1.49
1.03
1.39
2
1.16
1.46
1.14
1.40
3
1.19
1.41
1.20
1.40
4
1.13
1.47
0.82
1.11
AVG±SD 1.15 ± 0.02 1.48 ± 0.06 1.04 ± 0.15 1.33 ± 0.13
CORN
(ours)
0
1.09
1.55
0.95
1.37
1
1.09
1.53
0.90
1.32
2
1.01
1.45
1.07
1.49
3
1.12
1.51
1.05
1.47
4
1.03
1.46
0.78
1.14
Table S5 :
S5Prediction errors on the test sets (lower is better). A ResNet34 backbone was used for the MORPH-2 and AFAD image datasets. A multilayer perceptron backbone was used for the AES and Fireman tabular datasets. The class labels in all datasets were balanced. Best results are highlighted in bold. .73 ± 0.12 5.04 ± 0.20 3.28 ± 0.04 4.19 ± 0.06 0.80 ± 0.01 1.14 ± 0.01 .13 ± 0.09 4.23 ± 0.10 2.85 ± 0.03 3.48 ± 0.04 0.76 ± 0.01 1.08 ± 0.01 AVG±SD 2.99 ± 0.04 4.01 ± 0.03 2.99 ± 0.03 3.70 ± 0.07 0.82 ± 0.01 1.15 ± 0.01 AVG±SD 2.98 ± 0.02 3.99 ± 0.05 2.81 ± 0.02 3.46 ± 0.02 0.76 ± 0.01 1.08 ± 0.01Method
Seed
MORPH-2
AFAD
Fireman
MAE
RMSE
MAE
RMSE
MAE
RMSE
CE-NN
0
3.81
5.19
3.31
4.27
0.80
1.14
1
3.60
4.8
3.28
4.19
0.80
1.14
2
3.61
4.84
3.32
4.22
0.79
1.13
3
3.85
5.21
3.24
4.15
0.80
1.16
4
3.80
5.14
3.24
4.13
0.80
1.15
AVG±SD 3OR-NN
[14]
0
3.21
4.25
2.81
3.45
0.75
1.07
1
3.16
4.25
2.87
3.54
0.76
1.08
2
3.16
4.31
2.82
3.46
0.77
1.10
3
2.98
4.05
2.89
3.49
0.76
1.08
4
3.13
4.27
2.86
3.45
0.74
1.07
AVG±SD 3CORAL
[1]
0
2.94
3.98
2.95
3.60
0.82
1.14
1
2.97
4.03
2.99
3.69
0.83
1.16
2
3.01
3.98
2.98
3.70
0.81
1.13
3
2.98
4.01
3.00
3.78
0.82
1.16
4
3.03
4.06
3.04
3.75
0.82
1.15
CORN
(ours)
0
2.98
4
2.80
3.45
0.75
1.07
1
2.99
4.01
2.81
3.44
0.76
1.08
2
2.97
3.97
2.84
3.48
0.77
1.10
3
3.00
4.06
2.80
3.48
0.76
1.08
4
2.95
3.92
2.79
3.45
0.74
1.07
When k = 1, f k x [i] represents the initial unconditional probability f1 x [i] =P y [i] > r1 .
When training a neural network using backpropagation, instead of minimizing the K − 1 loss functions corresponding to the K − 1 conditional probabilities on each conditional subset separately, we can minimize their sum, as shown in the loss function we propose in Section 3.5, to optimize the binary tasks simultaneously.
https://www.faceaginggroup.com/morph/ 5 https://github.com/afad-dataset/tarball 6 http://www.di.unito.it/~schifane/dataset/beauty-icwsm15/ 7 https://github.com/gagolews/ordinal_regression_data
https://github.com/Raschka-research-group/corn-ordinal-neuralnet 9 https://github.com/Raschka-research-group/coral-pytorch
https://www.kaggle.com/septa97/100k-courseras-course-reviews-dataset 11 https://www.kaggle.com/andrewmvd/trip-advisor-hotel-reviews
This research was supported by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
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| [
"https://github.com/afad-dataset/tarball",
"https://github.com/gagolews/ordinal_regression_data",
"https://github.com/Raschka-research-group/corn-ordinal-neuralnet",
"https://github.com/Raschka-research-group/coral-pytorch"
] |
[
"Scalable Semiparametric Spatio-temporal Regression for Large Data Analysis",
"Scalable Semiparametric Spatio-temporal Regression for Large Data Analysis"
] | [
"Ting Fung ",
"M A ",
"Fangfang Wang ",
"Jun Zhu ",
"Anthony R Ives ",
"Katarzyna E Lewińska "
] | [] | [] | With the rapid advances of data acquisition techniques, spatio-temporal data are becoming increasingly abundant in a diverse array of disciplines. Here we develop spatio-temporal regression methodology for analyzing large amounts of spatially referenced data collected over time, motivated by environmental studies utilizing remotely sensed satellite data. In particular, we specify a semiparametric autoregressive model without the usual Gaussian assumption and devise a computationally scalable procedure that enables the regression analysis of large datasets. We estimate the model parameters by quasi maximum likelihood and show that the computational complexity can be reduced from cubic to linear of the sample size. Asymptotic properties under suitable regularity conditions are further established that inform the computational procedure to be efficient and scalable. A simulation study is conducted to evaluate the finite-sample properties of the parameter estimation and statistical inference. We illustrate our methodology by a dataset with over 2.96 million observations of annual land surface temperature and the comparison with an existing state-of-the-art approach highlights the advantages of our method. | 10.1007/s13253-022-00525-y | [
"https://arxiv.org/pdf/2111.15086v1.pdf"
] | 244,729,213 | 2111.15086 | e6656c3339aff2a86bc3db1fa9af10934aa5c858 |
Scalable Semiparametric Spatio-temporal Regression for Large Data Analysis
December 1, 2021
Ting Fung
M A
Fangfang Wang
Jun Zhu
Anthony R Ives
Katarzyna E Lewińska
Scalable Semiparametric Spatio-temporal Regression for Large Data Analysis
December 1, 2021Environmental StatisticsNon-Gaussian processSparse matrix operationsSpatio- temporal autoregression
With the rapid advances of data acquisition techniques, spatio-temporal data are becoming increasingly abundant in a diverse array of disciplines. Here we develop spatio-temporal regression methodology for analyzing large amounts of spatially referenced data collected over time, motivated by environmental studies utilizing remotely sensed satellite data. In particular, we specify a semiparametric autoregressive model without the usual Gaussian assumption and devise a computationally scalable procedure that enables the regression analysis of large datasets. We estimate the model parameters by quasi maximum likelihood and show that the computational complexity can be reduced from cubic to linear of the sample size. Asymptotic properties under suitable regularity conditions are further established that inform the computational procedure to be efficient and scalable. A simulation study is conducted to evaluate the finite-sample properties of the parameter estimation and statistical inference. We illustrate our methodology by a dataset with over 2.96 million observations of annual land surface temperature and the comparison with an existing state-of-the-art approach highlights the advantages of our method.
Introduction
With the rapid advances of data acquisition techniques, spatio-temporal data are becoming increasingly abundant in a diverse array of disciplines including the physical, biological, and social sciences (see, e.g., Cressie & Wikle 2011, Dutilleul 2011, Anselin 2013). Here we consider developing novel spatio-temporal regression methods for analyzing large amounts of spatially referenced data collected over time, motivated by environmental studies utilizing remotely sensed satellite data.
For illustration, we consider an environmental study of the land surface temperature (LST), which quantifies thermal energy flow among land surface, atmosphere and biosphere and thus, characterizes local ecological conditions. Changes in LST may have different causes, but the implications are critical for agriculture, biochemical processes, bioecology, economy, and health.
The 2001-2019 MOD11A2 version 6 data, containing the night LST, were resampled to the 8km spatial resolution and the annual averages were computed for the contiguous US (Wan et al. 2015). It is of importance to investigate the time trend of LST across the study region while accounting for environmental conditions such as elevation and latitude. For example, the left column of Figure to 2019, which arises from, among others, increase in global mean temperature (NOAA 2021) and change in land cover (Lazzarini et al. 2013, Fu & Weng 2016. In addition, it is of interest to examine LST in different ecoregions (Figure S.2) defined to be areas with similar landform, soil, vegetation, land use, wildlife, and hydrology (Omernik & Griffith 2014). Figure We could cast this research on LST in a spatio-temporal regression framework, regressing the response variable of LST on the predictor variables of time trend, ecoregion classes, and interactions between the time trend and ecoregions, as well as the environmental covariates of elevation and latitude. However, there are multiple challenges with using the existing spatiotemporal regression methods. First, the sample size of the dataset is large. With T = 19 years and N = 155, 900 image pixels per year, there are well over 2.96 million LST observations in the dataset. The traditional spatio-temporal regression models with a regression mean and a spatio-temporal covariance function would be infeasible to implement, as the computations are on the order of O(N 3 T 3 ) for evaluating the likelihood function and O(N 2 T 2 ) for memory usage (see, e.g., Cressie 1993, Cressie & Wikle 2011. There is ample room for innovations to reduce the computational burden and to make spatio-temporal regression analysis feasible for practical applications.
Second, although spatio-temporal statistics have advanced greatly in the past two decades, most of the state-of-the-art methods focus on the spatio-temporal dependence structure and the prediction of the underlying spatio-temporal processes. Raw data are often pre-processed by subtracting the mean function from the response (i.e., detrending) and the residuals are treated as the observed data. Even when the mean function is considered, the focus tends to be on the trend as a linear combination of the spatial coordinates and time points for the purpose of predicting the trend surface rather than the regression analysis (see, e.g., Wikle et al. 2019). Thus these techniques are not directly applicable for the purpose of the LST study, calling for further research on the statistical inference of the mean function.
Third, the distribution of the data is not necessarily Gaussian as is assumed by many existing spatio-temporal models. Indeed, the histograms depicted in Figure There has been much research on the development of statistical methodology for analyzing spatio-temporal data (see, e.g., Huang & Cressie 1996, Zhang et al. 2003, Johannesson et al. 2007, Lu et al. 2009, Cressie et al. 2010, Lee & Yu 2015, Zhang et al. 2015, Chu et al. 2019).
See also Cressie & Wikle (2011) and Wikle et al. (2019) for excellent reviews. For Gaussian processes, Cressie et al. (2010) proposed a fixed-rank filtering method for spatio-temporal data focusing on fast computation by dimension reduction spatially and fast smoothing, filtering, or forecasting over time, which in principle can be adapted to perform regression analysis but in practice is not quite feasible yet for the scale of our LST data. Guinness (2021) developed a Gaussian process (GpGp) method that scales up more readily and can be adapted to spatio-temporal regression analysis. GpGp type of methodology approximates the full likelihood of a Gaussian process by a product of conditional likelihoods on subsets, where the subsets are formed by reordering and grouping the data (see, e.g., Vecchia 1988, Guinness 2018, Katzfuss & Guinness 2021. For non-Gaussian processes, Chu et al. (2019) and Lee & Yu (2015) proposed semiparametric models without assuming Gaussian errors, which can be applied to spatio-temporal regression but both emphasized modeling the spatio-temporal dependence and the sample size needs to be kept at a modest size (in the thousands, not millions) for the methods to be computationally feasible. Alternatively, statistical modeling and inference are carried out under a Bayesian framework and the computational challenges are addressed by, for example, dimension reduction (Brynjarsdóttir & Berliner 2014), predictive processes (Banerjee et al. 2008) and
Laplace approximation (Rue et al. 2009(Rue et al. , 2017.
Although the aforementioned statistical methods are useful for many applications, they remain to be either infeasible when the sample size is on the scale of our LST data and/or not well suited for regression analysis involving spatio-temporal processes that are not necessarily Gaussian. Thus here we aim to develop a novel computationally scalable procedure that enables the regression analysis of large datasets, while guided by rigorous asymptotic theory and computational complexity analysis. For modeling and theoretical development, our proposed method is semiparametric in the sense that no explicit distributional assumption is made and only finitemoment conditions of the underlying spatio-temporal processes are assumed. We estimate the model parameters by maximizing a quasi-likelihood and establish the asymptotic properties of the parameter estimators. In addition, the existing literature in spatial statistics tends to directly model the spatio-temporal covariance function and provide its approximation (see, e.g., Cressie & Huang 1999, Gneiting 2002, Chu et al. 2019). Here we take an alternative approach to modeling the spatio-temporal dependence by autoregression. While the autoregression modeling idea is widely used particularly in econometrics, most existing methods are not computationally scalable to the size of our LST data (see, e.g., Lee & Yu 2015, Chi & Zhu 2019. Although Guo et al. (2016) and Gao et al. (2019) considered the estimation and inference of autoregressive models under the high-dimensional setting, the methodology focuses on the estimation of coefficient matrices for zero-mean autoregressive processes without addressing the regression nor computational complexity in detail. In terms of computation, our proposed procedure can be used for spatio-temporal regression with a large sample size N T as in the LST data example. In particular, we adopt advanced computational techniques including efficient data pre-processing, constrained sequential quadratic programming (SQP), and implicit parallel computing. These computational innovations provide a substantive improvement over the existing methods in the literature that tend to require tuning parameter estimation and/or approximating the spatio-temporal neighborhood structure for statistical inference (see, e.g., Rue et al. 2009, Bai et al. 2012, Guinness 2018, 2021, Zhao et al. 2021).
The remainder of this paper is organized as follows. Section 2 presents the model and its estimation. Section 3 establishes the asymptotic properties of the quasi maximum likelihood estimates of the model parameters. Section 4 provides a fast computational procedure for estimation and inference. The finite-sample properties of the estimators are assessed by simulation studies in Section 5 and the LST data example is given in Section 6. Proofs of the theoretical results and other technical details including additional computational aspects, tables, and figures are provided in the Supplementary Materials.
Model and Estimation
Model Specification
At time t ∈ Z, let Y t = (Y 1,t , . . . , Y N,t ) denote an N -dimensional vector that contains the response variables from N cells that partition the study region of interest in R 2 . Let X t denote an N × k design matrix of k nonstochastic predictor variables. We model the spatio-temporal evolution of Y t in relation to the design matrix X t through the following spatio-temporal regression
model Y t = X t β + U t , t ∈ Z,(1)
where β denotes a k×1 vector of regression coefficients. The spatio-temporal error U t is stochastic and modeled by a spatio-temporal dynamic process such that
U t = λW U t + ρW U t−1 + γU t−1 + V t ,(2)
where V t = (v 1,t , . . . , v N,t ) is an N × 1 vector of innovations that are assumed to be iid, not necessarily Gaussian, with mean zero and variance σ 2 I N and I N is the N × N identity matrix.
The spatio-temporal dependence parameters include the conventional temporal lag effect γ, the contemporaneous spatial interactions effect λ, and the effect of spatial diffusion that takes place Finally, the spatial weight matrix W is an N × N nonstochastic symmetric matrix with zero diagonals for a given spatial neighborhood structure (Cressie 1993). The symmetry of W has important implications on computation, which will be elaborated in later sections. Special cases of the spatial weight matrix W include the block-diagonal structure and commonly assumed first-or second-order neighborhood structures. For a block-diagonal structure, W = Diag{w 1 , . . . , w p },
where w i is a n i × n i matrix, with N = p i=1 n i (Case 1991). On a regular square grid, the first-order neighbors are the four nearest cells whereas the second-order neighbors are the eight nearest cells (Cressie 1993). In addition, the spatial weight matrix could be used to construct the design matrix X t in order to capture the spatial neighboring effects; for instance, let
X t = (1 N , Z 1t , Z 2t ), where Z 2t =W Z 1t
andW is a spatial weight matrix defined above.
Let θ = (λ, γ, ρ) denote the vector of the spatio-temporal dependence parameters. We define R(θ) = ρW + γI N , S(λ) = I N − λW , and A(θ) = R(θ)S(λ) −1 . We may then rewrite the spatio-temporal dynamic process (2)
as S(λ)U t = A(θ)S(λ)U t−1 + V t .
That is, the spatiotemporal error U t follows a vector autoregression model of order one and can be shown to be weakly stationary under the assumption that S(λ) is non-singular and the eigenvalues of A(θ) are all strictly less than one in magnitude.
Parameter Estimation
For the observed response vectors Y 1 , . . . , Y T modeled by (1), we define the vector of all the spatio-temporal errors U = (U 1 , U 2 , . . . , U T ) and its matrix operator
B(θ) = S(λ) 0 · · · 0 0 −R(θ) S(λ) · · · 0 0 . . . . . . . . . . . . . . . 0 0 · · · −R(θ) S(λ) N T ×N T (3) such that B(θ)U = ((S(λ)U 1 ) , V 2 , . . . , V T ) . The covariance matrix of B(θ)U is σ 2 Ω(θ),
where Ω(θ) = Diag(K(θ), I N , . . . , I N ) and K(θ) = ∞ j=0 A(θ) j A(θ) j . Let δ = (β , θ , σ 2 ) denote the vector of all the model parameters. To estimate δ, we propose the following quasi log likelihood function,
log L N T (δ) = − N T 2 log(2πσ 2 ) − 1 2 log det(K(θ)) + T log |det(S(λ))| − 1 2σ 2 (Y − Xβ) Σ(θ) −1 (Y − Xβ),(4)where Y = (Y 1 , . . . , Y T ) denotes the N T × 1 vector of all the response variables, X = (X 1 , . . . , X T ) is the corresponding N T × k design matrix, and Σ(θ) −1 = B(θ) (Ω(θ)) −1 B(θ)
is the precision matrix. Denote by δ the maximizer of the quasi log likelihood function log L N T (δ);
that is,
δ = arg max δ∈Θ δ log L N T (δ),
where Θ δ is the parameter space specified in Appendix A. Throughout this paper, we refer to δ as our quasi-maximum likelihood estimator (QMLE) of the model parameters δ.
As illustrated by the LST data example in Section 1, our primary interest is statistical inference of the regression coefficients β, while the estimation of the spatio-temporal dependence parameters θ is of secondary interest intended to account for spatio-temporal correlation when drawing the inference about β.
Asymptotics
Under suitable regularity conditions, we may establish the asymptotic properties of the QMLE δ, as the number of cells N → ∞ while the number of time points T can be either fixed or T → ∞.
Denote by δ 0 = (β 0 , θ 0 , σ 2 0 ) the vector of true model parameters. We first consider the case that T is fixed.
Theorem 1. Suppose that W has more than two distinct eigenvalues, and Assumptions (A.1)-(A.6) hold. Then, δ 0 is identifiably unique and δ p −→ δ 0 as N → ∞.
Theorem 1 establishes that the QMLE δ is a consistent estimator of the true parameter vector δ 0 in the sense that δ converges to δ 0 in probability, when N → ∞.
Next, under additional conditions about the higher-order properties of the quasi log likelihood function, we examine the asymptotic distribution of the QMLE δ.
Theorem 2. Suppose that the conditions in Theorem 1, and additional Assumptions (A.7) and
(A.8) are fulfilled. Then, √ N ( δ − δ 0 ) d −→ N 0, 4 Σ −1 1 + Σ −1 1 Σ 2 Σ −1 1 ,(5)
where
Σ 1 = lim N →∞ N −1 Σ 1,N , Σ 2 = lim N →∞ N −1 Σ 2,N , Σ 1,N = Diag(4σ −2 0 X Σ(θ 0 ) −1 X, 2Ω N ) with Σ(θ 0 ) −1 = B(θ 0 ) (Ω(θ 0 )) −1 B(θ 0 )
and Ω N defined in (11), and Σ 2,N is defined in (S.14)
in the Supplementary Materials.
Theorem 2 establishes that δ converges to a multivariate Gaussian distribution at the rate of √ N . The asymptotic covariance matrix involves two matrices Σ 1 and Σ 2 , which can be replaced by their consistent estimators for the evaluation of the asymptotic distribution of δ in practice.
Since the primary interest is in the statistical inference about the regression coefficients β, we present the asymptotic distribution of β and its relationship to the other parameter estimators θ andσ 2 in the following corollary.
Corollary 1. Suppose the conditions in Theorem 2 hold. Then, we have
√ N ( β − β 0 ) d −→ N 0, Σ −1 β 0 ,(6)
where
Σ β 0 = σ −2 0 lim N →∞ N −1 X Σ(θ 0 ) −1 X. Under the additional assumption that µ 3 = E(v 3 j,t ) = 0, β is asymptotically independent of θ andσ 2 .
Corollary 1 provides the basis for a computationally efficient approach to the statistical inference about the regression coefficients β, as we will detail in Section 4.2. Moreover, the asymptotic distribution of β remains unchanged regardless of the distribution of the spatio-temporal innovations. In particular, when the innovation is symmetric (i.e., µ 3 = 0, which is satisfied by many commonly used distributions including Gaussian and Student-T distributions), Corollary 1 establishes that β is asymptotically independent of the spatio-temporal dependence parameter estimators θ and the variance component estimatorσ 2 .
By Theorems 1 and 2, N −1 Σ 1,N and N −1 Σ 2,N with δ 0 replaced δ converge in probability to Σ 1 and Σ 2 , respectively, as N → ∞. Thus, a consistent estimator of the asymptotic covariance matrix of δ can be obtained from
4 Σ −1 1,N + (Σ 1,N ) −1 (Σ 2,N )(Σ 1,N ) −1(7)
evaluated at the QMLE δ. However, Σ 1,N and Σ 2,N are both challenging to compute when N is large. One major challenge is that the calculation of Σ 1,N and Σ 2,N requires solving large linear systems, or equivalently inverting large matrices, which is computationally expensive. On the other hand, the estimator of the upper-left block of Σ 1,N is readily available as σ −2 0 X Σ(θ 0 ) −1 X can be consistently estimated byσ −2 X Σ( θ) −1 X. Thus, a consistent estimator of the asymptotic covariance matrix of β isσ 2 (X Σ −1 ( θ)X) −1 , and its computation can in fact be made scalable (see Section 4).
In addition, the asymptotic results hold when T is either fixed or tends to infinity with N at an arbitrary rate. That is, Theorems 1, 2 and Corollary 1 can be readily extended to the case when N and T both tend to infinity, in which case the rate of convergence in Theorem 2 and Corollary 1 becomes Before closing this section, we remark on the regularity conditions (A.1)-(A.8) provided in the Appendix. As assumed in (A.1), the spatial weight matrix is symmetric; for example, it is common to set the weight between two distinct locations to one if the distance between the two locations are within a certain threshold and zero otherwise (see, e.g., Cressie 1993). The symmetry assumption plays a vital role in facilitating the computation, because it follows that S(λ)
√ N T instead of √ N ,
and R(θ) are symmetric as well. Further, the long-run covariance matrix of the process S(λ)U t ,
K(θ), can be written as (I N −A(θ) 2 ) −1 , thereby reducing S(λ)K(θ) −1 S(λ) to S(λ) 2 −R(θ) 2 ,
which enables efficient computation of the quasi log likelihood function (4) (see Section 4.2 for details). In contrast, without the symmetry assumption, the long-run covariance matrix involves infinite terms approximated by a sum of finite matrices by assuming that the process starts at some time point in the past instead of the infinite past (see, e.g., Lee & Yu 2015), which not only poses a challenge for evaluating the quasi log likelihood function in practice when the sample size is large, but also makes the resulting error process possibly non-stationary.
For Assumption (A.2), a sufficient condition for the matrix S(λ) = I N − λW being nonsingular and the eigenvalues of A(θ) being less than one in magnitude is that the parameters λ, γ, ρ satisfy the following inequality:
(λ 2 − ρ 2 )d 2 j − 2(λ + γρ)d j + (1 − γ 2 ) > 0, j = 1, . . . , r,(8)
where {d i , i = 1, . . . , r} are the non-zero eigenvalues of W with the smallest eigenvalue (d 1 ) and the largest eigenvalue (d r ) of W having opposite signs. For (8) to hold, it is sufficient to consider the following set,
(λ, γ, ρ) : −1 < γ < 1, 1 − γ d 1 < λ + ρ < 1 − γ d r , 1 + γ d 1 < λ − ρ < 1 + γ d r .(9)
In practice, we choose Θ θ as a compact subset of the above set.
Computation
In this section, we will develop a novel fast computation procedure and show that its computational complexity is on the order of O(N T ) for obtaining the QMLE δ and the variance estimate of β, which is scalable to the sample size of the LST data. The existing state-of-the-art methodology generally approximates the dependence structure for computational ease, while our approach does not require an approximation of the spatio-temporal dependence. Thus, our computational procedure provides a novel and scalable alternative to the existing spatio-temporal modeling and inference without approximating the likelihood function.
Computational Procedure
We obtain the QMLE, δ, and an estimate of its variance Var( δ) by bringing together a set of computational techniques for nonlinear optimization and sparse matrix operations. An overview of the procedure is visualized by a flowchart in Figure 1. Specifically, the procedure starts with the input of the spatial weight matrix W , the vector of the response variables Y , and the design matrix X. We then preprocess the data by applying the reverse Cuthill-McKee (RCM) algorithm (Gilbert et al. 1992). In particular, the RCM algorithm permutes the rows and columns of W , which is a symmetric, generally sparse matrix, into a symmetric sparse banded matrix with a small bandwidth. This effectively moves the non-zero elements of W towards the diagonal while preserving the spatial neighborhood structure. The underlying graph theory for the RCM algorithm views the spatial weight matrix as a graph with vertices (of spatial locations) and edges that connect spatial neighbors specified in the W matrix. We then reorder the rows of Y t and X t according to the Cuthill-McKee ordering of W for t = 1, . . . , T . For a given spatial weight matrix W , it is always possible to obtain a sparse banded W , without distorting the pre-specified spatial neighboring structure (Mafteiu-Scai 2015). Thus henceforth we assume that W is a pre-specified symmetric sparse banded matrix with bandwidth b, which eases the implementation of computational techniques for banded matrices and enables a more precise account of computational complexity.
Next, the parameter vector δ is estimated by maximizing the quasi log likelihood using an iterative SQP (i.e., fmincon() in MATLAB) (see Chapter 18 of Nocedal & Wright 2006). At each iteration, the quasi log likelihood function (4) and its gradient functions are evaluated for optimizing (4) subject to a set of constraints on the parameter space Θ δ . To ensure the scalability of SQP, however, care is needed in the evaluation of the quasi log likelihood function, as we will show in the next subsection. In addition, the constraints on the parameters need to be checked, which we will refer to as feasibility check. The standard feasibility check would require computational cost on the order of O(N 2.4 ). Here we apply the sufficient condition (9) developed in Section 3, which requires solving for the smallest (d 1 ) and largest (d r ) eigenvalues of W .
We thus preprocess W by the Krylov-Schur algorithm, which is an iterative method for solving eigenproblems with sparsity and belongs to the class of Krylov subspace methods (Stewart 2002 In addition, most of the computations can be parallelized and in particular, we enable the implicit parallelism through maxNumCompThreads(), which distributes the computation in multiple cores and utilizes the sparsity of matrices in our MATLAB code (Luszczek 2009
Computational Complexity
Direct evaluation of the quasi log likelihood function (4) requires O(N 3 T 3 ) operations and is computationally infeasible when N T is large. In the following we show that our computational procedure has the computational complexity of O(N T ) for the regression coefficient estimation and inference, which is linear to the sample size and thus is scalable to the size of the LST dataset.
Note that the matrix operator B(θ) in (3) and the covariance matrix Ω(θ) are involved in the quasi log likelihood function (4). Storing the entirety of B(θ) and Ω(θ) during the process of computing the precision matrix would require standard memory usage and operations to be on the order of O(N 2 T ). Instead, we partition the matrix operator B(θ) and the covariance matrix Ω(θ) in such a way that we only store the unique non-zero blocks of quadratic terms. More specially, we rewrite the last term of the quasi log likelihood function (4) as
(Y − Xβ) Σ(θ) −1 (Y − Xβ) = (Y − Xβ) B(θ) (Ω(θ)) −1 B(θ)(Y − Xβ) (10) = (Y 1 − X 1 β) S(λ)K(θ) −1 S(λ)(Y 1 − X 1 β) + T t=2 (Y t − X t β) S(λ) 2 (Y t − X t β) + T −1 t=1 (Y t − X t β) R(θ) 2 (Y t − X t β) − 2 T −1 t=1 (Y t − X t β) R(θ)S(λ)(Y t+1 − X t+1 β).
Note that the total number of non-zero elements ( The second and third terms of the quasi log likelihood function (4) involve the evaluation of two log determinants, log det(K(θ)) and log |det(S(λ))|, which is in general numerically unstable and computationally infeasible when the sample size N T is large. To overcome such challenges, we utilize the relationship between an LU decomposition and the determinant. Recall that K(θ), given by ∞ j=0 A(θ) j A(θ) j , is dense in general. Thus, it is computationally challenging to compute its log determinant and invert the matrix, as these operations involve solving large linear systems and infinite sum of matrices. Here, we overcome the difficulty by taking full advantage of the symmetric spatial weight matrix and noting the following identity:
S(λ)K(θ) −1 S(λ) = S(λ) 2 − R(θ) 2 .
After some algebra, we have, The first term of the quasi log likelihood function (4) would require O(N ) operations after profiling out σ 2 in (4). That is, by setting
log(det(K(θ))) = log det(S(λ) 2 ) − log det(S(λ) 2 − R(θ) 2 ),∂ log L N T (δ) ∂σ 2 = − N T 2σ 2 + 1 2σ 4 H(β, θ), to zero, we haveσ 2 = (N T ) −1 H(β, θ), where H(β, θ) = (Y − Xβ) Σ(θ) −1 (Y − Xβ).
Combining the results above, the overall computational complexity for evaluating the quasi log likelihood function (4) is O(bN T + kN T + b 2 N ).
To compute the gradient of (4), the computational cost is on the order O(kbN T + b 2 N ), because the partial derivative of (4) with respect to β has a closed form
∂ log L N T (δ) ∂β = 2 X 1 S(λ)K(θ) −1 S(λ)(X 1 β − Y 1 ) + T t=2 X t S(λ) 2 (X t β − Y t ) + T −1 t=1 X t R(θ) 2 (X t β − Y t ) + T −1 t=1 X t R(θ)S(λ)(Y t+1 − X t+1 β) + X t+1 R(θ)S(λ)(Y t − X t β)
and requires O(kbN T ) operations, due to the multiplication of sparse matrices. The computational complexity of calculating the partial derivative of (4) with respect to θ using the analyti- However, the computation of (7) is dominated by solving a large linear system (Gilbert et al. 1992) in the calculation of Σ 1,N and Σ 2,N , which requires at most O(N 2.4 T 2.4 ) computations using the Coppersmith-Winograd algorithm (Coppersmith & Winograd 1990). As such, for practical applications, it may be prudent to apply resampling to compute the standard errors of the spatio-temporal dependence parameter estimates in θ. For example, spatial subsampling may be applied to overlapping or non-overlapping spatial blocks and provide replications of θ for estimating the asymptotic covariance matrix (see, e.g., Sherman 1996, Nordman & Lahiri 2004.
Simulation Study
Simulation Setup
We conduct simulation experiments to assess the finite-sample properties of our proposed methodology and evaluate its computational efficiency. For the design matrix X, we let k = 2 including the intercept and a covariate sampled from the standard Gaussian distribution N (0, 1). Once generated, X is kept fixed. The true parameter vector δ 0 is set at (1, 0.5, 0.1, 0.05, 0.7, −0.03) .
The random innovations V t are sampled independently from the standard Gaussian distribution, t = 1, . . . , T . We also consider a two-dimensional spatial domain with the data taken at spatial coordinates { (1, 1), . . . , (1, n), . . . , (n, n)} and the spatial weight matrix W is under a first-order spatial neighborhood structure. To examine the effect of sample sizes, we consider N = n 2 ∈ {10 2 , 20 2 , 50 2 , 100 2 , 200 2 } and T ∈ {5, 10, 20, 50}. For each combination of N and T , 1000 simulations are generated.
The core computation is executed on an application server with dual Intel Xeon Silver 4116 2.1GHz 12-core (24 thread) processors and 512GB of RAM, running MATLAB R2020a.
Simulation Results
The QMLE δ of the model parameter vector is obtained from maximizing the quasi log likelihood (4). To evaluate the finite-sample properties of the parameter estimates, we compute the bias and mean squared error (MSE) by taking the sample average of the differences and the squared differences between the estimate δ and the true value δ 0 over the 1000 simulations for different ations of N and T (Table 1). Overall, both the bias and the MSE decrease gradually as N or T increases for each of the parameters in δ 0 .
Next, we compare various estimates of the standard errors (SE) of the regression coefficients β = (β 0 ,β 1 ) , which are of primary interest. Table 2 shows the sample standard deviation (SD) of the estimates among 1000 simulations, the asymptotic SD approximated by σ −2 0 N −1 X Σ(θ 0 ) −1 X, and the plug-in SE developed in Corollary 1 evaluated β. The sample SD can be viewed as the gold standard. Both the asymptotic SD and the plug-in SE are close to the sample SD for different combinations of N and T , supporting the results of Corollary 1.
We also evaluate the distributions of the estimated regression coefficients β. Note that both
σ −2 0 X Σ(θ 0 ) −1 X 1/2 ( β − β 0 ) and σ −2 X Σ( θ) −1 X 1/2 ( β − β 0 ) converge in distribution
to the standard bivariate Gaussian distribution by Corollary 1 and the Slutsky's theorem. Table 3 reports the coverage probabilities of the confidence intervals for β 0 and β 1 under the nominal level of 95% using the asymptotic SD and the plug-in SE. The confidence intervals for β 0 and β 1 achieve the nominal coverage well for different combinations of N and T . In addition, for different δ 0 and W , the results are similar and not shown here to save space.
The last column of Tables 1 and 2
Appendix A Notation and Assumptions
We first introduce some notations and conventions. Given an n × n matrix P = (p ij ) n×n , we use tr(P ) and det(P ) to denote the trace and determinant of a square matrix P , and we let vec D (P ) denote the column vector formed by the diagonal elements of P . The (i, j)th element of a matrix P is denoted by ent ij (P ). We define P 1 = max 1≤j≤n n i=1 |p ij | and P ∞ = max 1≤i≤n n j=1 |p ij |. We also let P 2 = {λ max (P P )} 1/2 and P F = {tr(P P )} 1/2 denote the spectral norm and the Frobenius norm, respectively. Let abs(P ) = (|p i,j |) n×n . A sequence of n × n matrix P n is said to be uniformly bounded (UB) in row and column sums, if sup n≥1 P n 1 < ∞ and sup n≥1 P n ∞ < ∞. We also use 0 and 1 to denote a matrix or a vector with all elements equal zero and one respectively.
For a real-valued function f (x), x = (X 1 , . . . , x k ) ∈ R k , we let ∇f (x) denote the gradient vector and let ∇ 2 f (x) denote the Hessian matrix. The partial derivative of f with respect to x j is
denoted by ∂ x j f (x) or ∂f (x)
∂x j , whereas the second partial derivative with respect to x j is denoted as ∂
x j x j f (x) (or ∂ 2 f (x) ∂x 2 j ).
In the following, we provide the regularity conditions for the establishing the large-sample properties of the QMLE δ.
A.1. The N × N spatial weight matrix W is non-stochastic, symmetric, and the diagonal elements are zeros.
A.2. The parameter space Θ δ of δ = (β , θ , σ 2 ) is compact and is the product space of Θ β , Θ θ and [σ 2 ,σ 2 ], where Θ θ is a compact set such that the matrices I N − λW are nonsingular and the eigenvalues of A(θ) are less than 1 in magnitude, while Θ β is a compact subset of R k .
The true value δ 0 = (β 0 , θ 0 , σ 2 0 ) lies in the interior of Θ δ .
A.3. The vector of innovations V t = (v 1,t , . . . , v N,t ) ∼ iid(0, σ 2 0 I N ) and E(|v j,t | 4+η ) < ∞ for some η > 0 for all j, t.
A.4. The precision matrix, infinite sum of power of A(θ 0 ), and the design matrix are uniformly bounded (UB).
(i) Σ(θ) −1 = B(θ) (Ω(θ)) −1 B(θ) and S(λ) −1 are UB, ∀θ ∈ Θ.
(ii) ∞ h=1 abs(A(θ 0 ) h ) is UB. (iii)
The N × k design matrix X t is nonstochastic with elements UB in N and t.
A.5. lim N →∞ 1 N X Σ(θ) −1 X = lim N →∞ 1 N X B(θ) (Ω(θ)) −1 B(θ)X is nonsingular ∀θ ∈ Θ.
A.6. Denote by λ j (θ), j = 1, . . . , N T , the distinct eigenvalues of Σ(θ) −1 Σ(θ 0 ) in non-increasing
order. Let f j (α) = − log(λ j (θ) σ 2 0 σ 2 ) + λ j (θ) σ 2 0 σ 2 where α = (θ , σ 2 ) , then lim inf N →∞ 1 N N T j=1 ∇ 2 f j (α) is nonsingular. A.7. Σ(θ), ∂ θ i (Σ(θ) −1 ), ∂ 2 θ i θ j (Σ(θ) −1 ), and ∂ 3 θ i θ j θ k (Σ(θ) −1 ) are UB in θ = (θ 1 , θ 2 , θ 3 ) ∈ Θ. A.8. lim N →∞ N −1 Ω N is nonsingular, where Ω N = tr(m 2 λ ) tr(m λ m γ ) tr(m λ m ρ ) − 1 σ 2 0 tr(m λ ) * tr(m 2 γ ) tr(m γ m ρ ) − 1 σ 2 0 tr(m γ ) * * tr(m 2 ρ ) − 1 σ 2 0 tr(m ρ ) * * * N T σ 4 0 ,(11)
with m λ , m γ , and m ρ defined in (S.12) in the Supplementary Materials. Table 3: Coverage probabilities of the confidence intervals for β 0 and β 1 under the nominal level of 95% using the asymptotic standard deviation (Asy SD) at β 0 and the average standard error (Plug-in SE) by Corollary 1 at β based on 1000 simulations.
S.1 in the Supplementary Materials shows that the LST has changed from 2001
S.3 in the Supplementary Materials shows the time series of LST in the ten largest Level III ecoregions, indicating that the LST time trend varies across different ecoregions.
S.4 in the SupplementaryMaterials suggest a possible departure of the LST distribution from Gaussian.
over time ρ (see, e.g., Anselin 2013, Lee & Yu 2015, Chi & Zhu 2019).
with corresponding adjustment of Assumptions (A.5), (A.6) and (A.8), and the asymptotic covariance matrices.
Assumption (A.3) assumes that the innovations are independent and identically distributed over time and across space and requires the existence of unconditional higher-order moments without assuming a specific distribution. Assumption(A.4) ensures that N −1 ( N (δ)−E N (δ)) = o p (1) where N (δ) = −2 log L N T (δ), which is weaker than the condition in the literature as we do not assume S(λ) to be positive definite (see, e.g.,Lee & Yu 2015). Assumption (A.5) ensures the non-singularity of Σ 1 when N → ∞. By assuming that W has more than two distinct eigenvalues along with Assumptions (A.5) and (A.6), lim inf N →∞ N −1 {E( N (δ)) − E( N (δ 0 ))} > 0 for δ = δ 0 , which implies δ 0 can be uniquely identified (see, e.g.,Gallant & White 1988).Assumption (A.7) regulates the gradient and the Hessian matrix of N (δ) around δ 0 , while Assumption (A.8) ensures that the asymptotic covariance matrix of δ is well-defined when N → ∞, both of which are standard regularity conditions.
. The Krylov-Schur algorithm first generates a sequence of subspaces containing the approximations of a subset of eigenvectors and eigenvalues of W . Then these approximations are extracted by applying the QR algorithm to the projection of W onto the subspaces and the subset of eigenvalues is approximated iteratively through the Arnoldi method. A reordering of the Schur decomposition in the previous step is considered to improve the standard Arnoldi method (see Chapter 3 of Kressner 2005). Based on d 1 , d r , and the sufficient condition (9), our feasibility check requires O(1) operations, which is a significant improvement over the O(N 2.4 ) operations and the O(N ) memory usage when a full eigen-decomposition of W is used for (8).
nnz) of W is O(bN ). Since the product of two N × N banded matrices each with bandwidth O(b) is still banded with bandwidth O(b), it follows that S(λ)K(θ) −1 S(λ), S(λ) 2 , R(θ) 2 , and R(θ)S(λ) in (10) are all banded matrices with bandwidth O(b). Thus, the computation of each quadratic form in the summand of (10) involves sparse matrix-vector multiplications and requires O(nnz) = O(bN ) operations. As a result, the computation of (10) has complexity O(bN T + kN T ).
which converts the computationally intensive task into sparse matrix multiplication and calculation of the (log-)determinant of two positive definite matrices with bandwidth O(b). Furthermore, incomplete LU (ILU) decomposition of banded matrix takes advantage of the sparsity pattern to speed up the LU factorization without compromising the accuracy (Saad 2003). This reduces the computational cost from the standard O(N 2.4 ) to O(b 2 N ) (see, e.g., Section 2 of Kilic & Stanica 2013) and ensures the numerical stability of the calculation of log determinant of banded matrices during the evaluation of quasi log likelihood.
cal form remains computationally expensive as it involves solving large linear system requiring O(N 2.4 ) operations. Thus, we use finite difference approximations in the gradient calculation, which reduce the computational cost from O(N 2.4 + kbN T ) to O(b 2 N + kbN T ). With the results above combined, the estimation of δ through numerical constrained optimization would require O(kbN T + b 2 N ) operations. In other words, the computational complexity of our method is linear to the total sample size (N T ) when k and b are fixed and hence, is computationally feasible for large datasets even on the order of millions. Last but not least, we turn to the computational cost involved in evaluating the estimate of Var( δ). By a similar argument in the evaluation of the quasi log likelihood, computinĝ σ 2 (X Σ −1 ( θ)X) −1 requires only O(kbN T ) operations and O(kN T + bN ) memory usage, as opposed to O(N 2 T ) operations and an extra O(N 2 T ) memory usage with the standard computation. Thus, our procedure facilitates the statistical inference about β with large sample size.
reports the average time (in second) required to obtain the QMLE δ and the various measures of the variation of β. The computation is reasonably fast. For example, when N is large (e.g., 200 2 ), the parameter estimation takes less than one minute per simulation. Moreover, the computational time is empirically linear to the spatial dimension N as the length T of the time series is relatively small. It is worthwhile to point out that the memory usage remains low (e.g., around 2GB when N = 200 2 and T = 50) in the computation.Overall, the simulation experiments corroborate the theoretical properties of δ and the computational complexity shown in Sections 3 and 4 respectively. 6 Data Example: Land Surface Temperature As described in Section 1, we regress the response variable of LST on the predictor variables of time trend, ecoregion classes, and interactions between the time trend and ecoregions, as well as the environmental covariates of elevation and latitude over T = 19 years and N = 155, 900image pixels per year. Thus, there are a total of k = 171 regression coefficients. To implement the proposed spatio-temporal regression method, we construct a binary spatial weight matrix,W = (w ii ) N ×N , such that w ii = 1 if cell i is a first-order neighbor of cell i, 0 otherwise. We then apply the computational procedure described in Section 4. The majority of the regression coefficients are significant after false discovery rate adjustments, suggesting that, as expected, the mean LST values are different among different ecoregion classes and the time trend in LST varies among ecoregions (Figure S.5). Left panel of Figure 2 maps the estimated time trend across ecoregions for the LST. Overall, there is an increasing time trend, especially in the southern and southeastern parts of the US, suggesting that these regions are subject to higher air temperatures than the rest of the continental US. This finding is consistent with previous findings that South and Southeast US seem to warm up the most in recent decades (Vose et al. 2017). Tables S.1 and S.2 give the estimated regression coefficients of elevation, latitude, and the intercept (with respect to water), as well as the time trend of the five largest and smallest ecoregions respectively. The LST tends to decrease with elevation and lat-itude, which are as expected. The estimates for σ 2 , λ, γ, and ρ are 0.6061, 0.0360, 0.7273, and −0.0247, respectively. For model diagnostics, we first assess the in-sample model fit. From Figure S.1, the estimated LST in 2001, 2019 and their difference are similar to the observed data, indicating that our method can recover the mean function well using the covariates. We then evaluate the out-of-sample prediction by fitting the 2001-2018 data and predicting the LST in 2019. The middle and right panels of Figure 2 suggest that the predicted LST for 2019 match up with the actual observations.Finally, we compare our method with GpGp with the same set of covariates (Guinness 2021).The default neighborhood structure and the exponential space-time covariance function are adopted for fitting models using the R package GpGp. The estimated LST values from GpGp seem to be quite different from the observed values (right panels ofFigure S.1), possibly due to numerical instability with the large sample size N T . While the computational complexity and the programming languages are not directly comparable between GpGp and our method, it took GpGp more than four days and our method within two hours to perform the regression analysis.
(Figure 1 :
1Cuthill-McKee) Reorder: , , (Krylov-Schur) Eigen problem: Numerical constrained optimization (Sequential Quadratic Programming) Objective function: log {( ′, ′,̂2) ′} -sparse matrix-vector/matrix multiplication -Incomplete LU decomposition (ILU) Gradient function: ∇ log {( ′, ′,̂2) ′} ∇ : sparse matrix-vector/matrix multiplication ∇ : finite-difference approximation Flowchart for carrying out the proposed spatio-temporal regression and inference.
Figure 2 :
2Estimated regression coefficients by ecoregion using data from 2001 to 2019 (left panel); Observed land surface temperature (LST) in 2019 (middle panel); and Predicted LST in 2019 based on model fitting with data from 2001 to 2018 (right panel).
)
). Similar techniques can also be implemented in R and Python, for example through the Basic Linear Alge-bra Subroutines (BLAS) or Linear Algebra Package (LAPACK) (Anderson et al. 1999, Blackford
et al. 2002, Buluc & Gilbert 2011).
Table 1 :
1Sample average bias (×10 −4 ) and mean squared error (MSE, ×10 −4 ) of δ based on 1000 simulations, and average computational time (in second) per simulation.Average bias ×10 −4
Sample MSE ×10 −4
Average
time
N T
β 0
β 1
λ
γ
ρ
σ 2
β 0
β 1
λ
γ
ρ
σ 2
10 2
5 -22.80
9.21
-5.24 -50.05 -20.98 -105.50 68.32 16.42 2.60 17.73 5.20 40.92
0.05
10 -14.23 -10.61 -11.03 -35.19
-6.29
-56.64 34.28
8.32 1.51
8.42 2.51 20.72
0.05
20 -17.43 12.26
1.40 -15.47
-7.07
-45.87 18.00
4.35 0.66
3.90 1.38
9.99
0.07
50
-9.01
-1.41
-2.64
1.08
-1.06
-10.59
7.99
1.51 0.27
1.46 0.48
3.99
0.11
20 2
5 -23.22 14.89
2.23
3.27 -12.63
-45.95 15.87
4.49 0.63
4.00 1.38 10.66
0.10
10 16.95
0.09
-0.69 -10.73
-1.35
-6.48
8.68
2.09 0.32
1.90 0.67
5.25
0.12
20
-0.21
1.18
1.11
-9.17
-1.73
-8.68
4.99
1.01 0.16
0.93 0.29
2.58
0.17
50
3.58
1.33
-0.19
-1.55
0.24
-2.55
1.89
0.41 0.07
0.40 0.10
0.94
0.24
50 2
5
-0.55
2.01
-0.32
0.82
0.00
-4.31
2.57
0.65 0.11
0.61 0.22
1.69
0.43
10
-0.85
-0.58
0.94
-0.26
-2.54
-9.74
1.34
0.34 0.05
0.31 0.09
0.83
0.54
20
-3.11
0.41
1.16
0.14
-1.53
-0.69
0.75
0.16 0.02
0.15 0.05
0.40
0.65
50
-1.87
-0.64
2.06
-0.45
-1.31
-1.51
0.29
0.06 0.01
0.06 0.02
0.16
0.96
100 2
5
5.11
1.19
1.16
-1.89
-0.57
-0.08
0.65
0.16 0.03
0.17 0.05
0.42
1.77
10
-0.85
0.27
1.86
0.73
-0.65
-1.17
0.34
0.08 0.01
0.08 0.02
0.19
2.12
20
1.58
1.03
1.52
0.21
-0.98
-0.60
0.19
0.04 0.01
0.04 0.01
0.10
2.67
50
1.81
0.43
1.78
0.59
-1.20
-1.27
0.08
0.02 0.00
0.01 0.00
0.04
4.41
200 2
5
3.33
-0.71
1.52
-0.53
-1.50
-0.42
0.17
0.04 0.01
0.04 0.01
0.11
10.13
10
-0.50
0.63
1.58
0.05
-1.35
-1.80
0.09
0.02 0.00
0.02 0.01
0.06
11.67
20
0.13
-0.37
1.75
0.15
-0.89
-1.61
0.05
0.01 0.00
0.01 0.00
0.02
13.84
50
-1.18
0.02
1.84
0.13
-0.88
-0.82
0.02
0.00 0.00
0.00 0.00
0.01
20.29
Table 2 :
2Sample standard deviation (Sample SD), asymptotic standard deviation (Asy SD) at β 0 , and average standard error (Plug-in SE) by Corollary 1 at β, based on 1000 simulations, and average computational time (in second) per simulation for Plug-in SE.N T
Sample SD
×10 −2
Asy SD
×10 −2
Plug-in SE
×10 −2
Average
time
β 0
β 1
β 0
β 1
β 0
β 1
10 2
5 8.266 4.053 7.979 4.067 7.907 4.044
0.002
10 5.856 2.884 5.954 2.936 5.897 2.930
0.003
20 4.241 2.083 4.331 2.097 4.323 2.094
0.004
50 2.826 1.228 2.793 1.244 2.792 1.243
0.008
20 2
5 3.978 2.116 4.001 2.085 3.992 2.080
0.005
10 2.943 1.447 2.984 1.424 2.980 1.423
0.006
20 2.235 1.005 2.170 0.999 2.168 0.999
0.011
50 1.373 0.638 1.397 0.634 1.397 0.634
0.021
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| [] |
[
"Tailoring Structure-borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review",
"Tailoring Structure-borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review"
] | [
"Mourad Oudich \nGraduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA\n\nUniversité de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54000NancyFrance\n",
"Nikhil Jrk Gerard \nGraduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA\n",
"Yuanchen Deng \nGraduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA\n",
"Yun Jing \nGraduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA\n"
] | [
"Graduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA",
"Université de Lorraine\nCNRS\nInstitut Jean Lamour\nF-54000NancyFrance",
"Graduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA",
"Graduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA",
"Graduate Program in Acoustics\nThe Pennsylvania State University\n16802University ParkPennsylvaniaUSA"
] | [] | In solid state physics, a bandgap (BG) refers to a range of energies where no electronic states can exist. This concept was extended to classical waves, spawning the entire fields of photonic and phononic crystals where BGs are frequency (or wavelength) intervals where wave propagation is prohibited. For elastic waves, BGs were found in periodically alternating mechanical properties (i.e., stiffness and density). This gave birth to phononic crystals and later elastic metamaterials that have enabled unprecedented functionalities for a wide range of applications. Planar metamaterials were built for vibration shielding, while a myriad of works focused on integrating phononic crystals in micro-systems for filtering, waveguiding, and dynamical strain energy confinement in optomechanical systems. Furthermore, the past decade has witnessed the rise of topological insulators which lead to the creation of elastodynamic analogs of topological insulators for robust manipulation of mechanical waves. Meanwhile, additive manufacturing has enabled the realization of three-dimensional (3D) architected elastic metamaterials which extended their functionalities. This review aims to comprehensively delineate the rich physical background and the stateof-the art in elastic metamaterials and phononic crystals that possess engineered BGs for different functionalities and applications, and to provide a roadmap for future directions of these manmade materials. | 10.1002/adfm.202206309 | [
"https://export.arxiv.org/pdf/2207.05234v4.pdf"
] | 252,090,001 | 2207.05234 | 0aaa8a7d89b2537e4db55e979d99ca277372aad0 |
Tailoring Structure-borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review
Mourad Oudich
Graduate Program in Acoustics
The Pennsylvania State University
16802University ParkPennsylvaniaUSA
Université de Lorraine
CNRS
Institut Jean Lamour
F-54000NancyFrance
Nikhil Jrk Gerard
Graduate Program in Acoustics
The Pennsylvania State University
16802University ParkPennsylvaniaUSA
Yuanchen Deng
Graduate Program in Acoustics
The Pennsylvania State University
16802University ParkPennsylvaniaUSA
Yun Jing
Graduate Program in Acoustics
The Pennsylvania State University
16802University ParkPennsylvaniaUSA
Tailoring Structure-borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review
1 * [email protected] ‡ [email protected] † MO and N JRK G contributed equally to the present work.
In solid state physics, a bandgap (BG) refers to a range of energies where no electronic states can exist. This concept was extended to classical waves, spawning the entire fields of photonic and phononic crystals where BGs are frequency (or wavelength) intervals where wave propagation is prohibited. For elastic waves, BGs were found in periodically alternating mechanical properties (i.e., stiffness and density). This gave birth to phononic crystals and later elastic metamaterials that have enabled unprecedented functionalities for a wide range of applications. Planar metamaterials were built for vibration shielding, while a myriad of works focused on integrating phononic crystals in micro-systems for filtering, waveguiding, and dynamical strain energy confinement in optomechanical systems. Furthermore, the past decade has witnessed the rise of topological insulators which lead to the creation of elastodynamic analogs of topological insulators for robust manipulation of mechanical waves. Meanwhile, additive manufacturing has enabled the realization of three-dimensional (3D) architected elastic metamaterials which extended their functionalities. This review aims to comprehensively delineate the rich physical background and the stateof-the art in elastic metamaterials and phononic crystals that possess engineered BGs for different functionalities and applications, and to provide a roadmap for future directions of these manmade materials.
Introduction
The concept of BGs was first introduced for classical waves in 1987 when Yablonovitch and John [1,2] demonstrated that a periodic arrangement of materials with contrasting optical indices could yield frequency intervals (BGs) that were forbidden to electromagnetic wave propagation. These BGs are analogous to those found in the energy band structure for crystalline materials. They can be characterized by evaluating the frequency of the propagating modes as a function of the wave vector magnitude in the different propagation directions along the Brillouin zone associated with the periodic configuration of the lattice. The optical structure that was employed for this realization was coined photonic crystal (PtC) and has ever since served as the cornerstone for modern research in wave physics and engineering optical devices [3,4]. Inspired by PtCs, phononic crystals (PnC) were subsequently introduced by Tamura, Hurley, and Wolfe [5] in 1988 for elastic waves in a one-dimensional (1D) periodic structure, which was followed by the pioneering studies of Sigalas and Economou [6,7] and Kushwaha et al. [8,9] in early 90 th where the existence of phononic BGs was demonstrated for elastic waves. A PnC is made of elastic or fluid units (or scatterers) that are periodically distributed in a host medium (either elastic or fluid) that has contrasting mechanical properties (density and elastic modulus or compressibility). The two decades following the introduction of PnCs have witnessed countless studies that put forward and examined diverse PnC designs alongside proposing modeling approaches specifically constructed to characterize their wave dispersion through numerical techniques like Plane Waves Expansion [8,9,[11][12][13]18,19,[21][22][23]26,27,29,30], Finite Difference Time Domain [16,20,24], Multiple Scattering Theory [14,17], and Finite Elements methods [25,26,28]. The primary objective of these early works was centered around the physics of mechanical wave dispersion in PnCs with the goal of unveiling the mechanisms behind the opening of the BGs. These works thus studied the influence of geometry, constituent materials, and periodicity on the BGs opening, their localization and width within the band structure. These methods thus not only facilitated the exploration of a large set of designs with structural complexity, but also helped demonstrate novel and remarkable mechanical wave phenomena such as slowing down the wave's group velocity [31,32], confining acoustic waves at structural defects [11,33,34], waveguiding [20,24,26,[35][36][37][38][39], and even setting up preferential directional wave propagation [40]. Nowadays, phononic BG holds an important place in modern research on wave physics and engineering mechanical wave-based devices and constitutes a fundamental means for the realization of wave confinement and filtering that are essential for high device performance in the fields of optomechanics, energy harvesting, mechanical vibrations shielding, and robust topological wave transport.
In the early PnC studies, the opening of the BG was attributed to the Bragg scattering mechanism caused by the periodicity of the lattice. Tamura et al. [5] were the first to explore the mechanism of acoustic Bragg reflection using a 1D PnC which displayed a barrier functionality for both longitudinal and transverse elastic waves at specific frequency intervals. Afterwards, numerous studies were conducted to evidence the Bragg BG in different phononic structures while exploring their dispersion. It was then well established that Bragg BGs strongly depend on the periodicity and the symmetry of the PnC, and that the operating wavelengths in the BG are on the order of magnitude of the periodicity. Meanwhile, in some other cases, BGs also may originate from another mechanism far different from Bragg scattering. Highlighted for the first time by Liu et al. [41] in 2000, this mechanism is generally associated with strong localized elastic resonances within the PnC which was labeled as locally resonant PnC (LRPC). In the structure proposed by Liu et al. [41], the units of periodicity are spherical resonators with low resonance frequencies and the BG is created from the coupling between the resonance of these spheres and the acoustic modes that propagates inside the PnC. Hence, the BG strongly depends on the mechanical properties of individual resonant units rather than the periodicity of the structure. Further, the resonators can be tailored to make the BG appear at wavelengths significantly larger than the periodicity of the LRPC which, hence, allows for the application of the homogenization theory to extract the dynamic effective properties of the structure. Under this homogenization assumption, it was revealed that the LRPC displays highly divergent negative effective mass density and/or elastic modulus at the BG frequency range [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. These LRPCs are also known as metamaterials, and their highly unconventional effective properties have attracted great attention within the communities working on PnCs at that time, and led to the proliferation of subsequent works that explored the effective dynamic properties of LRPCs, thereby giving birth to the sub-fields of acoustic and elastic metamaterials.
The concept of a metamaterial was originally introduced by Veselago in 1968 for optical waves as the theory for a prospective functional materials that could exhibit negative refractive index [59]. But the absence of natural materials and means for the realization of a demonstrator with such abnormal property have hindered the interest into this field. It was not until 1996 when Pendry et al. [60,61] proposed a realistic structure comprised of metal wires and displayed a negative effective permittivity, and another structure made of two concentric split-ring-resonators with negative effective magnetic permeability [62]. Then, Smith et al. [63] first constructed an optical structure capable of manifesting simultaneous negative electrical permittivity and magnetic permeability, which led to the first experimental demonstration of negative optical refraction [64]. This realization had sparked immense interest towards this kind of artificial materials with negative optical properties, called electromagnetic metamaterials. The structure proposed by Smith et al. [63] was made of a periodic distribution of resonators well-tailored to endow the structure with negative effective permittivity and/or permeability at certain frequency ranges [60][61][62][63][64][65]. This led to the realization of optical devices with unconventional capabilities such as negative-index materials [66,67], optical super-lenses [65,68], cloaking [69,70], wavelength demultiplexing [71], and reversed Doppler effects [72,73]. These outstanding efforts inspired parallel works that went on to demonstrate analogous functionalities for mechanical waves. Acoustic metamaterials were then constructed as periodic structures with mechanical resonators well designed to manifest negative effective mechanical properties, i.e., density and compressibility in the case of fluid-borne sound or bulk modulus in the case of structure-borne sound. Intuitively, the negativity is associated with the material's abnormal response to dynamic solicitations as it tends to expand for a compressional pressure in the case of negative compressibility, while its acceleration becomes opposite to the applied force for the case of negative density. These special behaviors reshaped and propelled the field of mechanical waves over the past two decades and enabled exotic functionalities like sub-wavelength sound mitigation [74,75], negative refraction [76], ultrasound superfocusing [77], and acoustic cloaking [78].
PnCs can be constructed through multiple possibilities of periodic structuration with different geometrical shapes and materials for the constituent units. For elastic waves, the design of PnC strongly depends on the kind of wave that is of interest. Early studies in PnCs mainly focused on a two-dimensional (2D) periodic distribution of infinite cylindrical inclusions in a hosting medium using isotropic materials. This design configuration not only presents less complexity in terms of the couplings between elastic modes, but also offers simplicity in solving the governing equations accommodating the limited computational capabilities at that time [7][8][9]11,20,29,36,37,79]. Three-dimensional (3D) periodic structures were also considered by adopting atomic-like arrangement (simple cubic, body-centered cubic, and face-centered cubic) of spherical inclusions in a host medium [6,10,16,17,41,[80][81][82][83][84][85][86][87], and efficient computational methods were introduced to precisely characterize the wave dispersion through the calculation of the band structure. Some studies also focused on using PnCs for the manipulation of surface acoustic waves (SAW) [12,13,21,22,24,38,[88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106] either in the microscale for SAW filtering [89,[91][92][93]95,96,98,[102][103][104] or for seismic waves shielding [97,[99][100][101]105,106], while other works drove their interest towards the manipulation of Lamb waves in plates whose wavelengths are in the order of the plate's thickness [18,23,[25][26][27][28]30,35,38,[107][108][109][110][111][112][113][114][115][116]. Meanwhile, the advances in additive manufacturing techniques have expanded the geometrical and material design space of elastic metamaterials and have facilitated the fabrication of lightweight and mechanically strong structures with great potential for wave propagation applications. In fact, architected materials with precise structural building based on connected rods and masses have enabled multiple 3D structures manifesting ultra-wide BG for mechanical waves [117][118][119][120][121][122][123][124][125][126][127][128][129]. These efforts paved the way to a new generation of architected elastic metamaterials for omnidirectional wide BG while being lightweight with high porosity.
In the last decade, elastic BG has become a fundamental knowledge for both understanding wave physics and creating high performance devices for mechanical wave control. In optomechanics, phoxonic crystals (PxC) which are dual PnCs and PtCs, have become a vital platform for efficient control of both electromagnetic waves and elastic waves through engineering simultaneous optical and elastic BGs . BG engineering has also been utilized to harvest mechanical energy carried by sound/vibration using a planar metamaterial endowed with subwavelength BG, and where a structural defect is introduced to allow for the existence of mechanical cavity modes at frequencies inside the BG [152][153][154][155][156][157][158][159][160]. For example, when an impinging sound wave reaches the metamaterial panel, it excites the phononic cavity modes with highly confined strain energy density, its energy can then be harvested using a piezoelectric material inside the defect. Moreover, PnCs have also served as an exotic platform for mimicking quantum phenomena in condensed matter physics, especially since the discovery of topological insulators. For elastic waves, these analogs includes Landau Level [161], quantum Hall based structures with robust chiral edge states [162], quantum spin Hall effect [163], elastic Valley-Hall edge states [164,165], and elastic higher-order topological insulator with topologically protected corner states [166]. These discoveries have enabled the creation of elastodynamic platforms where wave manipulation can be performed in a highly unusual but robust manner, with backscattering-free waveguiding that is immune to structural defects and fabrication imperfections. Figure 1 presents a timeline for phononic and metamaterial platforms hosting elastic BGs that were engineered for exotic wave functionalities. In this review, we browse through the establishment of the elastic BG in PnCs and elastic metamaterials and reveal its importance in wave physics and device engineering for wave manipulation applications spanning Hz to GHz frequency regimes. Here, we mainly focus on structure-borne sound (i.e., elastic waves) and specifically the physics of elastodynamic bandgaps and their applications. The term "structureborne sound" is used to indicate sound that propagates in solid elastic materials, namely elastic waves. Compared to acoustic functional materials for controlling air-borne sound, our review paper is motivated by the fact that elastic wave functional materials cater to an entirely different set of applications such as SAW-devices, optomechanics, plate-type devices, seismic shielding, etc. For recent review papers on acoustic functional materials for controlling air-borne sound, the reader may refer to [167][168][169]. The second section of this review paper describes the main historical works on PnCs and metamaterials with focus on the physics of BG opening. Then, section 3 discusses the important achievements on the structural design of PnCs and elastic metamaterials while summarizing different routes for enlarging the elastic BG for different application purposes. We particularly deal with the proposed 3D designs in literature and their architectural building enabled by advanced additive manufacturing techniques. In section 4, we focus on the importance of BG for specific applications that encompass optomechanics, topological elastodynamics, energy harvesting, sensing, and active metamaterials for either frequency tuning of the wave filtering capability or to realize non-reciprocal wave propagation. Further, we reveal the numerical inverse design methods and optimization approaches for BG engineering, with focus on topology optimization and machine learning algorithms. We finalize our review on presenting an outlook on future routes for elastic BG applications. Figure 2 gives the presented sections in this manuscript. The main body of the paper is divided into three main sections: Section 2 concerns the mechanism of BG opening; section 3 presents the structural designs for BG engineering; section 4 gives an overview on application avenues for elastic BG.
Fundamentals of elastodynamic bandgaps
The now long-standing interest in PnCs mainly emanates from their capability of producing BGs for acoustic and elastic waves. The interaction of the wave with the periodic structuration of the crystal causes internal wave reflections and interferences or resonances that lead to evanescent waves with spatial exponential decays as the wave propagates through the PnC. Since the birth of PnCs, extensive works have been devoted to understanding the underlaying physics behind the opening of BGs that later led to the building of numerous phononic structures and multiple modeling approaches for their designs and characterization. Based on the underlying mechanisms, BGs can be broadly classified as those that occur as a result of Bragg scattering or/and local resonances. The origin and features of these BGs are introduced in the following section.
Bragg bandgap
Historically, early works on PnCs focused on BGs that are created from the scattering of elastic wave by the periodic structuration of the PnC [5][6][7][8][9]. PnCs were constructed by considering periodic layers in 1D propagation [5] or solid inclusions in a hosting medium in 2D [6][7][8][9] with highly contrasting elastic properties between the inclusions and the medium, which is key for the creation of the BG. The physical mechanism of BG opening was attributed to wave interference known as Bragg scattering that arises from the wave interaction with successive periodic rows of inclusions inside the crystal. Consequently, the Bragg BG occurs at frequencies where the wavelength is on the order of magnitude of the periodicity, specifically when half of the wavelength closely matches the periodicity (bottom right panel in Fig. 3(a)). Examples of applications are discussed in section 4. The existence of phononic BG was initially evidenced theoretically for elastic waves [5][6][7][8]. The wave dispersion was characterized by calculating the band structure analog to the electronic band structure where instead of energy, the frequencies of the propagating modes are evaluated as function of the wave vector amplitude in the Brillouin zone associated with the lattice. An omnidirectional phononic BG is identified as the frequency region where no eigenmode can exist in all directions of the irreducible Brillouin zone. During the 90s and beginning of the 21 st century, acoustic and elastic band structures were investigated to understand the physical mechanism of Bragg BG in multiple PnC designs for different kinds of waves: bulk waves with 2D [6][7][8][9]15,[170][171][172][173] or 3D periodicity [6,10,16,17,80,81,83,85,87,[174][175][176][177][178], guided plate waves (Lamb waves) [18,23,[25][26][27][28]30,35,38,[107][108][109][110][111][112][113][114][115][116] (Fig. 3(a) upper middle panel), and SAW [12,13,21,22,24,[88][89][90][91]94,96] (Fig. 3(a) upper right panel). Since these Bragg BGs are created due to the destructive interference caused by the periodicity inside the PnCs, the frequency of the BG is proportional to the size of the PnC. Most of these works characterized the wave dispersion by displaying the band structure with the normalized frequency ⁄ where is the angular frequency, is the lattice constant of the PnC, and is the smallest shear wave velocity of the bulk medium. Large structures can be designed for low frequency BG while building small size PnC (at the micro and nano scales) lead to BG operating at very high frequency range (hundreds of MHz to few GHz) as it will be detailed in section 4 through some examples.
Meanwhile, if one creates a structural defect inside the PnC by changing either the geometry or the material composition of one scatterer or multiple scatterers along a line, elastic waves can be trapped or guided inside the defect where the wave energy becomes confined. These trapped or guided modes appear in the band structure as additional modes in the frequency range of the BG. The latter then became a fundamental physical tool for engineering phononic platforms for wave control for a wide variety of applications including filtering, waveguiding [20,24,26,[35][36][37], demultiplexing [179], optomechanics , and acoustic energy harvesting [152][153][154][155][156][157][158][159][160]. The physics of Bragg BG opening was also investigated by calculating the complex band structure for evanescent waves which put forth the wave dispersion complexity inside the PnC, especially in the BG frequency range [29,30]. Interestingly, the last decade has seen the revival of interests for Bragg BGs after the discovery of acoustic analogs of topological insulators. The topological features of the bands with regards to the PnC crystal symmetry was investigated to create trivial and non-trivial BGs that could be used to create robust guided interface modes between two PnC with different topologies. Furthermore, and recently, the Bragg BGs were exploited to realize nonreciprocal acoustic and elastic wave propagation. The properties of the PnC were modulated both in space and time to break the time-reversal symmetry and realize unidirectional BGs. These topics will be discussed in further detail in section 4.2. [170], micro holes in a plate [114], and holes in a mable quarry [88]. (bottom panel) band structure of a PnC plate with a BG (highlighted in cyan) [25] and wave propagation at a frequency inside the BG. (b)-(d) Locally resonant PnC : (b) unit cell of a 3D lattice [41], (c) pillar resonators on a plate [180], ans (d) micro-pillared surface [95]. (e)-(g) Negative dynamic properties of elastic metamaterials: (e) negative mass [50], (f) negative bulk modulus [51], and (g) a double negative elastic metamaterial leading to negative refraction [55].
Local resonance bandgap
BGs can also be created using strong mechanical resonances as first demonstrated by Liu et al. [41] in 2000. Their proposed phononic structure is made of a 3D lattice of spherical resonant units distributed periodically in a simple cubic configuration embedded in a host matrix of epoxy. Each resonator is made of a heavy core of lead coated with a very soft material (silicone rubber) ( Fig. 3(b)). The pair of lead core and silicone rubber coating behaves like a spring-mass resonator with low stiffness and heavy mass leading to resonance modes located at very low frequencies. The physical mechanism of the BG is based on the coupling between the resonance modes of the spherical units, which are localized, and the bulk modes inside the crystal. In the band structure, resonances appear as flat bands away from the Г point of the Brillouin zone ( Fig. 3(b)) indicating zero group velocity, and delimiting the lower edge of the BG. Further, the local resonance (LR) based BG is governed by the geometry and mechanical properties of the resonator, and does not depend on the periodicity nor the symmetry of the crystal [181]. The choice of materials is key in this case as low stiffness materials included in the resonators are used to enable the LR in a stiffer matrix (epoxy). Consequently, the LR mechanism can be tailored to occur at low frequencies where the wavelengths in the hosting media can be several orders of magnitudes larger than the periodicity of the crystal (deeply subwavelength). In the structure designed by Liu et al. [41], the BG was found at 380 Hz ( Fig. 3(b)) where the longitudinal wave wavelength in the epoxy matrix is around 300 times the periodicity. The deeply subwavelength BGs enabled by LRPC opened new routes for exploring the physics of wave dispersion at the microscopic scale towards the design of reduced size structures for wave shielding at the low audible frequency range. This also sparked the exploration of other 3D LRPC designs [43,84,182] alongside with 2D structures [183][184][185][186][187] based on soft inclusions in a stiff matrix. Other studies have proposed multiples elastic systems with LR BG for Lamb waves using plates with either soft inclusions [188,189] or resonant pillars [28,110,111] (Fig. 3(c)), and for SAW [93][94][95]190] (Fig. 3(d)). Considering the deeply subwavelength functionality of the LRPC, homogenization methods were used to study their effective dynamic behavior which led to the demonstration of anomalous dynamic properties that have attracted significant attention. In fact, at the resonance frequencies where the wavelengths in the hosting media are larger than the periodicity of the structure, Liu et al. [41] showed that their structure manifests a divergent and negative effective mass density at the regions of the LR BG. This groundbreaking discovery has served as the cornerstone for acoustic and elastic metamaterials that have reshaped the paradigm of mechanical wave propagation. Multiple theoretical studies were conducted to investigate the negativity of the effective density in elastic metamaterials [43,46,[48][49][50]53,54]. Furthermore, Li and Chan [42] demonstrated that acoustic metamaterials can be designed to display both negative effective mass density and bulk modulus using rubber spheres in water. Then, for elastodynamics, Milton and Willis [47] presented a generalized Newton second law to better describe the dynamics of LRPC especially in the extreme case of divergent effective mass density at the resonance. Afterwards, several studies demonstrated double negativity upon the effective mass density and the effective bulk and/or shear modulus in elastic metamaterials [44,45,51,52,[55][56][57][58]. Analytical formulations were developed based on the spring-mass model [50,51] to understand the concepts of negative effective mass and bulk modulus and their relationship to the opening of the BG (Fig.3 (e)-(g)). Recent works conducted by Dong et al. [56][57][58] used topology optimization to design double negative elastic metamaterials. Single negative density or bulk modulus leads to LR BG opening that causes total wave reflection at the resonance with values of the effective parameter tending to be infinite in a lossless system. However, simultaneous negative density and elastic modulus at the same frequency range allows for the appearance of a propagating band with the particularity of endowing the structure with negative effective index, leading to negative refraction of the wave ( Fig.3 (g)) [55,58]. The extensive works on these negative metamaterial behaviors [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58] led to unprecedented wave functionalities such as sub-wavelength sound mitigation [74,75], super-focusing [77], and acoustic and elastic cloaking [56,78,191].
Structural designs for bandgap engineering
The opening of an elastic BG and the control of its width and localization within the band structure has long been a subject of concern for researchers working on the topics of PnCs and elastic metamaterials. BG engineering has thus surfaced due to its usefulness for applications like vibration mitigation, seismic shielding, multidirectional wave cancellation, SAW filtering, long-life time phonon confinement for optomechanics, and shaping the topology of the bands to harbor topologically robust waveguiding and confinement that are immune to defects. In this section, we present examples of PnCs and elastic metamaterials engineered for the purpose of BG manifestation while focusing on wave manipulations in 2D and 3D space for different kinds of guided waves associated with specific applications. The road of studying elastic BG and their utilization has led to a number of PnC and metamaterials designs, especially for the case of 2D lattices. Particularly, the control of guided waves such as Lamb waves and SAW, has led to considering structures with planar periodicity.
Two dimensional phononic crystals
Early studies on elastic BG were conducted on 2D periodic infinite inclusions, mostly of cylindrical shape, in a hosting medium for bulk waves dispersion [7][8][9]15,170,172,173]. The objective of these works was not only to demonstrate theoretically and experimentally that a periodic mismatch in the mechanical properties of the material creates BGs, but also to investigate the Bragg scattering mechanism that is behind their origin. The aforementioned studies on 2D PnCs focused on elastic waves using solid inclusions in a solid matrix. Kushwaha et al. [8] adopted both Al inclusions in Ni and Ni inclusions in Al, but only considered the dispersion of pure shear waves with out-of-plane polarization (displacement along the axis of the cylinders). Shortly after, a first complete BG was evidenced by Sigalas and Economou using Au inclusions in Be [7]. Although the tools for experimental demonstration were easily accessible in acoustics using solid rods in air [170,172], BGs for elastic waves remained purely theoretical for quite a while since fabricating a prototype comprising solid inclusions in a solid matrix was challenging. It was not until 1998 when the first experimental realization of an elastic PnC was presented by Montero de Espinosa et al. [171]. They revealed the existence of a BG for longitudinal waves in PnC made of a thick aluminum plate with holes filled with mercury. The mercury was chosen to create high stiffness and density mismatches with aluminum while the values of the acoustic impedances are close between the two materials in order to reduce mode conversions from the wave scattering inside the structure. Shortly after, García-Pablos et al. [15] built a PnC made of an aluminum slab with periodic cylindrical holes filled with either air, mercury or oil. In 2001, Vasseur et al. [173] fabricated a composite PnC made of a triangular array of steel cylinders in epoxy and showed for the first time the existence of an absolute BG for bulk waves, independently from the direction of propagation. To gain more insight on the Bragg BG opening, Laude et al. [29] and Moiseyenko and Laude [192] developed a plane waves expansion method to calculate the complex band structure of 2D PnC made of infinite cylindrical inclusions in hosting material. They were able to precisely characterize the dispersion of evanescent waves inside the PnC by evaluation the complex value of the wave number in the frequency regions of the BGs for each elastic polarization. Veres et al. [193] also investigated the evanescent waves dispersion in 2D PnC made of square lattice of holes with different shapes.
Regarding SAW, Meseguer et al. [88] experimentally demonstrated for the first time the existence of BG for Rayleigh waves. They considered hexagonal and honeycomb lattices of drilled cylindrical holes on the surface of marble quarry ( Fig. 3(a), upper right panel). In the same year, Tanaka and Tamura [13] investigated theoretically the BG opening for Rayleigh waves using a triangular lattices of cylinders embedded in a background media. These works sparked growing interest on integrating PnCs into microelectromechanical systems and SAW devices. Wu et al. [89] [194][195][196] to track in real time the SAW interaction with a PnC in a silicon substrate, allowing a first experimental extraction of the band structure. In another study, Liu et al. [96] designed and fabricated a PnC with BG for Love waves. Their structure was made of a square lattice of holes etched on a silica guiding layer deposited on the surface of quartz substrate. Meanwhile, PnCs crystals were also considered at larger scale for seismic waves shielding. Brûlé et al. [97] conducted a first large scale experiment on Rayleigh waves scattering caused by holes dug in soil and demonstrated the existence of partial BGs for seismic waves.
Phononic Bragg BGs were also demonstrated for Lamb waves using engineered PnC plates. Wilm et al. [18] proposed PnC plates with 1D and 2D periodicities while considering the piezoelectric effect using quartz inclusions in the composition of the PnC. Later, Hsu and Wu [23] and Khelif et al. [25] have theoretically investigated the dispersion of Lamb waves in plates with periodic cylindrical inclusions to show that a full BG can be created in the whole Brillouin zone. Afterwards, Hsiao et al. [107] experimentally exposed the BG in a PnC slab made of spherical beads in a solid epoxy matrix. Bonello et al. [108] designed and fabricated a PnC plate at the microscale deposited at the surface of a silicon plate. It was made of cylindrical iron inclusions embedded into a copper background, where the authors experimentally characterized the band structure for the first time and showed that depending on the filling fraction, frequency gaps appear for the lowest-order symmetric and antisymmetric modes. At the same time, Morvan et al. [109] studied Lamb waves interaction with a phononic micro-grating and experimentally evidenced the BG at the ultrasonic regime. Shortly after, Djafari-Rouhani et al. [38] and Vasseur et al. [26] optimized the BGs in PnC plates and leveraged them to achieve waveguiding by creating a defect line inside the phononic slab. This line defect was introduced by creating a wide geometrical spacing between the cylindrical units along specific directions of the periodicity. In another study, Sun and Wu [113] used the FDTD method to study wave propagation in a 2D phononic plate consisting of cylindrical steel inclusions. With the existence of the BG, they achieved waveguiding by reducing the size of the steel inclusions along a line. Besides, it became obvious to used Lamb waves in microsystems for filtering since they have the advantage of being well confined within the plate, contrary to Rayleigh waves. In fact, the scattering of Rayleigh waves by PnC generates bulk waves that propagates into the substrate, which significantly reduces the amplitude of the transmitted surface wave through the PnC outside the BG regions. Mohammadi et al. [114] designed a PnC microstructure made of hexagonal lattice of holes in a silicon plate using micro-fabrication process based on semi-conductor technology ( Fig. 3(a) upper-central panel). The transmission measurements through the micro-PnC displayed high wave attenuation in the region of the theoretically predicted BG at 134 MHz with a bandwidth of 23%. Evanescent Lamb waves were also characterized by Oudich and Assouar [30] in phononic plates with inclusions made of different sets of materials. Other designs considered cross-like holes in a plate and showed that they can give rise to multiple complete BGs [115,197]. Miniaci et al. [116] proposed a phononic plate made of unit cells with cross holes at different hierarchical scales and evidenced the existence of BGs at multiple frequency ranges associated with the size of the constituent unit cells of each hierarchical order.
The interest in Bragg BG engineering has grown fast in the last decade thanks to its advantages for high performance optomechanical devices, alongside the realization of non-reciprocal wave propagation, and the creation acoustic and elastic states that are only governed by the topology of the bands. These subjects will be addressed in section 4.
Locally resonant elastic metamaterials
After the discovery and introduction of LRPC by Liu et al. [41], multiple studies focused on 2D ternary designs to reproduce the LR based BG at the subwavelength scale. Goffaux et al. [183,184] used the FDTD method to analyze the transmission spectrum of elastic waves through a lattice of cylindrical cores coated with a soft material, and investigated the effect of different materials of the core on the resonance. The LR BG was depicted in the transmission spectrum in the form of sharp asymmetric Fano like resonance. Similar ternary 2D systems were investigated using analytical and numerical methods [43,181,186,198]. Furthermore, Sheng et al. [181] considered non-periodic arrangement of inclusions to highlight the nondependence of the LR BG on the periodicity of the structure.
Meanwhile, it is not necessary to consider ternary materials in the phononic structure to access LR BG. Wang et al. [185] introduced for the first time a binary LR PC made of cylindrical inclusions of soft rubber in an epoxy host. The key for the creation of LR BG is the high contrast in the mass density and elastic moduli between the soft material of the resonant units and the stiff material of the hosting medium. However, as the resonance phenomena is generally narrow band, it was challenging to produce the LR BG with a relatively wide frequency range which is of great interest for sound and vibration shielding applications. An attempt for the enlargement of LR BG was conducted by Larabi et al. [187] who proposed a structural design made of coaxial multilayered cylindrical inclusions of alternating soft (rubber) and stiff materials (steel). They observed several resonance peaks in the transmission spectrum that were leveraged to further enlarge the LR BG.
Meanwhile, LR BGs were also evidenced for Lamb waves by Hsu and Wu [188] who used rubber cylindrical inclusions of square and hexagonal lattices in an epoxy plate. However, an experimental realization of a prototype made of embedded units in plates was challenging at that time. A more feasible design was proposed by Wu et al. [28] and Pennec et al. [110] who introduced for the first time a structure made of a plate decorated with square lattice of pillars to create LR BG from the coupling between the Lamb waves and individual resonances of the pillars. Then, Oudich et al. [111,180,199] adopted the same design to theoretically and experimentally demonstrate ultra-low frequency LR BG using homogeneous and composite cylindrical pillars made of soft rubber with lead cape, distributed on an aluminum plate ( Fig. 3(c)). The design of plates decorated with pillars has extended the geometrical parameter space of the resonators for tailoring the LR phenomena which offers better control over the BG width and frequency localization. It was found that the position of the BG is significantly affected by the height of the pillars and the mass of the heavy cape, while its width mainly depends on the pillar's radius [111]. Furthermore, in addition to the employed materials, the shape of the pillars and their contact with the plate is of great importance for the LR-wave coupling which hence directly affects the position and the width of the BG. This was demonstrated using pillars with neck [200,201], or having conical [202,203] or hollow shapes [204], or multi-concentric building design [205]. Consequently, metamaterial plates decorated with pillars are of great potential for real word application for both sound and vibration mitigation [74,75,189,[206][207][208].
Besides, LR BG were also engineered for SAW both for microsystems integration, and for large scale application to shield seismic waves. Khelif et al. [93] was the first to theoretically investigate Rayleigh waves dispersion by a square lattice of pillars decorating the surface of a silicon substrate. They demonstrated the existence of LR BG for Rayleigh waves, created from the resonance of the pillars under the sound line. In general, Rayleigh waves interaction with periodic structures often results in mode conversion to the bulk substrate. In the case of periodic holes with sufficient depth, the scattering of Rayleigh wave with the holes causes the leaking of the strain energy carried by the surface wave into the bulk causing a quick decay of the transmitted surface wave amplitude. The phenomena can be depicted in the linear dispersion curves where the SAW-BAW couplings occur at frequencies above the sound line. Conversely, in the case of BG created by the resonance of the pillars, the couplings between the resonance modes and SAW occur below the sound line, which therefore causes less energy leakage into the bulk [94]. At the microscale, Robillard et al. [209] performed a first experimental observation of individual resonances of cubic metallic pillars decorating the surface of a substrate, by employing an ultrafast laser pump and a probing technique through the photoelastic mechanism. Afterwards, Achaoui et al. [95] used an optical measurement technique to characterize the vibration of individual pillars interacting with Rayleigh waves, and observed the generated LR BG by a square lattice of cylindrical nickel pillars deposited on lithium niobate substrate ( Fig. 3(d)). Using the same design, Yudistira et al. [98] showed both theoretically and experimentally the simultaneous existence of Bragg and LR BGs. Furthermore, LR BG were also exposed experimentally using a lattice of metallic nano-discs [92] and pillars in holes [102] for Rayleigh waves. Oudich et al. [103,104] designed phononic pillars with BG that were designed to host cavity modes through structural defects, and used these pillars to realize wave confinement, Fano resonances, and elastic analog of electromagnetically induced transparency for Rayleigh waves.
For large scale SAW, LRPC were designed to provide potential solutions for shielding seismic waves. Colombi et al. [99,100] introduced the idea of using a forest as a lattice where each tree behaves as a resonator with flexural vibration to strongly attenuate seismic waves [105]. Other designs of metamaterial made of either 2D lattice of resonators [101] or circular inclusions [106] in soil were proposed for seismic waves mitigation.
Routes for enlarging the bandgap in 2D metamaterials
The design of periodic structures for elastic BG engineering was stimulated by the growing interest for real world applications centered around shielding and filtering undesirable vibration, either in the ultrasonic or audible frequency ranges. Although the main advantage of LR-based BG is the access to small size structures operating at the subwavelength scale, the challenge of widening the BG was to be faced. This engaged a race towards engineering metamaterials that could shield the wave at the largest frequency range possible. The evident route for BG enlargement is acting upon the geometrical layout of the constituent materials and the contrast of their mechanical properties that strongly affects the wave dispersion. The first study dealing with BG engineering for the purpose of frequency width maximization was initiated by Yilmaz et al. [210,211]. They considered a design of connected masses in a particular configuration that is engineered to promotes the mechanism of inertia amplification. This mechanism allowed them to achieve extremely wide BG reaching a relative width (the frequency width of the BG divided by its central frequency) of 109%. Their theoretical studies were shortly followed by an experimental realization of an architected metamaterial made of 2D lattice of beams well-constructed and connected for the inertia amplification mechanism to operate [212].
Besides, multiple studies have delt with BG enlargement via design engineering by constructing numerical and optimization methods applied to 1D and 2D PnC, and the first works focused on lattices of unit voids in a solid material. Hussein et al. [213] built a multi-objective genetic algorithm to optimize the design of a 1D binary PnC in order to achieve maximum wave attenuation via broadband BG. For 2D phononic designs, Diaz et al. [214] considered a structural design network of connected masses where the design variable is the addition of a unit mass. They sought for optimal choices of mass positioning for the creation of the BG, its central frequency localization, and the control of its width. Afterwards, Bilal et al. [215] also used a genetic algorithm to propose a 2D PnCs made of voids in silicon, and presented optimized structural unit-cells with complex design featuring out-of-plane, in-plane and combined out-of-plane and in-plane elastic wave BGs. They were able to attain a BG enlargement of 122.7% for out-of-plane waves. Shortly after, Dong et al. [56] considered material composition using Pb or Au in epoxy. They used topology optimization to engineer binary 2D PnC with enlarged BG and reached a relative bandwidth of 108%. Section 4.5 of this review discusses in detail the optimization methods based on topology optimizations and genetic algorithm engaged for engineering elastic BG through inverse design. Recently, Jia et al. [216] proposed a single-phase 2D PnC made of connected masses to reach 119% for the relative BG width, with both theoretical and experimental demonstrations ( Fig. 4(a)). They also extended their design into 3D lattice of connected spherical masses in a centered cubic configuration and highlighted an omnidirectional BG of 100% relative bandwidth. Section 3.4 discusses the 3D metamaterial designs proposed in literature for omnidirectional BG engineering.
Regarding guided waves in plates, the first study on BG width maximization was conducted by Halkjaer et al. [217] in 2006, using theoretical optimization and experimental validation to reach a BG width of 95% ( Fig. 4(b)). Then, Acar and Yilmaz [212] applied inertia amplification mechanism for engineering wide BG by designing a plate with connected beams. In another study, instead of pillars on one side of the plate, Assouar and Oudich [218] decorated both sides of the plate with a lattice of pillars to increases the coupling between Lamb waves and the resonances of the pillars. They found that the relative bandwidth of the BG can be increased from 27% for single side pillared plate to 51.4% for the double-side pillared plate. Shortly after, Bilal et al. [219,220] added holes in-between the pillars to lower the effective stiffness of the plate and increase the coupling mechanism ( Fig. 4(d)). By this geometrical configuration, they introduce the mechanism of trampoline resonance leading to BG widening from 19% for simple plate with pillars to 48% for the plate with pillars and holes in-between the pillars. BG enlargement was also realized by combining finite structures with overlapping BGs enabling a wider frequency range than what is allowed by individual unit cells [221,222]. Zhu et al. [221] showed that combining two different but well-designed resonators in a single unit-cell enables considerable enlargement of LR BG. They designed and fabricated a chiral metamaterial endowed with different resonators to access a BG width of 65% (Fig. 4(c)). Celli et al. [222] investigated BG widening using pillar resonators fixed on a plate, with different resonance frequencies Fig. 4(e).
Besides, BG enlargement was also realized for SAW, with a first attempt conducted by Rupp et al. [223] who applied topology optimization to pattern the surface of a silicon substrate for Rayleigh waves filtering in a wide frequency range. At the geophysical scale, Colombi et al. [100] proposed a spatially graded subwavelength resonators distributed on the soil surface, to create what is called a "seismic rainbow" effect. The rainbow effect is based on the coupling between Rayleigh waves and the resonances of stick resonators whose heights are varied gradually to create large LR BGs. This leads to a conversion of Rayleigh waves into bulk waves by deviating the mechanical energy into the earth depth to protect buildings at the surface. A forest with well distributed trees can be used as resonators in the real-world application to achieve the rainbow effect for broad band seismic shielding [100,224].
Three-dimensional structures
The structural design of PnC and metamaterials is manly dictated by the nature of the propagating waves.
For the case of guided waves such as Lamb waves and surface acoustic waves, the geometrical construction and material distribution are set to be along 2D space as the wave propagation operates in the plane. Nevertheless, the control of wave propagation along the 3D space is also of great importance, particularly for the case of creating omni-directional BG, known also as complete BG, for the application of vibration shielding. The road to achieve complete BG was enabled by PnC and metamaterials with geometrical and material distributions along the three directions of space.
Composite lattices with embedded units
The first theoretical investigation of band dispersion for elastic waves in a 3D lattice was conducted in 1992 by Sigalas and Economou [6]. They considered a face-centered cubic lattice of solid spheres in an elastic hosting medium (matrix) with contrasting longitudinal and shear wave velocities, by which they demonstrated the existence of a BG. The band dispersion exhibited flat bands at the edges of the BG frequency region which is associated with the local resonance of the spherical units. At that time, this BG was not identified as created by the LR mechanism. Shortly after, the same authors explored the influence of the volumetric filling fraction of the spheres and materials contrast between the spheres and the hosting medium in search for the optimum design of the face-centered cubic lattice for the creation of the BG [80]. Their conclusion held on considering heavy solid spherical units made of either gold or lead and distributed on a Be, or Si or SiO2 matrix with a lattice volume occupation of 10%. While exploring the maximization of the BG, the same group extended their investigations to include simple and body-centered cubic configurations using metallic spheres made of either W, Ni, Fe, Cu, Steel, or Ag inside the epoxy matrix [10]. They observed a relative bandwidth of the BG higher than 50%, with a maximum of 75% by using solid tungsten spheres. This was the first engineering study about BG width maximization in a 3D lattice. Following these works, other studies investigated similar lattices with different materials [81,87,174] or used unit-cells containing both spherical and cubic scatterers [176], or piezoelectric spherical units [177,178]. Further, Suzuki and Yu [82] characterized the dispersion of evanescent waves by developing a numerical model based on the plane wave expansion method to calculate the complex band structure of 3D PnC made of spherical tungsten scatterers embedded in polyethylene matrix.
It was not until the year 2000, that we witness a first experimental fabrication and characterization of a 3D lattice, namely the seminal work of Liu et al. [41]. Nevertheless, this classical design of spheres in a hosting media has seen a decrease in interest because embedding heavy spheres in a bulk material is not only challenging and onerous in fabrication with limited choice in materials, but the structure itself can be heavy for practical applications. It was not until 2015, that the field of elastic BG with 3D structural design was revived thanks to the rapid development and wide availability of additive manufacturing techniques.
Single phase architected metamaterials
There has been a growing interest in the search for constructing artificial materials that can have more than one functionality, such as combining the low density with mechanical strength. These two properties are highly desirable for aeronautic, aerospace, and automotive applications, mainly for the purpose of material longevity and energy saving. Interestingly, nature provides us with multitude examples of lightweight and strong materials with mind-blowing architectural designs based on highly ordered structuration with hierarchical building from the microscopic scale to the visible macroscopic size. Examples are wood [225], bones, exoskeletons, and spider silk [226]. Drawing inspiration from these biomaterials, researchers have developed and used contemporary fabrication techniques to mimic their micro-structure and access ultralow materials density with remarkable mechanical behavior. Particularly, additive manufacturing techniques have enabled the fabrication of architected materials with precise geometrical features at multiple orders of hierarchical building [227,228]. Most of these techniques use a single-phase material which means that the constituent material of the architected structure is a single material. This sparked a new generation of artificial materials made of 3D architected lattices with outstanding mechanical properties such as recoverability maximization under compression [229,230], super-elastic tensile behavior [227,228], and even decoupling the density and mechanical performance [231].
Such accurate control over the geometrical features and material composition at the micro-scale in the building process is of great interest from the perspective of wave propagation. 3D printing techniques were employed to create single-phase lightweight architected lattices with outstanding acoustic performance in terms of BG width. Most of these architected designs are of centimeter size lattice constant with low frequency BG (below 10 kHz). Taniker and Yilmaz [117] designed and manufactured a 3D octahedron structure using a polymer printer ( Fig. 4(f)), that leveraged inertia amplification to enable wide elastic BG. Their lattice constant is around 11cm to reach a very low frequency BG ranging from 70 to 200 Hz. Other works have employed embedded or connected masses to introduce LR BG with increased bandwidth ( Fig. 4(g)-(j), (m) ), but at the cost of increasing the total mass of the system which may hinder their applications where being lightweight is vital [119][120][121][122][123][124][125][126][127]129]. For instance, Matlack et al. [119] used steel cubic inclusions embedded in a 3D printed cubic frame where the size of a unit cell is 18.25 mm. The structure was fabricated using combined 3D printing and manual assembly by embedding the steel cubes into the printed polycarbonate frame at the middle of the printing process. This meta-structure with heavy masses led to the opening of BGs from 6 kHz to 10 kHz and between 2.15 kHz and 6.11 kHz using different material stiffnesses. In an attempt to reduce the total masse, McGee et al. [232] used connected hollow sphere units enabling wide BG for vibration attenuation. More interestingly, a lightweight design using only connected metallic curved beams without heavy masses was proposed by Warmuth et al. [233] to enable ultrasound wide BG (Fig. 3(k)). The structure was fabricated using elective Electron Beam Melting (SEBM) which is a powder based generative manufacturing technique. Moreover, auxetic designs with added heavy masses have shown to be a promising route for enabling wide BG at low frequency as it was demonstrated by D'Alessandro et al. [121] and Fei et al. [127] (Fig. 4(i), (j)). Very recently, Gerard et al. [128] proposed an ultralight auxetic single phase metamaterial made only with thin polymer rods, where trampoline mechanism is extended to the 3D space to create a wide omnidirectional BG (Fig. 4(l)). Their structure has a mass density as low as 0.034g/cm 3 with a BG width of 82.8% ranging from 2.3 to 5.55 kHz with a lattice constant of 2 cm, which is associated with negative and divergent effective bulk modulus coupled with a near-zero yet positive effective mass density. The fabrication of such auxetic structure was enabled via a high-resolution large area projection micro-stereolithography platform. In another work, Muhammad and Lim [129] proposed a 3D lattice made of connected cylindrical masses on a cubic hollow frame that was fabricated using 3D printing, and reached a BG width of 160% ranging from 1.25 to 11.32 kHz with a lattice constant of 5 cm.
In these architected lattices, not only the precision of the 3D printing process is important, but also knowing the mechanical properties of the constituent material is essential for the desired dynamic behavior of the metamaterial. In fact, despite the advance in additive manufacturing techniques involving a wide range of materials [229,234], polymers have been predominately used in building architected lattices endowed with elastic BG, though polymers also display a wide range of mechanical properties that present some challenges to control. For instance, the stiffness of polymer is dependent on the printing orientation as well as ambient conditions such as temperature. The polymer's Young's modulus and loss factor are also frequency-dependent. These factors have hindered a precise numerical prediction of the dynamic behavior of the printed meta-structure. On the other hand, the advance in additive manufacturing techniques have also enabled 3D printing based on non-polymers such as metals and ceramics [229,234], and even multimaterials printing [235]. These techniques can be leveraged to make elastic metamaterials that are potentially more predictable and with less dissipation, but have been largely unexplored. (b) An architected phononic plate with triangulate lattice of triangular holes [217]. (c) A chiral elastic metamaterial with resonant units [221]. (d) A trampoline metamaterial plate [220]. (e) A metamaterial plate decorated with low frequency resonators [222] .(f) A 3D metamaterial with induced inertial amplification-based vibration [117]. (g) A 1D metamaterial with integrated heavy mass units [119]. (h) A 3D mass based resonant metamaterial [125]. (i) A 3D anti-chiral auxetic metamaterial [127]. (j) A 3D auxetic metamaterial with connected masses [121]. (k) A 3D metallic metamaterial [233]. (l) A 3D lightweight auxetic metamaterial [128]. (m) A 3D resonating metamaterial [129].
(b) (c) (f) (g) (h) (k) (e) (d) (j) (l) (m) (a)(i)
Contemporary avenues for elastic bandgap engineering
The early works on elastic BGs laid the foundation for understanding the underlying mechanisms of BG and how they are associated with structural design of PnC and metamaterials. This enabled efficient mechanical wave control and have opened avenues exploring new routes for wave physics and engineering elastic metamaterials that aimed at new real-world applications. In this section, we review contemporary efforts where elastic BG engineering is at the heart of the structure dynamic performance. This includes the development of advanced optomechanical systems, elastic topological insulators for robust waveguiding and confinement, acoustic energy harvesters, PnC for sensing, active PnC and metamaterials, and inverse designs for BG engineering.
Optomechanical crystals
For more than half a century, the field of cavity optomechanics has flourished thanks to the plethora of applications in fundamental and applied physics [147,148,151] that hinge on the interaction between electromagnetic radiation and nano or micromechanical vibration. Phononic BG engineering is indispensable in the context of optomechanical device design and has enabled elastodynamic performance by increasing the phonon lifetime and enhancing optomechanical interactions. After the introduction of the concept of BG for classical waves at the end of 80s, we have witnessed an exponential rise in interests toward both photonic and phononic crystals. A combination of the two crystals was born, coined phoxonic crystal (PxC) (the "x" stands for both "t" and "n") which is a dual photonic and phononic crystal that exhibits simultaneously BGs for optical and acoustic or elastic waves [130,131,[134][135][136]139]. Maldovan and Thomas [130] first introduced PxC crystals made of 2D lattices of square and triangular periodicities of either solid cylinders in air or air holes in solid medium. For specific filling fractions, they demonstrated the presence of simultaneous BG for electromagnetic and elastic waves at the same scale of wavelength. This property attracted more attention when Eichenfield et al. [132,133] conducted a series of experiments where they designed and fabricated a one-dimensional micro-optomechanical crystal, with a periodicity of 362 nm, endowed with a well-tailored cavity for strong co-localization of 200 THz photons and 2 GHz phonons. Their optomechanical crystal has constituted a powerful photonic and phononic platform for enhanced acousto-optical interaction providing an original sensitive optical measurement of mechanical vibrations [132,133,137,138,[140][141][142]144,146]. The crystal is endowed with simultaneous photonic and phononic BGs with a well-tailored structural defect to host optical and elastic cavity modes inside their associated BGs, which produce highly confined electromagnetic and mechanical energies inside the defect for enhanced optomechanical interaction. A typical example of the device is presented in Fig. 5(a) which is made of a 1D PxC nano-beam with a tapered cavity constructed by gradually varying the hole diameters to enable high quality factor optical mode trapping. 2D PxCs with a periodicities from 400 to 600 nm were also introduced where mechanical BGs can be localized from 4 to 10 GHz to achieve high performance phononic and photonic cavities for strong optical excitation and measurement of localized phononic modes (Figs. 5(b)) [137,138,140,146]. These nano-optomechanical systems were fabricated from silicon wafers using cleanroom nanofabrication techniques where the lattices patterns were realized via electron beam lithography. At the same time, Fuhrmann et al. [236] experimentally characterized the periodic modulation of the optical mode's wavelength by a coherent acoustic phonons formed by SAW. Also Gavartin et al. [237] experimentally observed the optomechanical interaction in a defect cavity designed in twodimensional suspended on-ship PxC (Fig. 5(c)). Following these works, theoretical investigations were conducted to study the physical mechanism of optomechanical interactions and engineer the phononic and photonic BGs along with the design of the cavity defect in order to produce highly confined elastic and optical modes inside the BG with enhanced optomechanical interaction [143,145,149,150,238,239]. Further, in high precision experimental optomechanics, a phononic shield is designed using a well-tailored PnC with wide BG to suppress mechanical losses created from coupling between the modes due to symmetry breaking caused by fabrication imperfections at the nano-scale [140,142,144,146,240] (Figs. 5(a)). In addition, Fang et al. [240] used a nano-PnC (lattice constant of 480 nm) to design waveguides that wire two local optomechanical micro-cavities which allowed for direct phonon exchange (at a frequency around 6 GHz) between the cavities without dissipation (Fig. 5(d)). This cavity-optomechanical circuits could be potentially used for performing controlled coherent signal processing. More recently, inspired by the increased importance of optomechanical resonators in the context of sensing [241] and detection of single biological species [242] and molecules [243,244], Navarro-Urrio et al. [245] proposed a PxC for the detection of silica sub-micrometer particles with the capability of determining the position of these particles by leveraging a family of mechanical modes known as pinch modes. Another application for optomechanical crystals is the realization of quantum entanglement of mechanical states (Fig. 5(e)) [246][247][248] which could open a new route towards the development of quantum networks based on silicon optomechanical crystals. However, these optomechanical phenomena were observed under cryogenic environment, and has yet to be realized in room temperature for more practical applications.
Topological elastodynamics
The discovery of topological phase transitions and topological phases of matter by Thouless, Haldane, and Kosterlitz (Nobel Prize in 2016) has become a catalyst for the development of a new class of quantum materials known as topological materials (e.g., topological insulators). These materials, interestingly, display symmetry-dependent topological phase transitions and non-trivial topological characteristics such as non-zero Chern numbers. Over the past few years, photonic and phononic crystals [4,[249][250][251] have become a fertile playground for the exploration of the frontier of topological physics owing to their capability of mirroring some of the quantum-mechanical properties of condensed matter systems. Meanwhile, topological photonic and phononic crystals have opened new gateways for designing devices that can route classical wave energy in a highly unusual and useful way. Topological BGs are the The transition of the topological phase is often resulted from a certain symmetry-breaking. For example, as predicted by the Haldane model of graphene [252], breaking the time-reversal symmetry (TRS) will open a topological BG, giving rise to robust one-way edge states that are immune to defects. This phenomenon is associated with the QHE. The topological phase in this model could be characterized by a non-zero Chern number, which is obtained by an integration along the Brillouin zone: = 1 2 ∯ ( ) , where ( ) is the Berry curvature. Such a topological insulator is known as the topological Chern insulator. Unlike quantum systems, TRS cannot be broken by introducing a local magnetic field in elastic wave systems, and alternative means must be sought. In 2015, Wang et al. [253] theoretically studied an elastic topological Chern insulator with Coriolis force. Soon after, a gyroscopic PnC was designed by Wang et al. [254], in which the gyroscopic inertial effect breaks the TRS (Fig. 6(a)). The experimental realization of the gyroscopic phononic topological insulator was reported by Nash et al. [162] in the same year, with the observation of topological edge states ( Fig. 6(b)). The structure was constructed by considering a honeycomb lattice of magnetically coupled spinning resonators where each gyroscope consists of small dc motor spinning a cylindrical mass and suspended to a plate. Beyond the reported topological edge states, Mitchell et al. [255] showed that the tunability of the gyroscopic phononic lattice could bring about complex topology. Besides the periodic lattice, gyroscopic phononic systems even exhibit non-trivial topology in amorphous configurations, producing topological BGs [256]. Since Bragg BG is mostly involved in topological elastic insulators, its frequency range strongly depends on the considered size of the phononic lattice. Early works on demonstrating mechanical topological states used lattices constructed to operate at frequencies ranging from few to hundreds of kilohertz. The objective was mainly to provide an experimental proof of topological manifestations in elastodynamics, which was facilitated by centimeter size structures that can be easily fabricated and characterized [163,164,[257][258][259][260][261][262][263]. Other designs were later proposed at the micro-scale for the purpose of topological elastic waveguiding at high frequency (from hundreds of megahertz to few gigahertz) for microsystem integration [165,[264][265][266][267].
In contrast to QHE, the QSHE opens a path to a class of topological insulators without breaking the TRS. This approach hinges on the intrinsic 1/2 spin of the electrons and TRS, which collectively create the Kramers doublet. Elastic wave systems, however, are bosonic that naturally lack the 1/2 spin. In order to overcome this barrier, elastic topological insulators were designed to give rise to pseudospins ±1/2 and artificial Kramers pairs. This class of topological insulators are characterized by a Z2 topological invariant, thus the name Z2 topological insulators [268]. The elastic Z2 topological insulators were first realized by Süsstrunk and Huber using a mechanical oscillator lattice with mode polarizations at very low frequencies (between 2 and 3 Hz) [257]. This mechanism of mode polarizations was soon introduced to multiple elastic platforms such as elastic plates [163,258,259], spring-mass systems [269], and granular media [270], with the observation of helical topological edge states that are immune to certain defects such as cavities and bending. For instance, Miniaci et al. [258] designed and fabricated a topological PnC plate made of a lattice of triangular and circular holes producing two Dirac cones at around 100kHz with a lattice constant of 2 cm. This Dirac degeneracy was lifted by breaking the symmetry of the lattice which resulted in the manifestation of spin orbital coupling. They also created a nontrivial interface between two topologically different lattices, which hosts helical edge waves (Fig. 6(c)). Another method to design elastic Z2 topological insulator is to leverage modal hybridization [260][261][262][263][264], which forms a double Dirac degeneracy in the band structure. This double Dirac degeneracy can be lifted so that pseudospin states become separated, opening up a topological BG (Fig. 6(d)). Zone-folding mechanism provides a simple way to achieve modal hybridization, as two single Dirac degeneracies are folded from the K point to the point in the Brillouin zone [271]. This zone-folding mechanism provides a robust route for constructing Z2 topological insulators in elastic plates [260,[262][263][264][265] and SAW systems [266]. Recently, a 3D elastic meta-crystal that obtained both QHE and QSHE topological phases was explored [272], which showed simultaneously topological surface states and hinge states.
In addition to QHE and QSHE, the QVHE has also been employed to construct elastic topological phononic crystals. The honeycomb lattice of the graphene possesses a Valley degree of freedom, which can be manipulated to break the spatial inversion symmetry (SIS) so that the Dirac degeneracy at K point can be lifted to separate two different Valley states. These Valley states are proven to yield opposite 1/2 pseudospins, leading to robust edge states that are immune to certain defects. Local resonance structures have been used to break the SIS in lattices formed by mechanical beams [164,165,[273][274][275][276][277][278][279][280][281] (see example in Fig. 6(e)), and soft materials [282,283], where the SIS is broken by the variation of resonators in the different valleys. The valley eigenstates have been observed with opposites 1/2 spins, which are characterized by Valley-Chern numbers. These valley states could be arranged along the interface between two different domains and consequently give rise to topologically non-trivial edge states. In addition to the honeycomb lattice, Kagome lattices have also been shown to host topological valley edge states [284,285] by altering the coupling spring between the masses, which gives rise to topological Stoneley waves. In addition, topological valley phases have been found in hierarchical lattices in a recent work by Han et al. [286]. Another feasible method to create elastic topological Valley lattices is by tuning the strain field in materials, which is especially effective in truss-like lattices [287,288]. Such a method has the advantage of high tunability for elastic topological valley insulators. In addition to bulk elastic waves, Valley Hall effect has been realized in surface acoustic wave (SAW) systems [267,289], where pillars were placed in a manner of breaking the SIS of a C6 symmetry lattice. The topological valley phases can be also embedded in systems with other topological phases. For example, Qian et al. discovered a new topological regime identified as the Valley-Chern effect by connecting the QVHE system to the QSHE system [290]. Such an elastic lattice was constructed by magnetic spinners with the ability to engineer the Berry curvatures near the valleys, which provided stronger topological protection compared to the original valley Hall designs. In another instance, Mei et al. successfully demonstrated a topological beam splitter by an elastic lattice with topological edge states protected by both QSHE and QVHE topological phases [291].
In classical wave systems, non-trivial topological phenomena are usually featured by a reduced dimensional response. For instance, 1D edge states can be engineered in 2D lattices. Non-trivial topological phases in 2D materials bring not only bulk-edge (2D-1D) correspondence but also bulk-corner (2D-0D) correspondence, which could be manifested by zero-dimensional topological corner states. Such a phenomenon often requires a higher-order topology [292]. One possible way to realize the higher-order topology is by introducing a gauge flux to each plaquette in a square lattice. In 2018, this concept was demonstrated by Serra-Garcia et al. [293] via an elegantly designed hopping mechanism in a perturbative mechanical metamaterial. An alternative way is to establish a breathing hexagonal lattice that hosts a second-order topological BG. Fan et al. [166] demonstrated a hexagonal high-order elastic topological insulator via beam-coupled masses, followed by another experimental demonstration in elastic plates by Chen et al. [294]. High order topology has also been demonstrated via a square lattice in elastic plates [295] ( Fig. 6(f)). It has also been shown that elastic Kagome lattices can host second-order topological corners states as proved by Wu et al. [296] and Wang et al. [297]. Additionally, pumping of topological elastic edge states was demonstrated in a plate by modulating its effective stiffness via the thickness where the edge mode in the BG was pumped from one edge to another [298][299][300].
One of the advantages of exploiting bands topology for the realization of guided elastic waves along edges or interfaces is the immunity to structural defects. Several works investigated the sensitivity of the topological elastic states to structural defects by either creating point, line or arbitrary defects via the removal of several unit cells, or even deforming the lattice [162,163,165,254,255,260,261,265,266]. This ability of defect immunity pushed further for exploring mechanical edge states in amorphous insulators that includes hyperuniform structures, quasicrystals and even random distribution of lattice units.
Besides, topological elastic lattices have contributed to the exploration of Weyl physics as well. While Weyl semimetals are also characterized by lattice topology, they are featured not by a topological BG but rather a doubly degenerate linear band crossing in 3D momentum spaces. As such, the development of this field is beyond the scope of this review paper. The readers can refer to the following papers for more details [301][302][303].
Finally, exploring and leveraging the topological features of lattices to seek exotic wave manipulation is continuously thriving where BG is at the core of many topological phenomena. In elastodynamics, among multiples routes to be explored are the introduction of topological defects such as disclinations and dislocations for wave confinement and transport, and lattices endowed with engineered long range hopping between unit cells. However, there are still design challenges that need to be addressed such as to introduce chiral symmetry or long range inter-cells interactions. [264]. (e) Left: SEM photo of the QVHE topological insulator on a silicon chip. Right: Experimentally measured intensity of the edge states propagation via a zigzag interface [165]. (f) Left: The unit cell of an elastic high order topological insulator in a rectangular lattice. Right: Experimentally measured displacement field of the topological corner mode in the high-order topological insulator [295].
Acoustic Energy harvesting
The motivation to meet the world's need for sustainable and renewable energy has also sparked the interest for acoustic energy harvesting (AEH). The common objective of AEH is to collect and convert the energy carried by sound and vibration to electrical energy, that otherwise would be wasted through conversion to heat. The classical approach for achieving AEH is to use a resonant cavity with a vibrating elastic element equipped with a piezoelectric material to convert the dynamic strain energy into electrical power. The earliest propositions for acoustic energy harvesters used Helmholtz resonators with a vibrating membrane attached to a piezoelectric ring [304], or a cantilever [305], or a series of piezoelectric plates in a straight acoustic tube [306,307]. With the aforementioned development of PnC and acoustic metamaterials for efficient wave manipulation, these concepts also came to light in the context of energy harvesting. PnCs were used to achieve high wave energy confinement through the design of a cavity inside the structure. An acoustic or elastic BG can be engineered to be located at the frequency of interest and a well-tailored structural defect in the geometry can give rise to a cavity mode inside the BG that has a highly confined elastic energy. This energy can be then collected via the traditional utilization of piezoelectrical elements. Wu et al. [152] were the first to explore PnCs for AEH. They used a phononic lattice of cylindrical rods with a lattice constant of 4.9 cm, and a unit-cell defect was created by removing a single rod which created an acoustic cavity mode at 4.02 kHz inside the BG. A flexible piezoelectric PVDF film was inserted inside the cavity to harvest the acoustic energy and a maximum power output of approximately 35 nW was collected for a sound pressure level of 100 dB (Fig. 7(a)) [153]. Later, Yang et al. [154] placed an electromechanical Helmholtz resonator at the center of a PnC cavity and used their coupling to enhance the harvested acoustic energy and generate electrical power reaching 429 μW for a sound pressure level of 110 dB (Fig. 7(b)). Nevertheless, these works used Bragg BG so the size of the PnC lattice is relatively large since it is governed by the operating wavelength (17 × 17cm 2 with rods length of 10 cm at 5.5kHz [154]). These proposed designs thus still remain cumbersome, and their wavelength-size dependency hinders their miniaturization for integrated AEH devices operating at the audible frequency regime. To overcome this limitation, metamaterials have now become more suitable candidates for AEH due to their deeply subwavelength features. Ma et al. [308] proposed a membrane-type metamaterial made of a vibrating membrane with a back cavity, where the acoustic energy around 150 Hz can be absorbed using a resonance state at deep-subwavelength range (device thickness is only of 17mm while the wavelength is around 2.37m in air). The mechanical to electrical conversion is realized by magnet wires carried by the membrane and four pairs of neodymium magnets along the magnet wires to reach an acoustic-electric energy conversion efficiency of 23% (Fig. 7(c)). For harvesting the elastic energy from the vibration of a plate, Carrara et al. [309] presented a mechanical energy harvester based on elastic metamaterials made of well distributed resonating pillars. They proposed three structural designs with different energy harvesting strategies: pillars with parabolic distribution for wave focusing to realized AEH at the focal point, an acoustic cavity created from a structural defect introduced inside a square lattice of pillars, and a waveguide constructed by removing a line of pillars in the lattice for broadband energy harvesting (Fig. 7(d)). The periodicity of the lattice of pillars is 1cm for an operating frequency of 35kHz. Later, Li et al. [310] proposed a metamaterial with the dual functionality of sound absorption with over 20 dB of the sound transmission loss and acoustic energy harvesting with a maximum energy conversion efficiency of 15.3% (Fig. 7(e)). At the same time, Qi et al. [155] proposed a planar AM made of periodic distribution of low frequency pillar resonators over an aluminum plate (Fig. 7(f)) and theoretically demonstrated the possibility of AEH at the subwavelength scale. They designed a subwavelength cavity with a defect mode located at 2.26 kHz using a lattice constant of only 1 cm. Considering the same system, Zhang et al. [157] and Ma et al. [311] presented an experimental demonstration where they reached a maximum harvested power of several microwatts from 100 dB of sound ( Fig. 7(g)). In these pillared plates, AEH was carried out through a structural defect by removing several pillars to create a cavity mode inside the BG. The existence of the cavity mode is conditioned by the spacing as a result of removing the pillars, and this spacing has to be comparable to half of the wavelength of the flexural wave at the desire frequency. Consequently, a lower frequency of interest implies that more pillars need to be removed in order to create the cavity mode inside the BG which increases the size of the planar metamaterial. To overcome this constraint, Oudich and Li [156] proposed an alternative solution of changing the mechanical properties of the resonators rather than removing them and theoretically showed energy harvesting at deep-subwavelength scale (Fig. 7(h)). A defect mode was created in a centimeter size cavity inside the BG at 600 Hz. Though not based of BG engineering, some other periodic structure designs used coiled acoustic channels [312,313], coupled acoustic resonating membranes [314], or coupled defects inside an elastic PnC [315] for the purpose of AEH or elastic energy harvesting. Recently, Javadi et al. [316] constructed a PnC consisting of five steel slabs standing in air, with a middle scatterer made of triboelectric nanogenerator, and demonstrated enhanced AEH from the wave confinement caused by the cavity inside the BG. Further, with the advent of PnC that mimic topological insulators, it became possible to create robust elastodynamic edge and interface states inside the BG that are immune to defects, which can be leveraged for acoustic and vibration energy harvesting. Fan et al. [317] designed a topological interface state inside the BG created from two sonic crystals having different bands topologies, and used a piezoelectric cantilever for acoustic to electromechanical energy conversion. Very recently, Wen et al. [318] designed a phononic plate that harbors a zero-dimensional cavity with a Kekulé distorted topological elastic vortices that were used to realize robust mechanical energy harvesting with a collected electrical power of almost 5 mW (Fig. 7(i)). For additional details on this topic, the reader is referred to these previously published review papers [158][159][160].
Finally, although harvesting ambient mechanical waves is a fascinating idea, these proposed mechanical metamaterials still do not provide sufficient energy for powering large devices. In addition, the range of power shown may be sufficient for low-power devices such as micro-sensors but great efforts are to be made in the structural design of metamaterials to solve the challenge of miniaturization and on-chip integration.
Fig. 7. PnC and elastic metamaterials for acoustic energy harvesting. (a) A 2D
PnC with a cavity where PBDF curved piezoelectric membranes were used to collect the confined acoustic energy [153]. (b) A 2D PnC with a defect made by removing a rod, combined with a HR to enhance the AEH [154]. (c) A membrane-type metamaterial where the mechanical to electrical conversion is realized by magnet wires carried by the membrane and four pairs of neodymium magnets [308]. (d) A pillared PnC with point and line defects for vibration energy harvesting [309]. (e) A hybrid resonant metamaterial for AEH and absorption [310]. (f) A pillared planar metamaterial with a defect created by removing a finite number of pillars, and the band structure showing a defect state inside the BG [155]. (g) Experimental realization of the pillared planar metamaterial [157,311].
(h) A planar elastic metamaterial for AEH with defect resonators [156]. (i) Robust topological cavity in a plate metamaterial decorated with low frequency pillar resonators [318].
Phononic crystals for sensing
Acoustic sensing is also of great interest to the materials science community and could benefit various applications such as medical diagnosis, biosensing, food processing, and underwater detection. Several studies have explored the possibility of using phononic BG for sensing. The sensing mechanism is based on the utilization of localized cavity modes trapped inside a structural defect at frequencies inside the BG, which can be highly sensitive to acoustic and elastic perturbations. Early studies on sensor designs based on PnC utilized lattice constants and cavity sizes in the order of few centimeters for sensing the properties of liquids at operating frequencies raging from hundreds of kHz to few MHz. Lucklum and Li [319] presented a sensing proof of concept using a 1D PnC made of solid plates and liquid filled cavities, and a 2D PnC made of water holes inside an elastic matrix made of aluminum or tungsten. The PnC hosts a liquid filled cavity to analyze the variation of the concentration of 2-propanol in water, which is directly related to the change of the wave velocity in the fluid mixture. The change in the concentration is detected through the variation of the resonance frequency of the defect modes as a function of the molar ratio of propanol on water. The same PnC structure was utilized for the determination of the octane number of gasoline [320,321] (Fig. 8(a)). Salman et al. [322,323] conducted a numerical study on the determination of ethanol concentration via a linear waveguide made of a line of holes filled with liquid inside a PnC ( Fig. 8(b)). Amoudache et al. [324] used similar design for the detection of the molar ratio using simultaneous photonic and phononic cavity modes inside their associated BGs. Recently, PnC were designed at the microscale and integrated into SAW microsensors to enhance their sensitivity. Bonhomme et al. [325] designed a lattice of phononic micro-pillars (6 μm period) with BG tailored to endow the pillars with highly confined elastic modes, and investigated theoretically their potential for the detection of nano-particles in Love wave-bases platform at 250 MHz (Fig. 8(c)). They also fabricated a Love-wave based sensor hosting a square lattice of SU-8 micro-pillars (with 9 μm period) where their resonance at 34.47 MHz creates a very narrow LR BG that are highly sensitive to perturbations [326] (Fig. 8(d)). Their device was utilized for sensing temperature, mass load induced by micro-droplets, sugar concentration, and the detection of microbeads. Besides, Sadeghi et al. [327] designed and fabricated a clamped PnC membranes at the microscale made of silicon nitride with a structural cavity for thermal sensing. Using laser-based heating, they also observed a frequency shift of both the defect mode and the BG around 600 kHz using a lattice constant of 30 μm. They even observed the frequency tuning of the defect mode at a point where it leaves the BG. Pennec et al. [328] used a PnC made of hollow pillars where the interior can be filled with liquid to detect its properties using whispering gallery modes inside the BG (Fig. 8(e)). Lately, PnCs made of a lattice of holes in a thick plate were used as a sensing platform by utilizing a honeycomb lattice with a liquid-filled single hole point defect, and a square lattice of liquid-filled holes with narrow BGs [329,330]. The concentration of the ethanol was characterized via either the shift of the defect mode inside the BG using the first phononic lattice, or the narrow BG shift in the second lattice (Fig. 8(f)).
The proposed sensing devices based on PnC with integrated cavity are at an early stage and still far from competing with optical sensing devices. However, it was recently shown that the integration of metamaterials such as an array of resonant micro-pillars at the micro-scale in a SAW device could enhance the sensitivity to temperature detection [326] beyond the limit allowed by the classical SAW devices. This could further open routes towards new generation on-chip SAW devices with integrated elastic metamaterials to push the boundary of acoustic sensing down to detecting ultralow-molecularweight [325], which has only been achieved by plasmonic metamaterials [331,332] and optomechanical devices [243].
Active materials for elastodynamics
Though the existence of BGs in PnCs and elastic metamaterials are now associated with a wide range of applications, functional limitations persist as the BG width and frequency localization are constrained by the geometrical and intrinsic properties that are permanently fixed once the structure is designed and fabricated for a specific application. Thus, the scientific community in this field is constantly exploring paths to extend the frequency spectrum of the BGs by incorporating active elements on PnCs and metamaterials. The field of active functional materials has thus been rapidly developed to realize exciting new functionalities like selective wave filtering and non-reciprocity by either statically or dynamically manipulating the intrinsic properties of the metamaterial both in space and time.
Temporally modulated PnC
There has been a rising interest in applying temporally modulated materials to break the time reversal symmetry in order to enable non-reciprocal wave behavior and subsequent active metamaterial-based applications. These time-modulated materials are realized by introducing a controlled time variation of their effective mechanical properties such as the stiffness which can be designed to break the wave reciprocity. Reciprocity is a fundamental property of classical waves in any linear time-invariant media, which stipulates that the measured frequency response of any point remains the same when the source and receiver are exchanged in the considered medium. Non-reciprocity is highly beneficial in the context of elastic wave propagation since it enables unconventional wave functionalities such as unidirectional wave propagation and breaking the time-reversal symmetry for the realization of Chern insulators in classical waves. The idea
(a) (b) (c) (d) (e) (f) A B
of temporally modulated periodic media was inspired from the early works of Oliner and Hessel [333] in 1959 and Cassedy and Oliner in 1963 [334]. They introduced a rigorous theoretical method to characterize the dispersion of electromagnetic waves in a medium where the electrical permittivity or the refractive index undergoes a spatiotemporal modulation (STM) in the form of a propagating wave. This type of STM process breaks the time reversal symmetry in the lattice and consequently enables a unidirectional BG. However, the technical capability of building an experimental demonstrator at that time hindered the interest for this class of active artificial materials. Later, in 1998, Winn et al. [335] theoretically demonstrated that the spatiotemporal variation of the dielectric constant enables optical band transitions analog to electronic ones in metals and semiconductors. It was not until 2008 that Dong et al. [336] first experimentally performed direct photonic transitions using ultrafast tuning of the refractive index where the time interval of this tuning was on the order of the inverse of the frequency difference between the optical modes. Shortly after, Yu and Fan [337] introduced an on-chip optical signal isolation that was achieved by the STM of the refractive index. These studies revived the interest towards time-varying mediums with the emergence of multiple studies on creating new photonic platforms with STM materials. Time-varying photonic meta-surfaces then emerged to show exotic optical functionalities such as unidirectional electromagnetic induced transparency [338], strong and broadband nonreciprocal wave transmission [339][340][341], frequency mixing [342], compact flat prism [343], and controlled linear frequency conversion [344]. Drawing inspiration from this progress in photonics, a flurry of spatiotemporal phononic designs were proposed in acoustics and elastodynamics. Fleury et al. [345] proposed an acoustic nonreciprocal isolator made of three ports system with a central ring cavity endowed with an internal circulating fluid flow to introduce an acoustic bias. Shortly after, they presented a compact acoustic circulator based on space-time modulation of the effective acoustic refractive index [346]. However, these pioneer acoustic systems do not involve the creation of BG based on phononic lattices. An acoustic analog of STM medium of Cassedy and Oliner [334] was proposed in elastodynamics by Trainiti and Ruzzene [347] who theoretically investigated the dispersion of elastic waves propagating in a beam with STM materials properties, for both longitudinal and bending waves. By spatiotemporally modulating the Young's modulus of the beam in a waveform shape, they demonstrated a new class of unidirectional BGs. In fact, starting from a simple PnC with a specific BG designed from a spatial modulation of the Young modulus, the introduced time variation shifts the BG to higher frequencies for wave vector directed along the propagating modulation while the same BG shifts to lower frequencies for wave vector in the opposite direction. This study sparked a host of theoretical investigations to explore the physics of non-reciprocal BGs in acoustic time-varying structures [348][349][350][351], while other works took the challenge of experimentally engineering mechanical materials with time and space modulated effective properties. These active materials have unit cell sizes in the order of few centimeters leading to operating frequencies ranging from 10 Hz to 20 kHz, since the objective was to mainly demonstrate non-reciprocal wave propagation. Wang et al. [352] were the first to experimentally achieve such STM using a 1D periodic coupled permanent magnets (period of 33.4 mm) and coils where the coupling can be varied in time via alternating current (AC) in the coils which dynamically modulates the magnetic force between these coils and the magnets ( Fig. 9(a)). The system here, could be modeled by a series of coupled masses connected to the ground by springs with time modulated effective stiffness. This magnet-coil based approach was proven to be effective to dynamically modulate the effective stiffness with high speed which led to non-reciprocal dispersion of longitudinal elastic waves observed around 20Hz. Using the same technique, the same group proposed an elastic metamaterial plate made of periodic modulated cantilever resonators distributed on a beam. Each resonator is built from a permanent magnet and a coil mounted on a flexible cantilever unit fixed on the beam, which lead to a demonstration of non-reciprocal propagation of flexural waves ( Fig. 9(b)) [353]. Another interesting modulating approach for the effective stiffness was proposed by Ruzzene et al., who showed unidirectional BG for flexural waves using piezoelectric elements periodically distributed on a beam and connected to controllable electrical circuit ( Fig. 9(c)) [354][355][356]. The experimental realization showed unidirectional wave propagation at 10.5 kHz with a periodicity of 24 mm [356]. The same approach was also investigated in depth by Yi et al. [357]. Meanwhile, an interesting yet different STM approach was introduced by Wallen and Haberman [358,359] who used nonlinear large mechanical deformations to spatiotemporally vary the effective stiffness in a periodic structure to enable non-reciprocity wave dispersion. Also, Wang et al. [360] considered a prestressed periodic structure made of prismatic tensegrity cells, and they modulated the pre-stress both in space and time to break the time reversal symmetry and achieve unidirectional BG. Attarzadeh et al. [361] used a series of local resonators in which their effective stiffness was modulated by varying the second area moment of inertia of each resonator's arm through dynamically changing its angular orientation ( Fig. 9(d)). Besides, non-reciprocity is also realized in the case of mechanical analog of Floquet topological insulator (FTI). Darabi et al. [362] built a FTI made of a hexagonal array of piezoelectric disks distributed over a plate surface and shunted through electrical circuits controlled to modulate their effective capacitance both in space and time in a way to break the time reversal symmetry and induce topological protected edges states with unidirectional propagation (Fig. 9(e)).
Non-reciprocal wave propagation was also demonstrated for SAW without relying on the presence of BG from the STM, but rather on altering the propagation characteristic by applying an electrical field [363], or an external magnetic field [364][365][366], or by leveraging the magnetoelastic coupling [367].
Stimuli-responsive metamaterials
Owing to the rich physics and applications associated with elastic wave BGs, several studies began exploring the possibility of tuning their widths and frequency of occurrence in real-time. Preliminary efforts towards this direction comprised the theoretical demonstration of BG control through varying the unit cell's shape [368][369][370] or the elastic properties [371][372][373][374][375][376]. However, enabling such geometrical or material property variation requires the direct or indirect influence of an external force which gave birth to the subfield of stimuli responsive phononic crystals and metamaterials [377][378][379]].
An early study by Goffaux and Vigneron [380] showed that the Bragg BG for a lattice consisting of solid square-section columns, could be tuned by simply rotating all the columns about their vertical axis. This inspired several subsequent works that theoretically demonstrated the tailoring of both Bragg and resonance-based BGs through simple geometrical modifications like varying the shape and symmetries of scatterers [381,382] or incorporating inclusions with elastic anisotropy [383,384], or adding an additional inclusion [385]. In this line of work, an interesting study by Bertoldi and Boyce [386] demonstrated that periodic elastomeric lattices could serve as a host for tailorable phononic BGs due to their ability to exhibit mechanically triggered pattern transformation. This finding sparked interests for instability/nonlinearityinduced and deformation-dependent BG [387][388][389][390] that were later also experimentally realized by Wang et al. [369], Shan et al. [391] and Babaee et al. [392] (for airborne sound). The first experimental study in this regard [369] proposed an adaptive elastic metamaterial whose unit cells comprise resonators that consist of metallic cores connected to easy-to-buckle elastomeric beams ( Fig. 10(a)). The metamaterial was fabricated using silicon rubber and a mold-casting process where the mold was built via 3D rapid prototyping. The BG was opened at around 100 Hz using a lattice constant of 50 mm. Such a system allows for the unit cell to deform systematically and change shape significantly, upon compression. This in turn tailors the BG and enables deformation dependent BGs in the band structure and transmission measurements. However, these works rely on large contact mechanical forces applied to the sample and hence had limited applications in their current form.
In parallel, other studies probed the alternative strategy of tuning the BGs by varying the intrinsic properties of the material constituting the phononic lattices. Although theoretically similar, this alternate approach requires the presence of an external field-like stimuli such as magnetic field [371,372] that does not deform the lattice and hence could be more promising for real-world scenarios and for other research that is also discussed in the previous sub-section of this manuscript. Early studies in this context, theoretically showed that BGs could be tuned by modifying the lattice to incorporate materials whose stiffnesses could be actively tuned via optical fields [393], electrorheological materials [394], electrical [395], piezoelectrical circuits [373,[396][397][398][399], external magnetic field [371], and temperature variation [400]. The first experimental validation of such an active metamaterial was put forward in the study by Bergamini et al. [401] (Fig. 10(b)) who considered a structure comprised of successive cylindrical stubs (1 cm period) with piezoelectric discs that were shunted through an inductive circuit in order to obtain frequency dependent stiffness elements between the substrate and the stubs. This in turn allows for tailoring the effective periodicity of the sample that could allow for the generation of a passband within the otherwise wide-band gap. Likewise, a study by Wang et al. [402] showed that similar behavior could be achieved by employing electromagnets in a 2D lattice that would attach or detach in the presence or absence an external magnetic field (Fig. 10(c)). This is highly beneficial since it implies that each unit cell can be controlled independently, and the BG can be readily turned on or off. The metamaterial is relatively large with a periodicity of 17 cm for an operating BG switching functionality from 5.5 to 12 Hz. Capitalizing on this convenient mechanism, the authors of this work demonstrated tunable digital metamaterials for low frequency elastic waveguiding and isolation.
Intrigued by these simple yet insightful phenomena demonstrated for active 2D periodic designs, researchers began extending these concepts to more complicated systems to enable new applications. For instance, Cha and Dario [395] proposed a novel nanoelectromechanical flexural phononic crystal ( Fig. 10(d)), that consists of free-standing nanomembranes with circular clamped boundaries. With a lattice constant of 7 μm, the system exhibits a BG that is shown in to be lowered in frequency from 18 to 14 MHz by means of applying a dc gate voltage that creates voltage dependent onsite potential. Further this study also showed that when a dynamic modulation of the voltage is employed, it triggers non-linear effects which induce the formation of a new BG that is analogous to the Peierls transition in condensed matter [403]. Subsequently, active metamaterials were also scaled upward with the advent of more advanced additive manufacturing techniques. To this end, Pierce et al. [404], employed a direct ink write fabrication method to produce a metamaterial whose struts are made up of a magnetoactive elastomer. This allows the sample's intrinsic elastic properties to be controlled via an external magnetic field (Fig. 10(e)), thereby enabling a BG shift that is dependent on the strength of the applied magnetic field. Similarly, Gliozzi et al. [405] fabricated a polymeric material via a UV polymerization method that employed a Methyl red as an azodopant, which served both as an optical absorber as well and active light responsive component of an elastic metamaterial. Exploiting this unique characteristic of the intrinsic material, the study here illustrated that the eigenmodes of the resonant pillars are affected by local illumination such that the overall BG can be significantly widened as function of time ( Fig. 10(f)). Using parallelepipedic pillars with 5 mm between two successive pillars, the BG located at 80 kHz was enlarged from 16 kHz bandwidth without illumination to 38 kHz with pillar illumination. Another realm of studies involved those that combined the aforementioned mechanisms by employing an external field upon the intrinsic material in order to induce a change in geometry of the metamaterial unit cell. The early studies in this regard were those of Foehr et al. [406], that leveraged the concept of spiral phononic plates which possess wide BGs that could be tailored based on their geometrical state. This concept was coupled with an external magnetic field in order to realize programmable phononic metasurfaces that have on and off states through flat and programmed scenarios respectively ( Fig. 10(g)). The presented design [407] consisted of Archimedean spirals with magnets in the center that enable a field-responsive geometry which results in the field-dependent BG ranging from 90 to 140 kHz for a lattice constant of 1.25 cm. Interestingly, this bi-stable metamaterial was also later extended to demonstrate a transistor like device capable of performing logic gate calculations and cascading elastic vibrations [408]. Such a design is highly desirable for the future of advanced computational systems. Likewise, similar concepts also emerged for higher order elastic metamaterials. Montgomery et al. [409] presented a magneto-tunable metamaterial that exhibits a considerable change in metamaterial shape through deformation mode branching. This change illustrates a variety of different BGs for various combinations of magnetic field and deformation. However, this study only characterized the static performance of these geometries and not the elastic wave transmission. A similar study conducted by Xu et al. [410], however, fabricated and experimentally characterized the elastic wave propagation through a minuscule 3D metamaterial whose intrinsic material was a magneto-elastomer ( Fig. 10(h)). The geometry under consideration was a well-known negative stiffness lattice that has configurable shapes for different magnetic fields, resulting in different transmission ratios. Additionally, it was demonstrated that for each of the shapes, the system had a different "ON" state which corresponds to the only frequency that is allowed to propagate within the lattice. [401]. (c) A 2D active metamaterial where each unit cell is made of two switching electromagnets between the attaching (1 bit) and detaching (0 bit) states to control the BG and the waveguiding functionality [402]. (d) A 1D nano-electromechanical phononic lattice made of periodic free-standing nanomembranes with clamped boundaries. The phononic BG can be shifted by the application of a static voltage [395]. (e) A 1D mechanical metamaterial made of magnetoactive elastomer that reacts to external magnetic field to continuously tune the elastic BG [404]. (f) A 1D metamaterial made of photosensitive parallelepiped pillars where illumination affects the BG [405]. (g) A Programmable PnC made of units of Archimedean spirals with magnets to realize phononic transistor-like device through switching and amplification of elastic vibrations [407,408]. (h) A shape reconfigurable elastic metamaterial through the application of magnetic field to control the attenuation of elastic waves [410].
BG engineering through inverse design
Owing to the fast development of computational capabilities that have emerged over the past decade, several recent studies on PnCs and elastic metamaterials have focused on harnessing new tools for their design and optimization. This has led to inverse-design based studies that start with the desired BG configuration or functionality and employ optimization approaches to arrive at the geometry and material that is required to achieve them. Most of these efforts largely rely on topology optimization [56,57,, genetic algorithms [215,432,437,438], or machine learning based approaches [439][440][441][442][443][444][445][446] and have unveiled unusual and hence previously inconceivable geometries that enable materials with enhanced BG characteristics. While these efforts are now burgeoning thanks to modern computational power, one of the earliest fruitful strides in this direction for elastic waves, can be attributed to the works of Sigmund and Søndergaard Jensen [411] in the early 2000s, who first put forward a theoretical framework showing that phononic BGs could be considerably enlarged by opening the design space of the unit cell geometry, while enforcing the boundary conditions as constraintsin other words, via topology optimization. This has led to a wide variety of subsequent works that later employed topology optimization for both Bragg [418] and local resonance-based BGs [416], and the merging of these efforts for exotic features like directional wave propagation [427,428], negative refraction [420], mode conversion [419] ultra-wide BGs through dielectric elastomers [414] and enhanced inertial amplification [415]. More recent works in this context have also extended the scope of topology optimization to 3D metamaterials [412,447] and by incorporating geometrical uncertainties [424,426,448] in their calculations in order to better capture the real-world scenarios. The calculations in all the above-mentioned studies can be broadly described as coupling of conventional wave analysis methods like multiple scattering theory, finite element, or plane wave expansion with gradient-based or non-gradient-based optimization techniques in order to achieve single or multiple desired objectives [449]. The entire process is thus two-stage -the first is the band structure analysis to
(a) (b) (c) (d) (e) (f) (g) (h)
extract the dispersion behavior of the periodic crystal, and the second is the mathematical solution of the optimization model that offers a new set of variables a step closer to the desired objective. The first and second stages are thus carried out repeatedly until the system reaches the required criterion. The difference between the two optimization techniques is that gradient descent-based methods [425,450] hinge on a smooth functions which work through continuous variables, while non-gradient-based methods [416,451,452] are often non-smooth and hence runs through somewhat random variables. Nongradients would thus be more precise and flawless at the cost of higher computational time since gradientbased methods could get stuck at an undesirable point in the local parameter space. Two popular nongradient optimization techniques are genetic algorithms and simulated annealing, and Figs. 11(a), (b) and (c), show representative examples of works that employed these for BG optimization in elastic waves. Figure 11(a) shows the work of Jung et al. [416], that designed a novel plate-type metamaterial with wide BGs for elastic waves via simulated annealing. The authors here demonstrated the benefit of using simulated annealing over gradient-based topology optimization by illustrating that it prevents the occurrence of realistic design issues and enables BGs at desirable frequency ranges via the unusual unit cell geometries shown here. Similar to this work, Bilal and Hussein [215] also showed that genetic algorithms could serve as a route to realize ultrawide BGs in 2D metamaterials, via the combination of in-plane and out-of-plane waves. This was made possible due to the unique geometrical configurations of the resultant unit cells, as can be seen in Fig. 11(b). Likewise, Fig. 11(c) shows the results from a work by Dong et al. [437] that carried out a multi-objective optimization of two-dimensional phononic crystals through a genetic algorithm. The two objectives in this work were to realize ultra-wide bandgaps and reduce the mass of the crystal.
Another optimization approach that has been equally popular for the inverse-design of BGs are those that are based on machine learning techniques (ML). These methods focus on acquiring vast collection of training data and utilizing artificial neural networks to classify them and arrive at the required parameters for the desired goal [453]. This route has become increasingly useful due to the high computational power that is now available and its ability to calculate band structures for a large number of geometries. While a majority of such inverse-design based studies are for electromagnetic [442,[454][455][456] and acoustic waves [457][458][459][460][461][462], some recent works have begun employing ML for mechanical metamaterials and elastic wave BG characterization as well. In the context of mechanical metamaterials with exceptional static properties, ML based inverse design approaches hold great value, since they allow designers to take geometric and material property variation and uncertainty into account in their training data set [463][464][465][466]. Inspired by such efforts, several recent studies have emerged centered around employing ML for the inverse-design of elastic metamaterials. It was recently shown in a work by Liu et al. [467] that ML could help design PnC with wide BGs, as shown in Fig. 11(d). This was made possible here by taking advantage of neural networks for both the eigenvalue problem (i.e., to calculate the dispersion behavior, as shown in Refs [443,468]) as well as the optimization step. Likewise, a recent study by Wu et al. [469] put forward the design of a modular metamaterial shown in Fig. 11(e), which was made possible by combining neural networks for the band structure calculation and genetic algorithms for the optimization, as illustrated. This allows them to realize on-demand wide and tunable BGs. Much more recently machine learning based designs have also found their way for the realization of topological edge states [470] through a recent effort that illustrated that these could be used to judiciously tailor the bandgap for flexural waves in plates.
For additional details on each of the optimization techniques discussed here, the reader is referred to the previously published review papers [413,441,449]. [416]. (Left to right, top to bottom panels) Iterations to minimize the objective function; multiple design solutions for BG maximization; Design solutions for five different target frequencies; Kinetic energy differences ΔKE and the BG for 5 designs.
(b) 2D PnCs designed using genetic algorithm for BG maximization [215]. (c) 2D PnCs designed with multi-objective optimization with simultaneously maximal BG width and minimal mass [437]. (d) 1D PnCs optimized for BG enlargement using neural network [467]. (e) 2D PnCs with meta-atom optimized for BG enlargement using combining genetic algorithm and machine learning [469].
Conclusion and perspectives
In conclusion, we have presented an overview of the history, including the latest development of PnCs and elastic metamaterials, focusing on their characteristics derived from frequency bandgaps. This review paper starts from the basic principles of the two types of bandgaps: Bragg and local resonance bandgaps. It then attempts to provide an extensive survey of existing 2D and 3D phononic crystal or metamaterial designs that possess bandgaps. Finally, we discuss the state-of-the-art of bandgap engineering, which encompasses GHz phononic crystals that can interact with photons, bandgaps that are topologically nontrivial, energy harvesting from vibration, sensing, active materials that can be modulated or reconfigured by external fields, and bandgap engineering via inverse algorithms.
Going forward, band gap engineering will likely remain a central topic in phononic crystals and elastic metamaterials, driving the advancement of structured materials for controlling elastic waves. In particular, it is expected that elastic bandgap engineering will continue to benefit from the exciting development of topological physics, which has already led to the realization of elastodynamic Z2 topological insulators, higher-order topological insulators, and Valley Hall edge states, among others. For example, topological defects such as disclination and dislocation can host bound states within the bandgap, that trap energy in the bulk rather than at the corner of the material. A recent paper demonstrated that disclinations in an acoustic lattice can give rise to degenerate zero-energy bound states [471]. It would be interesting to also demonstrate such symmetry-protected bound states in elastic wave systems for sensing, energy harvesting, or photon-phonon interaction. However, an elastodynamic structure that yields chiral symmetry must be first conceived.
Further, elastic BG engineering plays a vital role in the design of optomechanical crystals for mechanical modes trapping and waveguiding at the GHz regime. Thanks to the progress made in the field of nanofabrication, the field of optomechanics has seen a significant development in the last decade with significant discoveries and experimental achievements that have closed the gap between the quantummechanical and classical-wave worlds towards building advanced systems that elegantly unite photons and phonons [151]. This includes the realization of non-classical correlations between single photons and phonons [472], optomechanical entanglement [246], quantum transduction [473], optomechanical quantum teleportation [474], reconfigurable quantum phononic circuits [475], among others, which could facilitate the development of future quantum based devices for quantum communication, quantum memories, and quantum transducers. However, most of these recent quantum optomechanical demonstrations were performed in cryogenic environments in order to reduce thermal noise. A challenge to overcome in upcoming years will be to demonstrate the aforementioned quantum phenomena in roomtemperature for more practical applications [476].
On the other hand, architected metamaterials embody the promise for next-generation 3D bandgap engineering. While the vast design space of architected metamaterials could enable uncounted designs with intriguing dynamic properties, this advantage can also become a hurdle for traditional forward design approaches. This is further confounded by the fact that these materials' dynamic footprint cannot be adequately described by singular peak values such as transmission rate, but are in fact captured by their entire dynamic behaviors spanning a wide range of frequencies. These challenges, however, create a great opportunity for data-driven approaches to overcome difficulties that make forward design approaches extremely ineffective. 3D architected metamaterials can further benefit from progresses made in the additive manufacturing domain, such as to leverage multi-material printing [235] as well as piezoelectric material printing [477] to further expand the design space of these rationally designed structures.
Furthermore, non-periodic lattices such as quasicrystals and hyperuniform structures can also serve as interesting platforms for controlling elastic waves. Although the lack of periodicity hinders the possibility of precise dispersion characterization via the band structure, these aperiodic lattices can still display frequency ranges where the wave cannot propagate. For instance, using frequency response analysis, it was demonstrated that quasicrystals can display photonic and elastic bandgaps and host intrinsic optical and mechanical wave localizations without introducing any structural defect [478][479][480]. Also, the field of hyperuniform structure has long attracted significant attention in photonics for creating bandgaps, waveguides and energy localization [481][482][483]. However, few works have introduced these aperiodic lattices [256,478,479] in elastodynamics, and exploration of their dynamical performance with real experimental demonstration is lacking for potential applications such as optomechanics.
Recent development on bilayer phononic crystals has also pointed out a new direction to engineer the bandgap using interlayer coupling and potentially even the twist degree of freedom. A recent study displayed a twisted bilayer design for elastodynamics made of two lattices of pillars distributed on both sides of a thick plate. Surface acoustic waves can propagate on each side of the plate with coupled wave dispersions between the two phononic lattices [484]. It was demonstrated that at a specific twist angle that makes the bilayer phononic crystal have an even sublattice exchange symmetry (even SE), an elastic bandgap can open. This particular bandgap was shown to host high order topological state (corner modes) in a photonic bilayer system [485]. Furthermore, the interlayer coupling can be tailored to create an elastic bandgap that can host Valley Hall topological interface states at a frequency range unattainable by the monolayer phononic crystal. These findings suggested that bilayer phononic crystals can have extended dispersion capability via the twist and inter-layer coupling, which could open new routes towards a new class of phononic heterostructures for exotic wave phenomena.
Meanwhile, nonlinear waves including solitons could constitute another promising area of research that can benefit from elastic bandgap engineering. Particularly, granular crystals have been extensively explored as they host highly nonlinear waves, which have enabled the design of impact absorbers [486], lenses [487], switches [488], and nondestructive detection [489]. Until now, elastic metamaterials and PnCs have been mostly designed to control linear elastic waves. The ability to extend their functionality for manipulating nonlinear elastic waves is of high interest. Recently, a deformable metamaterial was built to support the propagation of vector solitons with two polarizations which was used to design an architected lattice with bandgaps for solitons [490,491]. These studies open the avenue towards a future generation of architected metamaterials with engineered nonlinearities to achieve amplitude-dependent control of vibration.
Another direction of exploration in elastodynamics is lattices designed in non-Euclidean space. For instance, hyperbolic lattices have been introduced in circuit quantum electrodynamics with the manifestation of exotic phenomena such as flat bands and localized eigenstates [492]. This has sparked the exploration of hyperbolic band theory [493] and crystallography in non-Euclidean geometries, which spanned the lattice space beyond the classical Bravais lattices [494]. Recently, theoretical explorations were conducted for wave dispersion in hyperbolic photonic lattice [495] as well as in elastodynamics [496]. This field presents several exciting avenues of research for exploring wave physics in curved space that is accessible via origami designs for instance, which could expand the design space toward curved elastic metamaterials.
Overall, as the field of bandgap engineering for elastic waves grows and evolves to be more multidisciplinary, we fully expect that this subject will continue to strive for decades to come.
Fig. 1 .
1Timeline for some milestone elastodynamic platforms endowed with elastic BG. These works include the emergence of phononic crystals and elastic metamaterials, optomechanical crystals, nonlinear and active metamaterial, topological elastic lattices, 3D architected lightweight metamaterials, non-reciprocal elastic metamaterial, and topological pumping. PnC: Phononic Crystal; LR: Local resonance; BG: bandgap; QSH: Quantum Spin Hall; TI: topological insulator; VH: Valley Hall.
Fig. 2 .
2Flow chart presenting the sections of this paper.
Fig. 3 .
3Phononic crystals and locally resonant metamaterials. (a) Examples of PnCs. (Top from left to right) 2D lattice of rods in air
Fig. 4 .
42D and 3D elastic architected metamaterials. (a) A phononic lattice made of connected masses for wide elastic BG[216].
Fig. 5 .
5Optomechanical crystals. (a) A phoxonic nanobeam with tapered optomechanical cavity[142]. (b) A 2D PxC with tapered waveguide based on snowflake PnCs[146]. (c) Optomechanical defect cavity[237]. (d) Routing two connected optomechanical cavities for phonon exchange (optomechanical circuit)[240]. (e) Phonon entanglement between two optomechanical nanobeams[246].
topological physics, and they could arise based on various types of mechanisms, such as the Quantum Hall Effect (QHE), Quantum spin Hall Effect (QSHE), and Quantum Valley Hall effect (QVHE).
Fig. 6 .
6Topological elastic lattices. Elastic non-trivial topological lattices (a) Left: Hexagonal lattice constructed by gyroscopes to break TRS. Right: Simulation results of the robust edge state propagation in the latticed shown left [254]. (b) Left: Experimental realization of the gyroscopic lattice. Right: Observed edge states in the gyroscopic lattice [162]. (c) Left: Photograph of a QSHE elastic topological insulator with a zigzag interface. Mid: The domain wall is highlighted in the top panel while the cross-sections of two domains are shown in the bottom panel. Right: Propagation of the topological edge states along the interface of the QSHE elastic topological insulator [258]. (d) Left: Topological 2D elastic nanoelectromechanical lattice. The top inset shows the unit cell, and the bottom inset shows an example flexural mode. Right: Experimental results of the dispersion curves along the topological waveguide
Fig. 8 .
8PnC and metamaterials for sensing. (a) (Left panel) A PnC made of holes in a thick slab with a slit defect filled with a mixture of water and propanol. (Right panels) Transmission coefficient showing a BG in which a cavity mode is manifested as a sharp peak. The cavity mode frequency is shown as a function of the molar ratio of the 1-propanol in water [320]. (b) (Upper panel) A PnC with a line defect of small holes filled with fluid which produces defect bands inside the BG [322] (Bottom left panel). (Right panel) Variation of the cavity mode frequency as a function of the ethanol concentration. (c) A phononic pillar made of alternating layers of tungsten and silica producing a BG that hosts edge modes that can be exited with Love wave [325]. (Upper right panel) Transmission spectrum showing sharp dips associated with the edge modes of the pillar. (Bottom right panel) Frequency shift induced by a femto-mass perturbation as function of its position at the top of the pillar. (d) A PnC made of square lattice of micro-pillars in which their resonance frequency is sensitive to temperature variation and micro-particles concentration[326]. (e) Hollow pillar in a plate with whispering-gallery-modes inside the BG used for the characterization of fluids filled into the pillar hole[328]. (f) Two PnCs of honeycomb and square lattices of holes in solid thick plates. The hexagonal lattice hosts a single hole filled with liquid while all holes in the square lattice are filled with liquid for the detection of propanol concentrations. The curves present the frequency response of each PnC as function of the ethanol concentration where we can depict the frequency shift of the defect mode inside the BG using the first phononic lattice, or the narrow BG shift for the second lattice[329].
Fig. 9 .
9Non-reciprocal PnC. (a) A nonreciprocal 1D PnC made of an array of permanent magnets and electromagnets to modulate the effective stiffness in space and time[352]. (b) A 1D metamaterial with periodic resonators made of coupled magnets and coils to modulate the local effective stiffness[353]. (c) A nonreciprocal phononic plate for flexural waves with spatiotemporally modulated effective bending stiffness via shunted piezoelectric periodic elements[356]. (d) A 1D metamaterial beam made of a series of resonators where the effective stiffness is modulated via the change in the second area moment of inertia of each resonator's arm through dynamical rotation[361]. (e) An elastodynamic Floquet topological insulator with one-way edge state propagation enabled via the modulation of the piezoelectric units of the lattice[362].
Fig. 10 .
10Active elastic metamaterials. (a) A metamaterial made of tunable resonating units. Each unit cell consists of a metallic core connected to a polymeric host via thin elastic beams. The buckling of the beams is exploited to control the elastic wave dispersion and the BG[369]. (b) A 1D PnC made of pillars with piezo-electric discs shunted through an inductive circuit in order to vary their effective stiffness
Fig. 11 .
11Inverse design of elastic metamaterials for BG engineering. (a) Plate type metamaterials designed by topology optimization
AcknowledgementsY. J. thanks the NSF for supports through CMMI 2119545, 1951221 and 2039463.Conflict of InterestThe authors declare no conflict of interest.Data Availability StatementData sharing not applicable to this article as no datasets were generated or analyzed during the current review.KeywordsPhononic crystals, Elastic metamaterial, Bandgap.
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| [] |
[
"Future Polarised DIS Fixed Target Experiments",
"Future Polarised DIS Fixed Target Experiments"
] | [
"E M Kabuß \nInstitut für Kernphysik\nUniversität Mainz\nBecherweg 45D 55099Mainz\n"
] | [
"Institut für Kernphysik\nUniversität Mainz\nBecherweg 45D 55099Mainz"
] | [] | New experiments in polarised deep inelastic scattering will mainly concentrate on the measurement of semiinclusive asymmetries. Especially, the upgraded HERMES experiment at DESY and the newly build COMPASS experiment at CERN will investigate the gluon polarisation via open charm and high pT hadron pair production, study in detail the flavour decomposition of the quark helicity distributions and measure the tranversity distributions with tranversely polarised targets. | 10.1016/s0920-5632(99)00793-8 | [
"https://export.arxiv.org/pdf/hep-ex/9906028v1.pdf"
] | 18,595,547 | hep-ex/9906028 | 283001ff01613a057de7c6f5244ebeb2da13064d |
Future Polarised DIS Fixed Target Experiments
16 Jun 1999
E M Kabuß
Institut für Kernphysik
Universität Mainz
Becherweg 45D 55099Mainz
Future Polarised DIS Fixed Target Experiments
16 Jun 19991
New experiments in polarised deep inelastic scattering will mainly concentrate on the measurement of semiinclusive asymmetries. Especially, the upgraded HERMES experiment at DESY and the newly build COMPASS experiment at CERN will investigate the gluon polarisation via open charm and high pT hadron pair production, study in detail the flavour decomposition of the quark helicity distributions and measure the tranversity distributions with tranversely polarised targets.
INTRODUCTION
The spin structure of the nucleon has been investigated in polarised deep inelastic scattering by a series of experiments at CERN, SLAC and DESY [ 1]. These experiments were initiated by the discovery of the EMC in 1987 that the contribution of the quark helicities to the proton spin is much smaller than expected originally [ 2]. The new experiments confirmed the original finding of the EMC also for the neutron that the singlet axial vector current matrix element is about 1/2 to 1/3 of the predicted value of 0.6 [ 3].
Up to now only the contribution of quark helicities to the nucleon spin was studied. Further insight into the spin structure of the nucleon can be gained by investigating the gluonic contribution and the contribution of angular momentum. In addition a measurement of the decomposition of the quark contribution into the different quark flavours will yield a deeper understanding of the nucleon spin puzzle.
There is a whole list of topics which need more detailed studies:
• The flavour decomposition of the polarised quark distributions can be extracted via the measurement of semi-inclusive asymmetries.
• The gluon polarisation can be measured with the help of open charm production or high p T hadron pairs. * Supported by the BMBF • Polarised fragmentation functions and spin transfer can be studied by measuring the Λ polarisation in the current and target fragmentation region.
• A new topis is the study of transversity in scattering on a transversely polarised target using the Collins effect. First signal were presented by SMC and HERMES during this workshop [ 4,5].
• The total angular momentum of quarks might be investigated via deeply virtual Compton scattering.
Further topics on the list refer to the measurement of off-forward parton distributions, vector meson production etc. The common feature of all these new measurements is the need to detect one or several hadrons in addition to the scattered lepton.
MEASUREMENT OF ∆G
Up to now the gluon polarisation has been investigated by indirect methods using NLO analyses of structure function data [ 1]. They indicate that integral ∆G is positive and of the order of 1 at Q 2 = 1 GeV 2 with fairly large errors while the functional form of ∆G(x) is completely unknown although there are some prejudices that ∆G(x) is largest around x ≈ 0.1.
The cleanest channel for a direct measurement of ∆G in polarised DIS is the photon gluon fusion process (PGF) as it depends in leading order on the gluon distribution (see fig. 1). In this context several methods are being discussed to extract ∆G/G:
• Open charm procduction:
γN → ccX → D 0 X PGF is signalled by the dectection of charmed particles in the final state, especially by D 0 and Λ c (close to threshold). The D 0 's are reconstructed via their decay to e.g. Kπ or µKπ. This process should yield a clear signal directly related to ∆G. Here, the hard scale is given by 2m c .
The cross section for open charm production σ γN→cc is large for quasi real photons and ∆σ γN→cc /σ γN→cc is largest for photon energies between 30 and 80 GeV. The asymmtry a LL for the hard subprocess γg → cc is about 1 at threshold (2m c ) while a LL = −1 for large energies. In this case a positive gluon polarisation will leaad to a negative photon nucleon asymmetry:
A cc γN = a LL ∆G/G . • Hidden charm production:
γN → ccX → J/ψX
Here the photon gluon process is signalled by the production of a J/ψ which is identified by its decay into a muon pair. While this is a very clean experimental signal the cross section is reduced considerably compared to open charm production. Moreover the relation of the signal to ∆G/G has to be done via the colour singlet or the colour octet model, a question which is not yet settled in unpolarised DIS.
• High p T hadron pairs:
γN → qqX → 2jets X
This third method tries to select all PGF events not only the cc production. The transverse momenta of the produced jets give the necessary hard scale.
At the moderate energies of fixed target experiments jets are not available but one can use fast hadrons instead [ 6]. Selecting oppositely charged high p T hadron pairs will enhance PGF events, but there is a considerable background especially from the QCD Compton process. Thus the measured asymmetry is given by
A HH LL ≈ a PGF LL ∆G G σ PGF σ tot + a COM LL ∆u u σ COM σ tot .
The hard asymmtries a PGF LL and a COM LL are large in the Q 2 range of the fixed target experiments and have opposite signs.
In addition one has to investigate the contribution due to resolved photons. Thus, the results from this method will be dominated by systematic effects due to large background subtractions. A first attempt to use the method was presented by the HERMES collaboration during this workshop [ 7].
FACILITIES
Up to now the experiments concentrated on inclusive measurements of the spin structure functions g 1 and g 2 . This era comes to an end with the present E155X experiment at SLAC [ 8] where data are being taken for a precise measurement of g p,d 2 . This effort will be continued at Jefferson Lab [ 9], where a high statistics measurement of the large x behaviour of g 1 and g 2 is being planned using a polarised 3 He target.
Several facilities will be available during the next years to measure semi-inclusive properties in polarised DIS:
• HERMES at DESY, which is in full swing measuring semi-inclusive asymmetries, has started a large upgrade program to attack several of the questions listed above.
• The COMPASS experiment is being setup at the CERN 100-200 GeV muon beam and will start data taking in 2000 focussing in the beginning on a measurement of ∆G.
• At MAMI in Mainz, ELSA in Bonn and Jefferson Lab measurements of the GDH sumrule and the generalized GDH sumrule will be continued.
In addition there are plans for future high luminosity maschines where a continuation and extension of the present spin program will be feasible, e.g. the ELFE proposal at CERN or DESY to study polarised DIS and the APPOLON at ELFE and the SLAC real photon beam proposal to investigate photoproduction. In the following I will concentrate on the HER-MES upgrade and the COMPASS experiment.
HERMES UPGRADE PROGRAM
The main aims of the upgrade program are • Particle identification in the full hadron momentum range, i.e. pion, kaon and proton separation,
• Enlarged muon acceptance and improved muon identification,
• Electron acceptance at very small scattering angles,
• Enlarged hadron acceptance covering also negative x F .
The first item was attacked with the installation of a dual RICH [ 10]. Due to the combination of an aerogel with a gas radiator π/K/p separation is achieved in the full momentum range up to 20 GeV. To yield high precision measurements of the Cerenkov rings the RICH is being red by an array of photomultiplier. The RICH is already installed and was succesfully operated in 1998. Currently particle identification is being implemented in the analysis chain.
The new muon filter system has been installed during the last shutdown [ 11]. It consists out of an iron absorber at the end of the spectrometer followed by a scintillator hodoscope.
The enlarged muon acceptance (for scattering angles above 170 mrad) will be made available by using tracks passing part of the magnet yoke. During the shutdown in May 1999 additional scintillators will be installed covering the region between 140 and 270 mrad behind the spectrometer magnet.
With a forward quadrupole spectrometer [ 11] the electron acceptance will be extended to smaller scattering angles. This is especially important for quasireal photoproduction events. Up to now only 10% of the scattered electrons were detected by the luminosity monitor. The new spectrometer will add another 16%.
For this, quadrupoles with larger apertures were installed in the last long shutdown. The electrons will be measured using a small vertical driftchamber installed inside the quadrupole. After the successfull test of a prototype chamber the system will be installed in May 1999.
The next topic is the enlarged hadron acceptance [ 12]. For this purpose a wheel of silicon detectors is being constructed to be positioned right after the target cell. This will enlarge the acceptance to x F < 0. Monte Carlo simulations showed that this improvement is especially important for measuring the Λ decay products. The installation of the system will start in May 1999.
The last project on the list is the recoil detector. It will consist out of a layer of double sided silicon detectors positioned below the target cell. It will be used to measure recoil particles from the target to identify diffractive events and measure tagged structure functions. This year a prototype detector was tested successfully. The installation of the full system is forseen for 2001 provided funding is available.
This upgrade will allow
• To study the flavour decomposition in more detail, e.g. measure strange quark polarisation ∆s(x)/s(x).
• A measurement of the gluon polarisation with several methods. Using open charm production and the D 0 decay into πK and µπK a precision of ∆G/G of 0.44 rsp. 0.40 can be reached with a luminosity of 80 pb −1 . The measurement of hidden charm yields δ(∆G/G) of about 0.69. In addition the enlarged hadron acceptance will improve the measurement via high p T hadron pairs.
• The measurement of the Λ polarisation in the current fragmentation region will yield a significant measurement of the polarised fragmentation function ∆D Λ u /D Λ u , while the spin correlation for strange quarks will be studied in the target fragmentation region.
• After 2000, measurements with a transversely polarised target will allow studies of azimuthal spin asymmetries to extract tranversity distributions.
THE COMPASS EXPERIMENT
Currently, the COMPASS experiment is being setup at the CERN M2 muon beam line to study polarised deep inelastic muon nucleon scattering. In addition a hadron program is planned e.g. to study charmed baryons and search for glue balls [ 13].
Compared to the previous muon experiment from SMC COMPASS will have an increased muon beam intensity of 2 · 10 8 µ/14.4 s with 100-200 GeV and 80% polarisation. Together with two oppositely polarised target cells of 60 cm length filled with 6 LiD or NH 3 a luminosity of about 2 fb −1 per year can be reached.
The spectrometer is optimized for large hadron acceptance and particle identification (see fig. 2). To achieve acceptance up to ±180 mrad a new target solenoid is being constructed with a large opening minimizing the multiple scattering for hadron tracks at large angles.
The spectrometer itself consists out of two stages. Each stage has a dipole spectrometer magnet surrounded by tracking chambers followed by a RICH detector, an electromagnetic and a hadronic calorimeter and a muon filter system.
A special feature of the muon beam is the 10% halo of muon tracks surrounding the muon beam up to a radius of 0.5 to 1 m. In addition, scattered muons have to be detected very close and in the muon beam for the measurement of quasireal photoproduction. Thus the tracking system has to be split into three regions.
The beam and scattered muons with very small angles will be measured by scintillating fiber hodoscopes. Small angle tracking will be performed with micromega chambers (stage 1) and GEM detectors (stage 2) while drift (stage 1), proportional (stage 2) and straw chambers will be used for the large angle tracking.
For the first year of data taking the detector will not be complete, especially the RICH in stage 2, the electromagnetic calorimeter electronics and part of the large angle tracking will be missing. Thus the measurements will concentrate on ∆G/G via quasireal photoproduction.
Using the above mentioned luminosity 82k charm events are expected with 1.2 D 0 per charm event. The D 0 's will be reconstructed via their πK decays. Due to MCS in the target it is not possible to reconstruct the decay vertex, thus the large combinatorial background has to be reduced by strict cuts on the D 0 kinematics. This should result in 900 charm events/day with a S:B of 1:3.9. Within 1.5 y with a 6 LiD target a precision of δA cc γN ≈ 0.05 could be reached translating to δ(∆G/G) ≈ 0.14 at x g = 0.14. Using e.g. additional decay channels or D ⋆ tagging will improve the results.
Alternatively high p T hadron pairs will be used to extract the gluon polarisation [ 6]. To reduce the background due to the QCD Compton process a series of cuts (opposite charge, high p T > 1 GeV, opposite azimuth, p T balance, positive x F ) have to be applied resulting in a good signal to background ratio of about 1:1. Due to the suppression of strangeness in the fragmentation process K + K − pairs yield an even cleaner signal.
The gluon polarisation can be studied in the range 0.04 < x g < 0.2 for 200 GeV muon energy. With 1 y of data taking a precision of δ(∆G/G) ≈ 0.05 should be achievable. The error of the gluon polarisation will then be dominated by systematic With the described spectrometer, especially with the full setup, all topics discussed in the introduction can be investigated like the flavour decomposition of the quark helicity distributions and polarised fragmentation functions. With a tranversely polarised target azimuthal asymmetries will be measured to extract transversity distributions. The possibility to study deeply virtual Compton scattering is currently being investigated.
SUMMARY
During the next years a rich experimental program is going on in fixed target polarised DIS. Experiments at DESY and CERN will do detailed studies of semi-inclusive processes to unravel more of the details of the nucleon spin structure.
Figure 1 .
1The photon fusion diagramm.
Figure 2 .
2Schematic top and side view of the COMPASS spectrometer effects for this analysis.
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| [] |
[
"Strategies for determining the cascade rate in MHD turbulence: isotropy, anisotropy, and spacecraft sampling",
"Strategies for determining the cascade rate in MHD turbulence: isotropy, anisotropy, and spacecraft sampling"
] | [
"Yanwen Wang \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n",
"Rohit Chhiber \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n\nHeliophysics Science Division\nNASA Goddard Space Flight Center\nUSA\n",
"Subash Adhikari \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n",
"Yan Yang \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n",
"Riddhi Bandyopadhyay \nDepartment of Astrophysical Sciences\nPrinceton University\nUSA\n",
"Michael A Shay \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n",
"Sean Oughton \nDepartment of Mathematics\nUniversity of Waikato\nHamiltonNew Zealand\n",
"William H Matthaeus \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n",
"Manuel E Cuesta \nDepartment of Physics and Astronomy\nUniversity of Delaware\nUSA\n"
] | [
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Heliophysics Science Division\nNASA Goddard Space Flight Center\nUSA",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Department of Astrophysical Sciences\nPrinceton University\nUSA",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Department of Mathematics\nUniversity of Waikato\nHamiltonNew Zealand",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA",
"Department of Physics and Astronomy\nUniversity of Delaware\nUSA"
] | [] | Exact" laws for evaluating cascade rates, tracing back to the Kolmogorov "4/5" law, have been extended to many systems of interest including magnetohydrodynamics (MHD), and compressible flows of the magnetofluid and ordinary fluid types. It is understood that implementations may be limited by the quantity of available data and by the lack of turbulence symmetry. Assessment of the accuracy and feasibility of such "third-order" (or Yaglom) relations is most effectively accomplished by examining the von Kármán-Howarth equation in increment form, a framework from which the third-order laws are derived as asymptotic approximations. Using this approach, we examine the context of third-order laws for incompressible MHD in some detail. The simplest versions rely on the assumption of isotropy and the presence of a well-defined inertial range, while related procedures generalize the same idea to arbitrary rotational symmetries. Conditions for obtaining correct and accurate values of the dissipation rate from these laws based on several sampling and fitting strategies are investigated using results from simulations. The questions we address are of particular relevance to sampling of solar wind turbulence by one or more spacecraft. | 10.3847/1538-4357/ac8f90 | [
"https://export.arxiv.org/pdf/2209.00208v2.pdf"
] | 251,979,728 | 2209.00208 | 9b05eaae212d79381170999658412fb09f574f30 |
Strategies for determining the cascade rate in MHD turbulence: isotropy, anisotropy, and spacecraft sampling
September 5, 2022
Yanwen Wang
Department of Physics and Astronomy
University of Delaware
USA
Rohit Chhiber
Department of Physics and Astronomy
University of Delaware
USA
Heliophysics Science Division
NASA Goddard Space Flight Center
USA
Subash Adhikari
Department of Physics and Astronomy
University of Delaware
USA
Yan Yang
Department of Physics and Astronomy
University of Delaware
USA
Riddhi Bandyopadhyay
Department of Astrophysical Sciences
Princeton University
USA
Michael A Shay
Department of Physics and Astronomy
University of Delaware
USA
Sean Oughton
Department of Mathematics
University of Waikato
HamiltonNew Zealand
William H Matthaeus
Department of Physics and Astronomy
University of Delaware
USA
Manuel E Cuesta
Department of Physics and Astronomy
University of Delaware
USA
Strategies for determining the cascade rate in MHD turbulence: isotropy, anisotropy, and spacecraft sampling
September 5, 2022Draft version Typeset using L A T E X default style in AASTeX631Interplanetary turbulence (830)-Solar wind (1534) -Magnetohydrodynamics (1964)
Exact" laws for evaluating cascade rates, tracing back to the Kolmogorov "4/5" law, have been extended to many systems of interest including magnetohydrodynamics (MHD), and compressible flows of the magnetofluid and ordinary fluid types. It is understood that implementations may be limited by the quantity of available data and by the lack of turbulence symmetry. Assessment of the accuracy and feasibility of such "third-order" (or Yaglom) relations is most effectively accomplished by examining the von Kármán-Howarth equation in increment form, a framework from which the third-order laws are derived as asymptotic approximations. Using this approach, we examine the context of third-order laws for incompressible MHD in some detail. The simplest versions rely on the assumption of isotropy and the presence of a well-defined inertial range, while related procedures generalize the same idea to arbitrary rotational symmetries. Conditions for obtaining correct and accurate values of the dissipation rate from these laws based on several sampling and fitting strategies are investigated using results from simulations. The questions we address are of particular relevance to sampling of solar wind turbulence by one or more spacecraft.
INTRODUCTION
Direct measures of cascade rates in turbulent systems often employ theoretical formulations related to Kolmogorov's "4/5" law (Kolmogorov 1941b;Frisch 1995) and its variants, in which the inertial range cascade rate is related to a signed third-order structure function. This so-called "exact" law is derived from the fluid equations without appeal to dimensional analysis, assumptions about scaling behavior, or any ansatz concerning time scales; however, this law does require time stationarity, spatial homogeneity, the existence of an inertial range, and a finite dissipation rate. The original formulation for isotropic incompressible hydrodynamics has been extended to magnetohydrodynamics (MHD) (Politano & Pouquet 1998a,b) and related models. The MHD version is frequently applied to in situ observations of plasma turbulence in the solar wind (Sorriso-Valvo et al. 2007;MacBride et al. 2008;) to obtain cascade rates that inform theories of heating and acceleration of the solar wind (Osman et al. 2011), providing ground truth for related approximations in space physics (Vasquez et al. 2007). Frequently a major issue in these applications is the use of formulations derived assuming isotropy in turbulence that is actually anisotropic (Verdini et al. 2015), this being the typical case for solar wind and magnetosheath turbulence. Usually this potential inconsistency is disregarded in favor of extensive averaging, whenever possible. Another more practical limitation is the challenging requirement of a sufficient volume of data (Podesta et al. 2009), a kinematic and statistical issue further complicated by potential sensitivity to the tails of the probability distribution of the fluctuations (Dudok de Wit 2004). Taking these challenges into account, we note that the ability to extract cascade rates from observational data is of increasing importance due to the centrality of fundamental questions relating to heating and dissipation in space and astrophysical plasmas (e.g., Kiyani et al. 2015). Therefore in the present study we revisit several related issues that are pertinent to the evaluation of third-order laws using single-point or multi-point measurements. We re-examine the issue of averaging by focusing on conditions for obtaining accurate results in both isotropic and anisotropic turbulence. The strategies we examine are implemented using data from three-dimensional (3D) MHD turbulence simulations. A motivation for this approach is that for such cases we have an unambiguous determination of the underlying turbulence symmetry as well as a straightforward method to quantify the absolute dissipation rate.
The remainder of the paper is structured as follows. In Section 2 we review relevant theoretical and observational studies that set the stage for the questions we address. Section 3 describes in detail the simulations used for the present study. Section 4 contains the results for the one-dimensional (1D) form of the third-order law using measurements for the isotropic and anisotropic cases. Section 5 delves into the direction-averaged 1D form of the third-order law in the inertial range and shows the effect of all terms in the von Kármán-Howarth equation. Section 6 gives an example of using the 1D form third-order law in a single spacecraft sample, and discusses the relative accuracy of this strategy to estimate the energy dissipation rate in observational measurements. Section 7 provides a summary of the results, and examines the relationship between the strategies used in this study and their potential applications to multi-point observations via a constellation of spacecraft.
BACKGROUND
Theory
We start from the 3D incompressible MHD equations (e.g., Biskamp 2003) ∂v ∂t
+ v · ∇v = −∇P + B · ∇B + ν∇ 2 v, ∂B ∂t + v · ∇B = B · ∇v + µ∇ 2 B,(1)
where v and B represent the local velocity and magnetic field (the latter in Alfvén speed units with B/ √ 4πρ → B and uniform mass density ρ), P is the total (thermal plus magnetic) pressure, ν is the kinematic viscosity, and µ is the resistivity. Note that B = B 0 + b where B 0 is the global mean field and b is the fluctuating field. As is well known, one may work with the Elsässer variables (Elsasser 1950), z ± (r) = v(r) ± b(r), instead of v and b. For situations where ν = µ, the incompressible MHD equations are then rewritten as:
∂ ∂t z ± = −(z ∓ · ∇)z ± − ∇P + ν∇ 2 z ± ± (B 0 · ∇)z ± .(2)
By taking the difference of the equation at z ± (r) and that at z ± (r + ), and assuming homogeneity and incompressibility, the pressure term and the term containing B 0 vanish. Let us define δz ± (r, ) = z ± (r + ) − z ± (r)
as the increment of the Elsässer variables. Taking now the dot product of δz ± with the equation for ∂(δz ± )/∂t and performing an ensemble average of the result yields the MHD von Kármán-Howarth equation (Politano & Pouquet 1998a,b):
∂ ∂t (δz ± ) 2 = −∇ · δz ∓ |δz ± | 2 + 2ν∇ 2 (δz ± ) 2 − 4 ± .(4)
Recall that because of homogeneity taking either an ensemble average (McComb 1990) or a spatial average (denoted as • ), means that the averaged increments δz ± (r, ) are only dependent on the lag , and similarly for other moments of the increments. Here ± = ν [∇z ± (r)] 2 are the mean dissipation rates associated with the Elsässer energies z ± · z ± /2 (not the increments). Since these dissipation rates involve real-space gradients (i.e., ∇ not ∇ ) and are independent of lag, they are constant over all length scales. The total energy dissipation rate of the system, also lag independent, is diss =
+ + − 2 = ν ω 2 + J 2 ,(5)
where the second form is in terms of the mean-square vorticity ω 2 = (∇ × v) 2 and mean-square electric current density J 2 = (∇ × b) 2 .
Eq. (4) is the fundamental equation of energy conservation on which all related results presented below will be based. The terms of Eq. (4) express four effects: time dependence, nonlinear transfer, dissipation (of the mean-square increments), and the exact dissipation rate of Elsässer energies, respectively from left to right. For convenience in referring to the first three of these terms, which are lag dependent, we designate ∂ ∂t (δz ± ) 2 = T ± , (δz ± ) 2 = G ± , and
Y ± ( ) = δz ∓ |δz ± | 2 .(6)
The quantities Y ± , the Yaglom fluxes, are the only third -order structure functions present. The MHD von Kármán-Howarth equation, Eq. (4), can then be rewritten as:
T ± + ∇ · Y ± − 2ν∇ 2 G ± = −4 ± .(7)
We will also make use of the sum of these equations which represents scale-by-scale conservation of the total (flow plus magnetic) energy rather than that of the Elsässer energies. Writing T = T + + T − , and similarly for Y and G, we have
T + ∇ · Y − 2ν∇ 2 G = −8 diss .(8)
Below we will refer to the ∇ · Y ± terms as the Yaglom term, recalling that it represents nonlinear transfer of energy across scales. In this scale-by-scale energy balance equation, T represents the time rate of change of energy at scales smaller than , while the term involving G is the dissipation at scales larger than .
Much of the remainder of this paper will examine various approximations and idealizations in which important information, especially the dissipation rates ± and diss , can be extracted easily, and to varying extents, accurately from Eq. (7) and (8). We emphasize the following properties of this central equation: (i) It is exact, subject to the assumptions of spatial homogeneity and incompressibility; (ii) It depends on the 3D structure functions of the various terms in vector lag ( ) space; (iii) It does not require very large Reynolds numbers; and (iv) It is much more general than the various forms of third-order laws (Politano & Pouquet 1998a,b) that are commonly implemented to estimate diss . The governing Eq. (7) is rather versatile as a starting point for determining the bookkeeping of energy at all scales, including its supply from large scales, its transfer across scales and its dissipation into heat. When appropriate the transfer across scales will be considered to be a cascade as will be discussed more precisely below. Equation (7) holds at every point in 3D lag space, so if the terms involving T ± , Y ± , and G ± are known at any point, then diss can be determined. However, this requires accurate determination of first and second derivatives in multiple independent lag directions, in general. Such information is formally available in high resolution 3D simulations, but since such derivatives will be evaluated approximately, averaging results over different 's-usually at constant = | |-is useful to achieve accuracy. To set the stage for subsequent results we begin with an illustrative example of this type, based on one of our simulations (discussed fully in the following section). We note that related recent studies have also implemented a direct evaluation of terms in the MHD von Kármán-Howarth equations, including direction averaging (Hellinger et al. 2018;Adhikari et al. 2021;Yang et al. 2022).
Shown in Figure 1 are the (direction-averaged) terms of the von Kármán-Howarth equation, Eq. (7), plotted versus magnitude of lag for a selected incompressible MHD simulation. Upon averaging over directions, the terms in Figure 1 only depend on lag length . It is readily seen that the (appropriate combinations of the) T ± , Y ± , and G ± terms quite precisely sum to the value of diss . This represents exact conservation of energy; specifically, for any chosen length scale, the sum of the rates of change of energy due to all processes is zero.
It is also apparent that the different terms T ± , Y ± and G ± in Eq. (7) make their principle contribution in different ranges of scales. In this particular simulation, such separation of scales is not perfect; this is discussed further below. This points to the most commonly encountered simplification of scale-dependent energy balance, namely the possibility of an inertial range. In the original hydrodynamic context, Kolmogorov (1941a) postulated that an inertial range is obtained asymptotically at an infinite Reynolds number.
The onset of an inertial range is expected when the large energy-containing eddies become well-separated from the scales at which dissipation occurs. This has been shown in some detail in high Reynolds number hydrodynamics experiments (e.g., Antonia & Burattini 2006). For a true inertial range, the Yaglom term −∇ · Y ± /4 is the only significant contribution to Eq. (7) over this range of lag (i.e., this term is the other terms) and we say that the Yaglom term determines the cascade rate. Ideally the contribution of the Yaglom term is flat over a span of lags. Fig. 1 exhibits only a hint of the emergence of such a clear scale separation and therefore the phrase inertial range is 7), evaluated using the anisotropic (B0 = 2ẑ) simulation III at t = 3.2. Here ∂ ∂t (δz ± ) 2 = T ± , δz ∓ |δz ± | 2 = Y ± , and (δz ± ) 2 = G ± . The green, red, and blue curves correspond to the three LHS terms in Eq. (8), averaged over many directions of vector lag . Lag is expressed in units of the simulation box length (2π). The orange horizontal dashed line indicates the exact dissipation rate diss = ν ω 2 + J 2 . The black curve is the sum of the three (LHS) terms: the large-scale energy supply, the nonlinear transfer rate to smaller scales, and the dissipation at scales larger than . As expected, they add up to the total dissipation rate.
only loosely applicable for that simulation. Note, however, that the presence or absence of an inertial range does not influence the accuracy of Eq. (7) in any way.
Third-order (aka Yaglom) laws
In a well established (i.e., large bandwidth) inertial range, both the time variation term (−T ± /4) and the scaledependent dissipation term (ν∇ 2 G ± /2) are small. If these terms become vanishingly small over an intermediate scale range then-without assuming isotropy-one obtains the simplified equation:
∇ · Y ± = ∇ · δz ∓ |δz ± | 2 = −4 ± .(9)
Such two-term specializations of the von Kármán-Howarth equation are called third-order or Yaglom laws. Herein we refer to Eq. (9) as the divergence form, or 3D form, of the MHD third-order law. A point of emphasis is that to obtain these forms requires that the Reynolds number are large enough that an effectively dissipation-free (inertial) range exists, and that the energy content of this range is steady. These assumptions are in addition to the requirements of homogeneity and incompressibility that are inherited from the developments leading to Eq. (7). Isotropy is not required.
Isotropy. An important historical development was the imposition of isotropy on Eq. (9), which leads to its further simplification. This was originally done for hydrodynamics (Kolmogorov 1941b) and later for MHD (Politano & Pouquet 1998a,b). Assuming then that the MHD turbulence is isotropic, Eq. (9) may be directly integrated to give a 1D form, or the isotropic form, for the third-order law (Politano & Pouquet 1998b;Osman et al. 2011):
Y ± = (ˆ · δz ∓ )|δz ± | 2 = − 4 3 ± ,(10)
sometimes called a "4/3" law. There are also equivalent forms which use just the longitudinal increments-sometimes called the "4/5" laws (Kolmogorov 1941b;Frisch 1995;Politano & Pouquet 1998a). Direction-averaging. Although isotropy can be established a priori only rarely, the isotropic or 1D form of the third-order law is nonetheless still often used in observational or experimental situations (Sorriso-Valvo et al. 2007;MacBride et al. 2008). The utility of this approach may be understood better when the technique is supplemented by direction averaging, carried out in an appropriate way. This very important and intuitively appealing idea has been developed in the hydrodynamics literature (Nie & Tanveer 1999;Taylor et al. 2003). The analogous result for incompressible MHD follows by direct extension of the hydrodynamic case. Below we present an abbreviated version of this straightforward derivation for MHD.
We proceed by direction-averaging the fundamental energy balance relation, Eq. (7). Carrying out a full integration over all directions, 1 and using an overbar to designate averaging over the full 4π solid angle, e.g., ∇ · Y ± = (4π) −1 S ∇ · Y ± dΩ = (4π) −1 π 0 2π 0 ∇ · Y ± sin θdφdθ, we find, without loss of generality, that
T ± + ∇ · Y ± − 2ν∇ 2 G ± = −4 ± .(11)
It is readily shown that the angular parts of the ∇ operators do not contribute when the averaging is taken into account. Some details are provided in Appendix A. The direction-averaged von Kármán-Howarth equation becomes:
T ± + 1 2 d d [ 2 Y ± ] − 2ν 2 d d 2 d d G ± = −4 ± or, T ± + D (1) Y ± − 2νD (2) G ± = −4 ±(12)
where each term on the left hand side depends on the scalar lag = | |. In Eq. (12), for convenience of presentation later, we have introduced the shorthand notation D (1) ≡ 1 2 d d [ 2 * ] for the radial contribution to the divergence operator, and D
(2) ≡ 1 2 d d 2 d d * for the radial part of the Laplacian operator, with dummy arguments * , and both defined in the lag space. We emphasize this explicitly to distinguish the form of Eq. (7) which involves three dimensional (3D) vector operations, while Eq. (12) involves only the reduced dimensional 1D operations.
This direction-averaged form of the von Kármán-Howarth equation, Eq. (12), remains quite general requiring only spatial homogeneity and incompressibility. If the system is also time stationary, or if suitable time averaging is performed (Taylor et al. 2003), the first term, T ± , may be safely neglected. Following the usual arguments, when the Reynolds number is sufficiently large (ν sufficiently small), the dissipative term involving G ± is also negligible over an intermediate range, which becomes the inertial range. In that case the remaining ordinary differential equation is
D (1) Y ± = 1 2 d d [ 2 Y ± ] = −4 ± ,(13)
which immediately integrates to the direction-averaged third-order law (cf. Appendix Eq. (A6))
Y ± = − 4 3 ± .(14)
Although this is structurally identical to the 1D form third-order law associated with an inertial range in isotropic cases, Eq. (10), we emphasize that there are several important distinctions between Eqs. (10) and (14). The former holds at any (inertial range) vector lag but only for isotropic turbulence. The latter holds for any rotational symmetry (or lack thereof) but requires averaging (the full 4π) solid angle. The third-order (Yaglom) laws for hydrodynamics and for MHD should be applied in situations in which one may reasonably assume that the conditions leading to these relations are actually attained. However, in general the most frequently quoted conditions -time-stationarity, homogeneity in space, and high Reynolds numbers -may not always hold and a pristine inertial range may not appear. In such cases the range over which a third-order law might be applied may be "polluted" by other terms in Eq. (7).
The exact statement of energy conservation in Eq. (4) or (7) provides a complete specification of the energy balance in homogeneous turbulence when evaluated over arbitrary regions of the (vector) lag space. When the full energy balance cannot be computed, it is common practice to resort (sometimes without demonstrating justification) to more compact third-order laws -such as Eqs. (9), (10), and (14) -all of which require that the time dependent terms T ± and dissipative terms G ± are negligible for a useful region of lag space.
Below we provide several examples of different approaches to approximately measure the inertial range transfer rate (sometimes loosely called the cascade rate) using MHD simulation data:
Method I. The unidirectonal 1D form, Eq. (10), is evaluated for a fixed lag direction, over a range of lag magnitudes. This is suitable for isotropic systems. The choice of direction is clearly not unique but should not matter for a truly isotropic system. There are several variations: Eq. (10) can be evaluated over a range of lags with a linear fit (through the origin) giving ± . Alternatively the equation can first be divided by and the result plotted, allowing esitmation of ± from a suitable flat range.
Method II. The 3D (or, divergence) form, Eq. (9), can be used to compute ± when 3D lag-space derivatives can be reliably determined in several directions. This method is based on a direct evaluation of (derivatives of) the Y ± terms in the von Kármán-Howarth equation (7) and is fully general in terms of turbulence symmetries when inertial range conditions are obtained. No direction averaging is required although averaging may reduce statistical inaccuracies. In this paper, the only occasion that the 3D lag-space derivatives are calculated is when obtaining the curves shown in Figure 1.
Method III. The direction-averaged 1D form, Eq. (14) is exact when integrated over the full spherical domain of lags, provided that inertial range conditions are established. However a full integration over direction may not be feasible with available measurements. In practice a discretized approximation to the continuum average is likely to be needed. This may be obtained, for example, by calculating Y ± for each of the (limited number of) available directions of and then forming the appropriately weighted average of these, see for example Eq. (A7).
All three of the above Methods, being based on third-order laws, require the existence of an inertial range, at least approximately. Otherwise, and more generally, when the T ± and G ± terms are significant, it is appropriate to use more complete forms of the von Kármán-Howarth equation, such as Eq. (7) or (12).
Observational Approaches and Limitations
Typical solar wind studies of turbulence are carried out with single spacecraft measurements and in a high speed flow for which time correlations can be interpreted as spatial correlations with reasonable accuracy (Jokipii 1971). In these circumstances observational analyses typically compute a cascade rate by employing 1D forms of the third-order law or its generalizations. In space plasma measurements, it is difficult to obtain the time variation and the dissipative terms in the von Kármán-Howarth equation Eq. (4) or (7), and therefore only the ∇ · Y ± terms (or really their integrals) can be calculated using in situ data from single spacecraft.
Most frequently Method I, Eq. (10), is employed for cascade rate estimation in solar wind (MacBride et al. 2005;Sorriso-Valvo et al. 2007;MacBride et al. 2008). Implicit in this approach is the assumption of isotropy, although the accuracy of this approximation has rarely been demonstrated. There have been attempts to adapt the 1D forms in order to refine the method (Stawarz et al. 2009;Coburn et al. 2015) by making various assumptions about the structure and symmetry of the Yaglom flux.
The 1D forms, essentially Eq. (10), have also been applied in the earth's magnetosheath ) and in Parker Solar Probe data near perihelia where turbulence is much more intense than at 1 au. It is noteworthy that there is considerable recognition that averaging is required, although this typically takes the form of a requirement for large data volumes and large data sets (Dudok de Wit 2004;Podesta et al. 2009) rather than a requirement for averaging over lag directions.
In single spacecraft measurements (with Taylor hypothesis) it is feasible to improve accuracy by averaging several measurements (MacBride et al. 2008). However, this averaging method is still not considered as the 3D form Eq. (9), and generally the weighting of different directions has not been considered. In particular, keeping in mind the results summarized in the previous section, finite sampling strategies generally do not guarantee a uniform distribution of lag directions on a sphere. In this regard a very important development is the recognition that the Yaglom flux varies systematically over the direction relative to the mean magnetic field (Verdini et al. 2015). However the assembly of measurements into a proper averaging over the unit sphere -essentially the result of Nie & Tanveer (1999) in hydrodynamics extended to the MHD third-order law (Politano & Pouquet 1998a) as in Eq. (14) -has apparently not been fully appreciated in previous MHD and space physics studies. There has been at least one study (Osman et al. 2011) employing multispacecraft Cluster data that accumulated the normal flux over a sphere in lag space, approximately carrying out the operations implied by Method III,Eq. (14). Such datasets are infrequently available in the solar wind.
As a consequence, it is crucial to know how accurately one might estimate the cascade or dissipation rate when computed from Method I, the 1D form Eq. (10), in various situations. This may be challenging in the solar wind, where the existence of a strong global magnetic field implies significant anisotropy. Proper direction averaging may not always be possible, unless very large ensembles are considered, as in, for example, recent ensemble average computations of correlation function that employed years of data and proper normalization of individual samples (Roy et al. 2021). However the intent of cascade rate estimation is often to understand more local conditions, so the emphasis may be on very much more local averaging. Such cases are severely constrained by the availability of single spacecraft data and the number of directions relative to the mean magnetic field that can be sampled. For an examination of the distribution of flow-magnetic field directions at 1 au, see the analysis of this question based on the MMS Turbulence Campaign in the solar wind by Chasapis et al. (2020). The remainder of this paper is largely devoted to exploring accuracy of energy transfer (cascade) rate measurements using von Kármán-Howarth equations and third-order laws in various forms.
SIMULATIONS
In order to study the energy cascade rate in MHD turbulence using third-order structure functions, we analyze data from several incompressible 3D MHD turbulence simulations. Key parameters of the simulations employed are shown in Table 1. For all simulations in the table, the domain is a three-dimensional periodic box with sides of length 2π, and the MHD equations are solved using a Galerkin spectral method (Orszag & Patterson 1972;Oughton & Matthaeus 2020). Each simulation is initialized with the condition that the fluctuation magnetic energy and fluid flow energy are equal, such that E m = E f = 0.5. Also, the viscous and resistive dissipation coefficients, ν and µ, are set equal. These parameters act, respectively, as reciprocal Reynolds number and magnetic Reynolds number. The normalized cross helicity,
σ c = v · b /(E f + E m )
, is known to be a significant factor in evolution of turbulent MHD (Pouquet et al. 1986). Run III has a substantial value of σ c , and this case will be qualitatively contrasted with the lower cross helicity case in run II. A full scan of parameters such as σ c will not be attempted in this study. Runs II and III are undriven and anistropic, with distinct values of B 0 . Run I is isotropic and driven; this case is included to minimize time dependent effects while examining residual transient dependence on direction, an effect that will be discussed further below.
The analyses presented here are carried out at the simulation times indicated in Fig. 2, which shows time evolution of mean square current density and mean square vorticity for each run. We can compute the exact value of the dissipation rate, diss , using Eq. (5). Table 1 lists the values of diss for each simulation at the time our analysis is performed.
Run
Type symmetry Resolution (3D) B0 δb/B0 k range ν = µ σc Table 1. Simulation parameters. The global magnetic field B0 is in theẑ direction. The rms magnetic fluctuation is δb = b 2 , and δb/B0 measures the initial relative strength of the fluctuations. Column "k range" indicates the wavenumber forcing band (Run I) or the initial conditions band (Runs II and III). Viscosity ν equals resistivity µ for each simulation. The normalized cross helicity σc, energy dissipation rate diss , and dissipation scale diss are computed at the respective times of analysis indicated in Fig. 2. In following sections, we carry out several types of analyses centered around the strategy of employing the third-order structure function to estimate the dissipation rate in the simulation. The exact dissipation rate, diss from Eq. (5), is used to study the accuracy of these various strategies.
RESULTS: 1D FORM THIRD-ORDER LAW
In this section we apply Eq. (10), a simplified and widely used 1D form of the third-order law referred to as Method I, to simulation data obtained from from both isotropic ( §4.1) and anisotropic cases ( §4.2). We examine the inertial ranges and dissipation rates estimated from this 1D form. We will consider estimates based on individual directions as well partial averages over directions, but we do not here attempt a full integration over solid angle, which is a requirement for Method III.
Isotropic case
We consider the driven B 0 = 0 isotropic simulation I. Figure 3 displays the curves for (Y + + Y − )/ as a function of , individually computed for each of 36 selected directions uniformly distributed on a sphere. The 3D trilinear interpolation method (Bai & Wang 2010) is used to calculate magnetic and velocity fields not located on grid points. Each curve represents a certain lag direction, and the value along the y axis is the average of −3Y + /(4 ) and −3Y − /(4 ). The peak value of −3(Y + + Y − )/(8 ) is used to estimate the dissipation rate. We observe that for all the curves the peak values are smaller than the actual dissipation rate, diss , which is explained in §5.2. It is also evident in Fig. 3 that the inertial ranges associated with the different lag directions are broadly consistent in extent, with some variation in the peak values. This indicates that at the instant of time of this analysis, even this nominally isotropic simulation admits some degree of variation over directions. It is reasonable to suppose that directional averaging might improve the estimates of dissipation rate in this case; we will take up this discussion in a later section. Figure 3. Estimating the dissipation rate using Method I, the 1D form third-order law (Eq. (10)), applied to data from isotropic simulation I. Recall Y ± = (ˆ · δz ∓ )|δz ± | 2 . Different curves represent results of 36 different lag directions uniformly distributed on a sphere. Each curve represents the average of the Y + / and Y − / terms for a fixed lag direction. The dark blue dashed horizontal line (at 0.795) indicates the actual energy dissipation rate. A standard procedure is to assume that the peak values provide estimates of the dissipation rate and the corresponding values locate the middle of the inertial range.
Anisotropic case
The anisotropic simulations we consider have a mean magnetic field, B 0ẑ , with B 0 = 1 or 2 and differing cross helicities (see Table 1). In this section, we only consider the anisotropic simulation III. To examine how the lag direction impacts the Yaglom term in anisotropic MHD, we evaluate the dissipation rate in simulation III using Method I, i.e., the 1D form of the third-order law, Eq. (10). Separate estimates are made using lags in each of 36 directions, uniformly spaced in co-latitude and azimuthal angles (∆θ = π 6 and ∆φ = π 3 ). The left panel of Fig. 4 demonstrates that the peak values associated with these different lag directions occur at different lags. The levels of the maxima also vary, in some cases exceeding the true dissipation rate. Similarly, the 'inertial ranges' associated with the different lag directions also vary in bandwidth and position.
In order to further study the effect of partial averaging in the presence of a global magnetic field, we group the lag directions by their corresponding polar angle θ. Recall that the global magnetic field is in +ẑ direction. For each Figure 4. A collection of estimates of dissipation rate using Method I, the 1D form third-order law (Eq. (10)), applied to data from the high σc, B0 = 2ẑ anisotropic simulation III. Left: Each curve represents the average of the Y + / and Y − / terms for a fixed lag direction. Dashed horizontal line indicates the actual energy dissipation rate. Peak values estimate the dissipation rate and (horizontal) location of a peak roughly locates the middle of the inertial range. Right: same data as left panel, normalized by diss , and averaged over azimuthal angle φ. Each curve is for a fixed polar angle θ, and averaged over six equally spaced azimuthal angles with ∆φ = π 3 .
polar angle, we average the Y ± / terms over 6 azimuthal directions (Fig. 4, right panel). We observe that the peak of each curve shifts to smaller lags as θ increases. Moreover, the peak value associated with the quasi-parallel case ( B 0 and θ = 0) provides a significantly lower estimate of diss than do the larger θ cases, which have peak values comparable to the actual dissipation rate. This reflects the well-known fact that energy transfer proceeds more rapidly perpendicular to an applied magnetic field (Shebalin et al. 1983).
An analysis similar to that of Fig. 4 is shown in Fig. 5 for simulation II, which has low σ c and B 0 = 1. Here, once again, larger θ values, more strongly perpendicular lag directions, are associated with inertial range behavior found at smaller lags. It is interesting to examine the degree of anisotropy of the energy transfer by looking at the disparity of the cascade rate estimates at varying angles in the two cases, simulation II with weaker B 0 and smaller σ c , and simulation III with stronger B 0 and larger σ c . The ratio of strongest estimate to weakest over angles is actually sightly greater in simulation II (ratio ∼ 1.4) than in simulation III (ratio ∼ 1.3), even though simulation III has the stronger mean magnetic field. Superficially this result appears to be anomalous; however, the significant contrast in cross helicity (σ c ) values is a likely explanation, as this is another factor that can influence nonlinear timescales, cascade strength, and anisotropy. Figure 5. Evaluating the dissipation rate using the 1D form third-order law (Eq. (10), Method I) for the low σc, B0 = 1 anisotropic simulation II. Curves correspond to lag directions with the indicated polar angle θ and averaged over six equally spaced azimuthal angles, φ. Each curve represents the average of the corresponding Y ± / terms, normalized by the true dissipation rate.
Comparison between isotropic and anisotropic cases using 1D form
The above Method I results for estimating diss indicate clear differences between the isotropic and anisotropic situations. In each system, we chose 36 different lag directions, distributed uniformly on a sphere, and employed the 1D form of the third-order law (Eq. (10)) to calculate ( + + − )/2 for each direction. For the driven (statistically steady) isotropic simulation we found that the inertial range and also the corresponding energy cascade rate determined this way are roughly independent of the lag directions (Fig. 3), as expected for isotropy. For the two anisotropic cases different lag directions vary much more in terms of the cascade rate estimates as well as location and bandwidths of the suggested inertial range(s). The conclusion is that it is difficult to accurately determine the energy cascade rate of an anisotropic system using the 1D form third-order law (Eq. (10)), especially with one or a small number of computed lag directions. For an isotropic system the situation is somewhat better, although some modest variation in estimated cascade rate is seen in varying lag directions.
RESULTS: FULL DIRECTIONAL AVERAGING
Given the variability we have seen in the above numerical experiments in both isotropic and anisotropic cases, we expect to obtain improved results starting from either the divergence (3D) form Eq. (9) (Method II), valid within a well-defined inertial range, or from the von Kármán-Howarth equation Eq. (4) or (7), which is broadly applicable even when an inertial range is not present. Another strategy, which we now explore, is to compute the dissipation rates ± using Method III, the fully direction-averaged 1D form of the third-order law, Eq. (14). 2 As emphasized above, based on the generalization of the result of Nie & Tanveer (1999) to the MHD von Kármán-Howarth equation, one finds that direction-averaging can reduce the problem to the 1D integration over the full 4π solid angle, see for example, Eq. (12). We can then consider just the radial component if the chosen lag directions completely cover the sphere (details of equations and discretization method can be found in Appendix A). In particular, when an inertial range is present, the direction-averaged MHD Yaglom law Eq. (14) emerges as an exact result.
Here we proceed numerically, employing a discretization method like Eq. (A7) to calculate the direction-averaged 1D form third-order law Eq.(14) in the inertial range, and a similar method for the direction-averaged von Kármán-Howarth equation Eq. (12) (details in Appendix A). Again, 3D trilinear interpolation is used for values of magnetic field and velocity not located on grid points. In order to get a uniform distribution, at any fixed lag length, we vary the direction of the lag. Let θ be the polar angle, and φ be the azimuthal angle, in such case, we keep ∆θ and ∆φ to be fixed, which are π/12 and π/6, respectively, for simulations with resolution 1024 (π/8 and π/4 for simulations with resolution 512).
Method III: Direction-Averaged 1D Form Third-Order Law
In Figs. 6 and 7, we show the directional average of 1D form third-order law, with its assumption of a well-defined inertial range, using data from simulations I and III. Recall that simulation III has a high cross helicity, leading to a large difference between '+' and '-' terms in Fig. 7. The dark blue curve is the average of the minus and plus terms. We see that the dissipation rate computed from the direction-averaged 1D form is slightly smaller than the exact dissipation rate in both isotropic and anisotropic cases. This indicates a relatively minor role of the time variation term and the dissipative term in the von Kármán-Howarth equation for both cases.
Direction-Averaged von Kármán-Howarth Equation
We now demonstrate estimation of energy transfer rates using all relevant terms in the direction-averaged von Kármán-Howarth equation Eq. (12). Note that in a driven system, the von Kármán-Howarth equation Eq. (4) should be extended to include a large-scale forcing term δz ± · F on the right hand side. The associated direction-averaged form can be written as
T ± + D (1) Y ± − 2νD (2) G ± = −4 ± + δz ± · F(15)
This analysis is first carried out for simulation I, a driven isotropic system. Fig. 8 displays these direction-averaged contributions to the von Kármán-Howarth equation, omitting the forcing term. We see that at small scales the sum of the (direction-averaged) time variation term (−T ± ), cascade term (−D (1) Y ± ), and dissipative term (2νD (2) G ± ) add up to the actual dissipation rate; evidently the driving force term does not play a role at these scales. On the Figure 6. Evaluating dissipation rates for isotropic simulation I using Method III, the direction-averaged 1D form third-order law, Eq. (14). Here the direction-averaged Y ± = (ˆ · δz ∓ )|δz ± | 2 term is designated as Y ± . The dark blue curve is the average of the two Y ± terms, and the orange horizontal line indicates the exact dissipation rate. Figure 7. Method III, direction-averaged 1D form third-order law (Eq. (14)), applied to data from anisotropic simulation III, which has high σc = 0.7 and B0 = 2. The same color code as Fig. 6 is used.
other hand, at large scales, the forcing term (not shown) is dominant, and the sum of the other three terms drops as increases. One may notice that when the Yaglom term −D (1) (Y + + Y − )/8 is at its peak of ∼ 86% of diss , the dissipative term νD (2) (G + + G − )/4 is ∼ 12% of diss , which is small although not quite negligible. Comparing this to the results shown in Fig. 6, obtained using Method III and Eq. (14), we see that the direction-averaged 1D form of the Yaglom law, which assumes inertial range conditions, actually produces a smaller estimate for the cascade rate, with a peak of ∼ 83% of diss . The equivalent results for the decaying anisotropic simulation III with global field B 0 = 2ẑ are shown in Fig. 9. Energy balance is again evident, in this case at all scales. Furthermore, the peak value of the −D (1) (Y + + Y − )/8 curve is ∼ 87% of diss , which is usefully close to the true value.
Comparison between anisotropic cases
We recall that two anisotropic simulations (runs II and III) are included, in part to explore the parameter variations that may influence the results. These simulations have different magnetic field strengths and different levels of cross The orange horizontal line indicates the actual dissipation rate. The black curve, representing the total transfer rate, is the sum of the three lines representing the Y , G and T terms. The forcing term, which contributes to the transfer rate on large scales, as in Eq. (15), is not plotted. Figure 9. Terms in the direction-averaged von Kármán-Howarth equations, same as Fig. 8, except using data from anisotropic simulation III (B0 = 2, high σc). Energy balance is obtained at all scales, as indicated by the comparison of the actual dissipation rate calculated from ν ω 2 + J 2 with the black line, which is the sum of the three lines representing the Y , G and T terms. helicity, both of which are known to influence the nature of the anisotropic cascade (Pouquet et al. 1986;Politano & Pouquet 1998a;Oughton et al. 2015).
In this section we make a comparison between the results from these two anisotropic cases, simulations II and III. First, as a comparison with Fig. 9 for simulation III, we show in Fig. 10 terms of the direction-averaged von Kármán-Howarth equation (Eq. (12)) for simulation II, which add up to the exact dissipation rate. However, for simulation II, due to the non-negligible effect of the time variation and dissipative terms, we observe that the peak value of the −D
(1) (Y + + Y − )/8 curve is much smaller than the exact dissipation rate (approximately 77%). Thus, one can also expect that using the direction-averaged 1D form third-order law (Method III), which only considers the Yaglom term Y ± , will yield a less accurate estimate for the cascade rate. As discussed in section 2, large Reynolds numbers are required in order to have an inertial range that is wellseparated from the dissipation range. Unfortunately, due to limited computing capabilities, this is not the case for our simulations. Furthermore, it has been assumed that the time variation term is negligible in Eq. (4), along with Figure 10. Terms in the direction-averaged von Kármán-Howarth equations, same as Fig. 9, except using data from anisotropic simulation II (B0 = 1, low σc). Energy balance is obtained at all scales. the dissipation term. Again, this is not the case for the anisotropic simulations we report on herein (see discussion of Figs. 9 and 10). Since our simulation II is not in a regime where the cascade terms (∇ · Y ± ) dominate over the time variation and dissipation terms, the transfer rates estimated using a third-order law are significantly less than the actual dissipation rate, even in lag directions perpendicular to B 0 , which usually give values closer to the exact energy transfer rate (see Fig. 5). Of the runs we consider simulation III is perhaps the least limited in this respect. In particular, as Fig. 9 indicates, the peak −D
(1) (Y + + Y − )/8 contribution is ∼ 13% smaller than diss . However, the situation is complicated since in some parts of the inertial range there is cancellation of the (negative at small ) time variation term −(T + + T − )/8 and the dissipative term νD (2) (G + + G − )/4.
CASCADE RATE AND SINGLE SPACECRAFT SAMPLING
Single spacecraft observations in the solar wind provide lags in only one direction, thus we can apply only the 1D form third-order law, usually that written as Eq. (10); and described as Method I, see §2.1. The same definition Y ± = (ˆ · δz ∓ )|δz ± | 2 is employed, but now z ± = v ± B/ µ 0 n i m i , where µ 0 is the magnetic permeability of free space and m i , n i are respectively the mass and (interval averaged) number density of the solar wind protons. Here we analyze measurements made by Parker Solar Probe from Nov. 3-8, 2018; the time cadence of the data is 1 s. The purpose is not to provide an exhaustive treatment of the solar wind cascade rates. Instead we present an example to inform and support our discussion.
The dissipation rate diss can be estimated by performing a linear fit in the inertial range, where the corresponding slope gives the value of diss . We choose the inertial range as the range separated from the correlation length and the estimated scale at which kinetic effects become important. The latter is typically a few ion inertial scales, as indicated in Fig. 11. Generally we also examine the energy spectra (not shown) to ensure that a reasonable power-law distribution is found in the selected range of scales. A linear least mean square method is employed over the selected inertial range for determining the best fitting slope to the computed values of Y ± . Note that Fig. 11 is plotted in log-log form, while the best fits are computed in linear-linear space. One should not be confused with the slope of the line in log-log form and the estimated dissipation rate . Plotted this way (log-log) the slope of the line indicates the power law in lag, which is expected to be linear, while the level of the line determines the estimated dissipation rate.
Third-order statistics are notoriously noisy in the solar wind and a single example will not be fully representative. The sample result shown can be meaningfully compared with other recent third-order analyses, including those that employ PSP data (Andrés et al. 2021(Andrés et al. , 2022). However, detailed comparisons are beyond the scope of the current study. Recent studies also indicate correlation anisotropy (Cuesta et al. 2022) and anisotropy of the energy spectrum Zhang et al. 2022) in the PSP datasets, which may suggest anisotropy of nonlinear energy transfer. More evidence has been shown in Andrés et al. (2022), which examines variations of third-order law with decomposition into parallel and perpendicular components using PSP data. Figure 11. The 1D form third-order law (Method I) is used for Parker Solar Probe observations to estimate the energy dissipation rate (data from Nov. 3-8 2018, with averaged helio-distance 37.9 solar radii; lags are in units of the ion inertial length, di). The indicated value of = 3.25 × 10 5 J/kg-s is obtained using a least mean square linear fit to −3(Y + + Y − )/8 = .
The inertial range in which we perform the fit to the 1D form is indicated by the two vertical dashed lines.
As discussed earlier, in anisotropic configurations, the dissipation rate we obtain from the 1D form third-order law Eq. (10) depends on the lag direction, as is evident in Fig. 4 for simulation III and in Fig. 5 for simulation II. In addition, the inertial range is also not uniquely determined in anisotropic cases and estimates of the optimal range of lags to associate with a putative inertial range vary with the lag direction as seen in the same figures. Therefore, the estimates of dissipation rate vary when analyzing different directions and assuming different inertial ranges. Nonetheless, the 1D form third-order law may provide reasonable approximations to the actual dissipation rate if the directional variability in values of estimated dissipation rate are acceptable. For example, cascade rates estimated from different peaks in Fig. 4 vary by about 50% of the exact dissipation rate.
The solar wind has a global magnetic field, and, in the inner heliosphere, a relatively high cross helicity. These features are similar to those of simulation III. Examining the results of our analysis for this run (Fig. 4, right panel), the error in the estimates of the cascade rate obtained from the 1D form third-order law can usually be assessed from the variation in the peak value and the variation in the location of the peak in the lag axis.
DISCUSSION AND CONCLUSIONS
We have examined the properties of several formulations for analyzing energy transfer in homogeneous MHD turbulence. The von Kármán-Howarth equations in increment form, Eq. (4), symbolically written as Eq. (7), provide the most complete treatment. These are exact equations and account for dissipation, time dependence, nonlinear transfer, and anisotropy. Quantitative evaluation of the several terms in the von Kármán-Howarth equations affords direct insight into the conditions required to identify a range of scales that can reasonably be viewed as an inertial range. Ideally in such a range, Kolmogorov's assumptions of steady dissipation-free transfer across scales can be realized, and the only term that makes substantial contribution is the one involving the third-order structure functions, Y ± . In that case various forms of the Kolmogorov-Yaglom (Frisch 1995) law become relevant, and specifically in MHD, the third-order law derived first by Politano & Pouquet (1998b).
When less information is available, and in particular when it is impractical to determine three-dimensional derivatives in lag space, researchers have traditionally adopted one of several approaches to simplify the estimation of the transfer rate, which in steady conditions is the cascade (or dissipation) rate. We have examined several issues that affect these familiar approximations.
In our study, we examined both the von Kármán-Howarth equation and a simpler form of the third-order law for incompressible MHD simulations of turbulence systems. In MHD under different global magnetic field conditions, one can compute the correct value for the cascade rate if each term of the von Kármán-Howarth equation, Eq. (4) or (7), can be computed exactly. With a numerical discretization only approximate values can be obtained. Nevertheless, averaging the von Kármán-Howarth equation in a sufficient number of independent directions (e.g., spanning a spherical surface), and keeping only the radial (ˆ ) components of the so-obtained Y ± , can provide accurate results. This directional-averaging strategy is based on a rigorous reduction of the problem to a one dimensional direction-averaged form, a direct extension to MHD of the hydrodynamic result due to Nie & Tanveer (1999). The direction-averaging approach may be particularly useful when adapted for use with multi-spacecraft datasets (for which there are typically only a small number of lag directions available) to estimate local cascade rates for space plasma turbulence. The accuracy of this method for different simulations is reported on in Section 5 and displayed visually in Figures 8, 9, and 10.
For further simplification, we require the existence of an inertial range of substantial length (i.e., very high Reynolds number) to justify the use of what we have called the third-order laws ( §2.1.1). Since the condition of infinite Reynolds number cannot be achieved, we are not able to observe a perfect inertial range, thereby leading to discrepancies between the actual cascade rate and the one determined from the third-order term. An advantage of these simplified forms is that they ignore dissipative and time variation effects, which are difficult to measure experimentally. The 3D form of the third-order law requires that at least some measurements are available in different directions and gives a reasonable estimate of 3D derivatives in lag space in simulations (Verdini et al. 2015). In the presence of time stationarity, the 3D form third-order law can provide an accurate estimate of energy transfer rate in the inertial range.
The futher assumption of isotropy is often used for in situ measurements of solar wind turbulence (e.g., Stawarz et al. 2009;Osman et al. 2011) and magnetic reconnection in Earth's magnetosheath (e.g., ).
In the presence of isotropy, the 3D form third-order law can be simplified to a 1D form, which employs only one lag direction. For isotropic MHD simulations, the 1D form third-order law provides a reasonable approximation, with some statistical variation with changing lag direction.
Issues regarding accuracy of energy transfer rate estimation become still more significant when anisotropy is induced by a mean magnetic field, a circumstance expected to be of significance in space and astrophysical plasmas. We report on two anisotropic simulations (Figs. 4 and 5) in Section 4.3 and (Figs. 9 and 10) in Section 5.3. From these results, it is apparent that error in estimation from the 1D form third-order law follows from a combination of effects due to the magnitude of the global magnetic field (anisotropy), the contribution of the dissipative term, and a lack of time stationarity. These errors can add up to be ∼ 50% of the exact dissipation rate in our simulations. We conclude that the (unaveraged) 1D form provides a correct order of magnitude result for the moderate levels of anisotropy found in the parameters we adopted.
Several aspects of the effects of anisotropy are summarized in Fig. 12, which includes an assessment of the directionaveraged von Kármán-Howarth equation Eq. (12). We intend to highlight variations of energy transfer for different polar angles θ, and therefore we do not show variations with varying azimuthal angle φ. In Fig. 12, we plot (green curves) the sum of the time variation (−(T + + T − )/8), the 1D Yaglom term (−D (1) (Y + + Y − )/8), and the 1D dissipative term (νD (2) (G + + G − )/4), each averaged over 12 azimuthal angles for fixed polar angles ranging over the interval 0 ≤ θ ≤ 90 • . Recall the definitions of the 1D operators D (1) and D (2) are given below Eq. (12). The average over φ is indicated by . . . φ in the legend. Note that the procedure is not equivalent to full 4π solid angle averaging, which is indicated by an overbar as in Eq. (12). Clearly averaging over φ is only a partial averaging and, in general, falls short of the effect of fully directional averaging. The average of the contributions from each θ can be weighted by sin θ and summed, to provide an approximation to a proper average over the sphere, which is the full 4π solid angle averaging indicated by the overbar as in Eq. (12). This sum is also shown in Fig. 12 (as the black line). We observe that this summation closely adds up to the exact dissipation rate, also shown (orange line). This demonstrates the approximate convergence that is expected based on the MHD generalization of the Nie & Tanveer (1999) exact result for hydrodynamnics, as we discussed in previous sections.
To delve somewhat further into this analysis of anisotropy, we note that since each partially averaged (green) curve in Fig. 12 is obtained at one only polar angle θ relative to the mean field, we see a large variance of the sum of the three (LHS) terms in the von Kármán-Howarth equation. At small scales, the top curve, having the largest estimated total transfer rate, corresponds to θ( , B 0 ) close to 90 • and the bottom curve, with lowest estimated transfer rate, corresponds to θ( , B 0 ) near 0 • , almost parallel to the mean field. In fact, the estimated transfer rate at small scales increases monotonically as we increase the polar angle (from 0 • to 90 • ). On the contrary, at large scales, the sum of the three terms is larger for smaller θ. As a consequence, the peaks of these curves shift from large scales to small scales as we increase θ. In addition, we may notice that these curves are intertwined in the middle range of lags, which is approximately the inertial range (see Fig. 9). There is actually a rather narrow range of lags near = 0.3 in which the estimates derived from different polar angles are close to one another, varying by only about ±10%. Overall, the observed variability at different values of polar angle θ emphasizes the necessity of a uniform angular coverage of the lag directions to compute an accurate energy transfer rate in anisotropic cases. It is also consistent with variability associated with the Y ± / term (i.e., Method I) that was demonstrated for the anisotropic cases (Figs. 4 and 5), and to a lesser extent even for the isotropic case (Fig. 3). Figure 12. Two sorts of angular averaging of the von Kármán-Howarth equation Eq. (8) applied to data from the B0 = 2, high σc simulation III. Terms with an overbar indicate an average over full 4π solid angle (all lag directions distributed uniformly on a sphere); see Eq. (12). Each green curve is calculated for a different fixed polar angle θ averaged over 12 azimuthal angles φ, with the averaging denoted by − φ . The black curve is the averaged value of all the green curves after weighting by sin θ. This paper has been developed with two main intentions. First it is intended to summarize pertinent analytical results relating to measurement of energy transfer rates beginning with the von Kármán-Howarth equations and leading to several reductions that are essentially third-order (Yaglom) laws, or for MHD, Politano-Pouquet thirdorder laws. The reduction to Yaglom laws becomes applicable when an inertial range is present, so that all terms but the Yaglom flux become negligible in contributing to the total energy balance over a range of lags in the inertial range. The second purpose has been to provide examples and caveats concerning the use of these methods by application to several moderate resolution MHD turbulence simulations.
The results are generally seen to be encouraging. Even with significant variation in estimates expected due to variation of lag direction in anisotropic cases, and due to "pollution" of the putative inertial range by time-dependent and dissipative effects, estimates can be broadly accurate within tolerances of ∼50%. Simulations can generally do better than this, but for single spacecraft observations this may be an acceptable estimate. It is clear that estimates will improve when three-dimensional derivatives in lag space are available, and when a large number of baseline directions are available so one might approach optimal direction-averaging. Some progress has been realized along these lines by exploiting these features in the four spacecraft Cluster mission (Osman et al. 2011). Significant advances in evaluating inertial range transfer rates will become available in the Helioswarm mission, comprising 9 spacecraft and currently under development (Spence 2019;Klein et al. 2019;Matthaeus et al. 2019) and a larger 24 point configuration envisioned in the MagneToRE approach (Maruca et al. 2021).
Finally we mention some limitations of the present study. We have not attempted any examination of the accuracy of third-order laws in differing interplanetary conditions and therefore have shown only a single case of solar wind cascade rate analysis. A complete study of solar wind situations would inevitably require examining variations of a number of interplanetary parameters including fluctuation and mean field strength, wind speed, cross helicity, turbulence age, etc. Such an effort is highly worthwhile, and the present study provides some guidelines regarding how such a major study might be undertaken, but for brevity and focus, we defer attempts at a comprehensive comparison with varying solar wind conditions to future research.
The present study is also limited to incompressible and simple MHD cases. Including compressibility, Hall effects and additional physical influences on energy transfer introduces considerable additional complexity to the estimation of the inertial range transfer rate, and requires much more extensive discussion. Some relevant observational results include some of these more general physical descriptions. For example, recent studies employing Parker Solar Probe, MAVEN, Cluster, Magnetosphere Multiscale and THEMIS data (Banerjee et al. 2016;Hadid et al. 2017Hadid et al. , 2018Andrés et al. 2019Andrés et al. , 2021 found moderate increases of compressible energy transfer rate with respect to the incompressible transfer rate. These results point the way to future studies that would generalize the simpler case that we have examined here.
This research supported in part by the NASA Parker Solar Probe Mission under a GI grant 80NSSC21K1765 and the ISOIS team (Princeton SUB0000165), by the IMAP project (Princeton SUB0000317), by the MMS mission under a Theory and Modeling grant 80NSSC19K0565, by NASA HSR grants 80NSSC18K1648 and 80NSSC19K0284, and by the US National Science Foundation NSFDOE program under grant PHY2108834. 4), let us integrate it over a spherical surface (i.e., over solid angle), using spherical polar coordinates with the "radius." See Nie & Tanveer (1999) and Taylor et al. (2003) for the closely related hydrodynamic case. As in Eq. (7), we adopt abbreviations: ∂ ∂t (δz ± ) 2 = T ± , δz ∓ |δz ± | 2 = Y ± , (δz ± ) 2 = G ± . The Y ± term of Eq. (7)
becomes S ∇ · Y ± dΩ = π 0 2π 0 1 2 ∂ ∂ ( 2 Y ± ) + 1 sin θ ∂ ∂θ (sin θY ± θ ) + 1 sin θ ∂ ∂φ Y ± φ sin θdφdθ = π 0 2π 0 1 2 ∂ ∂ ( 2 Y ± ) sin θ dφdθ + 2π 0 1 (sin θY ± θ ) π 0 dφ + π 0 1 (Y ± φ ) 2π 0 dθ.(A1)
By periodicity, the integrals of the θ and φ components vanish and therefore S ∇ · Y ± dΩ = π 0 2π 0 1 2 ∂ ∂ ( 2 Y ± ) sin θdφdθ.
Similarly, writing the Laplacian of G ± term in spherical polar coordinates also, and then integrating over a spherical surface, S 2ν∇ 2 l G ± dΩ = π 0 2π 0 2ν 2 ∂ ∂ 2 ∂G ± ∂ + 1 sin θ ∂ ∂θ sin θ ∂G ± ∂θ + 1 sin 2 θ ∂ 2 G ± ∂φ 2 sin θdφdθ
= π 0 2π 0 2ν 2 ∂ ∂ 2 ∂G ± ∂ sin θdφdθ(A3)
where the ∂ θ and ∂ φ dependent terms also vanish after integration; in a numerical (discretized) evaluation this vanishing relies on a proper distribution of lag directions. Using these results we can write the integral of Eq. (7) over solid angle as
2 ∂ ∂ ( 2 Y ± ) sin θdφdθ − π 0 2π 0 2ν 2 ∂ ∂ 2 ∂G ± ∂ sin θdφdθ = −16π ± ,(A4)
where 4 ± π 0 2π 0 sin θ dφdθ = 16π ± gives the last term. Abbreviating the average over full 4π solid angle by an overbar, this equation leads immediately to Eq. (12).
When analyzing simulation data, these integrals are evaluated using discrete approximations. To illustrate this we consider the simpler well-defined inertial range case, the 3D form of the third-order law, Eq. (9), integrated over solid angle. Using Eq. (A2) in Eq. (9), multiplying by 2 , and integrating over both dΩ and yields
If the average over full 4π solid angle is again denoted by an overbar, this result states that
Y ± = (ˆ · δz ∓ )|δz ± | 2 = − 4 3 ± .(A6)
This is written as Eq. (14) in the main text. With a simple discretization of θ and φ this becomes
− 3 4 i j Y ± (θ i , φ j ) sin θ i ∆θ i ∆φ j 4π = ± .(A7)
Similar discretization approaches are applied to the more general case, the direction-averaged von Kármán-Howarth equation, Eq. (12) or Eq. (A4).
Figure 1 .
1Terms of the von Kármán-Howarth equation, Eq. (
Figure 2 .
2Time evolution of J 2 and ω 2 for runs I (left), II (middle), and III (right). The vertical orange line indicates the time for performing analysis.
Figure 8 .
8Terms of the direction-averaged von Kármán-Howarth equation Eq. (12) in the driven isotropic simulation I at t = 8.
REDUCTION OF 3D TO 1D FORMS BY DIRECTIONAL AVERAGING Starting from Eq. (
θ dφdθ.
Using spherical polar coordinates.
We already showed results inFig. 1obtained from the von Kármán-Howarth equation (Eq. (4) or(7)) which was also direction-averaged as in Eq. (11). This averaging over directions inFig. 1was not required but increased accuracy of the computation.
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| [] |
[
"Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning",
"Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning"
] | [
"Marinos Poiitis [email protected] \nAristotle University of Thessaloniki\n\n",
"Pavlos Sermpezis [email protected] \nAristotle University of Thessaloniki\n\n",
"Athena Vakali [email protected] \nAristotle University of Thessaloniki\n\n"
] | [
"Aristotle University of Thessaloniki\n",
"Aristotle University of Thessaloniki\n",
"Aristotle University of Thessaloniki\n"
] | [] | Graph Representation Learning (GRL) has become essential for modern graph data mining and learning tasks. GRL aims to capture the graph's structural information and exploit it in combination with node and edge attributes to compute low-dimensional representations. While Graph Neural Networks (GNNs) have been used in state-of-the-art GRL architectures, they have been shown to suffer from over smoothing when many GNN layers need to be stacked. In a different GRL approach, spectral methods based on graph filtering have emerged addressing over smoothing; however, up to now, they employ traditional neural networks that cannot efficiently exploit the structure of graph data. Motivated by this, we propose PointSpectrum, a spectral method that incorporates a set equivariant network to account for a graph's structure. PointSpectrum enhances the efficiency and expressiveness of spectral methods, while it outperforms or competes with state-of-the-art GRL methods. Overall, PointSpectrum addresses over smoothing by employing a graph filter and captures a graph's structure through set equivariance, lying on the intersection of GNNs and spectral methods. Our findings are promising for the benefits and applicability of this architectural shift for spectral methods and GRL. | 10.1109/wi-iat55865.2022.00053 | [
"https://arxiv.org/pdf/2109.02358v2.pdf"
] | 237,420,876 | 2109.02358 | 1817e3bdb9c71261189d700f52120e8d03c6cbde |
Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning
Marinos Poiitis [email protected]
Aristotle University of Thessaloniki
Pavlos Sermpezis [email protected]
Aristotle University of Thessaloniki
Athena Vakali [email protected]
Aristotle University of Thessaloniki
Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning
Graph Representation Learning (GRL) has become essential for modern graph data mining and learning tasks. GRL aims to capture the graph's structural information and exploit it in combination with node and edge attributes to compute low-dimensional representations. While Graph Neural Networks (GNNs) have been used in state-of-the-art GRL architectures, they have been shown to suffer from over smoothing when many GNN layers need to be stacked. In a different GRL approach, spectral methods based on graph filtering have emerged addressing over smoothing; however, up to now, they employ traditional neural networks that cannot efficiently exploit the structure of graph data. Motivated by this, we propose PointSpectrum, a spectral method that incorporates a set equivariant network to account for a graph's structure. PointSpectrum enhances the efficiency and expressiveness of spectral methods, while it outperforms or competes with state-of-the-art GRL methods. Overall, PointSpectrum addresses over smoothing by employing a graph filter and captures a graph's structure through set equivariance, lying on the intersection of GNNs and spectral methods. Our findings are promising for the benefits and applicability of this architectural shift for spectral methods and GRL.
Introduction
Graphs are universal mathematical structures -usually accompanied by a plethora of features-that are extensively used to describe real-world data in various domains such as citation and social networks (Kipf and Welling 2016a;Liu et al. 2019), recommender systems (Zhang and Chen 2019), or adversarial attacks (Zhang and Zitnik 2020). Due to the complex structure of graphs, traditional machine learning models are insufficient for addressing graph-based tasks such as node and graph classification, link prediction and node clustering. This necessity has given rise to Graph Representation Learning (GRL) methods, which aim to capture the structure of input graph data and produce meaningful low-dimensional representations.
A main track of GRL methods is based on Graph Neural Networks (GNNs). GNNs and specifically Graph Convolutional Networks (GCN) (Kipf and Welling 2016a) have advanced the research on GRL leading to outstanding performance as they capture a node's neighborhood influence.
Nevertheless, each GNN layer considers only the local onehop node relations leading to unavoidably deep layer stacking to efficiently account for the global graph information. Notwithstanding, vanilla GNNs cannot be arbitrarily deep as they lead to over smoothing and information loss (Xu et al. 2018; Zhao and Akoglu 2019; Chen et al. 2020). To address over smoothing, spectral methods exploiting graph filters have been introduced (Wang et al. 2019;. However, so far, these methods have been used in conjunction with traditional neural networks (e.g., MLPs or CNNs) that cannot efficiently exploit the set equivariance property of graph data (i.e., equivariance to permutations in the input data). In the graph domain there is no implicit data ordering, and thus the existing spectral methods waste an important portion of their computational capacity. Contribution. In this work, we propose PointSpectrum, a GRL architecture that bridges the gap between GNN-based and spectral GRL methods (Section 2). PointSpectrum is based on Laplacian smoothing (used in spectral methods to alleviate the problem of over-smoothing), while it maintains the set equivariance property of GNN-based approaches.
Specifically, PointSpectrum is an unsupervised end-toend trainable architecture, consisting of the following components:
• Input: a low-pass graph filter (Laplacian smoothing) is applied on the input graph data that enables the computation of k-order graph convolution for arbitrarily large k without over smoothing node features • Encoder: the input data are fed to a set equivariant network that generates low-dimensional node embeddings • Decoder: a joint loss is employed to account for the reconstruction of the input data and their better separation in the embedding space through a clustering metric.
To the best of our knowledge this is the first work that introduces set equivariance in spectral GRL methods. Our experimental results (Section 3) show that: (i) Using set equivariant networks can increase the robustness (wrt. the model parameters) and efficiency (e.g., faster convergence, expressiveness) of spectral methods. (ii) PointSpectrum presents high performance in all benchmark tasks and datasets, outperforming or competing with the state-of-the-art (detailed in Section 4). Overall, our findings showcase a new direction for GRL: the combination of spectral methods with set Figure 1: PointSpectrum architecture. Encoder is a pointNetST network producing node embeddings in an equivariant manner, the pair-wise decoder reconstructs inputX and last ClusterNet is employed to better separate points in the embedding space.
equivariance. Incorporating set equivariant networks in existing spectral methods can be straightforward (e.g., replacing MLPs or CNNs), and our results are promising for the efficiency of this approach.
Methodology
Definitions. We consider a non-directed attributed graph
G = (V, E, X), where V = {v 1 , v 2 , ..., v n } is the node set with |V| = n, E is the edge set, and X = [x 1 , x 2 , . . . , x n ] T ∈ R n×l is the feature matrix consisting of feature vectors x i ∈ R l , ∀v i ∈ V. The structural infor- mation of graph G is represented by the adjacency matrix A = {a ij } ∈ R n×n
≥0 . Goals. The goal of GRL is to map nodes to low-dimensional embeddings, which should preserve both the structural (A) and contextual (X) information of G. We denote as Z ∈ R n×l , with l l, the matrix with the node embeddings. Approach overview. We propose a methodology to compute an embedding matrix Z, which combines structural and contextual information in a twofold way: on one hand, it uses Laplacian smoothing (based on A) of the feature matrix X (Section 2.1) and, on the other hand, by considering the node features as a set of points it exploits global information using a permutation equivariant network architecture (Section 2.2). Last, we leverage node clustering as a boosting component that further enhances performance by better separating nodes in the embedding space. The entire architecture that brings these components together is presented in Section 2.3.
Graph Convolution and Laplacian Smoothing
The most important notion in the prevalent GNN-based embedding methods, such as GCN (Kipf and Welling 2016a), is that neighboring nodes should be similar and hence their features should be smoother than non-neighboring nodes in the graph manifold. However, these methods capture deeper connections by stacking multiple layers, leading to deep architectures, which are known to overly smooth the node features . Over smoothing occurs as each layer repeatedly smooths the original features so as to account for the deeper interactions. To address this problem, the domain of graph signal processing has been used and in particular graph convolution by using Laplacian Smoothing filters Cui et al. 2020).
Specifically, "spectral methods" in GRL use a smoothed feature matrixX, instead of the original X, which corresponds to a k-th order graph convolution of X:
X = H k X(1)
where matrix H ∈ R n×n is the Laplacian Smoothing filter (discussed below). The multiplication of feature matrix X with filter H corresponds to a 1-order graph convolution (or 1-order graph smoothing). Stacking k filters together, i.e., k multiplications with H, leads to a k-order convolution as in Eq. 1. Thus, deep network interactions are captured by the power of filter H instead of repeated convolutions of the input features and therefore over smoothing is avoided. The Laplacian Smoothing Filter, H: The intuition behind k-order convolution of spectral methods is the following: Each column of feature matrix X(:, i) ∈ R n (i.e., the vector with the values of all nodes for a single feature) can be considered as a graph signal. The smoothness of a signal depicts the similarity between the graph nodes. Since neighboring nodes should be similar, to capture this similarity, we would need to construct a smooth signal based on X(:, i) that captures node adjacency and takes into account the features of the most important neighboring nodes. It can be shown (details in Appendix A.7) that the Laplacian Smoothing filter H as defined in Eq. 2 (Taubin 1995) can cancel high frequencies between neighboring nodes and preserve the low ones:
H = I − µL(2)
where µ ∈ R, I is the identity matrix and L is the graph Laplacian. The graph Laplacian is defined as L = D − A, where D = diag(d 1 , d 2 , . . . , d n ) ∈ R n×n is the degree matrix of A, with d i = vj ∈V a ij being the degree of node v i . In order for H to be low-pass, 1 − µλ should be a nonnegative degressive function. (Cui et al. 2020) showed that the optimal value of µ is 1/λ max , with λ max denoting the largest eigenvalue of L; in the remainder, we use this value for µ. Figure 2: Accuracy, ARI and NMI metrics of the PointSpectrum with PointNetST, MLP and CNN-based networks in the encoder solving the clustering task on the Cora dataset. For all metrics, the mean value and standard deviation of 10 experiment runs are depicted. The set equivariant PointNetST achieves higher performance and is more robust than the MLP and CNN variants, irrespective of the convolution order.
Renormalization trick: In practice the renormalization trick (Kipf and Welling 2016a), i.e., adding self-loops in the graph, has been shown to improve accuracy and shrink the graph spectral domain . Qualitatively, this means that for every node the smoothing filter also considers its own features alongside the ones from its neighbors. Thus, we perform the following transformation: self-loops are added to the adjacency:Ā = I+A; we then use the symmetric normalized graph LaplacianL sym =D − 1 2LD − 1 2 , whereD andL are the degree and Laplacian matrices of A. Finally, the resulting Laplacian smoothing filter becomes H = I − kL sym .
Equivariance and PointNetST
Neural network architectures can approximate any continuous function f given sufficient capacity and expressive power (Sonoda and Murata 2017;Sannai, Takai, and Cordonnier 2019;Yarotsky 2021); here, we denote a neural network as a function f : X → Y operating on the feature matrix X (and by extension onX). Sets and permutation equivariance. Conventional neural networks such as Multilayer Perceptrons (MLPs) or Convolutional Neural Networks (CNNs) act on data where there is an implicit order (e.g., adjacent pixels in images). However, graphs do not have any implicit order: nodes can be presented in different order but the graph still maintains the same structure (isomorphism). To this end, X can be seen as a set of points x ∈ R l , and f as a function operating on a set.
Definition 1 A function f : R n×l → R n×l acting on a set X ∈ R n×l is permutation equivariant when f (σ · X) = σ · f (X) for any permutation σ of the set X.
In other words, a permutation equivariant function f produces the same output (e.g., embedding, label) for each set item (e.g., node) irrespective of the order of the given data. This means that if a transformation (e.g., a permutation of the input matrix's rows) is applied to the input data, then the same transformation should be applied to the model's output as well.
Equivariance in GRL methods. Set equivariance is not captured by traditional neural networks such as MLPs (or CNNs), which are mainly used in the "spectral methods" in GRL Cui et al. 2020). More specifically, (Zaheer et al. 2017) proved that a function f is permutation equivariant iff it can be decomposed in the form ρ( x∈X φ(x)), where ρ and φ are suitable transformations. Also, each MLP or CNN layer can be seen as a transformation φ, and a deep network with m such layers can be expressed as f (X) = φ m • · · · • φ 1 (X). Hence, the necessary decomposition for permutation equivariance does not hold. On the contrary, GNNs can capture not only permutation equivariance but also graph connectivity, which makes them even more expressive. Nevertheless, as mentioned earlier, this high expressiveness comes with oversmoothing problems. In this work, we aim to bring the benefits of both approaches used in GRL: we employ a set equivariant network that accounts for the graph structure (through the matrixX) and avoids oversmoothing at the same time (as discussed in Section 2.1). This property is crucial for this work's model prevalence over the traditional neural networks, as it will be shown in Section 3. PointNetST. In this work, PointNetST (Segol and Lipman 2019) is employed as the permutation equivariant neural network architecture. While different choices can be made, such as a deep self-attention network (Wang et al. 2018) or the constructions of (Keriven and Peyré 2019) and (Sannai, Takai, and Cordonnier 2019), PointNetST is preferred as it is provably a universal approximator over the space of equivariant functions (Segol and Lipman 2019) and can be implemented as an arbitrarily deep neural network with the following form:
f (X) = φ m • σ · · · • σ • φ 1 (X)(3)
where σ is a non-linearity such as ReLU and φ i , i = 1, ..., m, is the DeepSet layer (Zaheer et al. 2017):
φ i (X) = XW 1 + 1 n 11 T XW 2 + 1w(4)
with W 1 , W 2 ∈ R li×li+1 and w ∈ R li+1 being the layer's parameters. Remark: while Eq. 4 is a generic form of an equivariant transformation, it is noted that PointNetST contains only a single layer with non-zero W 2 .
PointSpectrum
This work proposes PointSpectrum, an encoder-decoder architecture for GRL that consists of the following main components: (i) Laplacian smoothing of the feature matrix, (ii) PointNetST as the permutation equivariant neural network of the encoder, and (iii) a clustering module alongside the decoder. A schematic representation of PointSpectrum is depicted in Figure 1.
Input. The inputX of the encoder-decoder network is the korder graph convolution of node feature matrix X, which is computed using a Laplacian filter H as described in Section 2.1. The larger the convolution order k, the deeper node-wise interactions the model can capture; k is a hyperparameter of the model. Encoder. The encoder is the permutation equivariant Point-NetST, which generates the node embeddings Z, as described in Section 2.2. The embeddings are fed to two individual modules: the decoder and the ClusterNet. Decoder: The aim of the decoder is to reconstruct a pairwise similarity value between the computed node embeddings based onX. Different choices for the reconstruction loss function can be made (e.g. minimum squared error (Wang et al. 2019) or noise-contrastive binary cross entropy (Veličković et al. 2018)). However, as connectivity is incorporated in the smoothed signal, we use a pairwise decoder and cross entropy with negative sampling as the loss function:
L r = − i∈V (i,j)∈E log(z i z j ) + k∈Ni log(1 − z i z k ) (5)
where N i are the negative samples (i.e., non-existing edges) for node i. ClusterNet is a differentiable clustering module, which learns to assign nodes to clusters to better separate them in the embedding space (Wilder et al. 2019). ClusterNet learns a distribution of soft assignments Q, where q ij expresses the probability of node i belonging to cluster j, by optimizing KL-divergence loss function:
L c = KL(P ||Q) = i j p ij · log p ij q ij (6)
where P is a target distribution (updated in every epoch or using different intervals) that emphasizes the more "confident" assignments (Wang et al. 2019):
p ij = q 2 ij / i q ij j (q 2 ij / i q ij )(7)
Having computed Q, cluster centers can be extracted directly by averaging the soft assignments for each cluster. Overall, PointSpectrum optimizes the following joint loss of the Decoder and ClusterNet
L = α · L r + β · L c(8)
with α, β being hyper parameters that control the importance of each component.
Experimental Results
In this section, we demonstrate the efficiency of PointSpectrum in benchmark GRL datasets and tasks. We first present the datasets, the experimental setup, and the baseline methods we compare against to (Section 3.1). Then, we provide experimental evidence for the gains that are introduced by the equivariance component of PointSpectrum over the traditional deep learning architectures in terms of efficiency (Sections 3.2), complexity (Section 3.3) and robustness (Section 3.4). Finally, we compare the performance of PointSpectrum against baseline and state-of-the art methods in GRL (Section 3.5) and provide a qualitative visual analysis (Section 3.6).
Datasets & experimental setup
Datasets. We evaluated the performance of PointSpectrum on three widely used benchmark citation network datasets, namely Cora (McCallum et al. 2000), Citeseer (Giles, Bollacker, and Lawrence 1998) and Pubmed (Namata et al. 2012). The statistics of these data sources are presented and further discussed in Appendix A.1. Table 1: Clustering results based on the true labels. The input is randomly permuted in every epoch during training. The reported metrics result from the best clustering assignments of the original (not permuted) input. The results in the parenthesis refer to the models' performance when trained on the original input. PointSpectrum can capture data permutations on which it has not been explicitly trained, whereas the MLP/CNN variants perform poorly.
PointSpectrum setup. For all evaluation tasks, a PointSpectrum model is used for the encoder with a single PointNetST layer of dimension l = 100 (which is also the embedding dimension). For the clusterNet module, centers are randomly initialized and correspond to the number of distinct labels in each dataset. Last, weights are initialized according to (He et al. 2015) (He initialization). Details for the hyperparameter tuning are given in Appendix A.2. Baseline methods. We compare PointSpectrum to several baseline methods (see details in Section 4), which we distinguish in four categories: (i) feature-only (K-Means), (ii) traditional GRL (DeepWalk, DNGR, TADW), (iii) GNNbased (VGAE, ARVGA -and their simpler derivatives -and DGI, GIC), and (iv) Spectral (DAEGC, AGC, AGE). In Section 3.5, we compare against all these methods on clustering, and only the most prevalent ones on link prediction. Results are obtained directly from (Mavromatis and Karypis 2020); in particular, for AGE, K-means is used as the clustering method (instead of spectral clustering as in (Cui et al. 2020)) to enable a fair comparison. In Sections 3.2-3.4, the reported results correspond to the node clustering task, since it is the most natural task for PointSpectrum, considering its architecture.
Efficiency of set equivariance
We investigate the efficiency of using a set equivariant network (PointNetST) along with Laplacian smoothing (X), by comparing the PointSpectrum architecture ( Figure 1) against two variants with MLP and CNN neural networks in the encoder. Figure 2 shows the results for the different encoder types and different convolution orders (k) for the clustering task on the Cora dataset. 1 PointNetST achieves higher performance, is more robust (lower variance), and has less fluctuations with respect to the convolution order than the MLP and CNN variants.
This prevalence of PointNetST suggests that set equivariant networks are able to capture richer information when combined with Laplacian smoothing, and could be good candidates for replacing the conventional encoders of other Spectral methods as well (e.g., DAEGC, AGC, AGE).
Set equivariance vs. training convergence
Set equivariant networks can offer multi-faceted benefits. As shown in (Zaheer et al. 2017;Segol and Lipman 2019) and in the results of Section 3.2, they can capture richer structural information compared to the traditional MLP/CNN variants. In addition to these benefits, here we show that they also aid in training efficiency and computation complexity. Figure 3 depicts the value of the loss function during the model's training for the original PointNetST and the MLP/CNN variants. 2 In both components of the loss function (reconstruction, clustering) and the joint loss, PointNetST helps PointSpectrum to converge faster than the MLP/CNN variants.
Remark: PointNetST achieves a lower loss value in overall (Figure 3a). While the clustering loss is slightly lower for the MLP/CNN variants (Figure 3c) this difference is infinitesimal (second decimal point) compared to the reconstruction loss term (Figure 3b).
Performance on permuted data
As already shown, set equivariance offers performance and computational efficiency. In this section we focus on the main characteristic of set equivariance: its robustness on permuted inputs (which can be considered as a specific type of noisy/corrupted input data). To demonstrate this, we train PointSpectrum by randomly permuting the rows ofX (input data) in every epoch. Then, the trained model is evaluated on the originalX. Table 1 presents the results of this experiment, as well as the initial results without permuted data during training (in parentheses). PointNetST achieves the highest performance, but more importantly, the drop in performance due to the corrupted input is significantly smaller compared to the drop in the MLP/CNN variants. This highlights that PointSpectrum can capture data permutations on which it has not been explicitly trained (e.g., in case of graph isomorphism). On the contrary, the MLP/CNN variants perform poorly on permuted data (see, e.g., the NMI/ARI metrics in the Citeseer and Pubmed datasets).
Comparison against baselines
In this section, we compare PointSpectrum's efficiency in clustering and link prediction tasks against baseline and state-of-the-art methods.
We would like to stress that the main goal of this work is to introduce set equivariance in Laplacian smoothing (Spectral) GRL methods, and demonstrate the benefits it can bring. Hence, we do not extensively emphasize on the hyperparameter optimization of PointSpectrum. Here, we compare its performance against baselines for completeness and for demonstrating its efficiency compared to the state-of-theart in GRL. Nevertheless, the tested PointSpectrum implementation still outperforms Spectral methods, and achieves top or near top performance in the evaluation tasks. Clustering: The goal in clustering is to group similar nodes into m classes based on the computed embeddings. Similar to related literature, the number of classes is given, and in the evaluation the labels provided by the datasets are used. In Table 2 we report the best PointSpectrum results out of 10 experiment runs for 4 metrics: Accuracy (ACC), Normalized Mutual Information (NMI), Adjusted Randomized Index (ARI) and Macro-F1 score (F1).
Focusing first on the Spectral methods (bottom rows of Table 2), we see that the overall PointSpectrum performance (i.e., for the majority of metrics and datasets) is superior to Spectral methods. When compared to all GRL methods, PointSpectrum achieves state-of-the-art performance on most metrics in the Cora and Pubmed datasets, as well as on the accuracy metric in the Citeseer dataset (in which the GNN-based models GIC and DGI perform best for the other metrics). Link prediction: In link prediction, some graphs edges are hidden to the model during training and its goal is to predict these hidden interactions based on the computed node em-beddings. For this task 10% of positive and negative edges are used as test and 20% as the validation set. Table 3 presents the model performance on link prediction (mean value and standard deviation over 10 runs) as measured by the Area Under the Curve (AUC) and Average Precision (AP) metrics.
PointSpectrum outperforms all baselines by a significant margin in Cora (∼ 5.5% ) and Pubmed (∼ 1.5%) datasets, while in Citeseer it is the second best method after GIC (with less than 1% margin; also note that PointSpectrum has much lower variance than GIC). On one hand, we observe that PointSpectrum separates well the nodes in the embedding space as the training proceeds. On the other hand, the ClusterNet component enables the PointSpectrum to learn the cluster centers as well (denotes as 'x' marks), as it pushes them from a random initial Table 3: Link prediction performance. Area Under the Curve (AUC) and Average Precision (AP) are reported. Best results on each dataset are shown in bold, and second best in underlined. point towards each group's center. Last, since training is an iterative process, node embeddings and cluster centers attract each other in turns, explaining the formulation of these distinct clusters.
Qualitative Analysis
Related Work
GRL has gained a lot of attention due to the need for automated processes that can analyze large volumes of structured data, with graphs being the hallmark of such structures. Conventional graph embeddings: The first efforts exploited well-known graph mechanisms to calculate node representations. DeepWalk (Perozzi, Al-Rfou, and Skiena 2014) and Node2Vec (Grover and Leskovec 2016) utilize random walks to sample the graph and train a Word2Vec model on these samples to extract the embeddings. Also, TADW (Yang et al. 2015) applies non-negative matrix factorization on both the graph and node features to get a consistent partition. Last, DNGR (Cao, Lu, and Xu 2016) employs denoising auto encoders to find low dimensional representations and then reconstruct the graph adjacency. GNN-based methods: Graph Neural Networks are designed to capture graph structures, and thus have been used for learning node representations. VGAE (Kipf and Welling 2016b) uses GCNs to form a variational autoencoder which learns to generate node embeddings, while ARVGA (Pan et al. 2018) uses adversarial learning to train the graph auto encoder. DGI (Veličković et al. 2018) leverages both local and global graph information for representation learning using contrastive learning, while GIC (Mavromatis and Karypis 2020) extends DGI by forming node clusters to better separate nodes in the embedding space. In particular ClusterNet (Wilder et al. 2019) -the clustering process of GIC which is crucial for its superior performance-is also incorporated in PointSpectrum aiding in performance and validating GIC's design. Although efficient, to capture deep graph interactions GNN methods are inevitably led to over smoothing, where node representations converge to indistinguishable vectors Zhou et al. 2020). Spectral methods: On the other hand, spectral methods exploit graph filters to perform high-order graph convolution at once, thus bypassing GNNs' over smoothing. AGC uses Laplacian filtering with spectral clustering to cluster nodes into groups, while AGE (Cui et al. 2020) employs an auto encoder to produce node embeddings through Laplacian smoothing. Also, DAEGC (Wang et al. 2019) leverages an attentional network alongside soft-labeling for self supervision to construct the embeddings. While spectral methods address over smoothing, they have only used conventional neural networks (MLPs, CNNs) that cannot capture graph properties (e.g., equivariance) by design; they can only learn the structural information contained in the smoothed input signal.
PointSpectrum -although a spectral method-lies on the intersection of GNN-based and spectral methods, alleviating over smoothing through graph filtering and capturing structural information through set equivariant networks.
Conclusion
PointSpectrum is the first work to introduce the set equivariance property (typically, a property of GNN-based methods) into spectral methods. Set equivariance is important when learning on graph data, since it is inherently designed to exploit the nature of unordered data. Our work was motivated by this, and our experimental results clearly demonstrated the performance benefits of using a set equivariant network (PointNetST) over the MLP or CNN layers that are used in spectral methods.
We deem PointSpectrum as an initial effort (or as a proof of concept) in the direction of integrating set equivariance with Laplacian smoothing. This is why we adopted a simple design for the model architecture, without exhaustively over-engineering its modules or tuning its hyperparameters. Nevertheless, and despite this simplicity, we have shown that PointSpectrum can achieve state-of-the-art results in benchmark datasets. This brings a positive message for the efficiency, applicability and generalizability of our approach to other spectral or more generic GRL methods.
In particular, we identify the following as promising directions for future research: Extensions: Set equivariant networks (e.g., DeepSet or PointNetST) can easily be introduced to existing spectral methods (e.g., AGC, AGE or DAEGC) by replacing the MLP or CNN layers that they use. A more challenging direction is the extension of the proposed approach to generative models, such as VAE or GAN architectures. Generalization: As shown, PointSpectrum performs well even under data permutations. A deeper understanding (experimental/theoretical) of its capacity to generalize on noisy, corrupted or unseen data, could provide further insights on the mechanics of using set equivariant methods on graphs, as well as lead to the design of more efficient GRL methods.
Unification: PointSpectrum has a modular design, where a set equivariant network receives as input the smoothed ma-trixX. Unifying these two operations in a single component (e.g., a new GNN layer), if possible, could simultaneously aim at a higher performance and bypass over smoothing. Table 4 presents the statistics of the datasets used throughout the experimental process. These data contain a different number of labels, which are used as the oracle information to calculate the reported metrics (Accuracy, NMI, ARI and F1). All of them present a sparse graph structure, while Pubmed contains less rich information in terms of node features in comparison to Cora and Citeseer. To conduct the experiments and validate PointSpectrum's performance, hyper parameter tuning is needed. The values corresponding to each dataset are presented in Table 5. It should be noted that the best value for convolution order is smaller for PointSpectrum when compared to other methods that employ graph filtering (8, 6 and 8 for Cora, Citeseer and Pubmed respectively). This showcases the fact that set equivariant networks can capture structural information more easily and thus they do not need the whole information to be presented explicitly.
A.2 Hyperparameter setup
Regarding hyper parameters α and β, various configurations were investigated. Specifically, we point out the below behaviors:
• constant: the hyper parameter has a constant value throughout training • linear: the hyper parameter linearly increases (decreases) in every epoch to reach a maximum (minimum) value, which is also provided by the user • exponential: the hyper parameter exponentially increases (decreases) in every epoch to reach a maximum (minimum) value, which is also provided by the user. Specifically, given the maximum value s and the number of epochs e the following function is used: f (x) = exp ( log 1+s e x)−1 for a given epoch x. For the decreasing values, we sort this function's results in decreasing order.
A.3 Ablation study
To validate the efficacy of PointSpectrum's components an ablation study is conducted. First a conventional autoencoder is tested using either an MLP or a CNN encoder. Then PointNetST substitutes the conventional autoencoder and last ClusterNet is also employed alongside reconstruction objective. As it can be seen in Table 6, in all three datasets the holistic PointSpectrum model achieves the best results. Furthermore, regarding Cora and Citeseer both set equivariance (through PointNetST) and the clustering objective (through ClusterNet) increase the model's performance. However, for Pubmed although clustering is beneficial, set equivariance does not seem to help. This may relate to the low number of features when compared to the graph size, meaning that the information captured from the conventional neural networks is sufficient to characterize the data, while node features (and thus their permutations) are not that important.
To verify the above assumption we have also tested the three PointSpectrum variants on a reduced number of features. Specifically, Figure 5 depicts this experiment on Citeseer dataset where the number of features is large enough to enable us different degrees of features reduction. Although performance deterioration is not analogous to features reduction, a general trend is shown: PointNetST is heavily dependent on features, while conventional neural networks seem to pull closer to their full-features performance no matter the reduction. However, this may come from conventional neural networks' tendency to overfit to the specific permutation, therefore a more thorough investigation is needed which is out of this work's scope.
A.4 Additional visual results
Here the visual analyses of tSNE on Citeseer and Pubmed are presented. As depicted, PointSpectrum behaves simi-
A.5 Influence of convolution order on Citeseer and Pubmed
Influence of parameter k -the convolution order -is also presented with respect to Citeseer and Pubmed in Figure 7. Citeseer presents similar behavior to Cora as it has similar graph structure with many node attributes being present. However, on Pubmed training behavior is slightly different. As shown, PointNetST does not perform constantly better than its CNN variant. This can be explained by the fact that node attributes are limited in Pubmed, while graph structure is dominant. Thus, the conventional neural networks can overfit on this structure and depict equally high performance compared to set equivariant ones.
A.6 Training Convergence on Citeseer and Pubmed
Extending the discussion on training convergence, Figure 8 depicts the comparison between PointNetST encoder and MLP and CNN ones. Again, Citeseer behaves similar to Cora due to their similar structural and contextual information. Moreover, as already discussed Pubmed presents a different graph structure and less node information complicating set equivariance and restricting its performance. However, despite these complications PointNetST maintains its efficiency and accelerates the training convergence, although by a much smaller factor, nearly indistinguishable from the rest methods.
A.7 Laplacian Smoothing and Graph Convolution
The most important notion in the prevalent GNN-based embedding methods, such as GCN (Kipf and Welling 2016a), is that neighboring nodes should be similar and hence their features should be smoother -than that of irrelevant nodes -in the graph manifold. However, these methods capture deeper connections by stacking multiple layers, leading to deep architectures, which are known to overly smooth the node features . To address this problem, the domain of graph signal processing considers x ∈ R n as a graph signal, where each one of the n nodes is assigned a scalar. Then, the smoothness of signal x depicts the similarity between all of the graph nodes. To calculate smoothness, the Rayleigh quotient (Horn and Johnson 2012) over the signal and the graph Laplacian L -essentially the normalized variance of x -is employed:
R(L, x) = x T Lx x T x = (i,j)∈E(xi−xj ) 2 i∈V x 2 i(9)
Since neighboring nodes should be similar, a smoother signal is expected to have lower Rayleigh quotient. To find the relation between eigenvalues and Rayleigh quotient, one needs to calculate the eigendecomposition of the graph Laplacian, that is L = UΛU −1 with U ∈ R n×n being the matrix of eigenvectors and Λ = diag(λ 1 , λ 2 , . . . , λ n ) the diagonal matrix of eigenvalues. Then, the Rayleigh quotient of the eigenvector u i is:
R(L, u i ) = u i T Lu i u i T u i = λ i(10)
It can be seen that the lower Rayleigh quotients -and by extension the smoother eigenvectors -are correlated with low eigenvalues, meaning low frequencies. To employ these observations to every signal x, the decomposition of x on the basis of L is considered:
x = Up = n i=1 p i u i(11)
Consequently, as smooth signals are associated with smooth eigenvectors and low eigenvalues according to Eq. 10, the used filter should cancel high frequencies and preserve the low ones. Laplacian smoothing filters are selected for this purpose, as they combine high performance with low computational cost (Taubin 1995). Laplacian Smoothing Filter: Here, we consider the generalized Laplacian Smoothing filter as defined in (Taubin 1995) H = I − kL (12) where k ∈ R, I is the identity matrix and H is the filter matrix. Using Eq. 12, the filtered signal is:
x = Hx = U(I − kΛ)U −1 Up = n i=1
(1 − kλ i )p i u i What this suggests is that for H to be low-pass, 1 − kλ should always decline. It has been found that the optimal value of k is 1/λ max , with λ max denoting the largest eigenvalue (Cui et al. 2020).
Having defined the filter, one can introduce k-order smoothing -and thus graph convolution -by stacking k filters together. Ultimately, the overall smoothed feature matrix isX = H k X (13)
A.8 System Specifications
All of the experiments were conducted using a computing grid with an Intel Xeon E5-2630 v4 CPU, 32Gb RAM and an Nvidia Tesla P100 GPU. Also, to speed up some computations a personal computer with an Intel(R) Core(TM) i7-6700K CPU @ 4.00GHz, 32Gb RAM and an NVidia GeForce GTX 1070 GPU was also employed for specific experiments alongside the ones running on the grid.
Figure 3 :
3Total loss, reconstruction loss and clustering loss of PointSpectrum with PointNetST, MLP and CNN-based networks in the encoder solving the clustering task on the Cora dataset (mean values over 10 runs). The set equivariant PointNetST helps the model converge faster than the MLP and CNN variants.
We evaluate qualitatively PointSpectrum by visualizing the computed embeddings using t-SNE (Van der Maaten and Hinton 2008) for the Cora dataset inFigure 4(see Appendix A.4 for visualizations on the Citeseer and Pubmed datasets).
Figure 4 :
4PointSpectrum's training behavior on Cora dataset using t-SNE for visualization. ClusterNet's trainable centers are denoted as black 'x' marks.
Figure 5 :
5PointNetST, MLP and CNN-based methods' performance trained and evaluated on reduced number of features (depicted as pointNet, mlp and cnn respectively.
Figure 7 :Figure 8 :
78Accuracy, ARI and NMI metrics for PointNetST, MLP and CNN-based networks solving the clustering task on Citeseer and Pubmed datasets. For all three measures, the mean value and standard deviation of 10 experiment runs are depicted. Loss, reconstruction and clustering loss for PointNetST, MLP and CNN-based networks solving the clustering task on Citeseer and Pubmed datasets. For all three measures, the mean value and standard deviation of 10 experiment runs are depicted. PointNetST helps the model to converge faster than the MLP and CNN variants.
Table 2 :
2Clustering results based on the true labels. Horizontal lines discriminate feature-only, traditional GRL, GNN-based and Spectral methods. Underlined values indicate the best results among the spectral methods, and bold values the best results among all methods.
Table 4 :
4Dataset specifics
Table 5 :
5Hyper-parameter values for different datasets.α and
Table 6 :
6Ablation study. lar to Cora, separating the embeddings as training proceeds and forcing cluster centers towards the centers of the node groups that it creates. More concretely, in the case of Citeseer, nodes are well separated into distinct clusters and therefore the trained ClusterNet centers match to the actual centers. For Pubmed though, node separation is not trivial and nodes appear to be similar to nodes of other clusters. We suppose that the reason behind this phenomenon is the small number of available features for PointSpectrum to exploit, when compared to Cora and Citeseer. PointSpectrum's training behavior on Citeseer (top) and Pubmed (bottom) dataset using t-SNE for visualization. ClusterNet's trainable centers are denoted as black 'x' marks.(a) Initial
(b) Intermediate
(c) Final
Figure 6:
Copyright © 2022, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.
The corresponding results on the Citeseer and Pubmed datasets can be found in Appendix A.5; on Citeseer PointNetST performs even better than on Cora, while on Pubmed it performs similarly to MLP/CNN variants (due to the small number of features and simpler graph structure).
Similar findings hold for the Citeseer and Pubmed datasets; see Appendix A.6
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| [] |
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"Near-field imaging of optical nano-cavities in Hyperuniform disordered materials",
"Near-field imaging of optical nano-cavities in Hyperuniform disordered materials"
] | [
"N Granchi *[email protected] \nDepartment of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly\n",
"M Lodde \nDepartment of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL\n",
"K Stokkereit \nAdvanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK\n",
"R Spalding \nAdvanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK\n",
"P J Van Veldhoven \nDepartment of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL\n",
"R Sapienza \nDepartment of Physics\nThe Blackett Laboratory\nImperial College London\nUK\n",
"A Fiore \nDepartment of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL\n",
"M Gurioli \nDepartment of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly\n",
"M Florescu \nAdvanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK\n",
"F Intonti \nDepartment of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly\n"
] | [
"Department of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly",
"Department of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL",
"Advanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK",
"Advanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK",
"Department of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL",
"Department of Physics\nThe Blackett Laboratory\nImperial College London\nUK",
"Department of Applied Physics and Science Education\nEindhoven Hendrik Casimir Institute\nEindhoven University of Technology\nEindhovenNL",
"Department of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly",
"Advanced Technology Institute and Department of Physics\nUniversity of Surrey\nSurreyUK",
"Department of Physics and Astronomy and LENS\nUniversity of Florence\nSesto FiorentinoFIItaly"
] | [] | Hyperuniform disordered photonic materials have recently been shown to display large, complete photonic bandgaps and isotropic optical properties, and are emerging as strong candidates for a plethora of optoelectronic applications, making them competitive with many of their periodic and quasiperiodic counterparts. In this work, high quality factor optical cavities in hyperuniform disordered architectures are fabricated through semiconductor slabs and experimentally addressed by Scanning Near-field Optical Microscopy. The wide range of confined cavity modes that we detect, arise from carefully designed local modifications of the dielectric structure. Previous works on hyperuniform disordered photonic systems has previously identified several Anderson localized states spectrally located at the PBG edges with relatively high quality factors. In this work, by engineering the structural parameters of the cavity, we achieve an experimental quality factor of order 6000 (higher than the one of the Anderson states) and we demonstrate that three types of localized modes of different nature coexist within a small area and in a relatively narrow spectral window of the disordered correlated system. Their compatibility with general boundary constraints, in contrast with ordered architectures that suffer strict layout constraints imposed by photonic crystal's axes orientation, makes optical cavities in disordered hyperuniform patterns a flexible optical insulator platform for planar optical circuits.INTRODUCTION.Photonic crystal cavities (PCCs) are state-of-the-art devices to strongly localize electromagnetic fields in volumes below a cubic optical wavelength of light, acting as efficient nano-resonators with high Q factors ( ≈10 6 ). Thanks to the photonic band gap (PBG) confinement effect, the well-controlled photon mode profiles in PCCs are extremely important for quantum electro-dynamics (QED) devices and integrated photonic applications, from lasing to optical fibers [1][2][3][4][5]. However, the anisotropy of PBGs caused by the lower-order rotational symmetry in wave-vector k-space has often hampered practical implementations, driving many efforts in developing alternative platforms like quasiperiodic photonic crystals [5-9] and correlated disordered photonic media[10]. Indeed, applications that demand cladding of some different heterostructures (like different types of photonic crystals or quasicrystals) require very isotropic band gaps to engineer point and line defects that span over larger spectral ranges and that are more robust against imperfections [11][12]. One approach of achieving more isotropic PBGs is to increase the rotational symmetry of the underlying structure as in the case of quasicrystals, which maintain the strong scattering of light while increasing the long-range orientational order in the system [13] and provide more efficient and uniform in-plane confinement in all directions, which could be beneficial for achieving lasing properties of lower threshold and higher quality factors[14,15]. A different approach has emerged in the last years, which is to erase the precise rotation symmetry while still preserving the PBG by using a special class of disordered photonic materials called Hyperuniform Disordered (HuD) systems[16]. These systems located in-between random structures and perfectly ordered photonic crystals, have emerged as a valid alternative solution to the problem: in particular, the structures built around stealthy HuD systems, whose name relates to their property of being transparent to incident radiation for a certain range of wavevectors k, have recently been shown to display large isotropic band gaps comparable in width to band gaps found in photonic crystals[10,[16][17][18]. In the context of optical confinement applications, HuDs provide two main advantages : (i) the large band gaps found in these | 10.1103/physrevb.107.064204 | [
"https://export.arxiv.org/pdf/2302.12590v1.pdf"
] | 257,118,052 | 2302.12590 | 136a165fcf3f24f9fe876e2dba2583c95894f888 |
Near-field imaging of optical nano-cavities in Hyperuniform disordered materials
N Granchi *[email protected]
Department of Physics and Astronomy and LENS
University of Florence
Sesto FiorentinoFIItaly
M Lodde
Department of Applied Physics and Science Education
Eindhoven Hendrik Casimir Institute
Eindhoven University of Technology
EindhovenNL
K Stokkereit
Advanced Technology Institute and Department of Physics
University of Surrey
SurreyUK
R Spalding
Advanced Technology Institute and Department of Physics
University of Surrey
SurreyUK
P J Van Veldhoven
Department of Applied Physics and Science Education
Eindhoven Hendrik Casimir Institute
Eindhoven University of Technology
EindhovenNL
R Sapienza
Department of Physics
The Blackett Laboratory
Imperial College London
UK
A Fiore
Department of Applied Physics and Science Education
Eindhoven Hendrik Casimir Institute
Eindhoven University of Technology
EindhovenNL
M Gurioli
Department of Physics and Astronomy and LENS
University of Florence
Sesto FiorentinoFIItaly
M Florescu
Advanced Technology Institute and Department of Physics
University of Surrey
SurreyUK
F Intonti
Department of Physics and Astronomy and LENS
University of Florence
Sesto FiorentinoFIItaly
Near-field imaging of optical nano-cavities in Hyperuniform disordered materials
1
Hyperuniform disordered photonic materials have recently been shown to display large, complete photonic bandgaps and isotropic optical properties, and are emerging as strong candidates for a plethora of optoelectronic applications, making them competitive with many of their periodic and quasiperiodic counterparts. In this work, high quality factor optical cavities in hyperuniform disordered architectures are fabricated through semiconductor slabs and experimentally addressed by Scanning Near-field Optical Microscopy. The wide range of confined cavity modes that we detect, arise from carefully designed local modifications of the dielectric structure. Previous works on hyperuniform disordered photonic systems has previously identified several Anderson localized states spectrally located at the PBG edges with relatively high quality factors. In this work, by engineering the structural parameters of the cavity, we achieve an experimental quality factor of order 6000 (higher than the one of the Anderson states) and we demonstrate that three types of localized modes of different nature coexist within a small area and in a relatively narrow spectral window of the disordered correlated system. Their compatibility with general boundary constraints, in contrast with ordered architectures that suffer strict layout constraints imposed by photonic crystal's axes orientation, makes optical cavities in disordered hyperuniform patterns a flexible optical insulator platform for planar optical circuits.INTRODUCTION.Photonic crystal cavities (PCCs) are state-of-the-art devices to strongly localize electromagnetic fields in volumes below a cubic optical wavelength of light, acting as efficient nano-resonators with high Q factors ( ≈10 6 ). Thanks to the photonic band gap (PBG) confinement effect, the well-controlled photon mode profiles in PCCs are extremely important for quantum electro-dynamics (QED) devices and integrated photonic applications, from lasing to optical fibers [1][2][3][4][5]. However, the anisotropy of PBGs caused by the lower-order rotational symmetry in wave-vector k-space has often hampered practical implementations, driving many efforts in developing alternative platforms like quasiperiodic photonic crystals [5-9] and correlated disordered photonic media[10]. Indeed, applications that demand cladding of some different heterostructures (like different types of photonic crystals or quasicrystals) require very isotropic band gaps to engineer point and line defects that span over larger spectral ranges and that are more robust against imperfections [11][12]. One approach of achieving more isotropic PBGs is to increase the rotational symmetry of the underlying structure as in the case of quasicrystals, which maintain the strong scattering of light while increasing the long-range orientational order in the system [13] and provide more efficient and uniform in-plane confinement in all directions, which could be beneficial for achieving lasing properties of lower threshold and higher quality factors[14,15]. A different approach has emerged in the last years, which is to erase the precise rotation symmetry while still preserving the PBG by using a special class of disordered photonic materials called Hyperuniform Disordered (HuD) systems[16]. These systems located in-between random structures and perfectly ordered photonic crystals, have emerged as a valid alternative solution to the problem: in particular, the structures built around stealthy HuD systems, whose name relates to their property of being transparent to incident radiation for a certain range of wavevectors k, have recently been shown to display large isotropic band gaps comparable in width to band gaps found in photonic crystals[10,[16][17][18]. In the context of optical confinement applications, HuDs provide two main advantages : (i) the large band gaps found in these
Abstract: Hyperuniform disordered photonic materials have recently been shown to display large, complete photonic bandgaps and isotropic optical properties, and are emerging as strong candidates for a plethora of optoelectronic applications, making them competitive with many of their periodic and quasiperiodic counterparts. In this work, high quality factor optical cavities in hyperuniform disordered architectures are fabricated through semiconductor slabs and experimentally addressed by Scanning Near-field Optical Microscopy. The wide range of confined cavity modes that we detect, arise from carefully designed local modifications of the dielectric structure. Previous works on hyperuniform disordered photonic systems has previously identified several Anderson localized states spectrally located at the PBG edges with relatively high quality factors. In this work, by engineering the structural parameters of the cavity, we achieve an experimental quality factor of order 6000 (higher than the one of the Anderson states) and we demonstrate that three types of localized modes of different nature coexist within a small area and in a relatively narrow spectral window of the disordered correlated system. Their compatibility with general boundary constraints, in contrast with ordered architectures that suffer strict layout constraints imposed by photonic crystal's axes orientation, makes optical cavities in disordered hyperuniform patterns a flexible optical insulator platform for planar optical circuits.
INTRODUCTION.
Photonic crystal cavities (PCCs) are state-of-the-art devices to strongly localize electromagnetic fields in volumes below a cubic optical wavelength of light, acting as efficient nano-resonators with high Q factors ( ≈10 6 ). Thanks to the photonic band gap (PBG) confinement effect, the well-controlled photon mode profiles in PCCs are extremely important for quantum electro-dynamics (QED) devices and integrated photonic applications, from lasing to optical fibers [1][2][3][4][5]. However, the anisotropy of PBGs caused by the lower-order rotational symmetry in wave-vector k-space has often hampered practical implementations, driving many efforts in developing alternative platforms like quasiperiodic photonic crystals [5][6][7][8][9] and correlated disordered photonic media [10]. Indeed, applications that demand cladding of some different heterostructures (like different types of photonic crystals or quasicrystals) require very isotropic band gaps to engineer point and line defects that span over larger spectral ranges and that are more robust against imperfections [11][12]. One approach of achieving more isotropic PBGs is to increase the rotational symmetry of the underlying structure as in the case of quasicrystals, which maintain the strong scattering of light while increasing the long-range orientational order in the system [13] and provide more efficient and uniform in-plane confinement in all directions, which could be beneficial for achieving lasing properties of lower threshold and higher quality factors [14,15]. A different approach has emerged in the last years, which is to erase the precise rotation symmetry while still preserving the PBG by using a special class of disordered photonic materials called Hyperuniform Disordered (HuD) systems [16]. These systems located in-between random structures and perfectly ordered photonic crystals, have emerged as a valid alternative solution to the problem: in particular, the structures built around stealthy HuD systems, whose name relates to their property of being transparent to incident radiation for a certain range of wavevectors k, have recently been shown to display large isotropic band gaps comparable in width to band gaps found in photonic crystals [10,[16][17][18]. In the context of optical confinement applications, HuDs provide two main advantages : (i) the large band gaps found in these structures are facilitated by the hyperuniform geometrical properties of the underlying point-pattern template upon which the structures are built [11,19] and (ii) the intrinsic geometrical statistical isotropy of the stealthy HuD point patterns induces a consequent statistical isotropy of the photonic properties of the dielectric structures, and this is highly relevant for a series of novel photonic functionalities, as their compatibility with general boundary constraints can provide a flexible optical insulator platform for planar optical circuits [20]. Many experimental works have focused on light confinement in 3D and 2D correlated disorder photonic systems [21][22][23][24][25]. Another unique feature exhibited by HuD systems and recently demonstrated in luminescent HuD photonic networks is that they support modes more robust against local perturbations and fabrication induced disorder with respect to their disordered and ordered counterparts [20]; eventual flaws that could seriously degrade the optical characteristics of photonic crystal devices are likely to have less effect on disordered hyperuniform structures, therefore relaxing fabrication constraints. Consequently, the concept of optical cavities in HUD photonic materials proposed in Refs. [17,19] may offer a novel route that could address many of the roadblocks in the field of optical cavities and could provide many opportunities yet to be explored experimentally. In this work, we address this task by designing and fabricating optical cavities in HuD photonic networks on slab technology [26,27], and explore their properties by performing a Scanning Near-field Optical Microscopy (SNOM) study on cavities with different structural parameters in order to obtain accurate knowledge of the design configuration that can maximize their quality factor. The wide range of confined cavity modes detected here and previously identified only theoretically [19], is obtained through carefully designed local modifications of the HuD dielectric structure and accurate fabrication and experimental characterization of the localized mode properties. So far, the highest experimental value of obtained for not engineered localized modes in HuD systems on slab has been detected as 1500 [26]. Thanks to this analysis, we are able to engineer the structural parameters that allow to achieve an experimental maximum value of ≈ 6000.
THEORY AND DESIGN. The structures investigated comprise planar HuD dielectric networks designed through a tessellation protocol described in [10]. Analogously to the lattice constant in photonic crystals, we define a length scale = /√ , such that an N-point hyperuniform pattern in a square box of side length has a scatterer density of 1/ 2 . The point pattern employed here contains N = 500 points and belongs to the sub-category of HuD systems known as "stealthy" [28,29]. For stealthy hyperuniform patterns, the structure factor ( ) is statistically isotropic, continuous and precisely equal to zero for a finite range of wavenumbers smaller than a certain critical wavevector , i.e., ( < ) = 0. The stealthiness parameter is defined as the ratio between the number of k-vectors for which the structure factor ( ) is constrained to vanish and the total number of k-vectors. Analogously to the case photonic crystals, the photonic modal resonances supported by HuD structures can be tuned by changing two structural parameters [19,26,30]: the width of the decorating dielectric walls in the network, i.e. , and a global scaling factor , which multiplies the size and yields the final size of the HuD domain. We start with the 2D band structure analysis of the unperturbed HuD network with = 0.5 and = 380nm, shown in Fig.1a. The walls are decorated with dielectric material of index of refraction n=3.4, and their width is = 0.36 . In Fig.1b, we present the corresponding 2D photonic band structure, calculated using the planewave expansion software MPB [30]. The band structure calculations are done using a supercell approach, which results in a folding of the bands. However, the photonic band gap (white region) is not affected by the folding and is located between bands (red lines) N and N+1, where N is the number of scattering cells in the supercell (here N=500). Fig. 1c shows the cavity design, i.e. the HuD network of Fig.1a in which we placed an engineered defect at its center. The optical cavity is created firstly by filling one of the air domains, and after placing an inner circular air hole (of radius ) at its center. Then we employ a methodology for achieving optimal designs similar to the conventional one used for photonic crystal cavities [1] of smoothing the transition between the cavity and the HuD surround to minimize the leakage of the confined mode in the vertical direction. This involves slightly reducing the size of the adjacent holes and shifting their position outwards along the lattice directions. In this disordered case there are no lattice directions, and we instead shift the six shrunken cells, surrounding the defect, along the vector given by the center of mass of the cavity to the center of mass of the neighboring cell. The cells are shrunken to 52% of the size of a cell with infinitesimal walls and are shifted by 8% of the length of the vector outwards [19]. The central defect is designed such that it supports several modes, as predicted by the 2D bandstructure shown in Fig. 1d, where several bands appear inside the PBG. We focus on the first four modes, that we label accordingly with the 2D profiles of the magnetic field component (Fig.1e): dipole-like (D), hexapole-like (H), quadrupole-like (Q) and octupole-like (O) [6]. Interestingly, similar elementary localized modes have already been detected in photonic quasicrystal single cells [14], proving the main advantages related to the insensitivity to the propagation direction and to the long coherent-interaction-range order with respect to their periodic counterparts. 3D samples are obtained by adapting the 2D theoretical designs, and extruding them in the vertical direction to produce a slab of finite thickness employing a process described in ref. [26]. The effects of the finite height of the slab on the spectral position of the cavity's modes can be compensated by tuning an additional overall scaling factor ; is a multiplying factor to the original size of the unitary cell that as a result increases ( > 1) or decreases ( < 1) the final dimension of the cell. This determines a rigid shift of all the bandstructure in terms of PBG. Hence, In order to finely tune the cavity resonances and to find the best conditions that maximize the experimental we change the parameters , and (the last two parameters are highlighted in the upper right inset of Fig.1c). Consequently, we design several types of cavities with three different values of (0.36 , 0.4 and 0.44 ) and three different values of (0.2 , 0.3 and 0.4 ) that span over values of from 0.9 to 1.1.
EXPERIMENT AND DISCUSSION
The structures fabricated consist of GaAs-based heterostructures: high-density InAs QDs emitting at 1300 nm are grown by molecular beam epitaxy at the center of a 200-nm-thick GaAs membrane and cover the entire area of the samples. The HuD structures with optical cavities under consideration are patterned with electron-beam lithography, reactive-ion etching and subsequent selective etching of a 3 m-thick AlGaAs sacrificial layer [27]. Fig.2a shows a scanning-electron-microscopy (SEM) top view image of one of the fabricated samples, with nominal lattice constant = 380 nm, =0.9, wall thickness = 0.36 and radius of the central hole = 0.2 (same parameters as the design shown in Fig.1). A room temperature commercial SNOM (Twinsnom, OMICRON) is used in an illumination-collection geometry (see sketch of Fig.2b). The sample is excited with light from a diode laser (780 nm) coupled into a chemically etched optical fiber, which allows a direct measurement of the local density of states (LDOS) through the spectral shift maps [31,32]. It has been demonstrated that the tip-induced spectral shift is an excellent tool to measure the local electric field intensity, as the strength of the tip-induced spectral shift is proportional to the electric field intensity of the eigenmode itself. In our experiment, photoluminescence (PL) spectra from the sample are collected at each tip position through the same probe and the PL signal dispersed by a spectrometer is detected by a liquid nitrogen cooled InGaAs array with a high spectral resolution of 0.1 nm. This allow to reconstruct PL maps, from which we can extract the spectral shift maps to achieve a reliable imaging of the LDOS [31-,32].
In the inset graph of Fig.2b it is shown the PL emission spectrum of the QDs collected in an unpatterned area of the sample. In Fig.2b we report a typical PL spectrum acquired on top of the cavity, where four sharp peaks are detected. The morphological information given by the SNOM topography, combined with the SNOM optical maps, allows us to establish that the four resonances correspond to the four cavity modes predicted our theoretical investigation: the dipole-like mode D ( = 1307.4 nm), the quadrupole-like mode Q ( = 1304.5 nm), the hexapole-like mode H ( = 1291.7 nm) and the octupole-like mode O ( = 1258.2 nm).We note that these resonances are located in the PBG spectral region (see Fig.1a). For each peak of the PL spectrum, we performed a single Lorentzian fit. This allows us to reconstruct the spectral shift maps induced by the SNOM tip (overlapped in transparency with the SEM image of the sample and reported in Fig.2c) and achieve accurate knowledge of the LDOS spatial distribution of every resonance of the cavity [31,32]. With this method, from the Lorentzian fit of each peak we can calculate the ; notably, the four sharp peaks of Fig.2b exhibit a of 700, 1500, 2500 and 2150 (in order of increasing energy). We performed Finite Element Method (FEM) simulations of the GaAs slab patterned with the nominal design of Fig.1b. By comparing the experimentally inferred spectral shift maps to the FEM simulated maps of the electric field intensity distribution (reported in Fig.2d), we can conclude that the sub-micrometric details of the design are faithfully reproduced. Our investigation reveals not only the exact location each mode's hotspot, but also that the symmetry of Q and H modes is not entirely complete as their spatial distribution is unbalanced towards opposite directions instead of being distributed over the entire defect (an consequence of the intrinsic disorder in the HuD structure also evident in the simulated maps). Before exploring the quality factors of HuD optical cavities, we analyze the light localization mechanisms. In photonic crystals, light with frequencies above or below the band edges are propagating modes that are transmitted through them, but they can become localized in the case of 2D hyperuniform disordered structures [26]. Here, we take a step further by introducing a cavity in the same HuD environment. In Fig. 3a we plot the Inverse Participation Ratio ( ) as a measure of light localization [33][34][35]26] for all the modes of the HuD network slab (same structures as Fig.1 and 2) obtained with FEM simulations. Values of close to one indicate the electric field intensity is nearly constant throughout the system volume, while higher values are associated with the mode profiles confined in a small volume. The blue dots indicate modes above and below the PBG; the expected trend portrays dielectric (air) modes with high ( > 50, i.e., strongly localized modes) at the upper (lower) PBG edge. The value of the decrease and reach the value of ≈ 1 as the frequency increases and the optical modes enter a diffusive regime and become delocalized. The red dots (reported on a different scale) indicate cavity modes and the accidental mode localized over a topological defect (T) [10,26].
Clearly, these strongly localized modes ( ≈ 400-550) coexists with all the other modes supported by the HuD network. This remarkable behavior is further explored in FEM simulations: in Fig. 3b and 3c we respectively present FEM maps of the electric field intensity of the T mode and of one of the dielectric Anderson modes at the PBG upper edge; the location of the cavity is highlighted with a dashed white circle. We compare the FEM maps with the SNOM PL maps filtered around the central wavelengths of each mode, 1304 nm for the T mode (Fig.3d) and 1347 nm for the dielectric Anderson mode (Fig.3e). As shown in Ref. [26], the T mode, that is pushed inside the PBG for patterns with = 0.5, displays the tightest localization, as confirmed by the graph of Fig. 3a. In the experiment this mode is spectrally found between the cavity modes, and not at longer wavelengths as theory predicts; this is due to the extreme sensitivity of the accidental modes to disorder (in this case, induced by fabrication). Fig.3e shows the SNOM PL map of the Anderson mode. The map of this mode, localized in the dielectric part of the network and spatially located near the cavity, is definitely noisy: this is because of the low PL signal collected in a spectral region at wavelengths longer than the QDs PL first band. Nevertheless, the results shown in Fig. 2 and 3 clearly demonstrate the ability of the HuD structures to support a variety of localized modes: three types of modes coexist in a small area and in a relatively narrow spectral window: light is confined over a topological defect (T) of the network, undergoes Anderson localization, and, finally, can be trapped with a high in an engineered cavity. We now address the dependence of the cavity modes spectral positions on the variation of the two structural parameters previously introduced, the wall thickness and the central hole radius . We first analyze the spectral positions of each of the resonances and their behavior. Here, we focus on structures with a fixed value of = 1, since, as argued before, the global scaling factor, , determines the spectral positioning of the modes that achieve maximum values. We performed FEM simulations of nine cavities with three different values of and three different values of , specifically: =0.36 , 0.4 and 0.44 , and =0.2 , 0.3 and 0.4 . The theoretical trends of the spectral positions of the three higher order modes, O, H and Q, with respect to and are shown in Fig.4a.
On the horizontal axis we report the three values of , while different values of are highlighted in different colors (green for =0.36 , red for =0.4 and blue for =0.44 ). FEM maps of the electric field intensity are reported in the insets of the graphs. The simulations show an analogous behavior between the studied systems and analogous photonic crystal cavities; indeed, we note an overall trend in two directions, depending on the increase of and on the increase of : as the central hole radius increases, an "air defect" is formed [36] and the central wavelength of all modes decreases. In contrast, as the wall thickness is increased, we transition towards a "dielectric defect" and the spectral positions of the cavity modes are redshifted. Interestingly, while the O and H modes exhibit shifts that are comparable and approximately constant, the Q mode results more affected by the structural changes and displays non-monotonous shifts with the radius dimension (see the results for = 0.4 ), for which an unexpected redshift is observed. Next, we compare the theoretical results with the SNOM data, reported in Fig.4b. We have performed a scan over the nine cavities with different parameters and extracted the changes in the modes spectral position from Lorentzian fits. Three different topographies for the samples with =0.2 , 0.3 and 0.4 ( =0.36 ) are shown in Fig.4c, where the increasing of the air hole at the center of the cavity can be appreciated. The blueshift with the increase of and the redshift with the increase of previously found in simulations are faithfully reproduced here. We observe a good agreement between theory and experiment, with the SNOM data reproducing the trends reported in Fig. 4a. However, the O resonances for = 0.36 and = 0.4 are not found. It is also possible to observe a ≈ 20 nm shift between simulations and experiments. Such deviations between theory and experiment are typical of cavities even in photonic crystal systems and they can be attributed to several uncertainties regarding the difference between nominal and fabricated structural parameters (slab thickness, wall thickness, central hole radius etc). We now analyze the optical quality factors, of the localized modes identified. In Fig.5a we report the theoretical trends for the O, H and Q modes obtained with the same FEM simulations of Fig.4a (we adopt the same color and symbol convention).
The values of in FEM simulations are calculated through non-hermitian perturbation theory [37]. The behavior of the as a function of the structural parameters and is less intuitive compared to the spectral position of the modes. The simulation shows a common trend, for all the three modes, with the radius of the cavity hole: by increasing , decreases. This tendency can be explained by considering that for photonic cavities in a slab configuration, the out of plane losses play an essential important role in determining the overall quality factor; in order to prevent mode leakage in the vertical direction one needs to rely on index confinement [19,36]. As such, smaller values of translate into a higher effective index for the slab and hence better vertical confinement. On the contrary, the influence of on the values seems not to indicate an universal trend. This is most likely due to the fact that the thickness of the dielectric walls in the network affects both the modes associated with the HuD surround and the cavity modes. Either way, has much less impact on the values than , especially for H and Q modes. By analyzing the experimental s, extracted from the lorentzian fits of the SNOM measurements and reported in Fig. 5b, we note that generally the highest values are obtained for = 0.2 . However, experimental values show a reduction compared to simulated values, and, interestingly, the reduction is larger for the O modes than for the Q modes. Moreover, while the experimental overall maximum value obtained is 5600 for mode H in the configuration of = 0.2 and =0.36 , FEM simulations establish that mode O has the highest . These discrepancies between theory and experiment can be attributed to unavoidable fabrication induced disorder and tip perturbation effects [38,39], but also to the presence of the QDs. Specifically, QDs can be considered also as a channel of loss; this means that the results in terms of losses that we obtain for modes at wavelengths overlapping with the QDs emission bands can be affected by this. In the specific case of Fig.5b, mode O is the closest one [1320-1380] nm to the first QDs band (see graph of Fig.2b), and this might explain why we are not detecting the maximum as in simulations. Clearly, going from a structure with = 1 to = 0.9 a rigid blueshift of almost 100 nm in the spectra is obtained, and for this reason the modes shown in Fig.2 are not ideal candidate for estimations, despite their high signal to noise ratio. Among the reasons explaining the deviations from theory, one can consider also the roughness of the membrane which is not atomically flat due to the QDs density. The theoretical simulations reported in [19] had predicted engineered modes with as high as 20000; however, in this work, we deal with samples of finite size surrounded by dielectric unpatterned regions. The FEM results reported in Fig. 5, simulating the fabricated sample, confirm that a maximum of 10000 should still be expected. In addition, the presence of the SNOM tip perturbation effects and of the QDs further lowers the value of the measured .
In conclusion, we investigated experimentally the near-field properties of localized modes in an engineered optical cavity embedded in a hyperuniform disordered environment. We showed interesting features like the coexistence of Anderson localized modes at the PBG edges with cavity modes exhibiting an experimental quality factor as high as 6000. This is very relevant for practical applications, since the ability of optically characterizing localized modes of different symmetry and frequency in the same physical cavity and to detect light modes with different localization properties can have a great impact on all-optical switching, implementations of linear-optical quantum information processors and single photon sources. Following the insight provided by modified L3 cavities, this work could pave the way for realizing photonic cavities in HuD structures with even higher Q factors, for example by employing more than one engineered hole [1,20]. Moreover, the results presented here can open novel approaches for finely tuning the features of resonant cavity modes that, with respect to their periodic counterparts, are located in a more adaptable and flexible architecture, induced by the isotropy of HuD systems.
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| [] |
[
"Problems with the Current Cosmological Paradigm",
"Problems with the Current Cosmological Paradigm"
] | [
"T Shanks \nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamEngland\n"
] | [
"Department of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamEngland"
] | [
"Maps of the Cosmos, IAU Symp. 216 ASP Conference Series"
] | We note that despite the apparent support for the ΛCDM model from the acoustic peaks of the CMB power spectrum and the SNIa Hubble diagram, the standard cosmological model continues to face several fundamental problems. First, the model continues to depend wholly on two pieces of undiscovered physics, namely dark energy and cold dark matter. Then, the implied dark energy density is so small that it is unstable to quantum correction and its size is fine-tuned to the almost impossible level of one part in ≈ 10 102 ; it is also difficult to explain the coincidence between the dark energy, dark matter and baryon densities at the present day. Moreover, any model with a positive Λ also creates fundamental difficulties for superstring theories of quantum gravity. We also review the significant number of astrophysical observations which are now in contradiction with the ΛCDM model. On the grounds that the SNIa Hubble diagram is prone to evolutionary corrections and also that the CMB power spectrum may be contaminated by the effects of foreground ionised gas, we argue that the existence of such systematics could still allow more satisfactory, alternative, models to appear. We suggest that if H 0 ∼ < 50 kms −1 Mpc −1 then a simpler, inflationary model with Ω baryon = 1 might be allowed with no need for dark energy or cold dark matter. We note that the clear scale error between HST Cepheid and Tully-Fisher galaxy distances and also potential metallicity dependencies for both the Cepheid P-L relation and the SNIa Hubble diagram may mean that such a low value of H 0 cannot yet be ruled out. | 10.1017/s0074180900196834 | [
"https://export.arxiv.org/pdf/astro-ph/0401409v1.pdf"
] | 14,472,728 | astro-ph/0401409 | de5f797e2c6a800299627695cebeeb360c23a232 |
Problems with the Current Cosmological Paradigm
2004
T Shanks
Department of Physics
University of Durham
South RoadDH1 3LEDurhamEngland
Problems with the Current Cosmological Paradigm
Maps of the Cosmos, IAU Symp. 216 ASP Conference Series
2004Eds. M. Colless & L. Staveley-Smith
We note that despite the apparent support for the ΛCDM model from the acoustic peaks of the CMB power spectrum and the SNIa Hubble diagram, the standard cosmological model continues to face several fundamental problems. First, the model continues to depend wholly on two pieces of undiscovered physics, namely dark energy and cold dark matter. Then, the implied dark energy density is so small that it is unstable to quantum correction and its size is fine-tuned to the almost impossible level of one part in ≈ 10 102 ; it is also difficult to explain the coincidence between the dark energy, dark matter and baryon densities at the present day. Moreover, any model with a positive Λ also creates fundamental difficulties for superstring theories of quantum gravity. We also review the significant number of astrophysical observations which are now in contradiction with the ΛCDM model. On the grounds that the SNIa Hubble diagram is prone to evolutionary corrections and also that the CMB power spectrum may be contaminated by the effects of foreground ionised gas, we argue that the existence of such systematics could still allow more satisfactory, alternative, models to appear. We suggest that if H 0 ∼ < 50 kms −1 Mpc −1 then a simpler, inflationary model with Ω baryon = 1 might be allowed with no need for dark energy or cold dark matter. We note that the clear scale error between HST Cepheid and Tully-Fisher galaxy distances and also potential metallicity dependencies for both the Cepheid P-L relation and the SNIa Hubble diagram may mean that such a low value of H 0 cannot yet be ruled out.
Introduction
It is a recurrent recent theme that we live in a 'New Age of Precision Cosmology' to the point where we may even be witnessing 'the end of cosmology'. These views are prompted by the cosmic microwave background anisotropy results from Boomerang and WMAP (Netterfield et al., 2002, Hinshaw et al., 2003 on the one hand and the SNIa Hubble Diagram results on the other (Riess et al., 1998, Perlmutter et al., 1999. These results both appear to indicate that the Universe is dominated by Cold Dark Matter and Dark Energy. But both fundamental and astrophysical problems for ΛCDM remain. These are significant enough to suggest that continued inspection of the current cosmological data for ways out of the current 'concordance' model may still be worthwhile. Here, after considering the fundamental problem areas for the standard model, we shall 2 T. Shanks look at the CMB and SNIa results which are the main observational pillars of the model and suggest that they may be more susceptible to systematic error than currently emphasised. This shall prompt us to look at alternative models which drop the assumption of either cold dark matter or dark energy or both.
A new age of precision cosmology?
The idea that the age of precision cosmology has dawned, is based on the Boomerang and WMAP CMB anisotropy experiments' detections of the first acoustic Doppler peak at l = 220 (≈1-2 deg). Such a large spatial scale for the first peak is expected in a spatially flat, CDM Universe. The confirmation of the Boomerang results by the WMAP experiment has removed any doubt as to the observational reality of this detection. This observation is complemented by the evidence for an accelerated expansion seen in the SNIa Hubble Diagram. Jointly, these two observations appear to require a zero spatial curvature Universe with Ω Λ = 0.7 and Ω m = 0.3.
Although the argument for the standard model has undoubtedly been strengthened by the above two observations, fundamental problems still remain. For example, the standard ΛCDM model still relies on two pieces of undiscovered physics! The first is the CDM particle for which there is still no laboratory detection, some twenty years after it was first proposed (Blumenthal et al., 1982, Bond, Szalay & Turner, 1982, Peebles, 1982. For the optimists, the search for the CDM particle is likened to the search for the neutrino in the 1930's but for the pessimists the situation may be more like the search for the electro-magnetic ether at the end of the 19th Century. The second piece of undiscovered physics is dark energy. The invoking of dark energy also makes ΛCDM complicated and fine-tuned. There are two separate fine-tuning problems associated with dark energy, at least when it is represented as a cosmological constant. First, the vacuum energy term is small; after inflation it is only one part in 10 102 of the energy density in radiation. This small size means that the dark energy is unstable to quantum correction (e.g. Dvali, Gruzinov & Zaldarriaga, 2003). Second, there is the coincidence that it is only relatively close to the present day where Ω Λ ≈ Ω m ; there seems no clear reason why the present day should have this special status. Even for those who dislike fine-tuning arguments, to start with one fine tuning (flatness) problem and end up with several seems circular! Several solutions have been proposed to solve the Λ fine-tuning problems. For example, quintessence is the name given to the dark vacuum energy when it takes the form of a scalar field slowly rolling down a potential, usually from an initially high value, until the present day (Wetterich, 1988;Peebles & Ratra, 1988). Indeed, the initial value can be comparable to the radiation energy density after inflation, thus addressing the first Λ fine-tuning problem. However, it offers no solution to the second Λ fine-tuning problem of the coincidence with the matter energy-density at the present day.
Another solution is represented by the aptly-named Cardassian model (Chung & Freese, 2000, Freese & Lewis, 2002 where an extra term is added to the Friedmann equation so that H 2 = Aρ + Bρ n , with n < 2/3. (A related model is the brane-induced gravity model of Dvali, Gabadadze & Porrati, 2000). The extra power-law term could arise from gravitational effects caused by embedding the Universe as a 3(+1)-D brane in a higher dimensional entity. Here the accelerated expansion arises from the extra term associated with the matter density, ρ. This has the benefit of removing the need for dark vacuum energy and even cold dark matter and so could be said to reduce the dependence of the model on undiscovered physics. The removal of dark energy again addresses the first Λ problem but the second problem of why the acceleration only starts to dominate at the present day is again left unaddressed.
We note that a further problem has appeared for any model with a positive cosmological constant in that superstring theories of quantum gravity with compactified extra spatial dimensions are much more viable in models where Λ < 0 (Anti-de Sitter space) than in cosmologies where Λ > 0 (Banks, 2000, Witten, 2001, Deffayet, Dvali & Gabadadze, 2002. Although solutions have been suggested to this problem they appear highly contrived (Kachru et al., 2003). Thus there are many fundamental problems involved with the size and sign of the dark energy density required by the standard model. So unnatural does a small, positive cosmological constant appear to be that several authors have resorted to invoking the anthropic principle as the most likely hope for an explanation (Efstathiou, 1995, Martel, Shapiro & Weinberg, 1998.
Even without dark energy, further fundamental problems are inherent in any model based on CDM. First, as noted by Peebles (1984), any CDM model has some fine-tuning since Ω CDM ≈ Ω baryon . Attempts have previously been made to explain this coincidence if the cold dark matter particle has approximately the mass of the proton (Turner & Carr, 1986, priv. comm.), but the accelerator lower limit on the mass of the neutralino, for example, is now an order of magnitude higher than this. Second, baryonic dark matter is needed anyway since nucleosynthesis implies that Ω baryon ≈ 10 × Ω star . The baryonic candidate for the Ω 0 ≈ 0.1 dark matter may then be a contender also for the Ω = 1 dark matter candidate (see Section 5 below). Third, the dark matter in the Coma cluster has a significant baryon component with ≈20% of the virial mass of Coma now well known to be hot X-ray gas (Lea et al., 1973). The discovery of substantial amounts of X-ray gas in clusters such as Coma has reduced the Coma mass-to-light ratio from M/L≈60-600 to M/L≈5. If the Coma 'missing mass' problem is only at the level of M/L≈5 then it may be considered less plausible to invoke a cosmological density of exotic particles than if M/L≈60-600! If Zwicky had known about the X-ray gas in Coma, the question is whether he would have been inclined to introduce the term 'missing mass' at all!
Astrophysical problems for ΛCDM
There are several other problems for the ΛCDM model which might be classed more observational or astrophysical than fundamental. First, the mass profiles of low surface brightness galaxies appear to be less sharply peaked than predicted by CDM models (Moore et al, 1999a). Second, the large numbers of sub-haloes predicted in galaxy haloes may make spiral disks subject to tidal disruption on timescales of less than a Gigayear (Moore et al., 1999b). Third, the observed galaxy luminosity function is much flatter than the mass distribution predicted by CDM; attempts to suppress star-formation by invoking significant feedback Figure 1.
The 2-D spatial cross-correlation between QSOs and foreground APM/SDSS selected galaxy groups and clusters from Myers et al. (2003). The anti-correlation is the result expected if the foreground clusters are lensing the background QSOs in a high-density, Ω m ≈ 1 Universe. The similarity of the results shown by the squares and triangles show the anti-correlation is robust to whether the search is made for QSOs around clusters or vice-versa.
in low-mass haloes appear to create further problems at higher masses (Benson et al., 2003). Fourth, the slope of the galaxy correlation function is flatter than predicted by ΛCDM, suggesting that the galaxy distribution must be antibiased on scales r < 1h −1 Mpc. This means that a simple high peaks bias model is disallowed (Colin et al., 1999) -although this is not a problem in principle, it does mean that the bias model has to be relatively complicated. Fifth, the L X − T relation for galaxy clusters is not scale-free as predicted by hierarchical models (Lloyd-Davies et al. 2000). Some attempts have been made to fix things by suggesting that at small-scales, entropy might be increased by shocks created during the process of galaxy formation (Voit et al., 2003). However, the simpler explanation is that it is the mass distribution that is not scale free and this would represent a fundamental argument against hierarchical models such as ΛCDM.
Of course, any evidence that Ω m ≈ 1 could be taken as evidence against the standard ΛCDM model which requires Ω m ≈ 0.3. One such piece of evidence comes from the lensing of background QSOs in the 2dF QSO redshift survey by foreground galaxy groups and clusters (Croom & Shanks 1999, Myers et al., 2003. These authors find a high lensing mass per cluster which leads to a 2σ rejection of the Ω m = 0.3 model. Evidence for Ω m ≈ 1 even arises from the space abundances of galaxy clusters (Eke et al., 1998, Vauclair et al., 2003. The evolution of clusters is often quoted as vital evidence for the concordance model. But many of these estimates seem remarkably close to Ω m = 1. Vauclair et al. claim that the data support 0.8 < Ω m < 1. The best estimate of Eke et al. is Ω m = 0.45±0.25. Even in the latter case, it might be recalled Guth (1981) argued that the Ω m > 0.01 lower limit from nucleosynthesis left Ω m embarassingly close to unity and now even the estimate of Eke et al. lies within a factor of two of the Einstein-de Sitter value.
Escape routes: SNIa evolution + CMB foreground contamination
Given this collection of fundamental and astrophysical problems, it is worthwhile considering if there are any escape routes from the observations that underpin standard model. The escape route from the SNIa Hubble diagram is certainly clear; there is the obvious possibility that the SNIa maximum luminosity evolves with look-back time in a way that is not detectable in the SNIa spectra. The SNIa are ≈0.5mag fainter at z≈0.5 if Ω Λ = 0.7 and Ω m = 0.3 than in the Einstein-de Sitter case. Quite natural evolutionary mechanisms for SNIa certainly exist. For example, the metallicity of the SNIa progenitor stars at high redshift are likely to be lower than they are locally. Also the C/O abundance ratio of the White Dwarf will change as it awaits the accretion of mass which will trigger the explosion. These evolutionary corrections are likely to be comparable to the above effect of q 0 (Hoeflich et al., 2000).
In the case of the CMB power spectrum, the main escape route here is likely to be the CMB foregrounds. Although there are now quite good constraints from the CMB spectral index on contamination from Galactic synchrotron and dust, the WMAP results have suggested two other sources of foreground contamination. The excess TE polarisation detected by WMAP at large angular scales is interpreted as strong evidence for an early epoch of reionisation at 10 < z < 20 with optical depth, τ ≈ 0.17 (Kogut et al., 2003). Homogeneous reionisation with this optical depth reduces the amplitude of the temperature power spectrum peaks by ≈30%. Inhomogeneous reionisation could also alter the peak shapes. Although this is expected only to affect the smaller peaks, the largescale peaks could also be affected, depending on the model and the details of the reionisation process.
Another source of foreground contamination could be due to the SZ effect. Myers et al. (2004) have cross-correlated the Abell R ≥ 2, |b| > 40 deg, clusters with the WMAP 94GHz W band data and found signifiant anti-correlation which they interpret as due to the SZ effect. Similar signals were found in the groups and clusters detected in the APM and 2MASS catalogues. Interestingly, they found that in the case of the rich clusters the anti-correlation appeared to extend to scales larger than the 12. ′ 6 W-band beam size, out to scales of ≈ 1 deg (≈ 5h −1 Mpc) which could be caused by ionised supercluster gas. Although the significance of the extended signal is lower than on the beam-size, if it is real then there could be important implications. In particular, there could be a significant SZ contribution to even the first peak of the power spectrum on ≈ 1 − 2 deg scales. Thus on grounds of both the ionised gas at the epoch of Cross-correlation of WMAP 94 GHz W band data with ACO R ≥ 2 Abell clusters for combined ACO |b| > 40 deg N+S samples. The dashed, dotted and dot-dash lines are isothermal models for the SZ decrement as presented by Myers et al. (2004). reionisation at z ≈ 15 and the hot gas in clusters at lower redshift, the CMB signal may have come through more foreground 'traffic' than previously expected and the resulting contamination may have seriously compromised its primordial signal.
H 0 route to a simpler model
Given that the quintessence and Cardassian modifications to the standard model only represent partial solutions to the problems of dark energy and dark matter we next consider a previously suggested route via H 0 to a simpler model. Shanks (1985Shanks ( , 1991Shanks ( , 1999Shanks ( , 2000 suggested that if H 0 ∼ < 30kms −1 Mpc −1 then there might be no need to introduce either dark matter or dark energy. With a low value of H 0 , an inflationary model with Ω baryon =1 is then better placed to escape the baryon nucleosynthesis constraint, since Ω 0 = ρ 0 /ρ c and ρ c = 3H 0 2 /8πG. Simultaneously, the low value of H 0 means that the X-ray gas in the Coma cluster increases towards the Coma virial mass, since M gas /M virial ∝ H 0 −1.5 . Finally, the lifetime of an Einstein-de Sitter Universe increases as 1/H 0 to become compatible with the ages of the oldest stars. Given the historical uncertainty there has been in observational estimates of H 0 , the potential simplification in cosmology that this very simple model offers, removing the need for dark energy and cold dark matter, provides clear motivation to continue to investigate the distance scale and Hubble's constant. Tully-Fisher versus metallicity/incompleteness corrected HST Key Project Cepheid distances (Allen & Shanks, 2004). The TF relation underestimates Virgo galaxy distances by 34±6%. The least squares fit (dashed line) shows 3.5σ evidence for a TF scale error.
The value of Hubble's constant has been notoriously difficult to estimate. prior to the opening of the Palomar 5-m telescope in 1950, Hubble's value was H 0 ≈ 500 kms −1 Mpc −1 . Since then, estimates of H 0 have moved down to H 0 ≈ 70 kms −1 Mpc −1 . We now argue that the value of H 0 may fall yet further. Some 25 galaxies have had Cepheids detected by HST. Seventeen of these were observed by the HST Distance Scale Key Project (Freedman et al., 1994, Ferrarese et al., 2000. Seven were observed in galaxies with SNIa by Sandage and collaborators (eg Sandage et al., 1996) and M96 in the Leo I Group was observed by Tanvir et al. (1995). Allen & Shanks (2004) have used these data to update the comparison of I-band Tully-Fisher (TF) distances of Pierce & Tully (1992) with the published HST Cepheid distances. These authors find that TF distance moduli at the Virgo distance are underestimates by ≈22±5%. If the Key Project metallicity correction (see also Hoyle, Shanks & Tanvir, 2003) and the P-L incompleteness correction of Allen & Shanks is applied to the Cepheids then the TF moduli at the Virgo distance are now underestimates by 34±6% (see Fig. 3). This reduces Tully-Fisher estimates of H 0 from ≈85 to ≈65kms −1 Mpc −1 (Giovanelli et al., 1997, Shanks 1997, Shanks, 1999, Sakai et al., 1999. Of course, H 0 might be further reduced if the TF scale error persists to Coma. The correlation of Cepheid residuals with line-width suggests TF distances may be Malmquist biased -possibly implying a bigger TF scale error at larger distances. This clear problem for TF distances, which previously has been the 'gold standard' of secondary distance indicators, warns that errors in the extragalactic distance scale may still be seriously underestimated! Figure 4. The SNIa absolute magnitude-metallicity relation using the SNIa peak magnitudes of Gibson et al. (2000), now corrected for ∆m 15 and Cepheid metallicity/incompleteness (Allen & Shanks 2004). The least squares fit (solid line) shows 2σ evidence for a correlation.
Eight HST Cepheid galaxies also have SNIa distances. Correcting the Cepheid scale for metallicity and incompleteness bias after Allen & Shanks and then using these distances to derive peak luminosities using the SNIa data from Gibson et al. (2000) implies a possible correlation between Type Ia peak luminosity and metallicity (see Fig. 4). Such a scatter in SNIa luminosities could easily be disguised by magnitude selection (Malmquist) effects at moderate redshifts. At higher redshift the correlation is in the right direction to explain away the need for a cosmological constant in the Supernova Hubble Diagram results, since galaxies at high redshift might be expected to have lower metallicity. Thus the conclusion is that if Cepheids have strong metallicity dependence then so have SNIa and therefore SNIa estimates of q 0 and H 0 may require significant correction.
Conclusions
Our main conclusions are as follows:-
• ΛCDM gains strong support from the WMAP and Boomerang CMB peaks and also the SNIa Hubble diagram -but leaves a standard model which is fine-tuned to the almost impossible level of one part in 10 102 and based on two pieces of undiscovered physics, dark energy and cold dark matter.
• The size of the vacuum energy density implied by the SNIa Hubble diagram is so small that it is unstable to quantum corrections.
• Superstring models of quantum gravity which invoke compactified higher spatial dimensions are broadly incompatible with the positive cosmological constant of the ΛCDM model and prefer models with negative or no cosmological constant.
• ΛCDM also has astrophysical problems predicting galaxy mass profiles that are too cuspy at small scales and a galaxy luminosity function that is too steep. The model also has a problem with new results from QSO lensing that prefer a value of Ω m ≈ 1.
• The main escape routes to other models include the expectation that the SNIa Hubble diagram may require evolutionary corrections. Further, the precision of the CMB power spectrum may still be compromised by foreground contamination from the epoch of reionisation at z ≈ 15 and the SZ signal from galaxy clusters at z ∼ < 1.
• We have argued that if H 0 ∼ < 50 kms −1 Mpc −1 then it might allow a simpler, inflationary model with Ω baryon = 1 and with no need to invoke dark energy or cold dark matter.
• The strong scale error between HST Cepheid and TF distances and the potential metallicity dependencies for the maximum luminosity of SNIa and the Cepheid P-L relation suggests that there may still be systematic errors in the distance scale which may allow a significantly lower value of H 0 ; our very simple model with Ω baryon = 1 may therefore still not be ruled out.
Finally, we note that the fundamental weaknesses of the standard model make the conclusion that the Universe is CDM and dark energy dominated also vulnerable to the new higher-dimensional 'brane-world' cosmologies motivated by string theories (Randall & Sundrum 1999, Dvali, Gabadadze & Porrati, 2000, Freese & Lewis, 2002. These cosmologies offer a rich, new variety of terms to add to the standard Friedmann solution of the field equations. The resulting increased flexibility in observational cosmology will at least increase the chance of finding alternative cosmologies to ΛCDM. For example, there exist Cardassian models that fit the current CMB and SNIa data, assuming a baryon-dominated model. This model is still highly finely-tuned but no more than ΛCDM. Thus whether the increased flexibility in observational cosmology arises from this route or from the presence of systematic errors in the current cosmological data as argued here, it seems likely that a more satisfactory model than ΛCDM will at some stage appear and therefore that the rumours of the 'end of cosmology' may well be premature!
Figure 2. Cross-correlation of WMAP 94 GHz W band data with ACO R ≥ 2 Abell clusters for combined ACO |b| > 40 deg N+S samples. The dashed, dotted and dot-dash lines are isothermal models for the SZ decrement as presented by Myers et al. (2004).
Figure 3 .
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| [] |
[
"Prediction of tone detection thresholds in interaurally delayed noise based on interaural phase difference fluctuations",
"Prediction of tone detection thresholds in interaurally delayed noise based on interaural phase difference fluctuations"
] | [
"Mathias Dietz [email protected] ",
"Jörg Encke ",
"Kristin I Bracklo ",
"Stephan D Ewert ",
"\nDepartment für Medizinische Physik und Akustik and Cluster of Excellence \"Hearing4all\"\nUniversität Oldenburg\n26111Oldenburg, Germany\n",
"\nUniversität Oldenburg\n26111OldenburgGermany\n"
] | [
"Department für Medizinische Physik und Akustik and Cluster of Excellence \"Hearing4all\"\nUniversität Oldenburg\n26111Oldenburg, Germany",
"Universität Oldenburg\n26111OldenburgGermany"
] | [] | Differences between the interaural phase of a noise and a target tone improve detection thresholds. The maximum masking release is obtained for detecting an antiphasic tone (Sπ) in diotic noise (N0). It has been shown in several studies that this benefit gradually declines as an interaural delay is applied to the N0Sπ complex. This decline has been attributed to the reduced interaural coherence of the noise. Here, we report detection thresholds for a 500 Hz tone in masking noise with up to 8 ms interaural delay and bandwidths from 25 to 1000 Hz. When reducing the noise bandwidth from 100 to 50 and 25 Hz, the masking release at 8 ms delay increases, as expected for increasing temporal coherence with decreasing bandwidth. For bandwidths of 100 to 1000 Hz, no significant difference was observed and detection thresholds with these noises have a delay dependence that is fully described by the temporal coherence imposed by the typical monaurally determined auditory filter bandwidth. A minimalistic binaural model is suggested based on interaural phase difference fluctuations without the assumption of delay lines.3 | 10.1051/aacus/2021054 | [
"https://arxiv.org/pdf/2107.00320v1.pdf"
] | 235,694,475 | 2107.00320 | 8034434cdc9024ca0ab2948f7910040b31f0e531 |
Prediction of tone detection thresholds in interaurally delayed noise based on interaural phase difference fluctuations
Mathias Dietz [email protected]
Jörg Encke
Kristin I Bracklo
Stephan D Ewert
Department für Medizinische Physik und Akustik and Cluster of Excellence "Hearing4all"
Universität Oldenburg
26111Oldenburg, Germany
Universität Oldenburg
26111OldenburgGermany
Prediction of tone detection thresholds in interaurally delayed noise based on interaural phase difference fluctuations
1 a) Author to whom correspondence should be addressed.
Differences between the interaural phase of a noise and a target tone improve detection thresholds. The maximum masking release is obtained for detecting an antiphasic tone (Sπ) in diotic noise (N0). It has been shown in several studies that this benefit gradually declines as an interaural delay is applied to the N0Sπ complex. This decline has been attributed to the reduced interaural coherence of the noise. Here, we report detection thresholds for a 500 Hz tone in masking noise with up to 8 ms interaural delay and bandwidths from 25 to 1000 Hz. When reducing the noise bandwidth from 100 to 50 and 25 Hz, the masking release at 8 ms delay increases, as expected for increasing temporal coherence with decreasing bandwidth. For bandwidths of 100 to 1000 Hz, no significant difference was observed and detection thresholds with these noises have a delay dependence that is fully described by the temporal coherence imposed by the typical monaurally determined auditory filter bandwidth. A minimalistic binaural model is suggested based on interaural phase difference fluctuations without the assumption of delay lines.3
Abstract
Differences between the interaural phase of a noise and a target tone improve detection thresholds. The maximum masking release is obtained for detecting an antiphasic tone (Sπ) in diotic noise (N0). It has been shown in several studies that this benefit gradually declines as an interaural delay is applied to the N0Sπ complex. This decline has been attributed to the reduced interaural coherence of the noise. Here, we report detection thresholds for a 500 Hz tone in masking noise with up to 8 ms interaural delay and bandwidths from 25 to 1000 Hz. When reducing the noise bandwidth from 100 to 50 and 25 Hz, the masking release at 8 ms delay increases, as expected for increasing temporal coherence with decreasing bandwidth. For bandwidths of 100 to 1000 Hz, no significant difference was observed and detection thresholds with these noises have a delay dependence that is fully described by the temporal coherence imposed by the typical monaurally determined auditory filter bandwidth. A minimalistic binaural model is suggested based on interaural phase difference fluctuations without the assumption of delay lines. INTRODUCTION The human binaural system can exploit differences between the interaural phase of a masker noise and a target tone to improve detection thresholds [1]. The maximum masking release is obtained for detecting an antiphasic tone (Sπ) in diotic noise (N0). It has been shown in several studies that this benefit gradually declines as an interaural time difference (ITD) is applied [2,3].
Two different hypotheses have been proposed to account for the decline.
One hypothesis, proposed by Langford and Jeffress [2], attributed the reduction of the binaural masking level difference (BMLD) with increasing noise delay to what they referred to as "interaural correlation of the noise". A more contemporary wording for their quantity is "normalized cross-correlation coefficient", i.e. the value of the normalized cross-correlation function at τ = 0.
The other hypothesis, also proposed by Jeffress [4], but before the above, is that using internal time delays ("delay lines") the auditory system has access to more of the cross-correlation function than just the cross-correlation coefficient at τ = 0. Such circuitry has indeed been found in the barn owl, where left and right sided inputs propagate along counterdirected axons [5].
Coincidence detecting neurons along the axonal delays effectively cross correlate the inputs at different values of τ. An ideal delay line could perfectly compensate any external noise delay (τ=ITD) by an opposed internal delay (-τ), allowing for maximum BMLDs even at large noise delays [6]. It is, however, reasonable to assume that such a compensation mechanism introduces errors for increasing internal delays. This increase in error is commonly simulated as a decrease of the density of correlating elements with increasing internal delay. This relationship is captured by the p(τ) function [7,8]. Based on this second hypothesis, models use p(τ) as a fitting parameter, i.e. they estimate the delay line length and potency from the decline of the BMLD with noise delay [7][8][9][10].
For the first "cross-correlation coefficient"-based concept, however, this degree of freedom does not exist. In this case, the cross correlation, or more generally speaking the complex-valued temporal coherence of the analytical signal γ [11], ( ) = 〈 ( + ) * ( )〉 〈| ( )| 2 〉 is solely determined by the spectrum of the noise [2,6] and is proportional to the inverse Fourier transform −1 of its power spectral density n' (Wiener-Khinchin theorem):
( ) ∝ −1 ( ′( )).
(1)
This means that the bandwidth of the input signal determines the decay of | ( )|: the broader the spectrum, the shorter the temporal coherence. The maximum bandwidth observed by the binaural system is limited by some form of band-pass filter. As a consequence, the filter bandwidth effective at the input to the binaural interaction ultimately dictates how binaural unmasking depends on the noise delay. As a first hypotheses, it seems reasonable to assume that the effective bandwidth at the binaural stage matches those from monaural estimates (e.g. 79Hz ERB at 500Hz [12]). Some previous studies indeed already tried to determine the effective bandwidth based on binaural unmasking data. Rabiner et al. [6] found that their experimental data could be best accounted for using an 85-Hz wide (at -3 dB) triangular filter. Langford and Jeffress [2] coarsely estimated a 100 Hz bandwidth but without specifying the filter shape and bandwidth definition. Both estimates are a little larger but close to the monaural estimates.
The goal of this study is, to revisit if and to what extent the decline in binaural unmasking as a function of noise delay can be explained solely based on the decline of temporal coherence in a simple, "minimalistic" binaural model. The model uses fluctuation of the interaural phase difference [13,14] resembling a physiologically plausible feature that might be extracted from a neural representation of the signal [15]. This physiologically inspired IPD metric is directly related to the correlation coefficient i.e. the degree of coherence [16,17]. Early attempts [2,6] connecting noise delay and temporal coherence appear promising, but have not been followed up, to test if filter bandwidths and filter shapes that are most commonly used in more recent models can quantitatively account for the data. If the simulated decline is faster than the experimental decay, delay lines have to be in operation, compensating for the external delay and thus increasing the coherence at the level of binaural interaction.
To complicate the argument, the effective processing bandwidth of the binaural system has previously been discussed controversially, with some studies suggesting it is larger than the monaural filter bandwidth, at least for certain complex maskers [8,18,19]. However, there is growing consensus that at least for the simplistic band-pass filtered noise investigated in the present study the binaural filter bandwidth is not wider [9,10,[20][21][22][23].
II. Experiment
A. Participants
Ten young normal-hearing volunteers (21-33 years, median 24 years, 5 male, 5 female) were recruited -all university students. Most subjects had some experience in speech in noise tests.
To our knowledge, no subject had prior experience in dichotic tone-in-noise detection. All subjects received at least 90 min of training prior to data collection. All audiometric thresholds were equal to or less than 15 dB HL from 125 to 10,000 Hz, pure tone averages less or equal 5 dB HL, and differences across ears did not exceed 5 dB in both pure tone average and at 500 Hz. One reason for these relatively strict inclusion criteria is that it has been shown recently that subjects with a slight (sub-clinical) hearing loss have a reduced binaural release from masking [25].
C. Stimuli
Gaussian noise was band-pass filtered by cutting out spectral components outside the pass-band.
All noises were arithmetically centered at 500 Hz and the bandwidths were 25,50,100,150,200 or 1000 Hz. Noises had a duration of 380 ms including 20-ms raised cosine on-and offset ramps.
The noise level was kept at a constant spectrum level of 45.5 dB relative to 20 µPa. Fully correlated noises were presented with interaural delays (τ) of 0, 2, 4, or 8 ms, or interaurally uncorrelated. The delay was applied prior to gating. Delays were chosen in multiples of the cycle duration at the 500 Hz center frequency, ensuring zero interaural phase difference (IPD) at 500
Hz.
Target tones had a frequency of 500 Hz and a duration of 300 ms, again including 20-ms raised cosine on-and offset ramps. Tones were always presented temporally centered in the noise. They were either interaurally in phase (S0) or antiphasic (Sπ).
D. Procedure
A 3-interval, 3-alternative forced choice procedure was employed, with two noise-only reference intervals (Nτ) and one target interval including both signal and noise. Subjects selected an interval by pressing the respective number key on a computer keyboard. Feedback was provided.
The signal level, initially 65 dB SPL, was adaptively changed in a 2-down, 1-up staircase procedure, aiming at the 70.7% correct rate [27]. The step size of 4 dB was reduced to 2 and 1 dB after the second and fourth reversal, respectively. After a total of 10 reversals each run was terminated and the average was taken across the last 6 reversals. Three of the ten listeners were not able to obtain N0Sπ thresholds below +8 dB above the masker spectrum level in the 100-Hz bandwidth noise, while the seven other subjects had thresholds less or equal +2 dB in their formal measurements. Comparable thresholds from other studies are in the range of -3 dB [20] to + 3 dB [3]. Two of the less-sensitive listeners were further tested in a sensitivity-optimized threshold ITD task. Their >100 µs thresholds were larger than any of 52 untrained listeners [28] and thus considered outliers. 3 of 10 listeners with excellent audiograms performed quite poor and in fact worse than the group average of less-sensitive listeners in [25].
To be able to compare the data to previous studies and to meaningfully apply statistics based on normally distributed values, only data from the other seven subjects are reported.
The experimental data is shown in Fig. 1 [3], similar to those by van der Heijden and Trahiotis [8], but overall higher than in the majority of studies focused on thresholds of highly trained listeners obtained with diotic masker [20,30]. We speculate that mixing in experimental conditions with interaural delay makes it harder for the subjects to fully train on the particularly subtle cues with narrow-band, i.e. tonal maskers and zero delay. is only about 2 dB, in line with [2].
To further assess the effect of bandwidth in the NSπ data, a two-way repeated-measures ANOVA [noise delay (5) x bandwidth (6)
III. Model predictions
A. Model description
The front end of the model employed here is essentially identical to the IPD model [13,14] and illustrated in Fig. 2a. Peripheral band-bass filtering is mimicked with a 4 th -order
Gammatone filter centered at the target frequency of 500 Hz, with an equivalent rectangular bandwidths of 79 Hz [12] in the implementation of [31]. Haircell processing is coarsely modelled by half-wave rectification, compression by taking the signal to a power of 0.4 and subsequent low-pass filtering with a 5 th order Butterworth filter with a 770 Hz cutoff frequency). For IPD extraction, the phases of left and right signal must be known by definition. The phase is extracted from the (unipolar) haircell representation by applying a second, broader Gammatone filter (2 nd order, 167 Hz bandwidth), referred to as temporal fine-structure (TFS) filter, again centered at 500 Hz. From the complex-valued output of this TFS filter, g(t), the argument is the phase. The TFS filter effectively reverts some effects of the haircell nonlinearity, including turning the unipolar signal into a bipolar signal again. In principle, for the purpose of the present study, the phase could have been obtained directly from the first Gammatone filter, however, the haircell stage and the TFS filter were kept as in the IPD model [13,14] to stay in the conceptual framework of auditory pathway models.
The instantaneous IPD, ( ), can now be derived by subtracting the phases from the left and the right signal, or, equivalently, by first multiplying the left signal and the complex conjugate of the right signal and then taking the argument from the product. A phase jitter, , in the form of Gaussian noise is added to the IPD as a limiting factor of binaural sensitivity, qualitatively corresponding to the time equalization jitter introduced by Durlach [32] or a combined monaural and binaural time jitter [32]. Adding an Sπ tone to more intense diotic noise causes the instantaneous IPD to fluctuate around zero. This fluctuation has previously been suggested as the detection cue [13,17,34,35].
In more general terms, it is assumed in the model that the target can be detected if the average fluctuation of the IPD in the target interval can be discriminated from the average fluctuation of the IPD in the noise-alone intervals.
In the current study, the long term average IPD of both target and reference intervals are always zero: ⟨ ⟩ = 0. It was therefore decided to simplify the model to only detect deviations from zero IPD, i.e. temporal averaging of the modulus across the entire observation interval:
⟨| |⟩. This value ranges from 0 in case of a diotic stimulus and no internal noise, to π for an interaurally antiphasic stimulus. For interaurally uncorrelated noise an average value of π/2 is obtained, resulting from a uniform distribution of | | in the range 0 to π in that case. A cosine mapping, projects the values to the interval 1 for no IPD to -1 for an antiphasic stimulus, and to 0 for an interaurally uncorrelated noise. In this simplified version, which can only compare intervals with no offset IPD, the internal variable cos⟨| |⟩ is practically identical to the interaural correlation coefficient, commonly used for similar purposes [10,29]. This term is then Fisher-Z transformed (Fig. 2b), again identical to the comprehensive binaural detection model by Bernstein and Trahiotis [29]. Last, a detector noise xD, is added, yielding the decision variable D.
= ℎ( ⟨| |⟩) + (4)
The arctanh Fisher-Z transformation expands differences near 1 and -1 and has previously been employed for correlation-based decision metrics [36] as well as for dichotic tone in noise detection [28]. In combination with the fixed-variance detector noise, an increased sensitivity to interaural correlation differences near 1 and -1 is obtained in combination with decreased sensitivity close to 0 (uncorrelated) as observed in, e.g. [37].
The model back-end is an artificial observer that retrieves the same three intervals in the same adaptive procedure as the listeners and selects the interval with the smallest decision variable D [9,38]. The artificial observer ran 100 runs in each condition. Rabiner et al. [6] where the x-axis shows the group delay and the y-axis the associated threshold level relative to the = 0 condition.
C. Application of the model to literature data
The suggested model was additionally tested in comparison to literature data for interaural correlation discrimination [37] and binaural unmasking with arbitrary group delays [6,39].
The current model can be directly applied to interaural correlation discrimination tasks.
The internal noise was also kept unchanged. Results are shown in Fig. 4 (a), again with the model operating as an artificial observer. Following the approach of [37], the d's were calculated between selected interaural correlations as given in their Table 1. ′ for any of the measured values of interaural correlation and the fully correlated stimulus ( ′ ( ,1) ) were then calculated by systematically summing over the calculated ′ as described in [37]. This process resulted in multiple approximations of ′ ( ,1) (e.g. ′ (0,0.8) + ′ (0.8,1) = ′ (0,1) and ′ (0,0.5) + ′ (0.5,0.9) + ′ (0.9,1) = ′ (0,1) ) all of which are given in Fig. 4 (a). It can be observed that the model is mostly able to reproduce the data. Only for uncorrelated noise (right-hand data points) the model slightly outperforms the average experimental data. Additionally, the model was employed to simulate detection thresholds obtained with noise maskers with a time delay in the range of 0 to 7.8 ms and an additional phase shift to adjust the resulting interaural phase difference at the target frequency of 500 Hz to zero, i.e. a pure group delay [6]. Results are shown in Fig. 4b. While no adjustment of the model parameters was necessary to reproduce the data of [37], an increase in the standard deviation of the IPD noise to = 0.45 rad was required here to simulate this data set. This increase in internal noise might be attributed to a shorter stimulus duration (while no quantification of the stimulus duration is given in [6] the stimuli were described as "short").
Thirdly, [39] The model uses the average absolute value of the instantaneous IPD as its decision metric.
This metric is used with the underlying assumption that IPD fluctuations are a cue in binaural tone-in-noise detection tasks [34]. This limitation makes the present version functionally equivalent to cross-correlation-coefficient based models and especially to the model of Bernstein and Trahiotis [10,29], disregarding internal delays.
Bernstein and Trahiotis [29] accounted for almost all of the data from six seminal studies without requiring internal delays. They only employed internal delays for tone detection in delayed noise in contrast to the current study. In combination, [29] and the present study account for a total of nine data sets with an interaural coherence based model. It is beyond the scope of the present study to simulate data sets with non-zero mean IPDs, such as [2,8], but first attempts with a manual setting of the mean IPD appear encouraging. Equivalently, correlation coefficient-based models could account for the binaural advantage offered by such noise, without requiring a delay line if they would use the complex-valued correlation coefficient, i.e.
interaural coherence [11], rather than the real-valued correlation coefficient. Alternatively to the here suggested model approach, it can be assumed that the current data can also be explained using an imperfect delay line with p() function and a similar data set could be simulated in such a model, e.g. [10]. Nevertheless, the simplistic IPD model of the present study indicates that a delay line is not required to account for the noise delay dependence of binaural unmasking.
The near-equivalence of the present IPD model and interaural correlation or coherence has further been shown exemplarily by simulating interaural correlation discrimination data [37] without changing model parameters. For the present model it does not make a difference if the masker-inherent IPD fluctuation, or synonymously the reduction in masker coherence, is introduced by a delay or by adding uncorrelated noise, which is in line with [2,6]. Moreover, it was shown that the current model is able to account quantitatively for Sπ detection thresholds in noises with arbitrary group delays when the phase delay at 500 Hz is fixed at zero. For noises wider than the critical bandwidth it predicts a threshold increase of just under 3 dB per millisecond delay, a little more than observed in [6] and a little less than in 39]. For 50-Hz wide noise, the increase is closer to 2 dB/ms. Many delay-line-based approaches fail to account for this bandwidth dependence, because sensitivity is dictated by the bandwidth-independent delay line potency [9,39] rather than the bandwidth-dependent loss of coherence. However, the model employed in the present study is not expected to be the only model that can account for these data. All previous models that employ relatively narrow filters [10,29] or a correspondingly steep function [8] are expected to be isomorphic for the present data [40]. Our model is effectively the same as the analytic description and fitting from Rabiner et al. [6], their eq. 13
and, amongst others, conceptually identical to what was proposed by Langford and Jeffress [2].
Nevertheless, the present IPD-fluctuation-based model reflects a physiologically motivated concept to account for the delay dependence of tone detection with interaural phase differences using a minimum of model assumptions.
The model code has been made available in the auditory modeling toolbox.
V. CONCLUSIONS
The decline of binaural unmasking with increasing noise delay can be solely attributed to the noise coherence at the output of common auditory filters. This processing concept, first suggested by Langford and Jeffress [2], operates without any binaural-specific assumptions. An according minimalistic binaural model based on the physiologically plausible concept of extracting interaural phase fluctuations can be used to explain the data based on the monaurally derived auditory filter bandwidth without requiring delay lines. The results of the present study i) bridge between the mathematical concept of coherence and physiologically plausible auditory feature extraction and ii) help resolving the discrepancy between physiologic reports that cast doubt on neurons systematically compensating for larger delays [41][42][43] and psychoacoustically motivated models that appeared to require delay compensation [8][9][10]29].
Fig. 1 :
1S0 and Sπ conditions were separated into two independent experiments presented in blocks but without a specified order. For S0 conditions only noise delays of 0 and 8 ms as well as uncorrelated noise were measured. For both the S0 and Sπ conditions the same measurement order principles were applied: For each randomly chosen bandwidth block all noise delays were measured once in random order. To allow for an "acclimatization" to the new bandwidth, two training runs were included at the beginning of each bandwidth block. Once all conditions, i.e. all six bandwidth blocks were measured, the procedure was repeated with new random orders until all conditions were measured 4 times. The study was approved by the Ethics committee of the NS0 detection thresholds (upper panel) and dichotic NSπ thresholds (lower panel) as a function of noise delay in ms. The separated data points at the right-hand side are for uncorrelated noise (Nu). Different symbols (color online) are for the different noise bandwidths ranging from 25 Hz to 1000 Hz.
S0 thresholds are shown in the upper panel ofFig. 1. These data were not in the focus of the current study and was recorded only for delays of 0 and 8 ms as well as for the uncorrelated noise to be able to estimate the observed BMLD. The BMLD is on average 14.8 dB without noise delay independent of bandwidth and is reduced to about 6.3 dB and 4.7 dB for 25-Hz and 50-Hz bandwidth at 8-ms noise delay, respectively. For larger bandwidths the BMLD at 8-ms noise delay
Fig. 2 :
2(a) Schematic of the proposed model. (b) Example of the decision stage for N0S stimuli at different signal levels. The top left shows the absolute value of the instantaneous IPD over time. Increasing the target level (with an IPD of ) increases fluctuations and thus the mean absolute IPD (y-axis top right). Taking the cosine of the mean IPD results in smaller values with increasing fluctuation. The decision variable D results directly from the cosine, subject to a Fisher's-Z transform, so that lower values indicate a stronger signal prevalence (bottom right).
Fig. 3 :Figure 3a
33aModel predictions (solid lines with symbols) and experimental data (dashed lines). In the left panel, thresholds are plotted as in Fig. 1 and in the right panel, the same data is plotted as a function of noise bandwidth with different lines representing the different noise delays. No error bars are shown for the simulations, because the standard error of the mean was < shows the model results for the different noise bandwidths as a function of noise delay and for uncorrelated noise, whereas Fig. 3b shows the same data but as a function of noise bandwidth. For comparison, the experimental data is plotted in the background as dashed lines. In the model, the thresholds obtained at 100 Hz bandwidth (diamonds) still differ slightly from the 1000 Hz condition. Both absolute values and the frequency-dependent increase with noise delay are quite accurately captured by the model. Only at 50 Hz bandwidth for 4 ms delay and at 25 Hz the model underestimates the thresholds by up to 3 dB. Diotic conditions were not modeled, given that monaural cues are not captured by the purely binaural model. This also explains that the model thresholds are slightly too high for uncorrelated noise (right-hand side of Fig. 3a): subjects apparently use a combination of weak binaural cues and weak monaural cues in these conditions. The two internal noise parameters influence the model output in the following way: The internal IPD noise ( ) mostly determines the threshold at τ=0 and with this the slope and curvature of the threshold functions for increasing delay. The decision noise ( ) determines the overall performance, effectively parallel shifting the functions towards higher or lower thresholds. The standard deviations ( , ) of the two noises were determined by first adjusting to fit the data of the = 8 ms condition and consecutively adjusting to fit the data at = 0 ms which resulted in = 0.3 , = 0.4. The root mean square error is 1.35 dB.
Fig. 4 :
4(a) Comparison between the results of the correlation discrimination task of Culling et al. [36] (grey line shows their fit for the average subject) and the model performance in the same task (b) Comparison between the Model results (79 Hz filter bandwidth) and the data from
measured tone detection thresholds in 50 Hz and 400 Hz wide noise maskers with a fixed group delay of 1.5 ms. Relative to their respective N0Sπ reference, thresholds did not increase for the 50 Hz condition but they increased by about 5 dB in their 400 Hz condition. Their model could not account for their observation at all whereas the present model predicts a 2 dB increase for the 50 Hz condition and a 4 dB increase for the 400 Hz condition.IV. DISCUSSIONThe experimental data and simulations presented in this study, assess the effect of tone detection thresholds as a combined function of noise delay and noise bandwidth. The lower the bandwidth of the noise and the higher thus the temporal coherence of the noise, the smaller the impact of time delay on the threshold. This relationship is confirmed by the data points at bandwidths smaller than the critical band. For larger stimulus' bandwidths the data does not depend on bandwidth. The experimental data systematically extends the classical delay dependence for broadband noise[2] by adding five narrower bandwidths of 25 to 200 Hz. It also extends[10] both in terms of bandwidths tested (6 instead of 2) and in terms of delay range (8 ms and infinity instead of 3 ms). The data set provides a valuable systematic baseline for testing binaural models.In order to interpret the experimental results, a minimalistic phenomenological model based on the IPD model front-end[13] was employed. The most relevant aspects of the model are the monaurally-motivated 79-Hz wide 4 th -order Gammatone filters[12] and the extraction of instantaneous IPDs. The model has two parameters: (1) a phase jitter, and (2) a detection uncertainty, both modelled by constant-variance internal noises. The model reproduces all trends in the data and can quantitatively account for all but the 25-Hz bandwidth data. Another minor deviation is that simulated thresholds still increase a little when increasing the bandwidth from 100 to 150 Hz, due to the energy in the filter tails. The experimental data flattens out marginally earlier. Both deviations would be reduced if a slightly narrower auditory filter was chosen. However, given the high accuracy obtained with the established 79-Hz ERB filter, we decided against adding an additional degree of freedom in the model evaluation.
as a function of the interaural delay from 0 to 8 ms and for interaurally uncorrelated (Nu) noise. Symbols depict the median detection thresholds across the seven subjects and the error bars the interquartile range. Different symbols are used for different noise bandwidths. The lower panel shows thresholds for the Sπ condition. N0Sπ thresholds (left-hand data points) were similar to the masker spectrum level and for large bandwidths virtually identical to a large and consistent body of literature [20, 29]. As expected, Sπ detection thresholds increase with increasing noise delay. The thresholds obtained with 100 to 1000 Hz wide noise are virtually identical with each other and to those obtained by Langford and Jeffress [2]. Smaller bandwidths of 50 and 25 Hz result in slightly lower N0Sπ and NuSπ thresholds and increased slower with increasing noise delay. Qualitatively, all these observations were expected as a direct consequence of the increasing temporal coherence in the acoustic stimulus for decreasing bandwidth. For narrow bandwidths up to about 100 Hz, previously reported N0Sπ are less consistent across studies. The N0Sπ thresholds of the present study are lower than those reported by Bernstein and Trahiotis
Hz was different (p<0.05) from all other bandwidths except for 25 and 200 Hz. Such deviations in the pattern of differences from the marginal mean and between the different delays are the source of the significant interaction term. Taken together, the 25-Hz data, and the 50-Hz data to a lesser degree, are different from the data for the other bandwidths, while the data for 100 Hz bandwidth and above show no significant differences.] was performed, showing a significant main effect of noise delay
[F(4, 24)=464.30, p<0.001], bandwidth [F(5, 30)=21.53, p<0.001], and a significant interaction
[F(20, 120)=6.47, p<0.001]. Post-hoc pair-wise comparisons (Bonferroni corrected) for the
marginal means showed that all noise delays were significantly different (p<0.01) from each
other. For the bandwidths, only the data for 25 Hz was significantly different from all other
bandwidth (p<0.05), and the data for 50 Hz was significantly from 200 Hz (p<0.05). At the largest
delay of 8 ms, 25 Hz was different (p<0.05) from all other bandwidths, except for 50 Hz, and 50
The binaural comparator element(s) can thus distinguish between, e.g. interaurally uncorrelated inputs causing large IPD fluctuations and a constant IPD of 90°. For a conventional cross-correlation element both stimuli generate the same output: zero correlation. For the present dataset, however, where all delayed noises have a long-term average IPD of zero, a simplified backend stage which only considers IPD deviations from zero is sufficient.
Acknowledgements
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| [] |
[
"Deep Multi-View Spatial-Temporal Network for Taxi Demand Prediction",
"Deep Multi-View Spatial-Temporal Network for Taxi Demand Prediction"
] | [
"Huaxiu Yao [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Fei Wu \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Jintao Ke \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Xianfeng Tang [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Yitian Jia [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Siyu Lu [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Pinghua Gong [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Jieping Ye [email protected] \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Didi Chuxing \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n",
"Zhenhui Li \nPennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n\n"
] | [
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n",
"Pennsylvania State University\nHong Kong University of Science and Technology\nPennsylvania State University\nPennsylvania State University\n"
] | [] | Taxi demand prediction is an important building block to enabling intelligent transportation systems in a smart city. An accurate prediction model can help the city pre-allocate resources to meet travel demand and to reduce empty taxis on streets which waste energy and worsen the traffic congestion. With the increasing popularity of taxi requesting services such as Uber and Didi Chuxing (in China), we are able to collect large-scale taxi demand data continuously. How to utilize such big data to improve the demand prediction is an interesting and critical real-world problem. Traditional demand prediction methods mostly rely on time series forecasting techniques, which fail to model the complex non-linear spatial and temporal relations. Recent advances in deep learning have shown superior performance on traditionally challenging tasks such as image classification by learning the complex features and correlations from largescale data. This breakthrough has inspired researchers to explore deep learning techniques on traffic prediction problems. However, existing methods on traffic prediction have only considered spatial relation (e.g., using CNN) or temporal relation (e.g., using LSTM) independently. We propose a Deep Multi-View Spatial-Temporal Network (DMVST-Net) framework to model both spatial and temporal relations. Specifically, our proposed model consists of three views: temporal view (modeling correlations between future demand values with near time points via LSTM), spatial view (modeling local spatial correlation via local CNN), and semantic view (modeling correlations among regions sharing similar temporal patterns). Experiments on large-scale real taxi demand data demonstrate effectiveness of our approach over state-ofthe-art methods. | 10.1609/aaai.v32i1.11836 | [
"https://arxiv.org/pdf/1802.08714v2.pdf"
] | 3,600,371 | 1802.08714 | aa139d6a689da06328aa59af249e3b436181c04f |
Deep Multi-View Spatial-Temporal Network for Taxi Demand Prediction
Huaxiu Yao [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Fei Wu
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Jintao Ke
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Xianfeng Tang [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Yitian Jia [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Siyu Lu [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Pinghua Gong [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Jieping Ye [email protected]
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Didi Chuxing
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Zhenhui Li
Pennsylvania State University
Hong Kong University of Science and Technology
Pennsylvania State University
Pennsylvania State University
Deep Multi-View Spatial-Temporal Network for Taxi Demand Prediction
Taxi demand prediction is an important building block to enabling intelligent transportation systems in a smart city. An accurate prediction model can help the city pre-allocate resources to meet travel demand and to reduce empty taxis on streets which waste energy and worsen the traffic congestion. With the increasing popularity of taxi requesting services such as Uber and Didi Chuxing (in China), we are able to collect large-scale taxi demand data continuously. How to utilize such big data to improve the demand prediction is an interesting and critical real-world problem. Traditional demand prediction methods mostly rely on time series forecasting techniques, which fail to model the complex non-linear spatial and temporal relations. Recent advances in deep learning have shown superior performance on traditionally challenging tasks such as image classification by learning the complex features and correlations from largescale data. This breakthrough has inspired researchers to explore deep learning techniques on traffic prediction problems. However, existing methods on traffic prediction have only considered spatial relation (e.g., using CNN) or temporal relation (e.g., using LSTM) independently. We propose a Deep Multi-View Spatial-Temporal Network (DMVST-Net) framework to model both spatial and temporal relations. Specifically, our proposed model consists of three views: temporal view (modeling correlations between future demand values with near time points via LSTM), spatial view (modeling local spatial correlation via local CNN), and semantic view (modeling correlations among regions sharing similar temporal patterns). Experiments on large-scale real taxi demand data demonstrate effectiveness of our approach over state-ofthe-art methods.
Introduction
Traffic is the pulse of a city that impacts the daily life of millions of people. One of the most fundamental questions for future smart cities is how to build an efficient transportation system. To address this question, a critical component is an accurate demand prediction model. The better we can predict demand on travel, the better we can pre-allocate resources to meet the demand and avoid unnecessary energy consumption. Currently, with the increasing popularity of taxi requesting services such as Uber and Didi Chuxing, we are able to collect massive demand data at an unprecedented scale. The question of how to utilize big data to better predict traffic demand has drawn increasing attention in AI research communities.
In this paper, we study the taxi demand prediction problem; that problem being how to predict the number of taxi requests for a region in a future timestamp by using historical taxi requesting data. In literature, there has been a long line of studies in traffic data prediction, including traffic volume, taxi pick-ups, and traffic in/out flow volume. To predict traffic, time series prediction methods have frequently been used. Representatively, autoregressive integrated moving average (ARIMA) and its variants have been widely applied for traffic prediction (Li et al. 2012;Moreira-Matias et al. 2013;Shekhar and Williams 2008). Based on the time series prediction method, recent studies further consider spatial relations (Deng et al. 2016;Tong et al. 2017) and external context data (e.g., venue, weather, and events) (Pan, Demiryurek, and Shahabi 2012;Wu, Wang, and Li 2016). While these studies show that prediction can be improved by considering various additional factors, they still fail to capture the complex nonlinear spatial-temporal correlations.
Recent advances in deep learning have enabled researchers to model the complex nonlinear relationships and have shown promising results in computer vision and natural language processing fields (LeCun, Bengio, and Hinton 2015). This success has inspired several attempts to use deep learning techniques on traffic prediction problems. Recent studies (Zhang, Zheng, and Qi 2017;Zhang et al. 2016) propose to treat the traffic in a city as an image and the traffic volume for a time period as pixel values. Given a set of historical traffic images, the model predicts the traffic image for the next timestamp. Convolutional neural network (CNN) is applied to model the complex spatial correlation. Yu et al. (2017) proposes to use Long Short Term Memory networks (LSTM) to predict loop sensor readings. They show the proposed LSTM model is capable of modeling complex sequential interactions. These pioneering attempts show superior performance compared with previous methods based on traditional time series prediction methods. However, none of them consider spatial relation and tempo-ral sequential relation simultaneously.
In this paper, we harness the power of CNN and LSTM in a joint model that captures the complex nonlinear relations of both space and time. However, we cannot simply apply CNN and LSTM on demand prediction problem. If treating the demand over an entire city as an image and applying CNN on this image, we fail to achieve the best result. We realize including regions with weak correlations to predict a target region actually hurts the performance. To address this issue, we propose a novel local CNN method which only considers spatially nearby regions. This local CNN method is motivated by the First Law of Geography: "near things are more related than distant things," (Tobler 1970) and it is also supported by observations from real data that demand patterns are more correlated for spatially close regions.
While local CNN method filters weakly correlated remote regions, this fails to consider the case that two locations could be spatially distant but are similar in their demand patterns (i.e., on the semantic space). For example, residential areas may have high demands in the morning when people transit to work, and commercial areas may be have high demands on weekends. We propose to use a graph of regions to capture this latent semantic, where the edge represents similarity of demand patterns for a pair of regions. Later, regions are encoded into vectors via a graph embedding method and such vectors are used as context features in the model. In the end, a fully connected neural network component is used for prediction.
Our method is validated via large-scale real-world taxi demand data from Didi Chuxing. The dataset contains taxi demand requests through Didi service in the city of Guangzhou in China over a two-month span, with about 300,000 requests per day on average. We conducted extensive experiments to compare with state-of-the-art methods and have demonstrated the superior performance of our proposed method.
In summary, our contributions are summarized as follow:
• We proposed a unified multi-view model that jointly considers the spatial, temporal, and semantic relations.
• We proposed a local CNN model that captures local characteristics of regions in relation to their neighbors.
• We constructed a region graph based on the similarity of demand patterns in order to model the correlated but spatially distant regions. The latent semantics of regions are learnt through graph embedding.
• We conducted extensive experiments on a large-scale taxi request dataset from Didi Chuxing. The results show that our method consistently outperforms the competing baselines.
Related Work
Problems of traffic prediction could include predicting any traffic related data, such as traffic volume (collected from GPS or loop sensors), taxi pick-ups or drop-offs, traffic flow, and taxi demand (our problem). The problem formulation process for these different types of traffic data is the same. Essentially, the aim is to predict a traffic-related value for a location at a timestamp. In this section, we will discuss the related work on traffic prediction problems.
The traditional approach is to use time series prediction method. Representatively, autoregressive integrated moving average (ARIMA) and its variants have been widely used in traffic prediction problem (Shekhar and Williams 2008;Li et al. 2012;Moreira-Matias et al. 2013).
Recent studies further explore the utilities of external context data, such as venue types, weather conditions, and event information (Pan, Demiryurek, and Shahabi 2012;Wu, Wang, and Li 2016;Tong et al. 2017). In addition, various techniques have also been introduced to model spatial interactions. For example, Deng et al. (2016) used matrix factorization on road networks to capture a correlation among road connected regions for predicting traffic volume. Several studies (Tong et al. 2017;Idé and Sugiyama 2011;Zheng and Ni 2013) also propose to smooth the prediction differences for nearby locations and time points via regularization for close space and time dependency. These studies assume traffic in nearby locations should be similar. However, all of these methods are based on the time series prediction methods and fail to model the complex nonlinear relations of the space and time.
Recently, the success of deep learning in the fields of computer vision and natural language processing (LeCun, Bengio, and Hinton 2015; Krizhevsky, Sutskever, and Hinton 2012) motivates researchers to apply deep learning techniques on traffic prediction problems. For instance, designed a neural network framework using context data from multiple sources and predict the gap between taxi supply and demand. The method uses extensive features, but does not model the spatial and temporal interactions.
A line of studies applied CNN to capture spatial correlation by treating the entire city's traffic as images. For example, Ma et al. (2017) utilized CNN on images of traffic speed for the speed prediction problem. Zhang et al. (2016) and Zhang, Zheng, and Qi (2017) proposed to use residual CNN on the images of traffic flow. These methods simply use CNN on the whole city and will use all the regions for prediction. We observe that utilizing irrelevant regions (e.g., remote regions) for prediction of the target region might actually hurts the performance. In addition, while these methods do use traffic images of historical timestamps for prediction, but they do not explicitly model the temporal sequential dependency.
Another line of studies uses LSTM for modeling sequential dependency. Yu et al. (2017) proposed to apply Longshort-term memory (LSTM) network and autoencoder to capture the sequential dependency for predicting the traffic under extreme conditions, particularly for peak-hour and post-accident scenarios. However, they do not consider the spatial relation.
In summary, the biggest difference of our proposed method compared with literature is that we consider both spatial relation and temporal sequential relation in a joint deep learning model.
Preliminaries
In this section, we first fix some notations and define the taxi demand problem. We follow previous studies (Zhang, Zheng, and Qi 2017;) and define the set of non-overlapping locations L = {l 1 , l 2 , ..., l i , ..., l N } as rectangle partitions of a city, and the set of time intervals as I = {I 0 , I 1 , ..., I t , ..., I T }. 30 minutes is set as the length of the time interval. Alternatively, more sophisticated ways of partitioning can also be used, such as partition space by road network (Deng et al. 2016) or hexagonal partitioning. However, this is not the focus of this paper, and our methodology can still be applied. Given the set of locations L and time intervals T , we further define the following.
Taxi request: A taxi request o is defined as a tuple Demand: The demand is defined as the number of taxi requests at one location per time point, i.e., y i t = |{o : o.t ∈ I t ∧o.l ∈ l i }|, where |·| denotes the cardinality of the set. For simplicity, we use the index of time intervals t representing I t , and the index of locations i representing l i for rest of the paper.
Demand prediction problem: The demand prediction problem aims to predict the demand at time interval t + 1, given the data until time interval t. In addition to historical demand data, we can also incorporate context features such as temporal features, spatial features, meteorological features (refer to Data Description section for more details). We denote those context features for a location i and a time point t as a vector e i t ∈ R r , where r is the number of features. Therefore, our final goal is to predict
y i t+1 = F(Y L t−h,...,t , E L t−h,...,t ) for i ∈ L, where Y L t−h,.
..,t are historical demands and E L t−h,...,t are context features for all locations L for time intervals from t − h to t, where t − h denotes the starting time interval. We define our prediction function F(·) on all regions and previous time intervals up to t − h to capture the complex spatial and temporal interaction among them.
Proposed DMVST-Net Framework
In this section, we provide details for our proposed Deep Multi-View Spatial-Temporal Network (DMVST-Net) framework, i.e., our prediction function F. Figure 1 shows the architecture of our proposed method. Our proposed model has three views: spatial, temporal, and semantic view.
Spatial View: Local CNN
As we mentioned earlier, including regions with weak correlations to predict a target region actually hurts the performance. To address this issue, we propose a local CNN method which only considers spatially nearby regions. Our intuition is motivated by the First Law of Geography (Tobler 1970) -"near things are more related than distant things".
As shown in Figure 1(a), at each time interval t, we treat one location i with its surrounding neighborhood as one S × S image (e.g., 7 × 7 image in Figure 1(a)) having one channel of demand values (with i being at the center of the image), where the size S controls the spatial granularity. We use zero padding for location at boundaries of the city. As a result, we have an image as a tensor (having one channel) Y i
Temporal View: LSTM
The temporal view models sequential relations in the demand time series. We propose to use Long Short-Term Memory (LSTM) network as our temporal view component. LSTM (Hochreiter and Schmidhuber 1997) is a type of neural network structure, which provides a good way to model sequential dependencies by recursively applying a transition function to the hidden state vector of the input. It is proposed to address the problem of classic Recurrent Neural Network (RNN) for its exploding or vanishing of gradient in the long sequence training (Hochreiter et al. 2001).
LSTM learns sequential correlations stably by maintaining a memory cell c t in time interval t, which can be regarded as an accumulation of previous sequential information. In each time interval, LSTM takes an input g i t , h t−1 and c t−1 in this work, and then all information is accumulated to the memory cell when the input gate i i t is activated. In addition, LSTM has a forget gate f i t . If the forget gate is activated, the network can forget the previous memory cell c i t−1 . Also, the output gate o i t controls the output of the memory cell. In this study, the architecture of LSTM is formulated as follows: Figure 1(b) shows, the temporal component takes representations from the spatial view and concatenates them with context features. More specifically, we define:
i i t = σ(W i g i t + U i h i t−1 + b i ), f i t = σ(W f g i t + U f h i t−1 + b f ), o i t = σ(W o g i t + U o h i t−1 + b o ), θ i t = tanh(W g g i t + U g h i t−1 + b g ), c i t = f i t • c i t−1 + i i t • θ i t , h i t = o i t • tanh(c i t ). (3) −ℎ+1 � −ℎ+1 FC � � FC � +1 Loss +1 Embed FC −ℎ+1 −ℎ+2 −1� −ℎ+1 Conv FC Conv FC Conv −ℎ+1 ,0 −ℎ+2 ,0 ,0 −ℎ+2 , FC FC FC Conv Conv −ℎ+1 ,g i t =ŝ i t ⊕ e i t ,(4)
where ⊕ denotes the concatenation operator, therefore, g i t ∈ R r+d .
Semantic View: Structural Embedding
Intuitively, locations sharing similar functionality may have similar demand patterns, e.g., residential areas may have a high number of demands in the morning when people transit to work, and commercial areas may expect to have high demands on weekends. Similar regions may not necessarily be close in space. Therefore, we construct a graph of locations representing functional (semantic) similarity among regions.
We define the semantic graph of location as G = (V, E, D), where the set of locations L are nodes V = L, E ∈ V × V is the edge set, and D is a set of similarity on all the edges. We use Dynamic Time Warping (DTW) to measure the similarity ω ij between node (location) i and node (location) j. ω ij = exp(−αDTW(i, j)),
where α is the parameter that controls the decay rate of the distance (in this paper, α = 1), and DTW(i, j) is the dynamic time warping distance between the demand patterns of two locations. We use the average weekly demand time series as the demand patterns. The average is computed on the training data in the experiment. The graph is fully connected because every two regions can be reached.
In order to encode each node into a low dimensional vector and maintain the structural information, we apply a graph embedding method on the graph. For each node i (location), the embedding method outputs the embedded feature vector m i . In addition, in order to co-train the embedded m i with our whole network architecture, we feed the feature vector m i to a fully connected layer, which is defined as:
m i = f (W f e m i + b f e ),(6)
W f e and b f e are both learnable parameters. In this paper, we use LINE for generating embeddings (Tang et al. 2015).
Prediction Component
Recall that our goal is to predict the demand at t + 1 given the data till t. We join three views together by concatenatinĝ m i with the output h i t of LSTM:
q i t = h i t ⊕m i .(7)
Note that the output of LSTM h i t contains both effects of temporal and spatial view. Then we feed q i t to the fully connected network to get the final prediction valueŷ i t+1 for each region. We define our final prediction function as:
y i t+1 = σ(W f f q i t + b f f ),(8)
where W f f and b f f are learnable parameters. σ(x) is a Sigmoid function defined as σ(x) = 1/(1+e −x ). The output of Randomly select a batch of instance Ω bt from Ω;
S spa = [Y i t−h+1 , Y i t−h+2 , ..., Y i t ]; 6 S cox = [e i t−h+1 , e i t−
13
Optimize θ by minimizing the loss function Eq. (9) with Ω bt 14 until stopping criteria is met;
our model is in [0, 1], as the demand values are normalized. We later denormalize the prediction to get the actual demand values.
Loss function
In this section, we provide details about the loss function used for jointly training our proposed model. The loss function we used is defined as:
L(θ) = N i=1 ((y i t+1 −ŷ i t+1 ) 2 + γ( y i t+1 −ŷ i t+1 y i t+1 ) 2 ),(9)
where θ are all learnable parameters in the DMVST-Net and γ is a hyper parameter. The loss function consists of two parts: mean square loss and square of mean absolute percentage loss. In practice, mean square error is more relevant to predictions of large values. To avoid the training being dominated by large value samples, we in addition minimize the mean absolute percentage loss. Note that, in the experiment, all compared regression methods use the same loss function as defined in Eq. (9) for fair comparison. The training pipeline is outlined in Algorithm 1. We use Adam (Kingma and Ba 2014) for optimization. We use Tensorflow and Keras (Chollet and others 2015) to implement our proposed model.
Experiment Dataset Description
In this paper, we use a large-scale online taxi request dataset collected from Didi Chuxing, which is one of the largest online car-hailing companies in China. The dataset contains taxi requests from 02/01/2017 to 03/26/2017 for the city of Guangzhou. There are 20 × 20 regions in our data. The size of each region is 0.7km × 0.7km. There are about 300, 000 requests each day on average. The context features used in our experiment are the similar types of features used in (Tong et al. 2017). These features include temporal features (e.g., the average demand value in the last four time intervals), spatial features (e.g., longitude and latitude of the region center), meteorological features (e.g., weather condition), event features (e.g., holiday). In the experiment, the data from 02/01/2017 to 03/19/2017 is used for training (47 days), and the data from 03/20/2017 to 03/26/2017 (7 days) is used for testing. We use half an hour as the length of the time interval. When testing the prediction result, we use the previous 8 time intervals (i.e., 4 hours) to predict the taxi demand in the next time interval. In our experiment, we filter the samples with demand values less than 10. This is a common practice used in industry. Because in the real-world applications, people do not care about such low-demand scenarios.
Evaluation Metric
We use Mean Average Percentage Error (MAPE) and Rooted Mean Square Error (RMSE) to evaluate our algorithm, which are defined as follows:
M AP E = 1 ξ ξ i=1 |ŷ i t+1 − y i t+1 | y i t+1 ,(10)RM SE = 1 ξ ξ i=1 (ŷ i t+1 − y i t+1 ) 2 ,(11)
whereŷ i t+1 and y i t+1 mean the prediction value and real value of region i for time interval t + 1, and where ξ is total number of samples.
Methods for Comparison
We compared our model with the following methods, and tuned the parameters for all methods. We then reported the best performance.
• Historical average (HA): Historical average predicts the demand using average values of previous demands at the location given in the same relative time interval (i.e., the same time of the day). • Autoregressive integrated moving average (ARIMA):
ARIMA is a well-known model for forecasting time series which combines moving average and autoregressive components for modeling time series. • Linear regression (LR): We compare our method with different versions of linear regression methods: ordinary least squares regression (OLSR), Ridge Regression (i.e., with 2 -norm regularization), and Lasso (i.e., with 1norm regularization). • ST-ResNet (Zhang, Zheng, and Qi 2017): ST-ResNet is a deep learning based approach for traffic prediction. The method constructs a city's traffic density map at different times as images. CNN is used to extract features from historical images.
We used the same context features for all regression methods above. For fair comparisons, all methods (except ARIMA and HA) use the same loss function as our method defined in Eq. (9). We also studied the effect of different view components proposed in our method.
• Temporal view: For this variant, we used only LSTM with inputs as context features. Note that, if we do not use any context features but only use the demand value of last timestamp as input, LSTM does not perform well. It is necessary to use context features to enable LSTM to model the complex sequential interactions for these features.
• Temporal view + Semantic view: This method captures both temporal dependency and semantic information.
• Temporal view + Spatial (Neighbors) view: In this variant, we used the demand values of nearby regions at time interval t asŝ i t and combined them with context features as the input of LSTM. We wanted to demonstrate that simply using neighboring regions as features cannot model the complex spatial relations as our proposed local CNN method.
• Temporal view + Spatial (LCNN) view: This variant considers both temporal and local spatial views. The spatial view uses the proposed local CNN for considering neighboring relation. Note that when our local CNN uses a local window that is large enough to cover the whole city, it is the same as the global CNN method. We studied the performance of different parameters and show that if the size is too large, the performance is worse, which indicates the importance of locality.
• DMVST-Net: Our proposed model, which combines spatial, temporal and semantic views.
Preprocessing and Parameters
We normalized the demand values for all locations to [0, 1] by using Max-Min normalization on the training set. We used one-hot encoding to transform discrete features (e.g., holidays and weather conditions) and used Max-Min normalization to scale the continuous features (e.g., the average of demand value in last four time intervals). As our method outputs a value in [0, 1], we applied the inverse of the Max-Min transformation obtained on training set to recover the demand value.
All these experiments were run on a cluster with four NVIDIA P100 GPUs. The size of each neighborhood considered was set as 9 × 9 (i.e., S = 9), which corresponds to 6km × 6km rectangles. For spatial view, we set K = 3 (number of layers), τ = 3×3 (size of filter), λ = 64 (number of filters used), and d = 64 (dimension of the output). For the temporal component, we set the sequence length h = 8
Performance Comparison
Comparison with state-of-the-art methods. Table 1 shows the performance of the proposed method as compared to all other competing methods. DMVST-Net achieves the lowest MAPE (0.1616) and the lowest RMSE (9.642) among all the methods, which is 12.17% (MAPE) and 3.70% (RMSE) relative improvement over the best performance among baseline methods. More specifically, we can see that HA and ARIMA perform poorly (i.e., have a MAPE of 0.2513 and 0.2215, respectively), as they rely purely on historical demand values for prediction. Regression methods (OLSR, LASSO, Ridge, MLP and XGBoost) further consider context features and therefore achieve better performance. Note that the regression methods use the same loss function as our method defined in Eq. (9). However, the regression methods do not model the temporal and spatial dependency. Consequently, our proposed method significantly outperforms those methods. Furthermore, our proposed method achieves 18.01% (MAPE) and 6.37% relative improvement over ST-ResNet. Compared with ST-ResNet, our proposed method further utilizes LSTM to model the temporal dependency, while at the same time considering context features. In addition, our use of local CNN and semantic view better captures the correlation among regions.
Comparison with variants of our proposed method. Table 2 shows the performance of DMVST-Net and its variants. First, we can see that both Temporal view + Spatial (Neighbor) view and Temporal view + Spatial (LCNN) view achieve a lower MAPE (a reduction of 0.63% and 6.10%, respectively). The result demonstrates the effectiveness of Figure 2 shows the performance of different methods on different days of the week. Due to the space limitation, We only show MAPE here. We get the same conclusions of RMSE. We exclude the results of HA and ARIMA, as they perform poorly. We show Ridge regression results as they perform best among linear regression models. In the figure, it shows that our proposed method DMVST-Net outperforms other methods consistently in all seven days. The result demonstrates that our method is robust. Moreover, we can see that predictions on weekends are generally worse than on weekdays. Since the average number of demand requests is similar (45.42 and 43.76 for weekdays and weekends, respectively), we believe the prediction task is harder for weekends as demand patterns are less regular. For example, we can expect that residential areas may have high demands in the morning hours on weekdays, as people need to transit to work. Such regular patterns are less likely to happen on weekends. To evaluate the robustness of our method, we look at the relative increase in prediction error on weekends as compared to weekdays, i.e., defined as |wk −wd|/wd, wherewd andwk are the average prediction error of weekdays and weekends, respectively. The results are shown in Table 3. For our proposed method, the relative increase in error is the smallest, at 4.04%.
Performance on Different Days
At the same time, considering temporal view, only (LSTM) has a relative increase in error of 4.77%, while the increase is more than 10% for Ridge regression, MLP, and XGBoost. The more stable performance of LSTM can be attributed to its modeling of the temporal dependency. We see that ST-ResNet has a more consistent performance (relative increase in error of 4.41%), as the method further models the spatial dependency. Finally, our proposed method is more robust than ST-ResNet.
Influence of Sequence Length for LSTM
In this section, we study how the sequence length for LSTM affects the performance. Figure 3a shows the prediction er- ror of MAPE with respect to the length. We can see that when the length is 4 hours, our method achieves the best performance. The decreasing trend in MAPE as the length increases shows the importance of considering the temporal dependency. Furthermore, as the length increases to more than 4 hours, the performance slightly degrades but mainly remains stable. One potential reason is that when considering longer temporal dependency, more parameters need to be learned. As a result, the training becomes harder.
Influence of Input Size for Local CNN
Our intuition was that applying CNN locally avoids learning relation among weakly related locations. We verified that intuition by varying the input size S for local CNN.
As the input size S becomes larger, the model may fit for relations in a larger area. In Figure 3b, we show the performance of our method with respect to the size of the surrounding neighborhood map. We can see that when there are three convolutional layers and the size of map is 9 × 9, the method achieves the best performance. The prediction error increases as the size decreases to 5 × 5. This may be due to the fact that locally correlated neighboring locations are not fully covered. Furthermore, the prediction error increases significantly (more than 3.46%), as the size increases to 13 × 13 (where each area approximately covers more than 40% of the space in GuangZhou). The result suggests that locally significant correlations may be averaged as the size increases. We also increased the number of convolution layers to four and five layers, as the CNN needed to cover larger area. However, we observed similar trends of prediction error, as shown in Figure 3b. We can now see that the input size for local CNN when the method performs best remains consistent (i.e., the size of map is 9 × 9).
Conclusion and Discussion
The purpose of this paper is to inform of our proposal of a novel Deep Multi-View Spatial-Temporal Network (DMVST-Net) for predicting taxi demand. Our approach in- tegrates the spatial, temporal, and semantic views, which are modeled by local CNN, LSTM and semantic graph embedding, respectively. We evaluated our model on a large-scale taxi demand dataset. The experiment results show that our proposed method significantly outperforms several competing methods.
As deep learning methods are often difficult to interpret, it is important to understand what contributes to the improvement. This is particularly important for policy makers. For future work, we plan to further investigate the performance improvement of our approach for better interpretability. In addition, seeing as the semantic information is implicitly modeled in this paper, we plan to incorporate more explicit information (e.g., POI information) in our future work.
(o.t, o.l, o.u), where o.t is the timestamp, o.l represents the location, and o.u is user identification number. The requester identifications are used for filtering duplicated and spammer requests.
Figure 1 :
1The Architecture of DMVST-Net. (a). The spatial component uses a local CNN to capture spatial dependency among nearby regions. The local CNN includes several convolutional layers. A fully connected layer is used at the end to get a low dimensional representation. (b). The temporal view employs a LSTM model, which takes the representations from the spatial view and concatenates them with context features at corresponding times. (c). The semantic view first constructs a weighted graph of regions (with weights representing functional similarity). Nodes are encoded into vectors. A fully connected layer is used at the end for jointly training. Finally, a fully connected neural network is used for prediction.where • denotes Hadamard product and tanh is hyperbolic tangent function. Both functions are element-wise. W a , U a , b a (a ∈ {i, f, o, g}) are all learnable parameters. The number of time intervals in LSTM is h and the output of region i of LSTM after h time intervals is h i t . As
•
Multiple layer perceptron (MLP): We compare our method with a neural network of four fully connected layers. The number of hidden units are 128, 128, 64, and 64 respectively. • XGBoost (Chen and Guestrin 2016): XGBoost is a powerful boosting tree based method and is widely used in data mining applications.
Figure 2 :
2The Results of Different Days.
Figure 3 :
3(a) MAPE with respect to sequence length for LSTM. (b) MAPE with respect to the input size for local CNN.
= (V, E, D); Length of the time period h; Output: Learned DMVST-Net model 1 Initialization; 2 for ∀i ∈ L do Use LINE on G and get the embedding result m i ; for ∀t ∈ [h, T ] doAlgorithm 1: Training Pipeline of DMVST-Net
Input: Historical observations: Y L
1,...,t ; Context
features: E L
t−h,...,t ; Region structure graph
G 3
4
5
Append < {S spa , S cox , m i }, y i t+1 > to Ω bt ;h+2 , ..., e i
t ];
7
8
end
9 end
10 Initialize all learnable parameters θ in DMVST-Net;
11 repeat
12
Table 1 :
1Comparison with Different BaselinesMethod
MAPE RMSE
Historical average
0.2513 12.167
ARIMA
0.2215 11.932
Ordinary least square regression 0.2063 10.234
Ridge regression
0.2061 10.224
Lasso
0.2091 10.327
Multiple layer perceptron
0.1840 10.609
XGBoost
0.1953 10.012
ST-ResNet
0.1971 10.298
DMVST-Net
0.1616 9.642
(i.e., 4 hours) for LSTM. The output dimension of graph em-
bedding is set as 32. The output dimension for the semantic
view is set to 6. We used Sigmoid function as the activation
function for the fully connected layer in the final prediction
component. Activation functions in other fully connected
layers are ReLU. Batch normalization is used in the local
CNN component. The batch size in our experiment was set
to 64. The first 90% of the training samples were selected for
training each model and the remaining 10% were in the val-
idation set for parameter tuning. We also used early-stop in
all the experiments. The early-stop round and the max epoch
were set to 10 and 100 in the experiment, respectively.
Table 2 :
2Comparison with Variants of DMVST-Net Temporal view + Semantic view has a lower MAPE of 0.1708 and an RMSE of 9.789 compared to Temporal view only, demonstrating the effectiveness of our semantic view. Lastly, the performance is best when all views are combined.Method
MAPE RMSE
Temporal view
0.1721 9.812
Temporal + Semantic view
0.1708 9.789
Temporal + Spatial (Neighbor) view 0.1710 9.796
Temporal + Spatial (LCNN) view
0.1640 9.695
DMVST-Net
0.1616 9.642
considering neighboring spatial dependency. Furthermore,
Temporal view + Spatial (LCNN) view outperforms Tempo-
ral view + Spatial (Neighbor) view significantly, as the local
CNN can better capture the nonlinear relations. On the other
hand,
Table 3 :
3Relative Increase in Error (RIE) on Weekends to Weekdays Method RIDGE MLP XGBoost ST-ResNet Temporal DMVST-NetRIE
14.91% 10.71% 16.08%
4.41%
4.77%
4.04%
Mon
Tue
Wed
Thu
Fri
Sat
Sun
0.14
0.16
0.18
0.2
0.22
0.24
MAPE
Ridge
MLP
Xgboost
ST-ResNet
Temporal
DMVST-Net
t ∈ R S×S×1 , for each location i and time interval t. The local CNN takes Y i t as input Y i,0 t and feeds it into K convolutional layers. The transformation at each layer k is defined as follows:Y i,k t = f (Y i,k−1 t * W k t + b k t ),(1)where * denotes the convolutional operation and f (·) is an activation function. In this paper, we use the rectifier function as the activation, i.e., f (z) = max(0, z). W k t and b k t are two sets of parameters in the k th convolution layer. Note that the parameters W 1,...,K t and b 1,...,K t are shared across all regions i ∈ L to make the computation tractable.After K convolution layers, we use a flatten layer to transform the output Y i,K t ∈ R S×S×λ to a feature vector s i t ∈ R S 2 λ for region i and time interval t. At last, we use a fully connected layer to reduce the dimension of spatial representations s i t , which is defined as:s i t = f (W f c t s i t + b f c t ),(2)where W f c t and b f c t are two learnable parameter sets at time interval t. Finally, for each time interval t, we get theŝ i t ∈ R d as the representation for region i.
AcknowledgmentsThe work was supported in part by NSF awards #1544455, #1652525, #1618448, and #1639150. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies.
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| [
"https://github.com/fchollet/keras."
] |
[
"A refined factorization of the exponential law",
"A refined factorization of the exponential law"
] | [
"P Patie [email protected] \nDepartment of Mathematics\nUniversité Libre de Bruxelles\nB-1050BruxellesBelgium\n"
] | [
"Department of Mathematics\nUniversité Libre de Bruxelles\nB-1050BruxellesBelgium"
] | [
"Bernoulli"
] | Let ξ be a (possibly killed) subordinator with Laplace exponent φ and denote by I φ = ∞ 0 e −ξs ds, the so-called exponential functional. Consider the positive random variable I ψ 1 whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106], is determined by its negative entire moments as follows:In this note, we show that I ψ 1 is a positive self-decomposable random variable whenever the Lévy measure of ξ is absolutely continuous with a monotone decreasing density. In fact, I ψ 1 is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106] the following factorization of the exponential law e:where I ψ 1 is taken to be independent of I φ . We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that S(α) α is a self-decomposable random variable, where S(α) is a positive stable random variable of index α ∈ (0, 1). This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 2, 814-826. This reprint differs from the original in pagination and typographic detail. | 10.3150/10-bej292 | [
"https://arxiv.org/pdf/1005.4011v2.pdf"
] | 13,244,026 | 1005.4011 | 2e4ec2d87ab7391a1d9bb06fc3aeee5eae626f12 |
A refined factorization of the exponential law
2011
P Patie [email protected]
Department of Mathematics
Université Libre de Bruxelles
B-1050BruxellesBelgium
A refined factorization of the exponential law
Bernoulli
172201110.3150/10-BEJ292arXiv:1005.4011v2 [math.PR]exponential functionalLévy processesself-decomposable random variableself-similar Markov processStieltjes moment sequencessubordinator
Let ξ be a (possibly killed) subordinator with Laplace exponent φ and denote by I φ = ∞ 0 e −ξs ds, the so-called exponential functional. Consider the positive random variable I ψ 1 whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106], is determined by its negative entire moments as follows:In this note, we show that I ψ 1 is a positive self-decomposable random variable whenever the Lévy measure of ξ is absolutely continuous with a monotone decreasing density. In fact, I ψ 1 is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106] the following factorization of the exponential law e:where I ψ 1 is taken to be independent of I φ . We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that S(α) α is a self-decomposable random variable, where S(α) is a positive stable random variable of index α ∈ (0, 1). This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 2, 814-826. This reprint differs from the original in pagination and typographic detail.
Introduction
Let ξ = (ξ t , t ≥ 0) be a possibly killed subordinator starting from 0, that is, a [0, ∞)valued (∞ serves as absorbing state) Lévy process such that ξ 0 = 0. The law of ξ is well known to be characterized by its Laplace exponent φ, which admits the following Lévy-Khintchine representation for any u ≥ 0:
φ(u) = bu + ∞ 0 (1 − e −ur )ν(dr) + q,(1.1)
where q ≥ 0 is the killing rate, b ≥ 0 is the drift and the Lévy measure ν satisfies the integrability condition R + (1 ∧ r)ν(dr) < ∞. Note that functions of the form (1.1) are also named, in the literature, as Bernstein functions. We refer to the monographs [5,16] (resp., [4,14]) for a detailed account on Lévy processes (resp., Bernstein functions). Next, consider the so-called exponential functional associated to ξ, which is defined as
I φ = eq 0 e −ξs ds,
where e q is an independent exponential random variable with parameter q and we understand that e 0 = +∞. Note that, for any q ≥ 0, I φ < ∞ a.s. We refer to the survey paper of Bertoin and Yor [9] for a thorough description of the properties of this positive random variable and of the motivations for studying its law. In particular, we mention that the law of I φ has been determined through its positive entire moments by Carmona et al. [11] as follows:
E[I n φ ] = Γ(n + 1) n k=1 φ(k)
, n = 1, 2, . . . , (1.2) where Γ stands for the gamma function. Bertoin and Yor [6] (see also [9], Theorem 2, for the case q > 0) showed that there exists a positive random variable J whose law is determined by its positive entire moments as follows:
E[J n ] = n k=1
φ(k), n = 1, 2, . . . , such that, when J is taken to be independent of I φ , one has the following factorization of the exponential law:
I φ J (d) = e,(1.3)
where "
(d)
= " means identity in distribution and e = e 1 . Let us now point out that since the random variable J is defined on the half-line and its law is uniquely determined by its positive entire moments, the sequence (s n = n k=1 φ(k)) n≥0 corresponds to a determinate normalized Stieltjes moment sequence. In this direction, we should mention that Berg [1] generalizes the above fact by showing that for any c > 0, the sequence (s c n ) n≥0 associated to a measure on the half-line ρ c is also a Stieltjes moment sequence which is determinate for c ≤ 2. He then deduces that there exists a unique product convolution semigroup (ρ c ) c>0 such that the moments of ρ c are given by s c n for any c > 0. Moreover, in [2], Berg characterizes the set of normalized Stieltjes moment sequences for which this power stability property still holds. In the same vein, Berg and Durán [3] study a more general mapping which allows, in particular, the construction of a Stieltjes moment sequence of the form (s n ) n≥0 with the Bernstein function φ replaced by a completely monotone function.
The first aim of this note is to show that the random variable 1/J is actually a positive self-decomposable random variable, provided that the Lévy measure ν in (1.1) admits a monotone decreasing density. This will be achieved by identifying the random variable 1/J as the exponential functional of a spectrally negative Lévy process which we now introduce. Let Ξ = (Ξ t , t ≥ 0) be a conservative spectrally negative Lévy process with a non-negative mean m and starting from 0, that is, a Lévy process having only negative jumps such that 0 ≤ m = E[Ξ 1 ] < ∞. Its law is characterized by its Laplace exponent ψ which admits, in this case, the following Lévy-Khintchine representation for any u ≥ 0:
ψ(u) = σu 2 + mu + 0 −∞ (e ur − 1 − ur)Π(dr),(1.4)
where σ ≥ 0 is the Gaussian coefficient and the Lévy measure satisfies the condition 0 −∞ (|r| ∧ r 2 )Π(dr) < ∞. The exponential functional associated to Ξ, denoted by I ψ , is finite a.s. whenever m > 0. Its law has been determined through its negative entire moments by Bertoin and Yor [7], as follows:
E[I −n ψ ] = m n−1 k=1 ψ(k) Γ(n) , n = 1, 2, . . . ,(1.5)
with the convention that the right-hand side is m when n = 1. We now recall that Lamperti [18], interested in limit theorems for stochastic processes, shows, in particular, that for any x > 0, the process X = (X t , t ≥ 0), defined for any t ≥ 0 by
X t = x exp(Ξ A t/x ), A t = inf s ≥ 0;
s 0 e Ξu du > t , (1.6) starting from x at time 0, is a self-similar Feller process on (0, ∞) having only negative jumps. The Lamperti transformation is actually one-to-one and extends to any Lévy process. Bertoin and Yor [7], Proposition 1, shows that the family of probability measures (Q (ψ)
x ) x>0 of X, as defined in (1.6), converges as x ↓ 0, in the sense of finite-dimensional distributions, to a probability measure Q (ψ) 0 ; see also [10] for the weak convergence in the Skorokhod topology. Thus, X is a Feller process on [0, ∞) and Bertoin and Yor determine the law of the random variable J ψ = (X 1 , Q They also deduce, in the case where m > 0 and ξ is the ascending ladder height process of the dual process of Ξ (see, e.g., [5], Chapter VI), that the random variable J , in (1.3), is J ψ , that is,
I φ J ψ (d) = e. (1.8)
The second aim of this note is to relate, in a simple way, the law of J ψ , for any m ≥ 0, with the exponential functional of a spectrally negative Lévy process. Finally, as observed by Rivero [22], the study of the exponential functional is also motivated by its connection to some interesting random equations. Indeed, from the strong Markov property for Lévy processes, which entails that for any finite stopping time T in the natural filtration (F t , t ≥ 0) of ξ, the process (ξ t+T − ξ T , t ≥ 0) is independent of F T and has the same distribution as ξ, we readily deduce that the random variable I φ , in the case q = 0, is a solution to the random affine equation
I φ (d) = T 0 e −ξs ds + e −ξT I ′ φ , (1.9)
where, on the right-hand side, I ′ φ is an independent copy of I φ . Note that this type of random equation have been studied by Kesten [15] and Goldie [13]. By means of a similar argument, but using the absence of positive jumps of Ξ (see [21], Proposition 4, for more details), we get that I ψ is a solution to the random affine equation, for any y > 0,
I ψ (d) = Ty 0 e −Ξs ds + e −y I ′ ψ ,(1.10)
where T y = inf{s > 0; Ξ s ≥ y} and, on the right-hand side, I ′ ψ is an independent copy of I ψ . Hence, I ψ is a positive self-decomposable random variable and, in particular, its law is absolutely continuous and unimodal; see, e.g., [23] and [24] for an excellent account of this set of probability measures.
Main results
Factorization of the exponential law with exponential functionals
In this subsection, we suppose that ξ is a subordinator starting from 0 with Laplace exponent given by (1.1). We introduce the following hypothesis on the Lévy measure of ξ.
Assumption 2.1. There exists a monotone decreasing function f such that ν(dx) = f (x) dx.
We recall that, under this condition, −df (x) is a Stieltjes measure on (0, ∞). We also use the notation −df (−x) for the image of the positive measure −df (x) under the map
x → −x. For instance, if f is, in addition, differentiable, then −df (−x) = −f ′ (−x) dx.
We are now ready to derive our refinement of the factorization of the exponential law.
= {u ∈ C; Re(u) > −1}, with ψ 1 (−1) = −φ(0), given by ψ 1 (u) = bu 2 + φ(1)u + 0 −∞ (e ur − 1 − ur)Π(dr), u ∈ C, where Π(dr) = e r (f (−r) dr − df (−r)
) is a Stieltjes measure on (−∞, 0). Moreover, the law of the positive self-decomposable random variable I ψ1 is determined by its negative entire moments as follows:
E[I −n ψ1 ] = n k=1 φ(k), n = 1, 2, . . . .
The exponential law admits the following factorization:
I φ /I ψ1 (d) = e,(2.
1)
where e stands for an exponential random variable of parameter 1.
Conversely, if ψ is of the form (1.4) with m > 0 and is analytic in the domain C with ψ(−1) ≤ 0, then there exists an independent subordinator with Laplace exponent φ −1 given by
φ −1 (u) = −ψ(−1) + σu + ∞ 0 (1 − e −ur )e r Π(−∞, −r) dr, u ≥ 0,
such that
I φ−1 /I ψ (d) = e. (2.2) Remark 2.3.
(1) We have several comments to offer on the identity (2.1) when compared to (1.8). First, our hypotheses are slightly less restrictive. Indeed, it is well known (see, e.g., [5], Chapter VI) that the ascending ladder height process of the dual process of Ξ satisfies Assumption 2.1 and is a killed subordinator; thus, in (1.8), q is necessarily positive. More importantly, we have identified the mixture random variable of I φ in the factorization of the exponential law as the reciprocal of a positive self-decomposable random variable. Finally, our identity allows further explicit examples to be obtained for the law of the exponential functional of Lévy processes. All of these facts will be illustrated in Section 3.
(2) The analyticity property of ψ 1 means that the associated spectrally negative Lévy process Ξ 1 admits exponential moments of order u ≥ −1, that is, for any u ≥ −1, we have
E[e uΞ 1 1 ] < ∞.
We shall show, in Proposition 2.4 below, how to construct a spectrally negative Lévy process with such a property from any spectrally negative Lévy process with a nonnegative mean.
Proof of Theorem 2.2. Let us write ψ 1 (u) = uφ(u + 1). Then, recalling that φ is analytic in the right half-plane, we readily deduce that the mapping u → ψ 1 (u) is analytic in C, with ψ 1 (0) = 0 and ψ 1 (−1) = −φ(0). Let us now show that under Assumption 2.1, ψ 1 is the Laplace exponent of a spectrally negative Lévy process with a positive mean. On one hand, since r → f (r) is monotone decreasing on (0, ∞), it follows that Π(dr) = e r (f (−r) dr − df (−r)) is clearly a Stieltjes measure on (−∞, 0). On the other hand, by integration by parts and a change of variable, we have, for any u ≥ 0,
ψ 1 (u) = u b(u + 1) + ∞ 0 (1 − e −(u+1)r )f (r) dr + q = bu 2 + b + q + ∞ 0 (1 − e −r )f (r) dr u + u ∞ 0 (1 − e −ur )e −r f (r) dr = bu 2 + φ(1)u + ∞ 0 (e −ur − 1 + ur)e −r (f (r) dr − df (r)) = bu 2 + φ(1)u + 0 −∞ (e ur − 1 − ur)Π(dr).
Checking, by integration by parts, that 0 −∞ (|r| ∧ r 2 )Π(dr) < ∞, we get that ψ 1 is the Laplace exponent of a spectrally negative Lévy process with a positive mean since ψ 1 (0 + ) = φ(1) > 0. Then, by means of the identity (1.5), we obtain, for any n = 1, 2, . . . ,
E[I −n ψ1 ] = φ(1) n−1 k=1 ψ 1 (k) Γ(n) = φ(1) n−1 k=1 kφ(k + 1) Γ(n) = n k=1 φ(k),
where we have used the identity Γ(n) = n−1 k=1 k. The self-decomposability of I ψ1 was discussed in the Introduction and the factorization of the exponential law follows readily from the independence of the random variables I ψ1 and I φ and the identity (1.2). The converse follows by means of similar reasoning. We only need to check that φ −1 (u) = ψ(u−1) u−1 is the Laplace exponent of a subordinator. From the conditions imposed on ψ, we can easily deduce that φ −1 is well defined on R + with φ −1 (0) = −ψ(−1) ≥ 0. Moreover, by integrations by parts, we get
φ −1 (u) = σu − (σ − m) + 1 u − 1 0 −∞ (e (u−1)r − 1 − (u − 1)r)Π(dr) P. Patie = σu − (σ − m) − 0 −∞ (e (u−1)r − 1)Π(−∞, r) dr = σu − (σ − m) − 0 −∞ (e ur − 1)e −r Π(−∞, r) dr − 0 −∞ (e −r − 1 + r)Π(dr) = −ψ(−1) + σu + ∞ 0 (1 − e −ur )e r Π(−∞, −r) dr.
The proof of the theorem is thus completed.
As a direct consequence of Theorem 2.2, we have the following fact. By selfdecomposability, the law of I ψ1 is absolutely continuous and thus the random variable J in (1.3) admits a density with respect to the Lebesgue measure, which, according to Bertoin and Yor [6], is a 1-harmonic function for the self-similar process associated to ξ in the Lamperti mapping (1.6). Thus, writing p 1 for the density of I ψ1 , the mapping x → x −2 p 1 (x −1 ) on R + is a 1-harmonic function for the self-similar process associated to ξ.
We complete this part with the following observation. Let us suppose that there exists a subordinator with Laplace exponent φ such that
φ(u) φ(u) = u, u ≥ 0.
Under such a condition, φ is called a special Bernstein function and we refer to Kyprianou and Rivero [17], and references therein, for more information on this function. Note that such an identity occurs in fluctuation theory for Lévy processes and that sufficient conditions for φ to be a special Bernstein function are that Assumption 2.1 holds, d = 0 and the mapping x → f (x)( ∞ x f (y) dy) −1 is a decreasing function on (0, ∞). As remarked in [6], we have, in this case, with the obvious notation,
I φ I φ (d) = e.
We deduce readily, from Theorem 2.2, that if φ is a special Bernstein function satisfying Assumption 2.1, then
I φ (d) = 1/I ψ1 .
Moreover, if, in addition, φ also satisfies Assumption 2.1, then 1/I φ is a positive selfdecomposable random variable. Note that if φ(0) = 0, then I φ is the solution to the random affine equation (1.9) and thus, in this case, solving this equation reduces to solve the random affine equation with constant coefficient (1.10).
Exponential functionals and entrance laws J ψ
We now assume that Ξ is a spectrally negative Lévy process with a non-negative mean m. Its Laplace exponent has the form (1.4). We recall that the positive random variable J ψ is the entrance law at time 1 of the self-similar Feller process associated to Ξ via the Lamperti mapping (1.6). We also mention that, when m > 0, Bertoin and Yor [8] show that the distribution of 1/I ψ is the so-called length-biased distribution of J ψ , that is, using the identity E[1/I ψ ] = m,
E[g(J ψ )] = m −1 E[1/I ψ g(1/I ψ )]
for any measurable function g : R + → R + . We refine this connection in the following proposition.
Proposition 2.4. Let ψ be the Laplace exponent of a spectrally negative Lévy process with mean m ≥ 0. Then, the mapping defined by
ψ 2 (u) = u u + 1 ψ(u + 1)
is analytic in C = {u ∈ C; Re(u) > −1} and is the Laplace exponent of a spectrally negative Lévy process with a positive mean ψ(1). Moreover, the identity in distribution J ψ
(d) = 1/I ψ2
holds. Consequently, the law of J ψ is absolutely continuous for any m ≥ 0.
Proof. First, since it is well known that ψ is analytic in the right half-plane, it is clear that the mapping ψ 2 (u) = u u+1 ψ(u + 1) is analytic in C. Next, we recall that ψ has the form
ψ(u) = σu 2 + mu + 0 −∞ (e ur − 1 − ur)Π(dr),
where 0 −∞ (|r| ∧ r 2 )Π(dr) < ∞ and σ ≥ 0. Thus, by means of integration by parts and writing f (r) = Π(−∞, r) for the tail of the Lévy measure, we get
u u + 1 ψ(u + 1) = σu 2 + (m + σ)u + u u + 1 0 −∞ (e (u+1)r − 1 − (u + 1)r)Π(dr) = σu 2 + (m + σ)u − u 0 −∞ (e (u+1)r − 1)f (r) dr = σu 2 + (m + σ)u − u 0 −∞ (e ur − 1)e r f (r) dr + 0 −∞ (e r − 1)f (r) dr (2.3) = σu 2 + m + σ + 0 −∞ (e r − 1 − r)Π(dr) u P. Patie + 0 −∞ (e ur − 1 − ur)e r (f (r) dr + Π(dr)) = σu 2 + ψ(1)u + 0 −∞ (e ur − 1 − ur)e r (f (r) dr + Π(dr)),
where we recognize the Laplace exponent of a spectrally negative Lévy process. Finally, observing that lim u→0 d du u u+1 ψ(u + 1) = ψ(1) > 0 since ψ is increasing on (0, ∞), we have, from (1.5) and any n = 1, 2 . . . ,
E[I −n ψ2 ] = ψ(1) n−1 k=1 (k/(k + 1))ψ(k + 1) Γ(n) = n k=1 ψ(k) Γ(n + 1) = E[J n ψ ],
where the last identity follows from (1.7). The absolute continuity property of the law of J ψ follows from that of I ψ2 as a self-decomposable random variable.
We mention that the random variable J ψ appears in the study of the so-called Ornstein-Uhlenbeck process associated to X. Indeed, if one considers the stochastic process U = (U t , t ≥ 0) defined, for any t ≥ 0, by
U t = e −t X e t −1 ,
then U is a stationary Feller process on [0, ∞) and its unique invariant measure is the law of J ψ ; see, for instance, [19], Theorem 1.2. The above proposition tells us that the invariant measure is absolutely continuous for any m ≥ 0.
We also indicate that the transformation of the Laplace exponent of a spectrally negative Lévy process used in the proof of Proposition 2.4 is a specific instance of more general mappings of characteristic exponents of Lévy processes introduced and studied by Kyprianou and Patie [12].
Some examples
In this section, we will make use of the identities presented in Section 2 to obtain new explicit examples of the law of the exponential functional associated to subordinators or spectrally negative Lévy processes, to obtain a new factorization of the exponential law and to prove the self-decomposability property of some positive random variables.
In [6], the authors study the connection between the law of the exponential functional of some subordinators and the following factorization of the exponential law:
e α S(α) −α (d) = e,(3.1)
where α ∈ (0, 1) and S(α) is a positive α-stable random variable, independent of e. We split this example into two parts.
On the one hand, the authors show that
I φ (d) = S(α) −α with φ(u) = αΓ(αu + 1) Γ(α(u − 1) + 1) = ∞ 0 (1 − e −ur )f (r) dr, (3.2) where f (r) = e −r/α Γ(1 − α)(1 − e −r/α ) α+1 , r > 0. (3.3)
(1) We start by applying the first part of Theorem 2.2. It is easy to check that the mapping r → f (r) is decreasing on (0, ∞) and thus, from Theorem 2.2 and using the recurrence relation Γ(u + 1) = uΓ(u), u > 0, we get that
ψ 1 (u) = Γ(α(u + 1) + 1) Γ(αu) ,
which, after some easy calculations, yields ψ 1 (u) = αΓ(α + 1)u + 0 −∞ (e ur − ur − 1) (α + 1)e (α+1)r/α αΓ(1 − α)(1 − e r/α ) α+2 dr. (3.4) Hence, from (3.1), we deduce that
I ψ1 (d) = e −α .
This result, up to a multiplicative constant, was actually obtained by Patie in [20], Theorem 4.1, where it is shown that the law of I ψ1 is related to the distribution of the absorption time of the α-self-similar continuous-state branching process.
(2) Moreover, let us define, as in Proposition 2.4,
ψ 2 (u) = u u + 1 ψ 1 (u + 1) = αu Γ(α(u + 2) + 1) Γ(α(u + 1) + 1)
and, after observing that ∂ ∂r e (α+1)r/α (1 − e r/α ) α+1 = (α + 1)e (α+1)r/α α(1 − e r/α ) α+2 , we obtain, from (2.3) and (3.4),
ψ 2 (u) = Γ(2α + 1) Γ(α) u + 0 −∞ (e ur − ur − 1) e (2α+1)r/α Γ(1 − α)(1 − e r/α ) α+2
2α + 1 α − e r/α dr.
P. Patie
Hence, from (1.5), we get, for any n = 1, 2, . . . ,
E[I −n ψ2 ] = α n−1 Γ(2α + 1) Γ(α)
Γ(αn + α + 1) Γ(2α + 1) = α n Γ(αn + α + 1) Γ(α + 1) and, by moment identification, we have αI ψ2
(d) = G −α (α + 1),
where G(a) stands for a gamma random variable of parameter a > 0. We deduce that, for any α ∈ (0, 1), the random variable G −α (α + 1) is a positive selfdecomposable random variable. (3) Next, we apply the converse part of Theorem 2.2 to ψ 2 . To this end, we introduce the subordinator with Laplace exponent φ −1 defined by
φ −1 (u) = 1 u − 1 ψ 2 (u − 1) = ψ 1 (u) u = α Γ(α(u + 1) + 1) Γ(αu + 1) .
To get the Lévy-Khintchine representation of φ −1 , note, from (3.2), that
φ −1 (u) = φ(u + 1) − φ(1) + φ(1).
That is, φ −1 is the Laplace exponent of the Esscher transform of φ killed at rate φ(1). Thus, using the expression (3.3), φ −1 (u) = αΓ(α + 1) + ∞ 0 (1 − e −ur )e −r f (r) dr.
We have, from (1.2), E[I n φ−1 ] = α −n Γ(α + 1)Γ(n + 1) Γ(αn + α + 1) , n = 1, 2, . . . .
In order to characterize the law of I φ−1 , let us denote by U an uniform random variable on (0, 1) and by S −α 1 (α) a random variable distributed according to the length-biased distribution of S −α (α), that is, for any measurable function g : R + → R + , we have
E[g(S −α 1 (α))] = E[S −α (α)g(S −α (α))] E[S −α (α)] .
Recalling that for any n = 1, 2, . . . , E[S −αn (α)] = Γ(n + 1) Γ(αn + 1) , we get, by taking the random variable U independent of S −α 1 (α),
E[U n S −αn 1 (α)] = E[U n ]E[S −α(n+1) (α)] E[S −α (α)]
= Γ(α + 1)Γ(n + 2) (n + 1)Γ(αn + α + 1) = Γ(α + 1)Γ(n + 1) Γ(αn + α + 1) .
By moment identification, we deduce the following identity:
αI φ−1 (d) = U S −α 1 (α).
This yields the following factorization of the exponential law:
U S −α 1 (α)G α (α + 1) (d) = e,
where the three random variables on the left-hand side are assumed to be independent.
On the other hand, Bertoin and Yor [6] also observed that (1 − e −ur ) (1 − α) 2 e r/α αΓ(α + 1)(e r/α − 1) 2−α dr.
Easily verifying that the density is decreasing, we obtain, appealing to obvious notation, that ψ 1 (u) = u Γ(αu + 1) αΓ(α(u + 1)) , which, after some easy manipulations, yields ψ 1 (u) = Γ −1 (α + 1)u + 0 −∞ (e ur − ur − 1) (1 − α) 2 e −r/α α 2 Γ(α + 1)(e −r/α − 1) 2−α 1 − α + 2 − α (1 − e r/α ) dr.
Thus, from the identity (3.1) and Theorem 2.2, we deduce that
I ψ1 (d) = S(α) α .
Hence, S(α) α is a positive self-decomposable random variable.
, the entrance law of X at time 1, in terms of its positive entire moments as follows: E[J n ψ ] = n k=1 ψ(k) Γ(n + 1) , n = 1, 2, . . . . (1.7)
Theorem 2 . 2 .
22Let ξ be a subordinator with Laplace exponent φ given by(1.1). If Assumption 2.1 holds, then there exists an independent spectrally negative Lévy process with a positive mean and Laplace exponent ψ 1 , analytic in the domain C
AcknowledgementsThe author is grateful to an anonymous referee for a careful reading of this paper.
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