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2
Continuous LTI System
Discrete LTI System
Bilinear
Transform
Hankel Operator
Encoder
LTI layer #0
Nonlinearity
LTI layer #1
Nonlinearity
Decoder
Output
HOPE-SSM
Encoder
LTI layer #0
Nonlinearity
LTI layer #1
Nonlinearity
Decoder
Output
S4, S4D, S5,
System
Identification
Impulse
Response
Figure 1: There are many equivalent ways to represent an LTI system. While most of the
canonical SSMs use continuous LTI systems as their parameters, we propose to parameterize | [
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] |
canonical SSMs use continuous LTI systems as their parameters, we propose to parameterize
an SSM by the Markov parameters in its Hankel operator. The feedthrough matrix D is not
shown in the diagram, but it is also a parameter of the LTI layers in both the canonical SSMs
and our HOPE-SSM.
computational complexity as the S4D model. Moreover, it requires only 1/3 the number of
parameters of an LTI system in an S4D model to parameterize an LTI system in a HOPE-SSM. | [
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] |
parameters of an LTI system in an S4D model to parameterize an LTI system in a HOPE-SSM.
The practical benefits of our novel parametrization are improved robustness (with respect to ini-
tialization and training stability) and performance (with respect to model quality and parameter
count). Moreover, we show that the memory of our HOPE-SSM does not decay (see eq. (8)),
making it possible to solve tasks that involve even longer-range dependency. This addresses | [
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making it possible to solve tasks that involve even longer-range dependency. This addresses
the well-known issue that the LTI system of a canonical SSM suffers from exponentially de-
caying memory [2]. We remark that [2] also proposes a different model called spectral SSM
that achieves non-decaying memory using the so-called spectral filtering method. While they
show that a spectral SSM is exponentially close to an SSM, our HOPE-SSM is an actual SSM | [
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] |
show that a spectral SSM is exponentially close to an SSM, our HOPE-SSM is an actual SSM
containing LTI systems, and it only differs from a canonical SSM in the parameterization.
Contributions. Our main contributions are summarized as follows.
1. We show a strong relationship between the performance of an SSM and the Hankel singular
values of its LTI systems. We justify this relationship using ideas in ROM. We theoretically | [
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values of its LTI systems. We justify this relationship using ideas in ROM. We theoretically
prove that high-degree LTI systems need careful designs and are unstable over training.
This explains the difficulties in initializing and training an SSM.
2. We propose a new parametrization of LTI systems using the Hankel operator, which can
be implemented efficiently by nonuniformly sampling the transfer function and requires
1/3 the number of parameters in an S4D model for an LTI system. We prove that our | [
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] |
1/3 the number of parameters in an S4D model for an LTI system. We prove that our
HOPE-SSM can be (i) randomly initialized, (ii) stably trained, and (iii) has non-decaying
long memory.
3. We empirically demonstrate that our HOPE-SSM is robust using the sCIFAR-10 task
and that it has long-term memory using a noise-padded sCIFAR-10 dataset. We test the
performance of a full-scale HOPE-SSM on the Long-Range Arena (LRA) and observe that | [
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] |
performance of a full-scale HOPE-SSM on the Long-Range Arena (LRA) and observe that
its performance exceeds that of its S4 and S4D counterparts for most tasks.
3
2
Preliminaries
Let Γ = (A, B, C, D) be a continuous-time LTI system defined in eq. (1). One can take a bilinear
transform to obtain a discrete system Γ = (A, B, C, D) so that the underlying dynamics is given
by
xk+1 = Axk + Buk,
yk = Cxk + Duk.
(2)
The transfer functions of Γ and Γ are
G(s) = C(sI − A)−1B + D
and
G(z) = C(zI − A)−1B + D, | [
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(2)
The transfer functions of Γ and Γ are
G(s) = C(sI − A)−1B + D
and
G(z) = C(zI − A)−1B + D,
respectively.
The importance of the transfer functions is that they bring the inputs to the
outputs in the Laplace domain by multiplication, which reduces to the Fourier domain if we
assume that u is bounded and compactly supported:
ˆy(s) = G(is)ˆu(s),
ˆy = G(ω)ˆu,
(3)
where the hat of a function and a vector means the Fourier transform and the Fourier coefficients, | [
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] |
where the hat of a function and a vector means the Fourier transform and the Fourier coefficients,
respectively, and ω is the vector of roots of unity. Throughout this paper, we always assume
that our LTI systems are asymptotically stable, i.e., the eigenvalues of A have negative real
parts, or equivalently, the eigenvalues of A are contained in the open unit disk in the complex
plane.
Given a discrete LTI system Γ, its Hankel operator is defined by a doubly infinite Hankel matrix
H : ℓ2 → ℓ2,
Hi,j = CA | [
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] |
H : ℓ2 → ℓ2,
Hi,j = CA
i+jB,
i, j ≥ 0.
(4)
This discrete Hankel operator on ℓ2 is a bounded linear operator of rank ≤ n, the number
of latent states.
We denote its singular values by σ1(H) ≥ σ2(H) ≥ · · · ≥ σn(H) ≥ 0.
It maps the past inputs to the future outputs:
if uk = 0 for all k ≥ 0, then we have
�
y⊤
0
y⊤
1
· · ·
�⊤ = H
�
u⊤
−1
u⊤
−2
· · ·
�⊤. Analogously, one can define a continuous Hankel op-
erator H : L2([0, ∞)) → L2([0, ∞)) given a continuous-time LTI system Γ. If Γ and Γ are | [
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] |
erator H : L2([0, ∞)) → L2([0, ∞)) given a continuous-time LTI system Γ. If Γ and Γ are
related by the bilinear transform, then the singular values of H are equivalent to those of H,
i.e., σj(H) = σj(H). (See Appendix B for more details.) In this paper, we consider the decay
of the Hankel singular values. To quantitatively measure how “small” a singular value is, we
introduce the numerical rank of an LTI system. Given a small number ϵ ≥ 0, we define the
ϵ-rank of Γ to be | [
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ϵ-rank of Γ to be
rankϵ(Γ) = max{j | σj(H)/σ1(H) > ϵ}.
When ϵ = 0, we reproduce the exact rank of H, which is rarely < n in the floating-point
arithmetic. The ϵ-rank allows us to eliminate the small but perhaps positive singular values.
3
Unravel a Mystery: Hankel Singular Values in Initialization
and Training
We begin our exploration by presenting a mystery. We train an S4D model to learn the sCIFAR-
10 image classification task [19, 28]. We consider three initialization schemes of the LTI systems | [
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] |
10 image classification task [19, 28]. We consider three initialization schemes of the LTI systems
in the SSM: Γ1, Γ2, and Γ3, where Γ1 and Γ2 are two “random” systems described in section 3.2
and section 3.1, respectively, and Γ3 is based on the HiPPO-LegS framework. For an SSM
initialized by a certain LTI system Γj (1 ≤ j ≤ 3), we train it in two different ways: either by
freezing A, B, and C and only training the other model parameters or by assigning A, B, and | [
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] |
freezing A, B, and C and only training the other model parameters or by assigning A, B, and
C a small learning rate. The three initializations and two learning rate assignments comprise a
total of six combinations, summarized in Figure 2. Figure 2 naturally raises two questions:
4
0
20
40
60
80
100
30
40
50
60
70
80
90
Random
1
Random
2
HiPPO-LegS
3
Trained LTI Systems
Epochs
Accuracy
0
20
40
60
80
100
30
40
50
60
70
80
90
Random
1
Random
2
HiPPO-LegS
3
Untrained LTI Systems
Epochs
Accuracy | [
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Random
1
Random
2
HiPPO-LegS
3
Untrained LTI Systems
Epochs
Accuracy
Figure 2: Test accuracy of the SSMs on the sCIFAR task. The LTI systems are initialized in
three different ways and are either trained or untrained. We notice that when the LTI systems
are initialized using Γ1, training the LTI system is impairing the model accuracy. This is in
contrast to SSMs initialized by the HiPPO-LegS systems Γ3, where training the LTI system is
helping the performance. | [
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helping the performance.
(a) Why does one random initialization Γ1 outperform the other one Γ2?
(b) While freezing Γ1 and Γ3 leads us to similar performance, why does a small learning rate
foster the Γ3-initialized SSM but deteriorate the Γ1-initialized one?
To answer these questions, we study the Hankel singular values of the systems, but why are
they relevant? The reason is that the Hankel singular values measure the “complexity” of an | [
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they relevant? The reason is that the Hankel singular values measure the “complexity” of an
LTI system. The easiest way to see this is via ROM of the system Γ [1, 9]:
For any k < n, there exists a reduced-order approximation ˜ΓH = ( ˜AH, ˜BH, ˜CH, ˜DH) with
˜AH ∈ Ck×k, such that σ1(H − ˜H) ≤ σk+1(H), where ˜H is the Hankel operator of ˜ΓH.
In addition, there exists another reduced-order approximation ˜Γ∞ = ( ˜A∞, ˜B∞, ˜C∞, ˜D∞)
with ˜A∞ ∈ Ck×k, such that ∥G − ˜G∥∞ ≤ �n | [
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with ˜A∞ ∈ Ck×k, such that ∥G − ˜G∥∞ ≤ �n
j=k+1 σj(H) ≤ (n − k)σk+1(H), where ˜G is
the transfer function of ˜Γ∞ and ∥ · ∥∞ is the infinity norm over the imaginal axis.
In particular, if σk+1(H) is small, then that means given the same past input, the future outputs
of Γ and ˜ΓH must be similar. Moreover, if the sum of the trailing singular values �n
j=k+1 σj(H)
is small, then by eq. (3), Γ and ˜Γ∞ produce similar outputs on any input, which is a stronger | [
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is small, then by eq. (3), Γ and ˜Γ∞ produce similar outputs on any input, which is a stronger
notion of approximation than using the Hankel norm σ1. Hence, ROM says that if the Hankel
singular values decay fast, then we can replace the LTI system with a much smaller one without
too much loss; in other words, most states in x(t) are not properly used to memorize the input.
In our experiment, each SSM has 4 layers and 128 channels. These comprise 4 × 128 = 512 | [
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0.2664960026741028
] |
In our experiment, each SSM has 4 layers and 128 channels. These comprise 4 × 128 = 512
different copies of single-input/single-output Γj (j = 1, 2, or 3). Note that every HiPPO-LegS
system Γ3 is also distinct because C is initialized randomly. Every LTI has n = 64 latent states.
These make up 512 × 64 = 32768 relative Hankel singular values σj(H)/σ1(H) to consider.
In Figure 3, we show the histograms of all these relative Hankel singular values at two different | [
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] |
In Figure 3, we show the histograms of all these relative Hankel singular values at two different
stages of training. Note that the three histograms on the second row only apply to the case when
the LTI systems are trained with a small learning rate; when they are frozen, the distributions
always stay the same as those at initialization. We can explain the behaviors of the three SSMs
using their Hankel singular values: | [
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] |
using their Hankel singular values:
1. The system Γ1 initially has a high numerical rank. Hence, when the system is untrained,
Γ1 defines a random mapping that captures the rich content in the input data and yields
the rest of the work to other model parameters in the SSM. However, when the system is
trained, its Hankel singular values start to decay rapidly with only 27.87% of the singular
values satisfying that σj(H)/σ1(H) > 0.01, and the system can no longer handle a variety
5
At Initialization | [
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] |
5
At Initialization
Random Γ1
After 10 Epochs
σj(H)/σ1(H)
Random Γ2
σj(H)/σ1(H)
HiPPO-LegS Γ3
σj(H)/σ1(H)
Figure 3:
The distribution of all relative Hankel singular values σj(H)/σ1(H) of the LTI
systems in an SSM. For each initialization, the distribution is shown both at initialization and
after the SSM is trained for 10 epochs. The second row only applies to the case where the LTI
systems are not frozen.
of distinct inputs. This makes it harder to parse complicated images in the sCIFAR-10
dataset. | [
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of distinct inputs. This makes it harder to parse complicated images in the sCIFAR-10
dataset.
2. No matter whether trained or not, the Hankel singular values of Γ2 decay very fast, which
means Γ2 is almost equivalent to a much smaller system and cannot effectively capture
the complicated patterns in the input space. Hence, the SSMs with both trained and
untrained LTI systems could not learn the task quickly.
3. The system Γ3, trained or not, has a high numerical rank. In particular, over 87.82% of | [
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] |
3. The system Γ3, trained or not, has a high numerical rank. In particular, over 87.82% of
the singular values satisfy that σj(H)/σ1(H) > 0.01 after 10 epochs. In this case, training
the LTI systems allows us to further accelerate the optimization.
While the distribution of the Hankel singular values allows us to explain the mysterious behaviors
of SSMs initialized by different LTI systems, a perhaps more interesting question is why we | [
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] |
of SSMs initialized by different LTI systems, a perhaps more interesting question is why we
observe such certain patterns of the histograms. In the next two subsections, we theoretically
show why the random system Γ2 has fast-decaying Hankel singular values and why Γ1 is sensitive
to training.
3.1
Random LTI Systems Have Low Ranks
In our experiment, we sample Γ2 by assuming the diagonal entries of A = diag(a1, . . . , an) are
uniformly sampled on the unit disk and B ◦ C | [
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uniformly sampled on the unit disk and B ◦ C
⊤ is a random vector with each entry sampled
i.i.d. from a normal distribution N(0, 1). We then compute Γ2 from the discrete system Γ2
using the bilinear transform. In this section, we make a relaxed assumption that aj are sampled
i.i.d. with
P (|aj| > (1 − ρ)) = O(ρα)
as
ρ → 0+
for some α > 0. For example, if aj is uniformly distributed on the open unit disk, then we have
α = 1. With these assumptions, we formally show that the system has a low ϵ-rank with high | [
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] |
α = 1. With these assumptions, we formally show that the system has a low ϵ-rank with high
probability.
Theorem 1. Given any ϵ > 0, 0 < α ≤ 1, and 0 < δ ≤ 1, with probability at least 1 − δ, the
ϵ-rank of the system Γ2 = (A, B, C, D) is O(ln
�
δ−3/2ϵ−1n
�
nβ), where
β ≤
1
1 + α + ln(2 +
�
ln(1/δ)/2)
ln(n)
6
and the constant in O is universal.
We defer the proof to Appendix C. As explained in section 2, the ϵ-ranks of the continuous Γ2 | [
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] |
We defer the proof to Appendix C. As explained in section 2, the ϵ-ranks of the continuous Γ2
and the discrete Γ2 are the same as the two systems share the same Hankel singular values.
What Theorem 1 says is that if we ignore the logarithmic factors in the O-notation, then the ϵ-
rank of Γ scales like nβ as n → ∞, where β < 1. For example, when aj are uniformly distributed
on the unit disk, we have β = 1/2 plus a small number bounded by ln(2 +
�
ln(1/δ)/2)/ ln(n). | [
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] |
on the unit disk, we have β = 1/2 plus a small number bounded by ln(2 +
�
ln(1/δ)/2)/ ln(n).
In practice, we find that this term can almost be ignored (see Appendix F). Hence, unlike
a randomly initialized Hankel matrix (see section 4), the ϵ-rank of a random LTI system is
unfortunately sublinear in n.
3.2
LTI Systems are Unstable under Perturbations
In the last subsection, we explained how we sampled Γ2 and showed that a randomly initialized | [
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] |
In the last subsection, we explained how we sampled Γ2 and showed that a randomly initialized
system Γ2 has a low numerical rank. To sample a random Γ1, we create random samples of the
transfer function {(isj, Gj)}N
j=1, where sj ∈ R and Gj ∈ C. We then identify a system Γ1 whose
transfer function G satisfies that G(isj) ≈ Gj. In this section, we aim to understand why the
system Γ1 tends to lose many numerical ranks through training while the HiPPO-LegS system | [
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] |
system Γ1 tends to lose many numerical ranks through training while the HiPPO-LegS system
Γ3 does not. We prove a theorem that says the stability of a system under perturbation highly
depends on its parameters. We assume that A is a diagonal matrix and we perturb its diagonal
entries. This aligns with the structure of the S4D and S5 models [13, 27].
Theorem 2. Let Γ = (A, B, C, D) be a stable continuous-time LTI system, where A = | [
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] |
Theorem 2. Let Γ = (A, B, C, D) be a stable continuous-time LTI system, where A =
diag(a1, . . . , an) is diagonal. Let ˜Γ = ( ˜A, ˜B, ˜C, D) be a perturbed stable system with ˜A =
diag(˜a1, . . . , ˜an). Assume there exist ∆A, ∆B > 0 such that |aj − ˜aj| ≤ ∆A ≤ minj |Re(aj)|/2
and |bjcj − ˜bj˜cj| ≤ min(|bjcj|, ∆B) for all j = 1, . . . , n. Let G and ˜G be the transfer functions
of Γ and ˜Γ, respectively. Then, the following statements hold.
(a) We have
∥G − ˜G∥∞ ≤ 4n∆A max
j
|bjcj|
|Re(aj)|2 + n∆B max
j
1 | [
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] |
(a) We have
∥G − ˜G∥∞ ≤ 4n∆A max
j
|bjcj|
|Re(aj)|2 + n∆B max
j
1
|Re(aj)|.
(b) The upper bound is tight up to a factor of n. That is, given any Γ, ∆A ≤ minj |Re(aj)|/2,
and ∆B ≤ minj |bjcj|, there exist two systems ˜ΓA and ˜ΓB, with transfer functions ˜GA
and ˜GB, respectively, that satisfy the above perturbation conditions and have
∥G − ˜GA∥∞ ≥ ∆A max
j
|bjcj|
|Re(aj)|2 ,
∥G − ˜GB∥∞ ≥ ∆B max
j
1
|Re(aj)|.
The proof can be found in Appendix D. Theorem 2 says that when a diagonal LTI system starts | [
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] |
The proof can be found in Appendix D. Theorem 2 says that when a diagonal LTI system starts
to deviate, its stability depends on two things: the proximity of its poles aj to the imaginary
axis and the magnitudes of B and C. We first explain why each of these might cause a problem
and then use Theorem 2 to show why Γ1 is unstable during training.
As stated in [5], “[the Hankel singular values] decay more rapidly the farther the Λ(A) falls in | [
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] |
As stated in [5], “[the Hankel singular values] decay more rapidly the farther the Λ(A) falls in
the left half of the complex plane,” where Λ(A) is the spectrum of A, which is equivalent to
{a1, . . . , an} in the diagonal case. Hence, many diagonal LTI systems with a high ϵ-rank would
have eigenvalues aj close to the imaginary axis, making the system unstable. Moreover, one
may believe that ∥G∥∞ is highly related to |bjcj|, and hence, the fact that the system’s stability | [
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] |
may believe that ∥G∥∞ is highly related to |bjcj|, and hence, the fact that the system’s stability
depends on |bjcj| is not an issue. However, this is not true as a system with a small transfer
function may have an arbitrarily large system matrix B◦C⊤. Such a system can be numerically
hazardous to work with even without perturbation due to the so-called cancellation errors [34].
We can now understand why Γ1 is much more sensitive to perturbation than Γ3. In Figure 4, | [
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] |
We can now understand why Γ1 is much more sensitive to perturbation than Γ3. In Figure 4,
we show the locations of the poles aj and their associated magnitudes |bjcj|. Compared to Γ3,
7
10
20
30
40
50
60
10-6
10-4
10-2
100
A = 0
A
A = 1%
Random Γ1
Hankel Singular Values
j
σj/σ1
-102 -100 -10-2
-10
-5
0
5
10
10-2
100
102
104
Real Axis
Imaginary Axis
aj and |bjcj|
10
20
30
40
50
60
10-6
10-4
10-2
100
A = 0
A
A = 1%
HiPPO-LegS Γ3
Hankel Singular Values
j
σj/σ1
-102 -100 -10-2
0
50
100
150
200
250
300
350
400 | [
-0.11953974515199661,
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0.7758213877677917,
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0.27488216757774353,
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0.4817163646221161,
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] |
HiPPO-LegS Γ3
Hankel Singular Values
j
σj/σ1
-102 -100 -10-2
0
50
100
150
200
250
300
350
400
10-2
100
102
104
aj and |bjcj|
Real Axis
Imaginary Axis
Figure 4: A random perturbation to the imaginary part of A is added to the random system
Γ1 and the HiPPO-LegS system Γ3. The magnitude of the perturbation is set to 0.1% and 1%
of the original matrix A. For each system Γj, on the left, we show the relative Hankel singular | [
-0.23938573896884918,
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] |
of the original matrix A. For each system Γj, on the left, we show the relative Hankel singular
values σj/σ1 of the original and perturbed systems; on the right, we plot the location of each
aj in the complex plane and use the color to indicate the magnitude of its associated |bjcj|.
the poles of Γ1 are much closer to the imaginary axis whereas the values of |bjcj| are also much
larger. Hence, by Theorem 2, the system Γ1 should be much less stable than Γ3. This is indeed
seen in Figure 4.
4 | [
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] |
seen in Figure 4.
4
HOPE-SSM: A Rankful, Stable, and Long-memory Parame-
terization
Given the potential issues of an SSM discussed in section 3, we propose an entirely different
parametrization of the LTI systems called HOPE. We first describe the details of our HOPE-
SSM and then explain how it resolves the low-rank and unstable issues of an LTI system. In
addition, it also benefits from long-term memory. Our strategy is to use a Hankel matrix defined | [
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addition, it also benefits from long-term memory. Our strategy is to use a Hankel matrix defined
in eq. (4) to parameterize an LTI system. Therefore, instead of having A, B, C, and D as the
model parameters, we now have a vector h of length n and the skip connection D. The finite
Hankel matrix H ∈ Cn×n is then defined by the Markov parameters in h:
Hi,j = 1{i+j<n}hi+j.
(5)
This Hankel matrix corresponds to a discrete LTI system Γ, which is further associated with | [
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] |
(5)
This Hankel matrix corresponds to a discrete LTI system Γ, which is further associated with
a continuous-time system Γ via the bilinear transform. Notably, in our HOPE-SSM, it is the
continuous system Γ that takes the role of (A, B, C) in a canonical SSM, e.g., S4 and S4D.
If we set ∆t = 1 and discretize Γ, the convolutional kernel is exactly h padded with L−n zeros,
where L is the length of the sequential input. However, we usually want to discretize with a | [
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] |
where L is the length of the sequential input. However, we usually want to discretize with a
different ∆t. We can do so via resampling the transfer functions of Γ and Γ, which equal [25]
G(s) =
n−1
�
j=0
hj ((1 + s)/(1 − s))−j−1
⇔
G(z) =
n−1
�
j=0
hjz−j−1.
(6)
Let ω(L) =
�
ω(L)
0
· · ·
ω(L)
L−1
�⊤
be the vector of the Lth roots of unity. Given an input u ∈ CL×1
of length L, when ∆t = 1, the outputs can be evaluated as
y = iFFT(FFT(u) ◦ G(ω(L))),
G(ω(L)) =
�
G(ω(L)
0
)
· · ·
G(ω(L)
L−1)
�⊤
.
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0.39091992378234863,
0.31612110137939453,
-0.012369437143206596,
0.4018864333629608,
-0.05504995584487915,
0.1264902949333191,
-0.23707126080989838,
0.44279125332832336,
-0.7316057682037354,
0.08981314301490784,
0.7251902222633362,
0.5609636306762695,
0.3182145655155182,
-0.4504352807998657,
-0.5729271173477173,
-0.2261885553598404,
-0.6077632904052734,
0.5723572373390198,
-0.03600533306598663,
0.5103387832641602,
-0.3498033583164215,
-0.32307058572769165,
-0.5880438089370728,
0.44394925236701965,
0.19559594988822937,
-0.04709944128990173,
-0.34651002287864685,
0.21326200664043427,
-0.010013756342232227,
0.342832088470459,
-0.11616884917020798,
-0.6623198986053467,
-0.37021583318710327,
0.03847859054803848
] |
y = iFFT(FFT(u) ◦ G(ω(L))),
G(ω(L)) =
�
G(ω(L)
0
)
· · ·
G(ω(L)
L−1)
�⊤
.
8
For a different ∆t, one way to think of it is that we have compressed or dilated the time domain
of u by a factor of ∆t. Hence, the frequency domain of its Fourier transform ˆu is dilated or
compressed by a factor of 1/∆t. That is, we should relocate our samplers in the frequency
domain as
ω(L,∆t) =
�
ω(L,∆t)
0
· · ·
ω(L,∆t)
L−1
�⊤
,
ω(L,∆t)
j
=
1 + s(L)
j
/∆t
1 − s(L)
j
/∆t
,
s(L)
j
=
ω(L)
j
− 1
ω(L)
j
+ 1
,
where s(L)
j | [
0.06270008534193039,
-0.5184550285339355,
0.5867181420326233,
-0.3561631143093109,
-0.06135525181889534,
0.5771690011024475,
0.19393131136894226,
-0.371721476316452,
0.16275012493133545,
-0.5103887319564819,
0.09792265295982361,
0.5023081302642822,
-0.21972258388996124,
0.5524004101753235,
-0.0407075472176075,
1.1200530529022217,
0.4655552804470062,
0.16589224338531494,
-0.38829028606414795,
-0.7740496397018433,
0.48402732610702515,
-0.09620022028684616,
0.5572108030319214,
0.11933682858943939,
0.03563316538929939,
-0.0754501149058342,
0.2895849645137787,
0.1571650505065918,
0.6676620841026306,
0.20190048217773438,
-0.1824667751789093,
0.4965275824069977,
0.08981948345899582,
0.042797889560461044,
0.07522111386060715,
0.10075792670249939,
0.5268915891647339,
0.4110826849937439,
0.3234198987483978,
-0.1399673968553543,
-0.10852459818124771,
-0.14601458609104156,
0.022947531193494797,
0.6387537121772766,
-0.5098981857299805,
-0.5338702201843262,
-1.8581206798553467,
0.02439689449965954,
0.008127167820930481,
-0.220542773604393,
0.10149402916431427,
0.4239039123058319,
0.5617843270301819,
0.05673864856362343,
-0.6050655841827393,
0.5869927406311035,
-0.2431817501783371,
-0.445319801568985,
0.514446496963501,
-0.08161555230617523,
0.25965195894241333,
0.011624296195805073,
0.07852306216955185,
0.610651969909668,
-0.3765580356121063,
0.5924965143203735,
-0.19583190977573395,
0.2030722200870514,
-0.6235059499740601,
-0.40891680121421814,
0.40109947323799133,
0.17362381517887115,
-0.1968551129102707,
0.403988778591156,
-0.31301477551460266,
0.4264172911643982,
-0.7166529297828674,
-0.10400035977363586,
0.6034945249557495,
0.1133723333477974,
0.17422936856746674,
-0.33924809098243713,
0.5245330333709717,
-0.4777657389640808,
0.21952544152736664,
0.20681598782539368,
0.0034200556110590696,
0.5327022671699524,
-0.39656761288642883,
0.6079484820365906,
0.03026224486529827,
-0.8289228081703186,
-0.05728963017463684,
0.1501540094614029,
0.163227841258049,
0.6304434537887573,
0.2770015597343445,
-0.34870287775993347,
0.3690502345561981,
0.3486395478248596,
-0.4261971116065979,
-0.27879640460014343,
-0.21254798769950867,
-0.21083331108093262,
-0.5343683958053589,
-0.705984890460968,
-0.6937224268913269,
-0.7054160833358765,
-0.5817133188247681,
-0.003099494380876422,
-1.3357502222061157,
0.5333306789398193,
-0.312928169965744,
-0.2832528352737427,
-0.01793326810002327,
-0.2389206737279892,
-0.1880093365907669,
-0.2649843096733093,
0.3431270718574524,
0.6351673007011414,
-0.29167044162750244,
0.5405031442642212,
-0.027464954182505608,
0.2985481023788452,
0.28509649634361267,
0.008370138704776764,
0.5255252122879028,
0.5295068621635437,
-0.5820854902267456,
-0.16313830018043518,
0.6599641442298889,
0.8460313081741333,
0.5412493348121643,
0.1701008826494217,
-0.3861914277076721,
-0.861825168132782,
0.04247758537530899,
-0.5347021818161011,
-0.7248361706733704,
-0.20143809914588928,
-0.1797131896018982,
-0.42991894483566284,
-0.6552383899688721,
-0.19747984409332275,
0.3788541555404663,
-0.253055214881897,
-0.358830064535141,
0.1810973882675171,
-0.31985169649124146,
-0.31693413853645325,
0.13638372719287872,
0.4850753843784332,
-0.9663863182067871,
0.24303556978702545,
-0.3199771046638489,
-0.6416001915931702,
-0.4413815438747406,
0.03363997861742973,
-0.28822875022888184,
0.25140753388404846,
0.007052803412079811,
0.3693011999130249,
0.05107256397604942,
-0.5080904960632324,
0.39450424909591675,
-0.7196183800697327,
0.7461373805999756,
-0.21427999436855316,
-0.3391175866127014,
-0.0932454988360405,
-0.3042859137058258,
0.1266733855009079,
0.02290804497897625,
0.5393249988555908,
-0.21808889508247375,
-0.28320661187171936,
0.022167732939124107,
-0.20791690051555634,
0.5547161102294922,
0.19252176582813263,
-0.7105096578598022,
-0.15682147443294525,
0.1281597763299942,
0.07542773336172104,
0.44319385290145874,
0.3378388285636902,
0.4481566548347473,
0.09916314482688904,
0.4156789481639862,
0.03305879980325699,
-0.5532034039497375,
-0.07076109200716019,
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-0.8801797032356262,
1.1519877910614014,
-0.1142628937959671,
-0.1930348128080368,
0.18957394361495972,
-0.23181311786174774,
0.3209492564201355,
0.0817863941192627,
0.15880006551742554,
0.09663132578134537,
0.1801227480173111,
-0.39888954162597656,
0.1516040414571762,
-0.09134165197610855,
-0.14053921401500702,
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0.5421348810195923,
0.2976824641227722,
-0.2107706069946289,
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0.12113109976053238,
-0.25136569142341614,
0.10140841454267502,
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0.3285483121871948,
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0.94746994972229,
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0.4734075367450714,
0.40028664469718933,
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0.547998309135437,
0.8646610975265503,
0.2890431880950928,
0.4778619110584259,
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0.12464578449726105,
0.6102096438407898,
0.15550431609153748,
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0.38081368803977966,
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0.5768252015113831,
0.40950894355773926,
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0.156800776720047,
0.10918718576431274,
-0.22051605582237244,
-0.25349676609039307,
-0.02804889716207981,
-0.4096044600009918,
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0.29153817892074585,
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0.6137360334396362,
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0.04678074270486832,
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0.8203133344650269,
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0.06219681724905968,
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0.7333199977874756,
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] |
L−1
�⊤
,
ω(L,∆t)
j
=
1 + s(L)
j
/∆t
1 − s(L)
j
/∆t
,
s(L)
j
=
ω(L)
j
− 1
ω(L)
j
+ 1
,
where s(L)
j
/∆t and ω(L,∆t)
j
are the scaled samplers in the time domain and the angular domain,
respectively. Then, the output vector y can be computed as
y = iFFT(FFT(u) ◦ G(ω(L,∆t))),
G(ω(L,∆t)) =
�
G(ω(L,∆t)
0
)
· · ·
G(ω(L,∆t)
L−1
)
�⊤
.
(7)
We provide a detailed derivation of eq. (7) in Appendix G. With L processors, we can sample the | [
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] |
(7)
We provide a detailed derivation of eq. (7) in Appendix G. With L processors, we can sample the
transfer function G independently at L locations, each of which takes O(n) time. Computing
the FFT and iFFT takes O(L log L). Overall, the evaluation of the Hankel-parameterized LTI
system takes ˜O(L + n) time, which agrees with the complexity of the S4 and S4D models. To
make the model recurrent during the inference time, one can either identify a system (A, B, C) | [
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] |
make the model recurrent during the inference time, one can either identify a system (A, B, C)
whose Hankel operator is H and compute as if it is an S4D model [18, 3], or compute directly
the convolutional kernel based on eq. (7).
4.1
Advantage I: A HOPE-SSM has a High Numerical Rank
The properties of a random Hankel matrix have been studied in [8] and [24]. We can build upon
their work to prove the following result. | [
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] |
their work to prove the following result.
Theorem 3. Let h1, h2, . . . be a sequence of i.i.d. random variables with mean 0 and variance 1
that have finite third moments. We almost surely have that for any ϵ > 0, the (ϵ/
�
ln(n))-rank
of Hn is Ω(n), where Hn is the n × n Hankel matrix defined in eq. (5).
Hence, if we ignore the logarithmic factor of
�
ln(n), then the numerical rank of a random
Hankel matrix should be proportional to n as n → ∞. This means if we imagine the landscape | [
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] |
Hankel matrix should be proportional to n as n → ∞. This means if we imagine the landscape
of parameters, the high-rank LTI systems parameterized by the system matrices A, B, and C
are sparse islands in a big ocean, whereas those parameterized by the Markov parameters h are
large continents.
4.2
Advantage II: A HOPE-SSM is Stable under Perturbation
Unlike the LTI system, a Hankel matrix H is very stable under perturbation, as shown in the
following theorem. | [
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] |
following theorem.
Theorem 4. Let h ∈ Cn×1 be a vector and G be the transfer function defined in eq. (6).
Suppose we perturb h to ˜h and let ˜G be its transfer function. Then, we have ∥G − ˜G∥∞ ≤
√n∥h − ˜h∥2.
Therefore, the stability of the HOPE-SSM does not depend on the Markov parameters them-
selves. That is, the HOPE-SSM can be trained stably without reparameterization (see [30]).
4.3
Advantage III: A HOPE-SSM’s Memory Does Not Fade | [
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] |
4.3
Advantage III: A HOPE-SSM’s Memory Does Not Fade
While many LTI systems are tailored for long-term memory [29, 12], an asymptotically sta-
ble system must inevitably suffer from an exponential decay in its memory [2]. This can be
manifested by the Hankel matrix: since
�
y⊤
0
y⊤
1
· · ·
�⊤ = H
�
u⊤
−1
u⊤
−2
· · ·
�⊤, we can write
yj =
∞
�
k=1
Hj,k−1u−k =
∞
�
k=1
H0,k+j−1u−k,
j ≥ 0.
(8)
9
0
10
20
30
40
50
60
70
80
90
100
30
40
50
60
70
80
90
Accuracy
Epochs
Test Accuracy of HOPE-SSM | [
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] |
9
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50
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70
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90
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Accuracy
Epochs
Test Accuracy of HOPE-SSM
Hankel Singular Values
At Initialization
After 10 Epochs
σj(H)/σ1(H)
Figure 5: The test accuracy of the HOPE-SSM on the sCIFAR-10 task and the evolution of the
Hankel singular values of the model. The plots are to be compared with Figure 2 and Figure 3.
That means the effect of the input u−k on the output yj depends on the magnitude of H0,k+j−1.
For a discrete LTI system, H0,k+j−1 = CA
k+j−1B. | [
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For a discrete LTI system, H0,k+j−1 = CA
k+j−1B.
Since the spectrum of A is contained
in the open unit disk, this value decays exponentially as long as k + j − 1 > 0.
However,
if we parameterize the LTI system by the Markov parameters in the Hankel matrix, then
H0,k+j−1 = hk+j−1, which does not have to decay as long as k +j −1 < n. When k +j −1 ≥ n,
we have H0,k+j−1 = 0, which means the LTI system has essentially no memory after time n. | [
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we have H0,k+j−1 = 0, which means the LTI system has essentially no memory after time n.
However, this is not an issue because given a discrete sequential input, we can control ∆t to
fit all data inside the t ∈ [0, n] sliding window. We will elaborate on this discussion using an
experiment in section 5.
5
Experiments and Discussions
5.1
Experiment I: Singular Values of a HOPE-SSM
In this section, we implement a randomly initialized HOPE-SSM to learn the sCIFAR-10 task. | [
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] |
In this section, we implement a randomly initialized HOPE-SSM to learn the sCIFAR-10 task.
We use the same model hyperparameters as the S4D models in section 3. In particular, the
Hankel matrices in this model are 64-by-64. We randomly initialize the Hankel matrix and do
not set a smaller learning rate for the Hankel matrix entries h, i.e., all model parameters except
for ∆t have the same learning rate. In Figure 5, we show the test accuracy and the Hankel | [
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] |
for ∆t have the same learning rate. In Figure 5, we show the test accuracy and the Hankel
singular values of the HOPE-SSM. Compared to Figure 2, a random HOPE-SSM can be trained
to a high accuracy. In addition, training does not reduce the numerical rank of a Hankel matrix.
This corroborates our findings in Theorem 3 and Theorem 4.
5.2
Experiment II: HOPE-SSMs Have Long Memory
In section 4, we claim that a HOPE-SSM benefits from non-decaying memory. We show this | [
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] |
In section 4, we claim that a HOPE-SSM benefits from non-decaying memory. We show this
experimentally in this section. To do so, for each flattened picture in the sCIFAR-10 dataset,
which contains 1024 vectors of length 3, we append a random sequence of 1024 vectors of length
3 to the end of it. The goal is still to classify an image by its first 1024 pixels. We call this task
noise-padded sCIFAR-10. This task obviously requires a long memory so that the earlier data | [
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] |
noise-padded sCIFAR-10. This task obviously requires a long memory so that the earlier data
can still be retrieved after the noises are taken. We train both an S4D model and our HOPE-
SSM to learn this task, using a common discretization size of ∆t = 0.1 (see Appendix H.2 for
different values of ∆t). We see in Figure 6 that the HOPE-SSM significantly outperforms the
S4D model on this task. To understand this gap, we give the LTI systems in both the S4D model | [
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] |
S4D model on this task. To understand this gap, we give the LTI systems in both the S4D model
and our HOPE-SSM after training a unit impulse at t = 0. We watch how the impulse response
10
0
20
40
60
80
100
20
40
60
80
HOPE-SSM
S4D
Accuracy
Epochs
Test Accuracy on noisy-sCIFAR
t
|y(t)|
Memory Decay of Systems
Figure 6: Left: the test accuracies of the S4D model and our HOPE-SSM on the noisy-sCIFAR
task. Right: a unit impulse at t = 0 is acted on the LTI system. The plot shows the decay | [
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] |
task. Right: a unit impulse at t = 0 is acted on the LTI system. The plot shows the decay
of |y(t)| as t increases. Data are collected for all 512 LTI systems in a trained model. The
dark curve shows the median of |y(t)| over the 512 numbers. The darkly shaded region is from
the first quartile to the third quartile. The lightly shaded region is from the minimum to the
maximum.
Model
ListOps
Text
Retrieval Image Pathfinder Path-X
Avg.
DSS
57.60
76.60
87.60
85.80
84.10
85.00
79.45
S4++
57.30
86.28
84.82 | [
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0.09122377634048462,
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0.11728895455598831,
0.3195001482963562,
0.3781346380710602,
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0.1558862179517746,
-0.5888063311576843,
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0.5125094056129456,
0.2720346748828888,
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0.06321955472230911,
0.1551886945962906,
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0.35136687755584717,
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-0.28172165155410767,
0.48605406284332275,
-0.5053768754005432,
0.37908270955085754,
0.32520976662635803,
0.4707508087158203,
0.32148951292037964,
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-0.25521084666252136,
0.35940808057785034,
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0.19564035534858704,
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0.5673362016677856,
0.14051185548305511,
0.07033343613147736,
-0.22357603907585144,
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-0.3230895698070526,
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-0.20365869998931885
] |
Avg.
DSS
57.60
76.60
87.60
85.80
84.10
85.00
79.45
S4++
57.30
86.28
84.82
82.91
80.24
-
-
Regularized S4D
61.48
88.19
91.25
88.12
94.93
95.63
86.60
Spectral SSM
60.33
89.60
90.00
-
95.60
90.10
-
S4
59.60
86.82
90.90
88.65
94.20
96.35
86.09
S4D
60.47
86.18
89.46
88.19
93.06
91.95
84.89
HOPE-SSM
60.20
89.60
91.52
88.76
94.89
96.84
86.97
Table 1: Test accuracies in the Long-Range Arena of our HOPE-SSM and other models. The | [
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] |
96.84
86.97
Table 1: Test accuracies in the Long-Range Arena of our HOPE-SSM and other models. The
bold numbers indicate the best performance on a task and the underlined numbers indicate the
best performance within a comparable group. An entry is left blank if no result is found. The S4
and S4D results are from the original papers [14, 13]. We use the same model hyperparameters
as those in the S4D model. While our HOPE-SSM is a direct substitute for the S4 and S4D | [
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as those in the S4D model. While our HOPE-SSM is a direct substitute for the S4 and S4D
models, we also include some other S4-based models, where DSS is by [16], S4++ is by [26],
Regularized S4D is by [21], and Spectral SSM is by [2].
y(t) decays as time goes by. In Figure 6, we see that the memory of the trained S4D model
decays exponentially while that of the trained HOPE-SSM does not decay at all for t ∈ [0, 64],
where n = 64 is the size of the Hankel matrix. This is exactly what we expected in eq. (8). | [
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] |
where n = 64 is the size of the Hankel matrix. This is exactly what we expected in eq. (8).
5.3
Experiment III: Performance in the Long-Range Arena
Finally, we present the performance of HOPE-SSM on large datasets. We use the same model
architecture as in the S4D paper [13], except that we replace the LTI blocks with our HOPE
blocks. The specific model and training hyperparameters are reported in Appendix H. We show
the performance of our model in Table 1, where we see that our HOPE-SSM outperforms the | [
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] |
the performance of our model in Table 1, where we see that our HOPE-SSM outperforms the
comparable S4 and S4D models on most tasks.
6
Conclusion
In this paper, we presented a new theory based on the Hankel singular values to understand the
difficulties in initializing and training an SSM. We proposed a new parameterization scheme,
11
called HOPE, that is based on the Markov parameters in a discrete Hankel operator. We proved | [
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] |
11
called HOPE, that is based on the Markov parameters in a discrete Hankel operator. We proved
that a HOPE-SSM can be randomly initialized, stably trained, and has a long memory.
Acknowledgments
AY was supported by the SciAI Center, funded by the Office of Naval Research under Grant
Number N00014-23-1-2729. NBE would like to acknowledge NSF, under Grant No. 2319621,
and the U.S. Department of Energy, under Contract Number DE-AC02-05CH11231 and DE- | [
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] |
and the U.S. Department of Energy, under Contract Number DE-AC02-05CH11231 and DE-
AC02-05CH11231, for providing partial support of this work. We would also like to thank Alex
Townsend, Anil Damle, and Sarah Dean for some inspiring discussions.
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] |
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14
Supplementary Material
This is the supplementary material for the paper titled “There is HOPE to Avoid HiPPOs for | [
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] |
This is the supplementary material for the paper titled “There is HOPE to Avoid HiPPOs for
Long-memory State Space Models.” It is organized as follows. In Appendix A, we survey the
background of the bilinear transform of the LTI systems, the Mobius transform of the transfer
function domains, and the Hankel operators more thoroughly. In Appendix B, we survey the
basic properties of Hankel singular values and different strategies to compute them. The proofs | [
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] |
basic properties of Hankel singular values and different strategies to compute them. The proofs
of Theorem 1, Theorem 2, and Theorem 3-4 are given in Appendix C, Appendix D, and Ap-
pendix E, respectively. In Appendix F, we conduct some numerical experiments to corroborate
the four theorems presented in the main text. In Appendix G, we carefully derive how we can
use the nonuniform samples of the transfer function to enable a different discretization step size | [
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] |
use the nonuniform samples of the transfer function to enable a different discretization step size
∆t. Finally, the details of the experiments shown in the main text are given in Appendix H.
A
More background on the LTI system and Hankel operators
The purpose of this section is to provide a more thorough exposition of section 2, which leaves out
some concepts that are unnecessary in order to understand the main text. Let Γ = (A, B, C, D) | [
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some concepts that are unnecessary in order to understand the main text. Let Γ = (A, B, C, D)
be a continuous-time LTI system defined in eq. (1). One can take a bilinear transformation to
obtain a discrete LTI system:
A=(I+A)(I−A)−1,
B=(I+A)B/
√
2,
C=C(I+A)/
√
2,
D=D + C(I−A)−1B.
The bilinear transformation is invertible and defines a one-to-one correspondence between Γ
in eq. (1) and Γ = (A, B, C, D) given by
xk+1 = Axk + Buk,
yk = Cxk + Duk.
The transfer functions of Γ and Γ are
G(s) = C(sI − A)−1B + D | [
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] |
xk+1 = Axk + Buk,
yk = Cxk + Duk.
The transfer functions of Γ and Γ are
G(s) = C(sI − A)−1B + D
and
G(z) = C(zI − A)−1B + D,
respectively. The two transfer functions are equivalent via a Mobius transformation:
G(s) = G ((1 + s)/(1 − s))
⇔
G(z) = G ((z − 1)/(z + 1)) .
The importance of the transfer functions is that they bring the inputs to the outputs in the
Laplace domain by multiplication, which reduces to the Fourier domain if we assume that u is
bounded and compactly supported:
ˆy(s) = G(is)ˆu(s), | [
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bounded and compactly supported:
ˆy(s) = G(is)ˆu(s),
ˆy = G(ω)ˆu,
(9)
where the hat of a function and a vector means the Fourier transform and the Fourier coefficients,
respectively, and ω is the vector of roots of unity.
Given a continuous-time LTI system Γ, one can define its Hankel operator by
H : L2(0, ∞) → L2(0, ∞),
(Hv)(t) =
ˆ ∞
0
C exp((t + τ)A)Bv(τ)dτ.
The Hankel operator maps the past inputs to the future outputs, i.e., if u(t) = 0 for t ≥ 0 and | [
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The Hankel operator maps the past inputs to the future outputs, i.e., if u(t) = 0 for t ≥ 0 and
we define v(t) = u(−t), then we have y(t) = (Hv)(t). Analogously, the Hankel matrix of a
discrete LTI system Γ is a doubly infinite matrix defined by
H : ℓ2 → ℓ2,
Hi,j = CA
i+jB,
i, j ≥ 0.
The Hankel matrix has the similar physical interpretation: if uk = 0 for all k ≥ 0, then we have
�
y⊤
0
y⊤
1
· · ·
�⊤ = H
�
u⊤
−1
u⊤
−2
· · ·
�⊤. Both H and H are bounded linear operators of rank | [
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] |
�
y⊤
0
y⊤
1
· · ·
�⊤ = H
�
u⊤
−1
u⊤
−2
· · ·
�⊤. Both H and H are bounded linear operators of rank
≤ n, the number of latent states. In fact, the singular values σ1(H) ≥ σ2(H) ≥ · · · ≥ σn(H) ≥ 0
of H are equivalent to those of H, and they are called the Hankel singular values of Γ and Γ.
15
B
More background on Hankel singular values
In section 2, we define the Hankel singular values σ1, . . . , σn to be the singular values of the | [
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] |
In section 2, we define the Hankel singular values σ1, . . . , σn to be the singular values of the
finite-rank bounded linear Hankel operator on a separable Hilbert space. Throughout the paper,
we treated them as black boxes and used their distributions to understand the performance of
SSMs. The goal of this section is to open the black boxes and introduce more useful properties
of the Hankel singular values. Note that the proof of Theorem 1 to be presented in Appendix C | [
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] |
of the Hankel singular values. Note that the proof of Theorem 1 to be presented in Appendix C
heavily relies on the background discussed in this section. The concepts in this section can
be presented on a continuous-time LTI system (A, B, C, D) or analogously on a discrete LTI
system (A, B, C, D). We focus mainly on a continuous-time system for cleanliness. Since D
has no effect on the Hankel singular values, we further assume that D = 0 and write only
Γ = (A, B, C).
B.1 | [
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] |
Γ = (A, B, C).
B.1
Hankel singular values from balanced realization
One of the most popular ways to compute the Hankel singular values is via the so-called balanced
realization. Assume (A, B, C) is asymptotically stable, i.e. the spectrum of A is contained in
the open left half-plane. Then, there exist two Hermitian and positive semi-definite matrices
P ∈ Cn×n and Q ∈ Cn×n such that
AP + PA∗ + BB∗ = 0,
A∗Q + QA + C∗C = 0.
(10) | [
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P ∈ Cn×n and Q ∈ Cn×n such that
AP + PA∗ + BB∗ = 0,
A∗Q + QA + C∗C = 0.
(10)
The equations in eq. (10) are called the Lyapunov equations and P and Q are called the
controllability Gramian and the observability Gramian, respectively. They can be explicitly
expressed by the following matrix integrals:
P =
ˆ ∞
0
exp(At)BB∗exp(A∗t)dt,
Q =
ˆ ∞
0
exp(A∗t)C∗Cexp(At)dt.
(11)
The controllability Gramian P is positive definite if and only if the system is controllable, i.e., | [
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] |
The controllability Gramian P is positive definite if and only if the system is controllable, i.e.,
for any x0, x1 ∈ Cn and any T > 0, there exists an input u on [0, T] that makes x(T) = x1 when
x(0) = x0. Likewise, Q is positive definite if and only if the system is observable, i.e., for any
T > 0, the initial state x(0) can be determined by the input u and the output y on [0, T] [35].
The singular values of PQ turn out to be exactly the squares of the Hankel singular values,
i.e., σ2
1, . . . , σ2 | [
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i.e., σ2
1, . . . , σ2
n, of the system Γ. In general, P and Q are dense matrices. However, one can
use the so-called balanced realization algorithm [20] to compute an equivalent system Γb =
(Ab, Bb, Cb) = (V−1AV, V−1B, CV), where V ∈ Cn×n is an invertible matrix, so that its
Gramians Pb and Qb are equivalent and diagonal, i.e.,
Pb = Qb = diag(σ1, . . . , σn).
Balanced realization is an algebraic method that is based on singular value decompositions | [
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] |
Balanced realization is an algebraic method that is based on singular value decompositions
(SVDs). In practice, it is a very popular method to compute the Hankel singular values of a
system.
B.2
Hankel singular values as a rational approximation problem
The balanced realization gives us a good way to compute the Hankel singular values in practice.
However, it does not offer too much insight in theoretically analyzing them. The theory of | [
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] |
However, it does not offer too much insight in theoretically analyzing them. The theory of
Hankel singular values is usually derived via function approximation. Here, we introduce how
a Hankel singular value can be reframed as the solution to a rational approximation problem.
To this end, we let H∞
+ (resp. H∞
− ) be the Hardy space with functions h : C → Cm×p that are
16
bounded and analytic in the open right (resp. left) half-plane. The non-tangential limit of h | [
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Subsets and Splits