text: 37,102,1273460843548565504,96779364,Arnab Bhattacharyya,"New paper: ""Efficient Statistics for Sparse Graphical Models from Truncated Samples"" with Rathin Desai, @SaiGaneshNagar1, and Yannis Panageas. <LINK> A common issue in statistical inference is *selection bias*. This occurs when a dataset contains missing samples, and whether a sample is missing or not depends preferentially on its value. E.g., poor people may not want to report their income in a poll. In general, inference is impossible when there's selection bias. E.g., only millionaires may report their earnings skewing our mean income estimate. But what if we make parametric assumptions? Like, suppose the underlying distribution is gaussian. We look at truncated samples: a sample x is seen only when it satisfies some fixed property P and not seen otherwise. The property P can be arbitrary, but we assume that we can efficiently check whether an item satisfies P. The first result in our paper gives improved sample complexity bounds for learning a Gaussian graphical model from truncated samples. The # of samples scales with the sparsity of the precision matrix (unlike the previous work, <LINK>). The second result in our paper looks at high-dimensional sparse linear regression with truncated samples. Here, we make stronger assumptions to show that you only need O(k^2 log(d)) samples to recover the parameters upto bounded *entrywise* error (k=sparsity, d=dimension). For untruncated samples, the typical approach for both problems is to do l1-regularization, i.e. Lasso. We do the same. The main technical challenge is to show that the Lasso optimum falls in a region that is locally strongly convex. The big open question here is to make our results computationally effective. For gradient descent-type algorithms, it's not clear how to choose the initial point inside the strongly convex region. Another interesting direction is to remove the assumption that we can check the property P. In the low-dimensional setting, there has been exciting work along this line here: <LINK>. Of late, my work has been following a pattern :) The general problem is impossible, @yudapearl, @eliasbareinboim and company (<LINK>) give conditions for statistical identifiability, and we show stronger conditions that imply finite sample guarantees.",https://arxiv.org/abs/2006.09735,"In this paper, we study high-dimensional estimation from truncated samples. We focus on two fundamental and classical problems: (i) inference of sparse Gaussian graphical models and (ii) support recovery of sparse linear models. (i) For Gaussian graphical models, suppose $d$-dimensional samples ${\bf x}$ are generated from a Gaussian $N(\mu,\Sigma)$ and observed only if they belong to a subset $S \subseteq \mathbb{R}^d$. We show that ${\mu}$ and ${\Sigma}$ can be estimated with error $\epsilon$ in the Frobenius norm, using $\tilde{O}\left(\frac{\textrm{nz}({\Sigma}^{-1})}{\epsilon^2}\right)$ samples from a truncated $\mathcal{N}({\mu},{\Sigma})$ and having access to a membership oracle for $S$. The set $S$ is assumed to have non-trivial measure under the unknown distribution but is otherwise arbitrary. (ii) For sparse linear regression, suppose samples $({\bf x},y)$ are generated where $y = {\bf x}^\top{{\Omega}^*} + \mathcal{N}(0,1)$ and $({\bf x}, y)$ is seen only if $y$ belongs to a truncation set $S \subseteq \mathbb{R}$. We consider the case that ${\Omega}^*$ is sparse with a support set of size $k$. Our main result is to establish precise conditions on the problem dimension $d$, the support size $k$, the number of observations $n$, and properties of the samples and the truncation that are sufficient to recover the support of ${\Omega}^*$. Specifically, we show that under some mild assumptions, only $O(k^2 \log d)$ samples are needed to estimate ${\Omega}^*$ in the $\ell_\infty$-norm up to a bounded error. For both problems, our estimator minimizes the sum of the finite population negative log-likelihood function and an $\ell_1$-regularization term. ",Efficient Statistics for Sparse Graphical Models from Truncated Samples,10,"['New paper: ""Efficient Statistics for Sparse Graphical Models from Truncated Samples"" with Rathin Desai, @SaiGaneshNagar1, and Yannis Panageas. <LINK>', 'A common issue in statistical inference is *selection bias*. This occurs when a dataset contains missing samples, and whether a sample is missing or not depends preferentially on its value. E.g., poor people may not want to report their income in a poll.', ""In general, inference is impossible when there's selection bias. E.g., only millionaires may report their earnings skewing our mean income estimate. But what if we make parametric assumptions? Like, suppose the underlying distribution is gaussian."", 'We look at truncated samples: a sample x is seen only when it satisfies some fixed property P and not seen otherwise. The property P can be arbitrary, but we assume that we can efficiently check whether an item satisfies P.', 'The first result in our paper gives improved sample complexity bounds for learning a Gaussian graphical model from truncated samples. The # of samples scales with the sparsity of the precision matrix (unlike the previous work, https://t.co/mwvngTJjeY).', 'The second result in our paper looks at high-dimensional sparse linear regression with truncated samples. Here, we make stronger assumptions to show that you only need O(k^2 log(d)) samples to recover the parameters upto bounded *entrywise* error (k=sparsity, d=dimension).', 'For untruncated samples, the typical approach for both problems is to do l1-regularization, i.e. Lasso. We do the same. The main technical challenge is to show that the Lasso optimum falls in a region that is locally strongly convex.', ""The big open question here is to make our results computationally effective. For gradient descent-type algorithms, it's not clear how to choose the initial point inside the strongly convex region."", 'Another interesting direction is to remove the assumption that we can check the property P. In the low-dimensional setting, there has been exciting work along this line here: https://t.co/3XsEQhSA9m.', 'Of late, my work has been following a pattern :) The general problem is impossible, @yudapearl, @eliasbareinboim and company (https://t.co/puAVNWltqX) give conditions for statistical identifiability, and we show stronger conditions that imply finite sample guarantees.']",20,06,2252