Sparse linear regression with ill-conditioned Gaussian random covariates is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief. Recent work has shown that, for certain covariance matrices, the broad class of Preconditioned Lasso programs provably cannot succeed on polylogarithmically sparse signals with a sublinear number of samples. However, this lower bound only holds against deterministic preconditioners, and in many contexts randomization is crucial to the success of preconditioners. We prove a stronger lower bound that rules out randomized preconditioners. For an appropriate covariance matrix, we construct a single signal distribution on which any invertibly-preconditioned Lasso program fails with high probability, unless it receives a linear number of samples. Surprisingly, at the heart of our lower bound is a new robustness result in compressed sensing. In particular, we study recovering a sparse signal when a few measurements can be erased adversarially. To our knowledge, this natural question has not been studied before for sparse measurements. We surprisingly show that standard sparse Bernoulli measurements are almost-optimally robust to adversarial erasures: if $b$ measurements are erased, then all but $O(b)$ of the coordinates of the signal are identifiable.

Sparse linear regression with ill-conditioned Gaussian random covariates is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief. Recent work has shown that, for certain covariance matrices, the broad class of Preconditioned Lasso programs provably cannot succeed on polylogarithmically sparse signals with a sublinear number of samples. However, this lower bound only holds against deterministic preconditioners, and in many contexts randomization is crucial to the success of preconditioners. We prove a stronger lower bound that rules out randomized preconditioners. For an appropriate covariance matrix, we construct a single signal distribution on which any invertibly-preconditioned Lasso program fails with high probability, unless it receives a linear number of samples.

Sparse linear regression with ill-conditioned Gaussian random designs is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief, even in the form of examples that are hard for restricted classes of algorithms. Recent work has shown that, for certain covariance matrices, the broad class of Preconditioned Lasso programs provably cannot succeed on polylogarithmically sparse signals with a sublinear number of samples. However, this lower bound only shows that for every preconditioner, there exists at least one signal that it fails to recover successfully. This leaves open the possibility that, for example, trying multiple different preconditioners solves every sparse linear regression problem. In this work, we prove a stronger lower bound that overcomes this issue. For an appropriate covariance matrix, we construct a single signal distribution on which any invertibly-preconditioned Lasso program fails with high probability, unless it receives a linear number of samples. Surprisingly, at the heart of our lower bound is a new positive result in compressed sensing. We show that standard sparse random designs are with high probability robust to adversarial measurement erasures, in the sense that if $b$ measurements are erased, then all but $O(b)$ of the coordinates of the signal are still information-theoretically identifiable. To our knowledge, this is the first time that partial recoverability of arbitrary sparse signals under erasures has been studied in compressed sensing.

Sparse linear regression is a fundamental problem in high-dimensional statistics, but strikingly little is known about how to efficiently solve it without restrictive conditions on the design matrix. We consider the (correlated) random design setting, where the covariates are independently drawn from a multivariate Gaussian $N(0,\Sigma)$ with $\Sigma : n \times n$, and seek estimators $\hat{w}$ minimizing $(\hat{w}-w^*)^T\Sigma(\hat{w}-w^*)$, where $w^*$ is the $k$-sparse ground truth. Information theoretically, one can achieve strong error bounds with $O(k \log n)$ samples for arbitrary $\Sigma$ and $w^*$; however, no efficient algorithms are known to match these guarantees even with $o(n)$ samples, without further assumptions on $\Sigma$ or $w^*$. As far as hardness, computational lower bounds are only known with worst-case design matrices. Random-design instances are known which are hard for the Lasso, but these instances can generally be solved by Lasso after a simple change-of-basis (i.e. preconditioning). In this work, we give upper and lower bounds clarifying the power of preconditioning in sparse linear regression. First, we show that the preconditioned Lasso can solve a large class of sparse linear regression problems nearly optimally: it succeeds whenever the dependency structure of the covariates, in the sense of the Markov property, has low treewidth -- even if $\Sigma$ is highly ill-conditioned. Second, we construct (for the first time) random-design instances which are provably hard for an optimally preconditioned Lasso. In fact, we complete our treewidth classification by proving that for any treewidth-$t$ graph, there exists a Gaussian Markov Random Field on this graph such that the preconditioned Lasso, with any choice of preconditioner, requires $\Omega(t^{1/20})$ samples to recover $O(\log n)$-sparse signals when covariates are drawn from this model.

Modern machine learning-based recognition approaches require large-scale datasets with large number of labelled training images. However, such datasets are inherently difficult and costly to collect and annotate. Hence there is a great and growing interest in automatic dataset collection methods that can leverage the web. % which are collected % in a cheap, efficient and yet unreliable way. Collecting datasets in this way, however, requires robust and efficient ways for detecting and excluding outliers that are common and prevalent. % Outliers are thus a % prominent treat of using these dataset. So far, there have been a limited effort in machine learning community to directly detect outliers for robust classification. Inspired by the recent work on Pre-conditioned LASSO, this paper formulates the outlier detection task using Pre-conditioned LASSO and employs \red{unsupervised} transductive diffusion component analysis to both integrate the topological structure of the data manifold, from labeled and unlabeled instances, and reduce the feature dimensionality. Synthetic experiments as well as results on two real-world classification tasks show that our framework can robustly detect the outliers and improve classification.