Reconstruction from anisotropic random measurements

Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. The current paper discusses the recently introduced Restricted Eigenvalue (RE) condition, which is among the most general assumptions on the matrix, guaranteeing recovery. We prove a reduction principle showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces. This principle allows us to establish the RE condition for several broad classes of random matrices with dependent entries, including random matrices with subgaussian rows and non-trivial covariance structure, as well as matrices with independent rows, and uniformly bounded entries.Comment: 30 Page

RIPless compressed sensing from anisotropic measurements

Compressed sensing is the art of reconstructing a sparse vector from its inner products with respect to a small set of randomly chosen measurement vectors. It is usually assumed that the ensemble of measurement vectors is in isotropic position in the sense that the associated covariance matrix is proportional to the identity matrix. In this paper, we establish bounds on the number of required measurements in the anisotropic case, where the ensemble of measurement vectors possesses a non-trivial covariance matrix. Essentially, we find that the required sampling rate grows proportionally to the condition number of the covariance matrix. In contrast to other recent contributions to this problem, our arguments do not rely on any restricted isometry properties (RIP's), but rather on ideas from convex geometry which have been systematically studied in the theory of low-rank matrix recovery. This allows for a simple argument and slightly improved bounds, but may lead to a worse dependency on noise (which we do not consider in the present paper).Comment: 19 pages. To appear in Linear Algebra and its Applications, Special Issue on Sparse Approximate Solution of Linear System

Post-Selection Inference for Generalized Linear Models with Many Controls

This paper considers generalized linear models in the presence of many controls. We lay out a general methodology to estimate an effect of interest based on the construction of an instrument that immunize against model selection mistakes and apply it to the case of logistic binary choice model. More specifically we propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest $\alpha_0$, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. These methods allow to estimate $\alpha_0$ at the root-$n$ rate when the total number $p$ of other regressors, called controls, potentially exceed the sample size $n$ using sparsity assumptions. The sparsity assumption means that there is a subset of $s<n$ controls which suffices to accurately approximate the nuisance part of the regression function. Importantly, the estimators and these resulting confidence regions are valid uniformly over $s$-sparse models satisfying $s^2\log^2 p = o(n)$ and other technical conditions. These procedures do not rely on traditional consistent model selection arguments for their validity. In fact, they are robust with respect to moderate model selection mistakes in variable selection. Under suitable conditions, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper

Trimmed Density Ratio Estimation

Density ratio estimation is a vital tool in both machine learning and statistical community. However, due to the unbounded nature of density ratio, the estimation procedure can be vulnerable to corrupted data points, which often pushes the estimated ratio toward infinity. In this paper, we present a robust estimator which automatically identifies and trims outliers. The proposed estimator has a convex formulation, and the global optimum can be obtained via subgradient descent. We analyze the parameter estimation error of this estimator under high-dimensional settings. Experiments are conducted to verify the effectiveness of the estimator.Comment: Made minor revisions. Restructured the introductory section

Sparse Signal Recovery under Poisson Statistics

We are motivated by problems that arise in a number of applications such as Online Marketing and explosives detection, where the observations are usually modeled using Poisson statistics. We model each observation as a Poisson random variable whose mean is a sparse linear superposition of known patterns. Unlike many conventional problems observations here are not identically distributed since they are associated with different sensing modalities. We analyze the performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting involves a non-linear optimization but yet is computationally tractable. We derive fundamental sample complexity bounds for sparse recovery when the measurements are contaminated with Poisson noise. In contrast to the least-squares linear regression setting with Gaussian noise, we observe that in addition to sparsity, the scale of the parameters also fundamentally impacts sample complexity. We introduce a novel notion of Restricted Likelihood Perturbation (RLP), to jointly account for scale and sparsity. We derive sample complexity bounds for $\ell_1$ regularized ML estimators in terms of RLP and further specialize these results for deterministic and random sensing matrix designs.Comment: 13 pages, 11 figures, 2 tables, submitted to IEEE Transactions on Signal Processin