Reconstruction from anisotropic random measurements

TL;DR

This paper introduces a reduction principle that simplifies verifying the Restricted Eigenvalue condition for various classes of random matrices, broadening the scope of matrices suitable for sparse recovery.

Contribution

It establishes a reduction principle linking the RE condition to restricted isometry on low-dimensional subspaces, enabling analysis of dependent and structured random matrices.

Findings

RE condition verified for matrices with subgaussian rows and covariance

Applicable to matrices with independent rows and bounded entries

Broadens understanding of matrix properties for sparse recovery

Abstract

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.