The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing

TL;DR

This paper investigates the fundamental limits of sparsity pattern recovery in compressed sensing, demonstrating that small error fractions are achievable with simple estimators and providing bounds on measurement rates relative to SNR and signal parameters.

Contribution

It establishes that recovery with a small, constant error fraction is possible and derives measurement rate bounds, showing near-optimality for various signal models.

Findings

Recovery with a small, constant error fraction is achievable.

Bounds on measurement rate depend on SNR and signal parameters.

Simple estimators can be near-optimal in certain settings.

Abstract

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models.