On oracle-type local recovery guarantees in compressed sensing

TL;DR

This paper improves sampling complexity bounds for stable sparse recovery in compressed sensing, especially for structured signals and measurements, showing near-oracle performance with adaptive sampling strategies.

Contribution

It provides a unified analysis for block-structured measurements and arbitrary structured sparsity, achieving bounds comparable to oracle estimators under certain conditions.

Findings

Sampling bounds match oracle least-squares requirements.

Variable density sampling improves recovery performance.

Numerical results confirm effectiveness in imaging and polynomial approximation.

Abstract

We present improved sampling complexity bounds for stable and robust sparse recovery in compressed sensing. Our unified analysis based on l1 minimization encompasses the case where (i) the measurements are block-structured samples in order to reflect the structured acquisition that is often encountered in applications; (ii) the signal has an arbitrary structured sparsity, by results depending on its support S. Within this framework and under a random sign assumption, the number of measurements needed by l1 minimization can be shown to be of the same order than the one required by an oracle least-squares estimator. Moreover, these bounds can be minimized by adapting the variable density sampling to a given prior on the signal support and to the coherence of the measurements. We illustrate both numerically and analytically that our results can be successfully applied to recover Haar wavelet coefficients that are sparse in levels from random Fourier measurements in dimension one and two, which can be of particular interest in imaging problems. Finally, a preliminary numerical investigation shows the potential of this theory for devising adaptive sampling strategies in sparse polynomial approximation.