Performance analysis for sparse support recovery

TL;DR

This paper analyzes the performance limits of support recovery for jointly sparse signals, deriving bounds that relate measurement matrix properties to recovery success, with implications for practical applications like DOA estimation.

Contribution

It introduces bounds on support recovery error probability based on measurement matrix properties, providing new insights into sparse signal recovery performance.

Findings

Performance depends on measurement matrix incoherence and total measurement gain.

Necessary and sufficient conditions for error-free support recovery are derived for Gaussian matrices.

Existing bounds in Compressive Sensing may be insufficient at higher noise levels.

Abstract

The performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space is analyzed. Support recovery is formulated as a multiple-hypothesis testing problem. Both upper and lower bounds on the probability of error are derived for general measurement matrices, by using the Chernoff bound and Fano's inequality, respectively. The upper bound shows that the performance is determined by a quantity measuring the measurement matrix incoherence, while the lower bound reveals the importance of the total measurement gain. The lower bound is applied to derive the minimal number of samples needed for accurate direction-of-arrival (DOA) estimation for a sparse representation based algorithm. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions for a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into sparse signal recovery. For example, as far as support recovery is concerned, the well-known bound in Compressive Sensing with the Gaussian measurement matrix is generally not sufficient unless the noise level is low. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in Compressive Sensing and other application areas exploiting signal sparsity.