Accuracy guaranties for $\ell_1$ recovery of block-sparse signals

TL;DR

This paper introduces a framework for structured signal recovery, focusing on verifiable conditions for accurate block-sparse signal reconstruction, and proposes an efficient algorithm with theoretical guarantees and numerical validation.

Contribution

It develops verifiable conditions for block-sparse signal recovery, enabling error bound optimization and introduces a computationally efficient Block Matching Pursuit algorithm.

Findings

Verifiable conditions lead to improved recovery error bounds.

The proposed algorithm performs comparably to traditional methods.

Numerical results demonstrate effectiveness of block regularizations.

Abstract

We introduce a general framework to handle structured models (sparse and block-sparse with possibly overlapping blocks). We discuss new methods for their recovery from incomplete observation, corrupted with deterministic and stochastic noise, using block-$\ell_1$ regularization. While the current theory provides promising bounds for the recovery errors under a number of different, yet mostly hard to verify conditions, our emphasis is on verifiable conditions on the problem parameters (sensing matrix and the block structure) which guarantee accurate recovery. Verifiability of our conditions not only leads to efficiently computable bounds for the recovery error but also allows us to optimize these error bounds with respect to the method parameters, and therefore construct estimators with improved statistical properties. To justify our approach, we also provide an oracle inequality, which links the properties of the proposed recovery algorithms and the best estimation performance. Furthermore, utilizing these verifiable conditions, we develop a computationally cheap alternative to block-$\ell_1$ minimization, the non-Euclidean Block Matching Pursuit algorithm. We close by presenting a numerical study to investigate the effect of different block regularizations and demonstrate the performance of the proposed recoveries.