Block-Sparse Recovery via Convex Optimization

TL;DR

This paper addresses the problem of block-sparse signal recovery using convex optimization, proposing new methods with theoretical guarantees and demonstrating improved face recognition performance with less training data.

Contribution

It introduces convex relaxations for block-sparse recovery, derives conditions for their equivalence to non-convex formulations, and applies these methods to enhance face recognition accuracy.

Findings

Convex programs outperform non-convex ones under certain coherence conditions.

Proposed methods improve face recognition accuracy by 10%.

Achieve similar or better results with only 25% of training data.

Abstract

Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a block-sparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and computer vision applications, such as face recognition, we consider the block-sparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a block-sparse representation of a signal, we propose two classes of non-convex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NP-hard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original non-convex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a block-sparse recovery problem improves the state-of-the-art results by 10% with only 25% of the training data.