Sparse Recovery for Overcomplete Frames: Sensing Matrices and Recovery Guarantees

TL;DR

This paper surveys the theory and guarantees of sparse recovery using convex optimization in overcomplete frames, highlighting recent advances in measurement design and recovery guarantees, especially for frame-sparse signals.

Contribution

It provides a comprehensive overview of recovery guarantees for overcomplete frames, including new insights into measurement conditions and proof techniques, with a focus on frame-sparse signals.

Findings

Recovery guarantees for orthonormal basis sparsity are well-established.

Few heavy-tailed random measurements can satisfy recovery guarantees.

New restricted isometry-like property related to frames is presented.

Abstract

Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core question is, by exploiting the intrinsic low dimensional structure of the signal, how to design the sensing process and decoder in a way that the number of measurements is essentially close to the complexity of the signal set. This chapter provides a survey of important results in answering this question, with an emphasis on a basis pursuit like convex optimization decoder that admits a wide range of random sensing matrices. The results are quite established in the case signals are sparse in an orthonormal basis, while the case with frame sparse signals is much less explored. In addition to presenting the latest results on recovery guarantee and how few random heavier-tailed measurements fulfill these recovery guarantees, this chapter also aims to provide some insights in proof techniques. We also take the opportunity of this book chapter to publish an interesting result (Theorem 3.10) about a restricted isometry like property related to a frame.