Sparse Signal Recovery in Hilbert Spaces

TL;DR

This paper consolidates and generalizes coherence-based sparse signal recovery results in Hilbert spaces, providing a unified theory that applies to various signal types and improves existing bounds with simplified proofs.

Contribution

It introduces a unified theoretical framework for sparse signal recovery in Hilbert spaces, encompassing multiple signal models and enhancing previous results.

Findings

Unified recovery thresholds for diverse signal types

Improved bounds and simplified proofs for coherence-based recovery

Applicability to infinite-dimensional and union of subspace models

Abstract

This paper reports an effort to consolidate numerous coherence-based sparse signal recovery results available in the literature. We present a single theory that applies to general Hilbert spaces with the sparsity of a signal defined as the number of (possibly infinite-dimensional) subspaces participating in the signal's representation. Our general results recover uncertainty relations and coherence-based recovery thresholds for sparse signals, block-sparse signals, multi-band signals, signals in shift-invariant spaces, and signals in finite unions of (possibly infinite-dimensional) subspaces. Moreover, we improve upon and generalize several of the existing results and, in many cases, we find shortened and simplified proofs.