On recoverability properties of fixed measurement matrices

TL;DR

This paper extends recoverability conditions for sparse signal representations from dictionaries to general fixed measurement matrices, providing a new method with potentially broader applicability.

Contribution

It introduces a novel sufficient condition for unique sparse representations in arbitrary fixed matrices, generalizing previous results limited to dictionaries.

Findings

Derived a new recoverability condition for fixed matrices

Method is at least as effective as previous dictionary-based approaches

Applicable to a wider class of measurement matrices

Abstract

The purpose of this paper is to extend a result by Donoho and Huo, Elad and Bruckstein, Gribnoval and Nielsen on sparse representations of signals in dictionaries to general matrices. We consider a general fixed measurement matrix, not necessarily a dictionary, and derive sufficient condition for having unique sparse representation of signals in this matrix. Currently, to the best of our knowledge, no such method exists. In particular, if matrix is a dictionary, our method is at least as good as the method proposed by Gribnoval and Nielsen.