On Sparse Representation in Fourier and Local Bases

TL;DR

This paper introduces a polynomial-time algorithm for finding all sparse representations of signals in Fourier and local bases, extending to other basis pairs, and bridging the gap between uniqueness and computational feasibility.

Contribution

The paper presents a new polynomial-time algorithm for sparse representation in Fourier and local bases, improving upon previous methods and applicable to various basis pairs.

Findings

The algorithm finds all $K$-sparse representations when $K<rac{ ext{sqrt}(2)}{ ext{mutual coherence}}$.

It solves the unique sparse representation problem in polynomial time for $K<1/ ext{mutual coherence}$.

The method extends to multiple basis pairs, including Fourier, cosine, and Gaussian bases.

Abstract

We consider the classical problem of finding the sparse representation of a signal in a pair of bases. When both bases are orthogonal, it is known that the sparse representation is unique when the sparsity $K$ of the signal satisfies $K<1/\mu(D)$, where $\mu(D)$ is the mutual coherence of the dictionary. Furthermore, the sparse representation can be obtained in polynomial time by Basis Pursuit (BP), when $K<0.91/\mu(D)$. Therefore, there is a gap between the unicity condition and the one required to use the polynomial-complexity BP formulation. For the case of general dictionaries, it is also well known that finding the sparse representation under the only constraint of unicity is NP-hard. In this paper, we introduce, for the case of Fourier and canonical bases, a polynomial complexity algorithm that finds all the possible $K$-sparse representations of a signal under the weaker condition that $K<\sqrt{2} /\mu(D)$. Consequently, when $K<1/\mu(D)$, the proposed algorithm solves the unique sparse representation problem for this structured dictionary in polynomial time. We further show that the same method can be extended to many other pairs of bases, one of which must have local atoms. Examples include the union of Fourier and local Fourier bases, the union of discrete cosine transform and canonical bases, and the union of random Gaussian and canonical bases.