An Open Problem on Sparse Representations in Unions of Bases

TL;DR

This paper proves tight lower bounds on the spark for sparse representations in unions of orthonormal bases, solving an open problem and leveraging concepts from quantum information theory.

Contribution

It provides constructive proofs that establish the tightness of known bounds on the spark for unions of orthonormal bases, addressing an open problem.

Findings

Lower bounds on spark are tight for unions of orthonormal bases.

Constructive proofs use mutually unbiased bases from quantum information.

Answers an open problem posed by Gribonval and Nielsen.

Abstract

We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsen's open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutually unbiased bases which first appears in quantum information theory.