Improved mutual coherence of some random overcomplete dictionaries for sparse repsentation

TL;DR

This paper introduces a method to reduce the mutual coherence of overcomplete random matrices, such as Gaussian and Bernoulli, improving sparse representation performance with high probability and demonstrating effectiveness through numerical experiments.

Contribution

It proposes a novel approach to lower the mutual coherence of overcomplete random dictionaries, enhancing sparse coding capabilities.

Findings

Mutual coherence can be significantly reduced with high probability.

The method improves coherence for Gaussian, Bernoulli, and uniform random matrices.

Numerical results confirm the effectiveness of the proposed reduction.

Abstract

The letter presents a method for the reduction in the mutual coherence of an overcomplete Gaussian or Bernoulli random matrix, which is fairly small due to the lower bound given here on the probability of the event that the aforesaid mutual coherence is less than any given number in (0, 1). The mutual coherence of the matrix that belongs to a set which contains the two types of matrices with high probability can be reduced by a similar method but a subset that has Lebesgue measure zero. The numerical results are provided to illustrate the reduction in the mutual coherence of an overcomplete Gaussian, Bernoulli or uniform random dictionary. The effect on the third type is better than a former result.