A Weighted Generalized Coherence Approach for Sensing Matrix Design

TL;DR

This paper introduces weighted generalized coherence criteria for designing sensing matrices that improve signal recovery in compressive sensing, incorporating training data and proposing an optimization algorithm.

Contribution

It generalizes mutual coherence to bi- and tri-coherence, incorporating weights and training data for enhanced sensing matrix design.

Findings

Weighted coherence improves sensing matrix quality.

The proposed algorithm effectively optimizes the coherence criteria.

Empirical results demonstrate better signal recovery performance.

Abstract

As compared to using randomly generated sensing matrices, optimizing the sensing matrix w.r.t. a carefully designed criterion is known to lead to better quality signal recovery given a set of compressive measurements. In this paper, we propose generalizations of the well-known mutual coherence criterion for optimizing sensing matrices starting from random initial conditions. We term these generalizations as bi-coherence or tri-coherence and they are based on a criterion that discourages any one column of the sensing matrix from being close to a sparse linear combination of other columns. We also incorporate training data to further improve the sensing matrices through weighted coherence, weighted bi-coherence, or weighted tri-coherence criteria, which assign weights to sensing matrix columns as per their importance. An algorithm is also presented to solve the optimization problems. Finally, the effectiveness of the proposed algorithm is demonstrated through empirical results.