On block coherence of frames

TL;DR

This paper analyzes block coherence in matrices, providing bounds, constructions, and algorithms to optimize it, which is crucial for improving block compressed sensing recovery performance.

Contribution

It introduces new bounds, deterministic constructions, and a flipping algorithm to enhance block coherence metrics in matrices.

Findings

Deterministic matrix constructions meet lower bounds on worst-case block coherence.

The flipping algorithm improves average block coherence without affecting worst-case coherence.

Numerical results show the proposed methods perform well in block compressed sensing applications.

Abstract

Block coherence of matrices plays an important role in analyzing the performance of block compressed sensing recovery algorithms (Bajwa and Mixon, 2012). In this paper, we characterize two block coherence metrics: worst-case and average block coherence. First, we present lower bounds on worst-case block coherence, in both the general case and also when the matrix is constrained to be a union of orthobases. We then present deterministic matrix constructions based upon Kronecker products which obtain these lower bounds. We also characterize the worst-case block coherence of random subspaces. Finally, we present a flipping algorithm that can improve the average block coherence of a matrix, while maintaining the worst-case block coherence of the original matrix. We provide numerical examples which demonstrate that our proposed deterministic matrix construction performs well in block compressed sensing.