Frame Coherence and Sparse Signal Processing

TL;DR

This paper examines frame coherence parameters to evaluate the suitability of deterministic sensing matrices for sparse signal processing, providing bounds, algorithms, and probabilistic guarantees for detection and reconstruction.

Contribution

It introduces new coherence bounds, an algorithm to reduce average coherence, and demonstrates near-optimal guarantees without relying on the Restricted Isometry Property.

Findings

Frames with small spectral norm and coherence are identified.

A new lower bound on worst-case coherence is established.

An algorithm effectively reduces average coherence without affecting spectral norm.

Abstract

The sparse signal processing literature often uses random sensing matrices to obtain performance guarantees. Unfortunately, in the real world, sensing matrices do not always come from random processes. It is therefore desirable to evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse signals. To this end, the present paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first provide several examples of frames that have small spectral norm, worst-case coherence, and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, we propose an algorithm that decreases the average coherence of a frame without changing its spectral norm or worst-case coherence. Finally, we use worst-case and average coherence, as opposed to the Restricted Isometry Property, to garner near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from arbitrary frames, and which further assume independence across the nonzero entries of the signal---in a sense, requiring small average coherence replaces the need for such an assumption.