Two are better than one: Fundamental parameters of frame coherence

TL;DR

This paper explores two key parameters of frame coherence, deriving probabilistic guarantees for sparse signal tasks, providing nearly tight frames, establishing a new lower bound, and presenting an algorithm to reduce average coherence.

Contribution

It introduces a comprehensive analysis of worst-case and average coherence, including new bounds, a frame catalog, and an algorithm to optimize frame coherence properties.

Findings

Derived near-optimal probabilistic guarantees for sparse signal detection and reconstruction.

Provided a catalog of nearly tight frames with small coherence parameters.

Established a new lower bound on worst-case coherence and an algorithm to reduce average coherence.

Abstract

This paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first use worst-case and average coherence to derive near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. Next, we provide a catalog of nearly tight frames with small worst-case and average coherence. Later, we find a new lower bound on worst-case coherence; we compare it to the Welch bound and use it to interpret recently reported signal reconstruction results. Finally, we give an algorithm that transforms frames in a way that decreases average coherence without changing the spectral norm or worst-case coherence.