Low-Coherence Frames from Group Fourier Matrices

TL;DR

This paper introduces a method for constructing low-coherence frames from group Fourier matrices, useful in compressive sensing and coding, achieving near-optimal coherence and small alphabet entries.

Contribution

It presents a novel technique for selecting group representations to build low-coherence, tight frames with few inner product values, including cases matching optimal Welch bounds.

Findings

Constructed frames with low coherence close to Grassmanian bounds.

Identified cases where frames meet the Welch lower bound.

Frames with entries from small alphabets.

Abstract

Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications. In this work, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a finite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmanian frame. We identify special cases in which our construction yields some previously-known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of a Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and find regimes in which they satisfy the Strong Coherence Property described by Mixon, Bajwa, and Calderbank.