On some deterministic dictionaries supporting sparsity

TL;DR

This paper introduces the oscillator dictionary, a new deterministic incoherent dictionary based on finite group representations, supporting sparse representations with low mutual coherence.

Contribution

It presents a novel construction of an incoherent dictionary using finite group theory and provides an explicit algorithm for generating a large subset.

Findings

Dictionary size of p^5 vectors in p-dimensional space

Maximum inner product magnitude of 4/√p

Explicit construction algorithm provided

Abstract

We describe a new construction of an incoherent dictionary, referred to as the oscillator dictionary, which is based on considerations in the representation theory of finite groups. The oscillator dictionary consists of order of p^5 unit vectors in a Hilbert space of dimension p, where p is an odd prime, whose pairwise inner products have magnitude of at most 4/sqrt(p). An explicit algorithm to construct a large portion of the oscillator dictionary is presented.