The Road to Deterministic Matrices with the Restricted Isometry Property

Abstract

Abstract. The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.

Fingerprinting With Equiangular Tight Frames

“…The specific class of ETFs additionally requires [44]. Whether scales as versus is an important distinction between random and deterministic RIP matrices in the compressed sensing community [47], since users seek to minimize the number of measurements needed to sense -sparse signals. However, this difference in scaling offers no advantage for fingerprinting.…”

Game of Sloanes: best known packings in complex projective space

“…Consider the problem of packing n lines through the origin of F d such that they are maximally geometrically spread. This problem has a number of applications in fields such as compressed sensing [1], digital fingerprinting [2], quantum state tomography [3,4], and multiple description coding [5,6,7]. Such configurations are also a cornerstone of discrete geometry [8].…”

Linear programming bounds for cliques in Paley graphs

“…The Paley graph of order p can be used to construct an optimal packing of lines through the origin of R (p+1)/2 , known as the corresponding Paley equiangular tight frame, [3][4][5] and these packings have received some attention in the context of compressed sensing. 6,7 For a simple, undirected graph G = (V, E), we say C ⊆ V is a clique if every pair of vertices in C is adjacent, and we define the clique number of G, denoted by ω(G), to be the size of the largest clique in G. It is a famously difficult open problem to determine the order of magnitude of ω(G p ) as p → ∞. The best known closed-form bounds that are valid for all primes p ≡ 1 (mod 4) are given by…”