The Road to Deterministic Matrices with the Restricted Isometry Property

Afonso S. Bandeira, Matthew Fickus, Dustin G. Mixon, Percy Wong

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1123-1149
Number of pages27
JournalJournal of Fourier Analysis and Applications
Volume19
Issue number6
DOIs
StatePublished - Dec 1 2013

Keywords

  • Compressed sensing
  • Equiangular tight frames
  • Restricted isometry property

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

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