The Restricted Isometry Property of Subsampled Fourier Matrices

TL;DR

This paper proves that randomly sampling rows from a Fourier matrix yields a matrix satisfying the restricted isometry property for sparse vectors, with fewer samples than previously known, advancing compressed sensing theory.

Contribution

It demonstrates that subsampled Fourier matrices satisfy the RIP with optimal sample complexity, improving prior bounds and theoretical understanding.

Findings

Sample complexity is $O(k \, \log^2 k \, \log N)$ for RIP

Random subsampling from Fourier matrices achieves RIP with high probability

Improves previous bounds on the number of samples needed for RIP

Abstract

A matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).