Restricted Isometries for Partial Random Circulant Matrices

TL;DR

This paper proves that partial random circulant matrices used in compressed sensing have small restricted isometry constants with fewer samples than previously known, improving theoretical understanding of convolution-based data acquisition.

Contribution

It establishes a new bound showing that the restricted isometry constant is small when the number of samples scales as (s log n)^{3/2}, improving on prior quadratic bounds.

Findings

Restricted isometry constant is small when m ≳ (s log n)^{3/2}.

Improves theoretical bounds for convolution-based measurement matrices.

Enhances understanding of compressed sensing with circulant matrices.

Abstract

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.