Improved bounds for the RIP of Subsampled Circulant matrices

TL;DR

This paper improves the theoretical bounds on the number of measurements needed for partial random circulant matrices to satisfy the restricted isometry property, which is crucial for compressed sensing.

Contribution

It provides tighter bounds on the RIP for partial random circulant matrices, reducing the number of measurements required compared to previous results.

Findings

Partial random circulant matrices satisfy RIP with fewer measurements.

New bounds improve upon previous logarithmic factors.

Results hold for subgaussian generators with independent entries.

Abstract

In this paper, we study the restricted isometry property of partial random circulant matrices. For a bounded subgaussian generator with independent entries, we prove that the partial random circulant matrices satisfy $s$-order RIP with high probability if one chooses $m\gtrsim s \log^2(s)\log (n)$ rows randomly where $n$ is the vector length. This improves the previously known bound $m \gtrsim s \log^2 s\log^2 n$.