Derandomized compressed sensing with nonuniform guarantees for $\ell_1$ recovery

TL;DR

This paper constructs explicit partial Fourier matrices with nonuniform guarantees that ensure sparse signal recovery via -minimization with high probability, extending previous derandomization techniques.

Contribution

It introduces a method to explicitly build partial Fourier matrices with nonuniform recovery guarantees for sparse signals, improving derandomization in compressed sensing.

Findings

Existence of explicit random partial Fourier matrices with near-optimal measurements.

High probability of unique -minimization recovery for s-sparse signals.

Use of decoupling techniques to analyze singular values of submatrices.

Abstract

We extend the techniques of H\"{u}gel, Rauhut and Strohmer (Found. Comput. Math., 2014) to show that for every $\delta\in(0,1]$, there exists an explicit random $m\times N$ partial Fourier matrix $A$ with $m=s\operatorname{polylog}(N/\epsilon)$ and entropy $s^\delta\operatorname{polylog}(N/\epsilon)$ such that for every $s$-sparse signal $x\in\mathbb{C}^N$, there exists an event of probability at least $1-\epsilon$ over which $x$ is the unique minimizer of $\|z\|_1$ subject to $Az=Ax$. The bulk of our analysis uses tools from decoupling to estimate the extreme singular values of the submatrix of $A$ whose columns correspond to the support of $x$.