Compressed sensing of data with a known distribution

TL;DR

This paper improves compressed sensing recovery thresholds by leveraging known data distributions, using weighted $ ext{l}_1$ minimization and geometric algorithms to enhance performance and provide failure probability bounds.

Contribution

It introduces a weighted $ ext{l}_1$-norm minimization approach that incorporates data distribution knowledge to reduce measurement thresholds in compressed sensing.

Findings

Reduced measurement threshold for perfect recovery.

Development of Monte Carlo algorithms for geometric analysis.

Bounded failure probability using new geometric methods.

Abstract

Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted $\ell_1$-norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.