Partially deterministic sampling for compressed sensing with denoising guarantees

TL;DR

This paper introduces a partially deterministic sampling scheme for compressed sensing that combines random and deterministic row selection, leading to improved theoretical guarantees and practical performance in image reconstruction.

Contribution

It proposes an optimized sampling scheme blending deterministic and random row selection, with enhanced guarantees and empirical results in compressed sensing.

Findings

Improved sample complexity bounds over previous methods.

Measurable performance gains in image compressed sensing.

Theoretical denoising guarantees in the proposed sampling scheme.

Abstract

We study compressed sensing when the sampling vectors are chosen from the rows of a unitary matrix. In the literature, these sampling vectors are typically chosen randomly; the use of randomness has enabled major empirical and theoretical advances in the field. However, in practice there are often certain crucial sampling vectors, in which case practitioners will depart from the theory and sample such rows deterministically. In this work, we derive an optimized sampling scheme for Bernoulli selectors which naturally combines random and deterministic selection of rows, thus rigorously deciding which rows should be sampled deterministically. This sampling scheme provides measurable improvements in image compressed sensing for both generative and sparse priors when compared to with-replacement and without-replacement sampling schemes, as we show with theoretical results and numerical experiments. Additionally, our theoretical guarantees feature improved sample complexity bounds compared to previous works, and novel denoising guarantees in this setting.