Partially deterministic sampling for compressed sensing with denoising guarantees

2026-04-06

AI Summary

This work addresses a key challenge in compressive sensing: reconciling theoretical performance guarantees with the practical need for deterministic selection of critical sampling rows. The authors propose an optimized Bernoulli sampling scheme that rigorously integrates random and deterministic strategies, explicitly identifying and prioritizing crucial rows in unitary measurement matrices. By incorporating both sparsity and generative prior models, the method yields tighter sample complexity bounds and novel denoising guarantees in theory. Experimental results demonstrate that, in image compressive sensing tasks, the proposed approach achieves significantly higher reconstruction quality compared to conventional sampling methods with and without replacement.

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.

Problem

• compressed sensing
• deterministic sampling
• random sampling
• unitary matrix
• denoising guarantees

Innovation

• partially deterministic sampling
• compressed sensing
• denoising guarantees
• sample complexity
• unitary matrix sampling