Partially deterministic sampling for compressed sensing with denoising guarantees
This work addresses the practical need in compressed sensing for combining random and deterministic sampling to enhance reconstruction quality, though it is incremental as it builds on existing sampling frameworks.
The paper tackles compressed sensing with sampling from a unitary matrix by developing an optimized sampling scheme that combines random and deterministic row selection, leading to improved image reconstruction for generative and sparse priors and providing better sample complexity bounds and denoising guarantees.
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.