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.

翻译：我们研究了采样向量取自酉矩阵行时的压缩感知问题。在现有文献中，这些采样向量通常随机选取；随机性的运用促进了该领域理论与实证的重大进展。然而，实际应用中常存在某些关键采样向量，此时实践者会偏离理论，以确定性方式采样这类行。本文针对伯努利选择器推导出一种优化采样方案，该方案自然融合了行的随机与确定性选择，从而严格确定了哪些行应被确定性采样。通过理论结果与数值实验，我们证明了相比有放回与无放回采样方案，该采样方案在图像压缩感知中（基于生成先验与稀疏先验）均能带来可量化的性能提升。此外，我们的理论保证在样本复杂度界上优于现有研究，并首次提出该场景下的新型去噪保证。