Denoising guarantees for optimized sampling schemes in compressed sensing
This provides theoretical denoising guarantees for compressed sensing, addressing noise reduction in signal recovery, though it is incremental as it extends existing optimized sampling frameworks.
The paper tackles the problem of measurement noise in compressed sensing with optimized sampling schemes, proving that the error from Gaussian noise vanishes as measurements increase and demonstrating empirical rates matching theory for sparse vectors and generative neural network priors.
Compressed sensing with subsampled unitary matrices benefits from \emph{optimized} sampling schemes, which feature improved theoretical guarantees and empirical performance relative to uniform subsampling. We provide, in a first of its kind in compressed sensing, theoretical guarantees showing that the error caused by the measurement noise vanishes with an increasing number of measurements for optimized sampling schemes, assuming that the noise is Gaussian. We moreover provide similar guarantees for measurements sampled with-replacement with arbitrary probability weights. All our results hold on prior sets contained in a union of low-dimensional subspaces. Finally, we demonstrate that this denoising behavior appears in empirical experiments with a rate that closely matches our theoretical guarantees when the prior set is the range of a generative ReLU neural network and when it is the set of sparse vectors.