Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors

This paper studies quantized corrupted sensing where the measurements arecontaminated by unknown corruption and then quantized by a dithered uniformquantizer. We establish uniform guarantees for Lasso that ensure the accuraterecovery of all signals and corruptions using a single draw of the sub-Gaussiansensing matrix and uniform dither. For signal and corruption with structuredpriors (e.g., sparsity, low-rankness), our uniform error rate for constrainedLasso typically coincides with the non-uniform one [Sun, Cui and Liu, 2022] upto logarithmic factors. By contrast, our uniform error rate for unconstrainedLasso exhibits worse dependence on the structured parameters due toregularization parameters larger than the ones for non-uniform recovery. Forsignal and corruption living in the ranges of some Lipschitz continuousgenerative models (referred to as generative priors), we achieve uniformrecovery via constrained Lasso with a measurement number proportional to thelatent dimensions of the generative models. Our treatments to the two kinds ofpriors are (nearly) unified and share the common key ingredients of (global)quantized product embedding (QPE) property, which states that the dithereduniform quantization (universally) preserves inner product. As a by-product,our QPE result refines the one in [Xu and Jacques, 2020] under sub-Gaussianrandom matrix, and in this specific instance we are able to sharpen the uniformerror decaying rate (for the projected-back projection estimator with signalsin some convex symmetric set) presented therein from O(m^-1/16) toO(m^-1/8).