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