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

SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1909-1977, September 2024.
Abstract.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 nonuniform one up to logarithmic factors, indicating that the uniformity costs very little. 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 nonuniform recovery. These results complement the nonuniform ones recently obtained in Sun, Cui, and Liu [IEEE Trans. Signal Process., 70 (2022), pp. 600–615] and provide more insights for understanding actual applications where the sensing ensemble is typically fixed and the corruption may be adversarial. 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. We present experimental results to corroborate our theories. From the technical side, our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of a (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves the inner product. As a by-product, our QPE result refines the one in Xu and Jacques [Inf. Inference, 9 (2020), pp. 543–586] under the 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 [math] to [math].