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 $O(m^{-1/8})$.

翻译：本文研究量化腐败感知问题，其中测量值被未知腐败污染，随后通过抖动均匀量化器进行量化。我们建立了Lasso的均匀保证，确保使用单次抽取的次高斯感知矩阵和均匀抖动即可准确恢复所有信号和腐败。对于具有结构化先验（例如稀疏性、低秩性）的信号和腐败，我们约束Lasso的均匀误差率通常与非均匀误差率[Sun, Cui and Liu, 2022]在对数因子范围内一致。相比之下，我们无约束Lasso的均匀误差率由于正则化参数大于非均匀恢复所需的参数，在结构化参数上表现出更差的依赖性。对于位于某些Lipschitz连续生成模型（称为生成式先验）值域内的信号和腐败，我们通过约束Lasso实现均匀恢复，所需测量数与生成模型的潜在维度成正比。我们对这两种先验的处理方式（几乎）是统一的，共享（全局）量化乘积嵌入（QPE）性质这一关键要素，该性质表明抖动均匀量化（普遍地）保持内积。作为副产品，我们的QPE结果在次高斯随机矩阵下改进了[Xu and Jacques, 2020]中的结论，并在此特定情形下，将其中（针对凸对称集中信号的投影反投影估计器）的均匀误差衰减率从$O(m^{-1/16})$提升至$O(m^{-1/8})$。