  Corrupted sensing concerns the problem of recovering a high-dimensional
structured signal from a collection of measurements that are contaminated by
unknown structured corruption and unstructured noise. In the case of linear
measurements, the recovery performance of different convex programming
procedures (e.g., generalized Lasso and its variants) is well established in
the literature. However, in practical applications of digital signal
processing, the quantization process is inevitable, which often leads to
non-linear measurements. This paper is devoted to studying corrupted sensing
under quantized measurements. Specifically, we demonstrate that, with the aid
of uniform dithering, both constrained and unconstrained Lassos are able to
recover signal and corruption from the quantized samples when the measurement
matrix is sub-Gaussian. Our theoretical results reveal the role of quantization
resolution in the recovery performance of Lassos. Numerical experiments are
provided to confirm our theoretical results.
