This work is concerned with the problem of recovering high-dimensional
signals $\mathbf{x} \in \mathbb{R}^n$ which belong to a convex set of
low-complexity from a small number of quantized measurements. We propose to
estimate the signals via a convex program based on rectified linear units
(ReLUs) for two different quantization schemes, namely one-bit and uniform
multi-bit quantization. Assuming that the linear measurement process can be
modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both
schemes near-optimal uniform reconstruction guarantees by adding well-designed
noise to the linear measurements prior to the quantization step. In the one-bit
case, we show that the program is robust against adversarial bit corruptions as
well as additive noise on the linear measurements. Further, our analysis
quantifies precisely how the rate-distortion relationship of the program
changes depending on whether we seek reconstruction accuracies above or below
the noise floor. The proofs rely on recent results by Dirksen and Mendelson on
non-Gaussian hyperplane tessellations. Finally, we complement our theoretical
analysis with numerical experiments which compare our method to other
state-of-the-art methodologies.

本研究旨在针对高维信号恢复问题，提出了基于修正线性单元的一比特和均匀多比特量化的凸规划算法，在线性测量过程中添加设计良好的噪声，从而获得最优近似重构保证，并证明了该算法对位坏的鲁棒性。其中，Dirksen 和 Mendelson 对非高斯超平面镶嵌的最新结果作为证明依据。