Robust One-Bit Recovery via ReLU Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis

Abstract: We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector $\theta_0\in\mathbb{R}<sup>d$ \textit{uniformly} via $m$ quantized noisy measurements. Specifically, we consider a new framework for this problem where the sparsity is implicitly enforced via mapping a low dimensional representation $x_0 \in \mathbb{R}<sup>k$ through a known $n$-layer ReLU generative network $G:\mathbb{R}<sup>k\rightarrow\mathbb{R}<sup>d$ such that $\theta_0 = G(x_0)$. Such a framework poses low-dimensional priors on $\theta_0$ without a known sparsity basis. We propose to recover the target $G(x_0)$ solving an unconstrained empirical risk minimization (ERM). Under a weak \textit{sub-exponential measurement assumption}, we establish a joint statistical and computational analysis. In particular, we prove that the ERM estimator in this new framework achieves a statistical rate of $$m=\widetilde{\mathcal{O}}(kn \log d /\varepsilon<sup>2)$$ recovering any $G(x_0)$ uniformly up to an error $\varepsilon$. When the network is shallow (i.e., $n$ is small), we show this rate matches the information-theoretic lower bound up to logarithm factors of $\varepsilon<sup>{-1}$. From the lens of computation, we prove that under proper conditions on the network weights, our proposed empirical risk, despite non-convexity, has no stationary point outside of small neighborhoods around the true representation $x_0$ and its negative multiple; furthermore, we show that the global minimizer of the empirical risk stays within the neighborhood around $x_0$ rather than its negative multiple under further assumptions on the network weights.