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We study the robust one-bit compressed sensing problem whose goal is todesign an algorithm that faithfully recovers any sparse target vector$\theta_0\in\mathbb{R}^d$ \textit{uniformly} via $m$ quantized noisymeasurements. Specifically, we consider a new framework for this problem wherethe sparsity is implicitly enforced via mapping a low dimensionalrepresentation $x_0 \in \mathbb{R}^k$ through a known $n$-layer ReLU generativenetwork $G:\mathbb{R}^k\rightarrow\mathbb{R}^d$ such that $\theta_0 = G(x_0)$.Such a framework poses low-dimensional priors on $\theta_0$ without a knownsparsity basis. We propose to recover the target $G(x_0)$ solving anunconstrained empirical risk minimization (ERM). Under a weak\textit{sub-exponential measurement assumption}, we establish a jointstatistical and computational analysis. In particular, we prove that the ERMestimator in this new framework achieves a statistical rate of$m=\widetilde{\mathcal{O}}(kn \log d /\varepsilon^2)$ 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 upto logarithm factors of $\varepsilon^{-1}$. From the lens of computation, weprove that under proper conditions on the network weights, our proposedempirical risk, despite non-convexity, has no stationary point outside of smallneighborhoods around the true representation $x_0$ and its negative multiple;furthermore, we show that the global minimizer of the empirical risk stayswithin the neighborhood around $x_0$ rather than its negative multiple underfurther assumptions on the network weights.

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