  We study the robust one-bit compressed sensing problem whose goal is to
design an algorithm that faithfully recovers any sparse target vector
$\theta_0\in\mathbb{R}^d$ \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 $x_0 \in \mathbb{R}^k$ through a known $n$-layer ReLU generative
network $G:\mathbb{R}^k\rightarrow\mathbb{R}^d$ such that $\theta_0 = G(x_0)$.
Such a framework poses low-dimensional priors on $\theta_0$ without a 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
$m=\widetilde{\mathcal{O}}(kn \log d /\varepsilon^2)$ 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 $\varepsilon^{-1}$. 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.
