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

We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector $\theta_0\in\mathbb{R}^d$ \emph{uniformly} from $m$ quantized noisy measurements. Under the assumption that the measurements are sub-Gaussian, to recover any $k$-sparse $\theta_0$ ($k\ll d$) \emph{uniformly} up to an error $\varepsilon$ with high probability, the best known computationally tractable algorithm requires\footnote{Here, an algorithm is ``computationally tractable'' if it has provable convergence guarantees. The notation $\tilde{\mathcal{O}}(\cdot)$ omits a logarithm factor of $\varepsilon^{-1}$.} $m\geq\tilde{\mathcal{O}}(k\log d/\varepsilon^4)$. In this paper, we consider a new framework for the one-bit sensing problem where the sparsity is implicitly enforced via mapping a low dimensional representation $x_0$ through a known $n$-layer ReLU generative network $G:\mathbb{R}^k\rightarrow\mathbb{R}^d$. Such a framework poses low-dimensional priors on $\theta_0$ without a known basis. We propose to recover the target $G(x_0)$ via an unconstrained empirical risk minimization (ERM) problem under a much weaker \emph{sub-exponential measurement assumption}. For such a problem, we establish a joint statistical and computational analysis. In particular, we prove that the ERM estimator in this new framework achieves an improved statistical rate of $m=\tilde{\mathcal{O}} (kn\log d /\epsilon^2)$ recovering any $G(x_0)$ uniformly up to an error $\varepsilon$. Moreover, from the lens of computation, we prove that under proper conditions on the ReLU 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. Our analysis sheds some light on the possibility of inverting a deep generative model under partial and quantized measurements, complementing the recent success of using deep generative models for inverse problems.