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Robust One-Bit Recovery via ReLU Generative Networks: Improved Statistical Rate and Global Landscape Analysis
Shuang Qiu · Xiaohan Wei · Zhuoran Yang
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, despite non-convexity, we prove that the objective of our ERM problem has no spurious stationary point, that is, any stationary point is equally good for recovering the true target up to scaling with a certain accuracy. 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.

Author Information

Shuang Qiu (University of Michigan)
Xiaohan Wei (University of Southern California)
Zhuoran Yang (Princeton University)

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