Robust One-Bit Recovery via ReLU Generative Networks: Improved Statistical Rates and Global Landscape Analysis
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We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector θ_0∈R^d uniformly m quantized noisy measurements. Under the assumption that the measurements are sub-Gaussian random vectors, to recover any k-sparse θ_0 (k≪ d) uniformly up to an error ε with high probability, the best known computationally tractable algorithm requires m≥Õ(klog d/ε^4) measurements. 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 ∈R^k through a known n-layer ReLU generative network G:R^k→R^d. Such a framework poses low-dimensional priors on θ_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 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=Õ(kn log d /ε^2) recovering any G(x_0) uniformly up to an error ε. 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 are equally good for recovering the true target up to scaling with a certain accuracy. Furthermore, our analysis also shed lights 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.
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