Timezone: »

Robust One-Bit Recovery via ReLU Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis
Shuang Qiu · Xiaohan Wei · Zhuoran Yang

Tue Jul 14 10:00 AM -- 10:45 AM & Tue Jul 14 09:00 PM -- 09:45 PM (PDT) @
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 on $\varepsilon^{-1}$. 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 weights.

Author Information

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

More from the Same Authors