Robust One-Bit Recovery via ReLU Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis
Shuang Qiu, Xiaohan Wei, Zhuoran Yang
Abstract
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} 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 \emph{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 weights.
BibTeX
@InProceedings{pmlr-v119-qiu20a,
title = {Robust One-Bit Recovery via {R}e{LU} Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis},
author = {Qiu, Shuang and Wei, Xiaohan and Yang, Zhuoran},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {7857--7866},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/qiu20a/qiu20a.pdf},
url = {https://proceedings.mlr.press/v119/qiu20a.html},
abstract = {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} 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 \emph{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 weights.}
}