Robust Compressed Sensing using Generative Models
This addresses the robustness issue in compressed sensing for applications with noisy or corrupted data, representing an incremental improvement over existing methods.
The paper tackles the problem of robust compressed sensing with heavy-tailed or outlier-corrupted measurements by proposing a Median-of-Means (MOM)-based algorithm, which guarantees recovery with the same sample complexity as empirical risk minimization under sub-Gaussian assumptions and demonstrates robustness in experiments.
The goal of compressed sensing is to estimate a high dimensional vector from an underdetermined system of noisy linear equations. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the vector is represented by a deep generative model $G: \mathbb{R}^k \rightarrow \mathbb{R}^n$. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is sub-Gaussian. However, when the measurement matrix and measurements are heavy-tailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the Median-of-Means (MOM). Our algorithm guarantees recovery for heavy-tailed data, even in the presence of outliers. Theoretically, our results show our novel MOM-based algorithm enjoys the same sample complexity guarantees as ERM under sub-Gaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavy-tailed and/or corrupted data, while our approach exhibits the predicted robustness.