Robust Compressed Sensing using Generative Models

TL;DR

This paper introduces a robust compressed sensing algorithm based on Median-of-Means that effectively recovers signals from heavy-tailed and corrupted measurements, matching classical guarantees under ideal conditions.

Contribution

The paper proposes a novel MOM-inspired algorithm for compressed sensing with generative models, providing robustness to heavy-tailed noise and outliers while maintaining theoretical guarantees.

Findings

The MOM-based algorithm succeeds with heavy-tailed data and outliers.

It matches sample complexity guarantees of classical ERM under sub-Gaussian assumptions.

Experimental results show improved robustness over existing methods.

Abstract

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.