Lipschitz Learning for Signal Recovery

TL;DR

This paper develops a theoretical framework using Lipschitz conditions to characterize when signals can be robustly recovered from transformed observations via machine learning, offering a more general alternative to compressive sensing.

Contribution

It introduces a Lipschitz-based condition for signal recovery, providing a necessary and sufficient criterion that surpasses the restricted isometry property in generality.

Findings

Lipschitz condition is necessary and sufficient for robust recovery.

Lipschitz condition is more general than restricted isometry property.

Proposes an ML method with minimal output dimension for linear observations.

Abstract

We consider the recovery of signals from their observations, which are samples of a transform of the signals rather than the signals themselves, by using machine learning (ML). We will develop a theoretical framework to characterize the signals that can be robustly recovered from their observations by an ML algorithm, and establish a Lipschitz condition on signals and observations that is both necessary and sufficient for the existence of a robust recovery. We will compare the Lipschitz condition with the well-known restricted isometry property of the sparse recovery of compressive sensing, and show the former is more general and less restrictive. For linear observations, our work also suggests an ML method in which the output space is reduced to the lowest possible dimension.