Analysis of Recurrent Linear Networks for Enabling Compressed Sensing of Time-Varying Signals

TL;DR

This paper investigates how recurrent network dynamics can improve the recovery of time-varying sparse signals from limited and noisy observations, providing theoretical bounds and insights into network characteristics that facilitate dynamic compressed sensing.

Contribution

It introduces bounds on recovery performance based on network dynamics and explores conditions for exact recovery, highlighting how different network traits influence dynamic compressed sensing.

Findings

Network dynamics can enhance signal recoverability.

Bounds on recovery performance are derived.

Tradeoffs in network characteristics affect sensing capabilities.

Abstract

Recent interest has developed around the problem of dynamic compressed sensing, or the recovery of time-varying, sparse signals from limited observations. In this paper, we study how the dynamics of recurrent networks, formulated as general dynamical systems, mediate the recoverability of such signals. We specifically consider the problem of recovering a high-dimensional network input, over time, from observation of only a subset of the network states (i.e., the network output). Our goal is to ascertain how the network dynamics lead to performance advantages, particularly in scenarios where both the input and output are corrupted by disturbance and noise, respectively. For this scenario, we develop bounds on the recovery performance in terms of the dynamics. Conditions for exact recovery in the absence of noise are also formulated. Through several examples, we use the results to highlight how different network characteristics may trade off toward enabling dynamic compressed sensing and how such tradeoffs may manifest naturally in certain classes of neuronal networks.