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

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.