Granger causality for compressively sensed sparse signals

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.