Multivariate Granger Causal Discovery for Compressively Sensed Signals

Abstract

Analysis of networks with multiple variables using causal inference tools has become very popular recently because of the development of mathematical tools and algorithms for causal discovery. For systems where temporal measurements are available, the pioneering concept of Granger Causality as well as its many extensions are being employed. These methods help to capture the interactions between network variables where exact model discovery is difficult. Contemporaneously, the field of compressed sensing (CS) has seen an increasing interest in the number and scope of its applications for the acquisition, storage and transmission of signals. While conventional signal acquisition protocols require that the signals are sampled at atleast the Nyquist frequency to keep the signal information intact, CS allows for the possibility of using far fewer number of measurements to acquire and represent the signal. The only requirement of CS is that the signal being sensed be `sparse’ in some domain, which is satisfied by many naturally occurring signals. Hence, CS has been adopted rapidly in the design of signal and image acquisition strategies, e.g., magnetic resonance imaging, scanning tunneling microscopy, quantum state tomography, analog to digital conversion technologies. Currently, the compressed signals are required to be reconstructed in order to make causal discovery analysis from compressively sensed signals possible. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically Circulant and Toeplitz, preserve Granger causality relationships in the compressed signal domain. This theorem is then verified on bivariate and multivariate sparse neuronal spike train simulations compressed using the recommended structured matrices. The approach is also applied to real neuronal spike trains from a rat prefrontal cortex to infer network causal connectivity. Demonstration of the possibility to make causal inferences in the compressed domain is useful to reduce computational-cost, to correctly apply the well-known Granger-Causality and to design appropriate sensing matrices. Authors: Aditi Kathpalia, Nithin Nagaraj Acknowledgements: This study is supported by the Czech Science Foundation, Project No. GA19-16066S and by the Czech Academy of Sciences, Praemium Academiae awarded to M. Paluš. Nithin Nagaraj gratefully acknowledges the financial support of Cognitive Science Research Initiative, Dept. of Science & Tech., Govt. of India (CSRI-DST) Grant No. DST/CSRI/2017/54(G).