Structured Compressed Sensing Matrices for Causality Detection

Compressed sensing enables sparse signals to be stored and transmitted in a linearly compressed fashion and finds applications in magnetic resonance imaging, photography, transmission electron microscopy etc. Linear compression is achieved by matrix multiplication of the input sparse signals with a random sensing matrix (for e.g., independent Gaussian entries) and recovery is enabled by nonlinear optimization techniques. However, causal relationships between input signals are no longer preserved by such a compression. In this work, we design structured sensing matrices having Toeplitz and Circulant structure that preserve causality (as measured by Granger causality index) between sparse autoregressive coupled input signals in the compressed domain. A potential application of causality detection in the compressed domain is in neural signal processing.