Granger causality for compressively sensed sparse signals.

Area of Science:

• Signal Processing
• Complex Systems Analysis
• Computational Neuroscience

Background:

• Compressed sensing (CS) enables efficient data acquisition for sparse signals, widely used in imaging and signal processing.
• Causal inference is crucial for understanding interactions in complex systems across scientific disciplines.
• Direct causal analysis on compressed data avoids costly reconstruction, but challenges exist for sparse temporal signals.

Purpose of the Study:

• To mathematically prove that structured compressed sensing matrices preserve causal relationships.
• To verify the preservation of Granger causality (GC) in compressed sparse signals.
• To demonstrate computational advantages for causal inference from compressed data.

Main Methods:

• Mathematical proof for circulant and Toeplitz compressed sensing matrices preserving Granger causality.
• Simulations of bivariate and multivariate sparse signals compressed with structured matrices.
• Application to real-world sparse neural spike train recordings for network causal connectivity estimation.

Main Results:

• Structured compressed sensing matrices (circulant, Toeplitz) mathematically proven to preserve Granger causality.
• Empirical verification on simulated sparse signals and real neural data.
• Demonstrated computational time savings for causal inference from compressed signals.

Conclusions:

• Structured compressed sensing matrices are effective for Granger causality estimation from sparse signals.
• The proposed strategy offers significant computational advantages over traditional methods.
• Enables efficient causal inference in fields utilizing sparse signal acquisition.