Statistical Mechanical Assessment of a Reconstruction Limit of Compressed Sensing: Toward Theoretical Analysis of Correlated Signals

We provide a scheme for exploring the reconstruction limits of compressed sensing by minimizing the general cost function under the random measurement constraints for generic correlated signal sources. Our scheme is based on the statistical mechanical replica method for dealing with random systems. As a simple but non-trivial example, we apply the scheme to a sparse autoregressive model, where the first differences in the input signals of the correlated time series are sparse, and evaluate the critical compression rate for a perfect reconstruction. The results are in good agreement with a numerical experiment for a signal reconstruction. Introduction. – Compressed sensing (CS) is a novel technique for data compression and has been drawing a lot of attention recently from the viewpoints of both theory and application. The key idea behind CS is to utilize the sparsity of the original input signals as the prior knowledge during the signal reconstruction stage, which can significantly reduce the number of signal measurements required for a perfect reconstruction. This setup is realistic because we often have to face situations where we have to handle sparse signals in the real world. A lot of effort has been paid and significant progress has been made in investigating the properties of CS [1–3]. After the pioneering works, contribution to CS problem from statistical mechanics analysis is now growing rapidly [4–10]. The measurement process of CS is summarized in the following linear equation: