Compressibility Analysis of Asymptotically Mean Stationary Processes

Abstract

This work provides new results for the analysis of random sequences in terms of $\ell_p$-compressibility. The results characterize the degree in which a random sequences can be approximated by its best k-sparse version under different rates of significant coeffi- cients (compressibility analysis). In particular, the notion of strong $\ell_p$-characterization is introduced to denote a random sequence that has a well-defined asymptotic limit (sample- wise) of its best k-term approximation error when a fixed rate of significant coefficients is considered (fixed-rate analysis). The main theorem of this work shows that the rich family of asymptotically mean stationary (AMS) processes has a strong lp-characterization and we present results for the characterization and analysis of its lp-approximation error function. Furthermore adding ergodicity in the analysis of AMS processes, we have a theorem that shows that its approximation error function is constant and determined in closed-form by the stationary mean of the process. The results and analysis presented in this paper offer a contribution to the theory and understanding of discrete-time sparse processes and, on the technical side, confirm how instrumental the point-wise ergodic theorem is to determine the compressibility expression of discrete-time processes even when stationarity and ergodicity assumptions are relaxed.

Avatar
Jorge F. Silva
Professor of Electrical Engineering

My research interests include learning, information theory, estimation and detection, universal source coding, and compressed sensing.