Optimal compressed sensing for mixing stochastic processes

TL;DR

This paper establishes the mean information dimension as the fundamental limit for compressed sensing of $\psi^*$-mixing processes, proving that fewer measurements than this threshold make almost lossless recovery impossible.

Contribution

It introduces the correlation dimension rate as a new lower bound and precisely characterizes the measurement threshold for universal recovery of stochastic processes.

Findings

Mean information dimension is the fundamental limit for compressed sensing.

Fewer measurements than the mean information dimension prevent almost lossless recovery.

Introduces correlation dimension rate as a new lower bound.

Abstract

Jalali and Poor introduced an asymptotic framework for compressed sensing of stochastic processes, demonstrating that any rate strictly greater than the mean information dimension serves as an upper bound on the number of random linear measurements required for (universal) almost lossless recovery of $\psi^*$-mixing processes, as measured in the normalized $L^2$ norm. In this work, we show that if the normalized number of random linear measurements is strictly less than the mean information dimension, then almost lossless recovery of a $\psi^*$-mixing process is impossible by any sequence of decompressors. This establishes the mean information dimension as the fundamental limit for compressed sensing in this setting (and, in fact, the precise threshold for the problem). To this end, we introduce a new quantity, related to techniques from geometric measure theory: the correlation dimension rate, which is shown to be a lower bound for compressed sensing of arbitrary stationary stochastic processes.