Rate-Distortion Dimension of Stochastic Processes

TL;DR

This paper establishes that the rate-distortion dimension of an analog stationary process equals its information dimension, providing a new operational method to evaluate the process's complexity and linking it to compressed sensing limits.

Contribution

It proves the equivalence of rate-distortion dimension and information dimension for stationary processes, extending previous results and offering a practical evaluation approach.

Findings

RDD equals twice the asymptotic ratio of R(D) to log(1/D) as D approaches zero

The relation between RDD and ID is demonstrated for a piecewise constant process

Provides an operational method to evaluate the information dimension of processes

Abstract

The rate-distortion dimension (RDD) of an analog stationary process is studied as a measure of complexity that captures the amount of information contained in the process. It is shown that the RDD of a process, defined as two times the asymptotic ratio of its rate-distortion function $R(D)$ to $\log {1\over D}$ as the distortion $D$ approaches zero, is equal to its information dimension (ID). This generalizes an earlier result by Kawabata and Dembo and provides an operational approach to evaluate the ID of a process, which previously was shown to be closely related to the effective dimension of the underlying process and also to the fundamental limits of compressed sensing. The relation between RDD and ID is illustrated for a piecewise constant process.