On the Information Dimension of Multivariate Gaussian Processes

TL;DR

This paper extends the concept of information dimension rate from univariate to multivariate Gaussian processes, showing it equals the average rank of the spectral distribution derivative, thus generalizing previous results.

Contribution

It introduces a formula for the information dimension rate of multivariate Gaussian processes based on the spectral distribution function's derivative.

Findings

Information dimension rate equals the average rank of the spectral derivative.

Scale and translation invariance properties extend to stochastic processes.

Generalizes univariate Gaussian process results to multivariate cases.

Abstract

The authors have recently defined the R\'enyi information dimension rate $d(\{X_t\})$ of a stationary stochastic process $\{X_t,\,t\in\mathbb{Z}\}$ as the entropy rate of the uniformly-quantized process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $m\to\infty$ (B. Geiger and T. Koch, "On the information dimension rate of stochastic processes," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, June 2017). For Gaussian processes with a given spectral distribution function $F_X$, they showed that the information dimension rate equals the Lebesgue measure of the set of harmonics where the derivative of $F_X$ is positive. This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function $F_{\mathbf{X}}$. It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{\mathbf{X}}$. As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes.