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On the Information Dimension of Multivariate Gaussian Processes
- Source :
- IEEE Trans. on Information Theory 65(10):6496-6518. (C) IEEE 2019
- Publication Year :
- 2017
-
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.<br />Comment: This work will be presented in part at the 2018 International Zurich Seminar on Information and Communication
- Subjects :
- Computer Science - Information Theory
Subjects
Details
- Database :
- arXiv
- Journal :
- IEEE Trans. on Information Theory 65(10):6496-6518. (C) IEEE 2019
- Publication Type :
- Report
- Accession number :
- edsarx.1712.07863
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1109/TIT.2019.2922186