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On the Information Dimension of Multivariate Gaussian Processes
Information Theory · Submitted 21 Dec 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{\mathbf{X}}$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.99 | 0.26 | 0.03 | asserted 0.98 |
| s1 | Koch, "On the information dimension rate of stochastic processes," in Proc. IEEE Int. Symp. Inf. | background 1.00 | 0.06 | 0.01 | unclear 0.97 |
| s2 | Theory (ISIT), Aachen, Germany, June 2017). | background 1.00 | 0.05 | 0.02 | unclear 0.94 |
| s3 | 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. | background 0.91 | 0.18 | 0.06 | asserted 1.00 |
| s4 | This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function $F_{\mathbf{X}}$. | contribution 0.99 | 0.76 | 0.10 | asserted 0.98 |
| s5 | It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{\mathbf{X}}$. | result 1.00 | 0.93 | 0.91 | asserted 1.00 |
| s6 | As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes. | result 0.99 | 0.88 | 0.03 | asserted 1.00 |
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Original 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.
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What it cites · 12 sentences without a confident label
Builds on 1 paper
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An important ingredient in the proof is the following generalization of [ref] .
On the Information Dimension of Stochastic Processes · Information Dimension of Gaussian Processes · 0.93
Uses the method or tool of 1 paper
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The proof follows along the lines of [ref] .
On the Information Dimension of Stochastic Processes · Proof of the Main Result · 0.97
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