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Rate distortion dimension and ergodic decomposition for $\mathbb{R}^d$-actions

Masaki Tsukamoto

TL;DR

This work investigates how the rate distortion dimension, a bridge between information theory and dynamical mean dimension, behaves for $\mathbb{R}^d$-actions under ergodic decomposition. By deriving a convex-combination formula for the rate distortion function and leveraging Wasserstein-transport ideas, the authors prove that the upper (resp. lower) rate distortion dimension is convex (resp. concave) under convex mixtures of invariant measures, provided the upper metric mean dimension is finite. A corollary shows exact averaging of the rate distortion dimension when it exists almost everywhere along ergodic components; the paper also constructs sharp counterexamples demonstrating the necessity of hypotheses and the possible strictness of the inequalities. Two explicit constructions illustrate both irregular behavior and possible strictness of convexity, highlighting subtle interactions between rate distortion, ergodicity, and mean dimension in nonergodic dynamical systems. Overall, the results deepen the connection between rate distortion theory and mean dimension, offering tools for analyzing lossy compression limits in nonergodic, high-complexity dynamical settings.

Abstract

Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has an intimate connection to Gromov's theory of mean dimension of dynamical systems. This paper studies the behavior of rate distortion dimension of $\mathbb{R}^d$-actions under ergodic decomposition. Our main theorems provide natural convexity and concavity of upper and lower rate distortion dimensions under convex combination of invariant probability measures. We also present examples which clarify the validity and limitations of the theorems.

Rate distortion dimension and ergodic decomposition for $\mathbb{R}^d$-actions

TL;DR

This work investigates how the rate distortion dimension, a bridge between information theory and dynamical mean dimension, behaves for -actions under ergodic decomposition. By deriving a convex-combination formula for the rate distortion function and leveraging Wasserstein-transport ideas, the authors prove that the upper (resp. lower) rate distortion dimension is convex (resp. concave) under convex mixtures of invariant measures, provided the upper metric mean dimension is finite. A corollary shows exact averaging of the rate distortion dimension when it exists almost everywhere along ergodic components; the paper also constructs sharp counterexamples demonstrating the necessity of hypotheses and the possible strictness of the inequalities. Two explicit constructions illustrate both irregular behavior and possible strictness of convexity, highlighting subtle interactions between rate distortion, ergodicity, and mean dimension in nonergodic dynamical systems. Overall, the results deepen the connection between rate distortion theory and mean dimension, offering tools for analyzing lossy compression limits in nonergodic, high-complexity dynamical settings.

Abstract

Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has an intimate connection to Gromov's theory of mean dimension of dynamical systems. This paper studies the behavior of rate distortion dimension of -actions under ergodic decomposition. Our main theorems provide natural convexity and concavity of upper and lower rate distortion dimensions under convex combination of invariant probability measures. We also present examples which clarify the validity and limitations of the theorems.

Paper Structure

This paper contains 11 sections, 30 theorems, 182 equations, 2 figures.

Table of Contents

  1. Main results
  2. Mutual information and rate distortion function
  3. Measure theoretic details
  4. Mutual information
  5. Rate distortion function
  6. Proofs of the main theorems
  7. Wasserstein metric and rate distortion function
  8. Rate distortion function and convex combination of measures
  9. Proofs of Theorems \ref{['theorem: upper rate distortion dimension and convex combination']} and \ref{['theorem: lower rate distortion dimension and convex combination']}
  10. Construction of an example: Proof of Theorem \ref{['theorem: irregular behavior of upper rate distortion dimension']}
  11. Another example

Key Result

Theorem 1.1

Let $T\colon \mathbb{R}^d\times \mathcal{X}\to \mathcal{X}$ be a continuous action of $\mathbb{R}^d$ on a compact metric space $(\mathcal{X}, \mathbf{d})$. Suppose the upper metric mean dimension $\overline{\mathrm{mdim}}_{\mathrm{M}}\left(\mathcal{X}, T, \mathbf{d}\right)$ is finite. Let $\lambda$ Then we have

Figures (2)

  • Figure 1: A $\mathbb{Z}^2$-invariant triangulation of the plane. The vertexes are points of the standard lattice $\mathbb{Z}^2$.
  • Figure 2: The graph of the function $h$. We have $c_k = k a_k$ and $k c_k < b_k < a_{k+1}$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1
  • Lemma 2.2: Data-Processing inequality
  • proof
  • Proposition 2.3: $I(\mu, \nu)$ is concave in $\mu$ and convex in $\nu$
  • proof
  • ...and 47 more