Mutual Dimension

TL;DR

This paper introduces mutual dimensions as a measure of shared algorithmic information between points in Euclidean space, establishing their properties and how they transform under computable Lipschitz functions.

Contribution

It defines mutual dimensions in Euclidean space, proves a data processing inequality for them, and analyzes how these measures change under computable Lipschitz transformations.

Findings

Mutual dimensions satisfy key properties of mutual information.

The data processing inequality holds for mutual dimensions under Lipschitz functions.

Conditions are established for functions to preserve or alter mutual dimensions.

Abstract

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is computable and Lipschitz, then the inequalities $mdim(f(x):y) \leq mdim(x:y)$ and $Mdim(f(x):y) \leq Mdim(x:y)$ hold for all $x \in \mathbb{R}^m$ and $y \in \mathbb{R}^t$. We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.