Information Properties of a Random Variable Decomposition through Lattices

TL;DR

This paper explores how a random variable can be decomposed into wrapped and quantized parts using lattices, analyzing the information-theoretic properties of this decomposition and its generalization to topological groups.

Contribution

It introduces a novel framework for understanding the information properties of random variables decomposed via lattice-based operations, extending to topological groups.

Findings

Analysis of entropy, mutual information, and Fisher information for the decomposition

Demonstration of the framework's generalization to locally compact topological groups

Insights into the structure of random variables through lattice-based decompositions

Abstract

A full-rank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on $\mathbb{R}^n$, two operations can be induced by considering the quotient of the space by such a lattice: wrapping and quantization. For a lattice $\Lambda$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through $\Lambda$, the wrapped distribution over the quotient is obtained by summing the density over each coset, while the quantized distribution over the lattice is defined by integrating over each fundamental domain translation. These operations define wrapped and quantized random variables over $D$ and $\Lambda$, respectively, which sum up to the original random variable. We investigate information-theoretic properties of this decomposition, such as entropy, mutual information and the Fisher information matrix, and show that it naturally generalizes to the more abstract context of locally compact topological groups.