Rate-Distortion Problems of the Poisson Process based on a Group-Theoretic Approach

TL;DR

This paper investigates the rate-distortion characteristics of a Poisson process using a novel group-theoretic framework, linking geometric covering problems with classical rate-distortion functions.

Contribution

It introduces a group-theoretic approach to analyze rate-distortion problems of Poisson processes, unifying multiple known problems under a geometric covering perspective.

Findings

Established rate-distortion as hyperoctahedral group covering problems.

Unified three existing Poisson rate-distortion problems with Laplacian-ℓ₁ case.

Connected distortion measures with geometric realizations in hypercube and hyperoctahedron.

Abstract

We study rate-distortion problems of a Poisson process using a group theoretic approach. By describing a realization of a Poisson point process with either point timings or inter-event (inter-point) intervals and by choosing appropriate distortion measures, we establish rate-distortion problems of a homogeneous Poisson process as ball- or sphere-covering problems for realizations of the hyperoctahedral group in $\mathbb{R}^n$. Specifically, the realizations we investigate are a hypercube and a hyperoctahedron. Thereby we unify three known rate-distortion problems of a Poisson process (with different distortion measures, but resulting in the same rate-distortion function) with the Laplacian-$\ell_1$ rate-distortion problem.