Logarithmic Voronoi cells

TL;DR

This paper investigates the geometric structure of logarithmic Voronoi cells in statistical models, revealing conditions under which they are polytopes and exploring their algebraic and numerical properties.

Contribution

It characterizes when logarithmic Voronoi cells are polytopes in various algebraic models and computes non-polytopal cells using numerical methods.

Findings

Logarithmic Voronoi cells are polytopes for finite, ML degree 1, linear, and toric models.

The algebraic moment map has polytopal fibers and images on the simplex.

Finite models' Voronoi polytopes are dual to Lie type A root polytopes.

Abstract

We study Voronoi cells in the statistical setting by considering preimages of the maximum likelihood estimator that tessellate an open probability simplex. In general, logarithmic Voronoi cells are convex sets. However, for certain algebraic models, namely finite models, models with ML degree 1, linear models, and log-linear (or toric) models, we show that logarithmic Voronoi cells are polytopes. As a corollary, the algebraic moment map has polytopes for both its fibres and its image, when restricted to the simplex. We also compute non-polytopal logarithmic Voronoi cells using numerical algebraic geometry. Finally, we determine logarithmic Voronoi polytopes for the finite model consisting of all empirical distributions of a fixed sample size. These polytopes are dual to the logarithmic root polytopes of Lie type A, and we characterize their faces.