Logarithmic Voronoi polytopes for discrete linear models

TL;DR

This paper investigates the structure of logarithmic Voronoi polytopes in linear statistical models, revealing their combinatorial properties and providing explicit descriptions for interior and boundary points.

Contribution

It introduces a novel geometric framework for analyzing logarithmic Voronoi cells in linear models, linking them to dual vector configurations and co-circuits.

Findings

Logarithmic Voronoi cells are polytopes with vertices described by co-circuits.

All interior points on a linear model share the same combinatorial type of Voronoi cell.

The paper extends the analysis to boundary points and partial linear models.

Abstract

We study logarithmic Voronoi cells for linear statistical models and partial linear models. The logarithmic Voronoi cells at points on such model are polytopes. To any $d$-dimensional linear model inside the probability simplex $\Delta_{n-1}$, we can associate an $n\times d$ matrix $B$. For interior points, we describe the vertices of these polytopes in terms of co-circuits of $B$. We also show that these polytopes are combinatorially isomorphic to the dual of a vector configuration with Gale diagram $B$. This means that logarithmic Voronoi cells at all interior points on a linear model have the same combinatorial type. We also describe logarithmic Voronoi cells at points on the boundary of the simplex. Finally, we study logarithmic Voronoi cells of partial linear models, where the points on the boundary of the model are especially of interest.