Logarithmic Voronoi cells for Gaussian models

We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in lognormal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.(c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by-nc -nd /4 .0/).

Automated summary

In this study, the theory of logarithmic Voronoi cells is extended to Gaussian statistical models. The properties of logarithmic Voronoi cells are analyzed for models of ML degree one and linear covariance models. The decomposition theory of logarithmic Voronoi cells is introduced for the latter family. The characteristics of logarithmic Voronoi cells in covariance models are also studied.