A computational approach to maximum likelihood thresholds for colored Gaussian graphical models
Abstract
Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.
Related
- Penalized Likelihood Inference in Highly Multivariate Sparse Spectral Graphical Gaussian Processes
- Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator
- The AECM Algorithm for Deterministic Maximum Likelihood Direction Finding in the Presence of Gaussian Mixture Noise
- Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
- Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design
BibTeX
@misc{homs2026computational,
title = {{A computational approach to maximum likelihood thresholds for colored Gaussian graphical models}},
author = {Roser Homs and Olga Kuznetsova and Bernadette J. Stolz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02382},
eprint = {2609.02382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02382},
}
RIS
TY - UNPB
TI - A computational approach to maximum likelihood thresholds for colored Gaussian graphical models
AU - Roser Homs
AU - Olga Kuznetsova
AU - Bernadette J. Stolz
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02382
UR - https://arxiv.org/abs/2609.02382
AB - Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.
ER -