A computational approach to maximum likelihood thresholds for colored Gaussian graphical models

2026-09-02

AI Summary

本文通过几何方法和拓扑数据分析解决彩色高斯图模型的最大似然阈值计算问题，减少样本需求并克服传统方法的瓶颈。

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.

Problem

• Gaussian graphical models
• maximum likelihood threshold
• colored Gaussian graphical models
• high-dimensional
• small-sample

Innovation

• computational approach
• maximum likelihood threshold
• colored Gaussian graphical models
• topological data analysis
• geometric formulation