有色高斯图模型的最大似然阈值的计算方法
A computational approach to maximum likelihood thresholds for colored Gaussian graphical models
- Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学（巴塞罗那科技大学））
- Aalto University（阿尔托大学）
- Max Planck Institute of Biochemistry（马克斯·普朗克生物化学研究所）
- Munich Center for Machine Learning（慕尼黑机器学习中心）
- École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对有色高斯图模型，建立统一理论框架并引入符号算法，结合拓扑数据分析开展计算研究，以解决其最大似然阈值的计算问题，突破传统方法的计算瓶颈。
AI中文摘要：
高斯图模型（GGMs）是可解释结构学习的重要工具。然而在高维小样本场景下，可用数据往往不足以让最大似然估计量存在。有色高斯图模型（CGGMs）通过图着色施加对称约束缓解了这一限制，减少了所需样本量。保证估计量几乎必然存在的最小观测数被定义为最大似然阈值（MLT）。本文针对CGGMs的MLT计算问题，聚焦其几何形式：寻找样本协方差矩阵的最小秩，使其投影几乎必然落在充分统计量锥的内部。我们建立了统一理论框架，将结果从无色模型扩展到有色模型，并引入新的符号算法。此外，我们开展了结合采样与拓扑数据分析（TDA）的计算研究，以探究充分统计量锥的局部几何。结果表明，TDA在分析CGGMs的似然几何时，具备克服传统符号代数方法（尤其是格罗比纳基计算）计算瓶颈的潜力。
英文摘要：
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.
