Colored Markov Random Fields for Probabilistic Topological Modeling

2025-12-03

AI Summary

Traditional probabilistic graphical models struggle to capture conditional and marginal dependencies among variables residing on topological spaces, as their fixed graph structures impose inherent limitations on statistical expressiveness. To address this, we propose the Colored Markov Random Field (CMRF), the first framework integrating Hodge theory into the Gaussian Markov Random Field paradigm. CMRF introduces a “link coloring” mechanism that jointly encodes conditional independence via connectivity and marginal independence via color labels—thereby unifying topological and statistical constraints. This design substantially enhances expressivity of topological priors. By bridging topological signal processing with probabilistic graphical modeling, CMRF enables principled incorporation of higher-order structural information. We further develop a distributed estimation algorithm for scalable inference. Empirical evaluation on distributed physical network estimation tasks demonstrates that CMRF consistently outperforms baseline methods across varying strengths of topological priors, confirming its superior modeling accuracy and statistical representational power.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs well-suited for analyzing complex systems and supporting decision-making. Recent advances in topological signal processing highlight the importance of variables defined on topological spaces in several application domains. In such cases, the underlying topology shapes statistical relationships, limiting the expressiveness of canonical PGMs. To overcome this limitation, we introduce Colored Markov Random Fields (CMRFs), which model both conditional and marginal dependencies among Gaussian edge variables on topological spaces, with a theoretical foundation in Hodge theory. CMRFs extend classical Gaussian Markov Random Fields by including link coloring: connectivity encodes conditional independence, while color encodes marginal independence. We quantify the benefits of CMRFs through a distributed estimation case study over a physical network, comparing it with baselines with different levels of topological prior.

Problem

• Model dependencies on topological spaces for complex systems
• Extend Gaussian Markov Random Fields with link coloring
• Enhance probabilistic modeling using Hodge theory foundations

Innovation

• Introduces Colored Markov Random Fields for topological modeling
• Uses link coloring to encode marginal and conditional dependencies
• Extends Gaussian Markov Random Fields with Hodge theory foundation