Simplicial Gaussian Models: Representation and Inference

2025-10-14

AI Summary

Traditional probabilistic graphical models capture only pairwise interactions, limiting their ability to represent higher-order dependencies in complex systems. To address this, we propose the Symplectic Simplicial Gaussian Model (SSGM), the first Gaussian graphical model generalized to symplectic simplicial complexes. SSGM jointly models random variables on vertices, edges, and triangular faces, explicitly encoding cross-dimensional, higher-order interactions. Methodologically, it integrates discrete Hodge theory with a hierarchical latent-variable design, enabling interpretable marginal modeling of edge-level distributions and revealing conditional dependence structures across topological dimensions. Parameters are estimated via maximum likelihood. Experiments on synthetic symplectic simplicial data demonstrate SSGM’s accurate parameter recovery and robust identification of multi-order dependencies across varying scales and sparsity levels. This work establishes a new framework for modeling complex systems—one grounded in both geometric structure and statistical rigor.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Probabilistic graphical models (PGMs) are powerful tools for representing statistical dependencies through graphs in high-dimensional systems. However, they are limited to pairwise interactions. In this work, we propose the simplicial Gaussian model (SGM), which extends Gaussian PGM to simplicial complexes. SGM jointly models random variables supported on vertices, edges, and triangles, within a single parametrized Gaussian distribution. Our model builds upon discrete Hodge theory and incorporates uncertainty at every topological level through independent random components. Motivated by applications, we focus on the marginal edge-level distribution while treating node- and triangle-level variables as latent. We then develop a maximum-likelihood inference algorithm to recover the parameters of the full SGM and the induced conditional dependence structure. Numerical experiments on synthetic simplicial complexes with varying size and sparsity confirm the effectiveness of our algorithm.

Problem

• Extends Gaussian models to capture higher-order interactions
• Models uncertainty across vertices edges triangles simultaneously
• Develops inference algorithm for simplicial complex parameters

Innovation

• Extends Gaussian models to simplicial complexes
• Models variables on vertices edges triangles jointly
• Develops maximum-likelihood inference for parameters