Simplicial Gaussian Models: Representation and Inference

TL;DR

The paper addresses the limitation of probabilistic graphical models to pairwise interactions by introducing the Simplicial Gaussian Model (SGM), a Gaussian framework on vertex, edge, and triangle signals built from incidence relations in a simplicial complex. It derives an edge-level marginal distribution by enforcing latent contributions from nodes and triangles and develops a convex, maximum-likelihood inference algorithm to jointly estimate the parameters $k$, $\mathbf{d}_V$, and $\mathbf{d}_T$ and recover the induced conditional dependencies across all simplex orders. Key contributions include a principled SGM formulation grounded in discrete Hodge theory, an edge-focused ML objective with a reparameterization that enables efficient block-coordinate optimization, and empirical validation on synthetic 2D complexes showing accurate triangle detection and low NMSE. This approach enables principled modeling and learning of higher-order interactions in complex networks, with potential impact on topological signal processing and related domains.

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.

Figures (2)

• Figure 1: F1-score for triangle detection after thresholding the estimated parameters $\hat{\mathbf{d}}_T$ at three different levels ($0.01$, $0.05$, $0.1$).
• Figure 2: NMSE for $(k,\mathbf{d}_V,\mathbf{d}_T)$ vs number of vertices.

Theorems & Definitions (2)

• Proposition 1
• proof