Simplicial Gaussian Models: Representation and Inference
This work addresses the problem of modeling higher-order interactions in high-dimensional systems for researchers in statistics and machine learning, representing an incremental advancement by extending existing Gaussian models to simplicial complexes.
The paper tackles the limitation of probabilistic graphical models to pairwise interactions by proposing the simplicial Gaussian model (SGM), which extends Gaussian models to simplicial complexes to jointly model variables on vertices, edges, and triangles, and develops a maximum-likelihood inference algorithm validated on synthetic data.
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.