将高斯图模型拓展到多体交互，支持顶点、边、三角形联合建模。
Simplicial Gaussian Models: Representation and Inference
- 基于离散霍奇理论，用单一高斯分布联合建模顶点、边、三角形上的随机变量
- 在不同规模和稀疏度的合成复形上，参数恢复准确，能有效推断条件依赖结构
- 适合研究复杂系统中高阶依赖关系的科研人员，尤其关注拓扑数据分析者
概率图模型（PGM）是高维系统中表示统计依赖的强大工具，但仅限于成对交互。本文提出单纯形高斯模型（SGM），将高斯PGM扩展至单纯形复形。SGM通过单个参数化高斯分布，联合建模定义在顶点、边和三角形上的随机变量。模型基于离散霍奇理论，在每个拓扑层级引入独立随机分量以刻画不确定性。受应用驱动，我们聚焦边级别的边缘分布，将节点与三角形级别变量视为隐变量。随后，我们开发了最大似然推断算法，以恢复完整SGM的参数及诱导的条件依赖结构。在不同规模与稀疏度的合成单纯形复形上的数值实验验证了该算法的有效性。
原文摘要 · Abstract (English)
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.
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