提出彩色马尔可夫随机场，用拓扑结构提升概率建模能力
Colored Markov Random Fields for Probabilistic Topological Modeling
- 在拓扑空间上构建彩色马尔可夫随机场，区分条件与边际独立性
- 在物理网络分布式估计中表现优于传统模型，拓扑先验越强效果越优
- 适合处理具有复杂拓扑结构的信号建模问题，如传感器网络
概率图模型（PGMs）通过图结构（节点为变量，边为依赖关系）对随机变量间的条件依赖进行编码，并将联合分布分解为低维分量，适用于复杂系统分析与决策支持。近年来，拓扑信号处理的发展凸显了定义在拓扑空间上的变量在多个应用领域的重要性。在此类场景中，底层拓扑结构决定统计关系，限制了经典PGMs的表达能力。为此，我们提出彩色马尔可夫随机场（CMRFs），在拓扑空间上对高斯边变量的条件与边际依赖进行建模，理论基础源于霍奇理论。CMRFs通过引入边着色扩展经典高斯马尔可夫随机场：连接表示条件独立性，颜色表示边际独立性。通过在物理网络上的分布式估计案例研究，量化了其优势，并与不同拓扑先验水平的基线模型对比，验证了其有效性。
原文摘要 · Abstract (English)
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.
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