证明了图模型的变分自由能临界点在同伦等价变形下保持不变。
Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs
- 通过同伦理论分析因子图变形对自由能临界点的影响
- 在链长不超过1的超图与偏序集中，临界点一一对应
- 适用于研究变分推断稳定性的理论工作者
在图模型、因子图及更一般的能量模型中，变量间的相互作用由图、超图或偏序集（poset）编码。由于底层结构中存在环，这类概率模型的精确推断难以实现，通常依赖于近似变分推断，即优化Bethe自由能。Bethe自由能的临界点对应于相关信念传播算法的不动点。对于具有有限变量的通用图、超图和偏序集，这些临界点的完整表征仍是开放问题。本文证明：对于链长至多为1的超图与偏序集，将交互的偏序集变形为同伦类型相同的另一偏序集时，其对应的自由能临界点之间存在双射关系。该结果推广并统一了经典中假设特定可收缩性以证明Bethe自由能临界点唯一性的结论。
原文摘要 · Abstract (English)
In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.
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