提出用于拓扑数据的交叉熵方法，可区分相同熵的图并用于知识蒸馏。
Persistent Cross Entropy
- 用相似性与权重映射构建跨图概率，解决事件空间不一致问题
- 新方法能区分相同持久熵的拓扑图，且在动力系统中识别因果方向
- 适用于拓扑对比学习与知识蒸馏，具理论稳定性保障
持久熵是基于持久图上概率测度的香农熵。但由于两个持久图通常具有不同事件空间，其交叉熵难以自然定义。为此，我们结合相似性函数与持久性加权，定义一种诱导概率，使一个图的信息能映射到另一图的事件空间，并将未解释的概率质量分配给未解释事件。基于此诱导概率，我们将交叉熵推广至持久图，称为持久交叉熵（PCE）。我们建立了诱导概率和PCE的主要性质，并证明了两者稳定性定理。通过三个数值实验，我们表明PCE能够区分具有相同持久熵的图，在无需构造联合持久图的情况下分离动力系统的因果方向，并可用作知识蒸馏中的定向拓扑损失。
原文摘要 · Abstract (English)
Persistent entropy is the Shannon entropy of a persistence-based probability measure defined on a persistence diagram. However, its cross-entropy version is not naturally defined because two persistence diagrams generally have different event spaces. To bridge these event spaces, we combine a similarity function with persistence weighting to define an induced probability. The induced probability reflects information from one diagram on the event space of the other diagram and assigns unexplained probability mass to the unexplained event. Using the induced probability, we extend cross entropy to persistence diagrams, called persistent cross entropy (PCE). We establish the main properties of both the induced probability and PCE and prove stability theorems for both. Through three numerical studies, we show that PCE distinguishes diagrams with the same persistent entropy, separates causal directions in dynamical systems without constructing a joint persistent diagram, and can be used as a directional topology loss for knowledge distillation.
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