Persistent Cross Entropy

2026-08-25

AI Summary

为了解决不同事件空间的持续熵无法直接比较的问题，通过结合相似函数和持久加权定义诱导概率，并基于此提出了持久交叉熵（PCE），用于区分和分离拓扑结构。

Abstract

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.

Problem

• Persistent Entropy
• Cross Entropy
• Persistence Diagrams
• Event Spaces

Innovation

• Persistent Cross Entropy
• Induced Probability
• Persistence Diagrams