持久交叉熵

中文摘要 AI 辅助

持久熵是定义在持久图上的基于持久度的概率测度的香农熵。然而，其交叉熵版本无法自然定义，因为两个持久图通常具有不同的事件空间。为桥接这些事件空间，我们结合相似度函数与持久度加权来定义诱导概率。该诱导概率反映一个图在另一个图事件空间上的信息，并将未解释的概率质量分配给未解释事件。利用该诱导概率，我们将交叉熵扩展到持久图，称为持久交叉熵（PCE）。我们确立了诱导概率和PCE的主要性质，并证明了两者的稳定性定理。通过三项数值研究，我们表明PCE可区分具有相同持久熵的图、在不构建联合持久图的情况下分离动力系统的因果方向，且可作为知识蒸馏的定向拓扑损失。

英文摘要

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.

发表机构

• Artificial Intelligence Graduate School, Pohang University of Science and Technology（浦项科技大学人工智能研究生院）
• Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学系）

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