用持久熵检测相变，揭示了拓扑结构变化的触发条件。
Persistent Entropy as a Detector of Phase Transitions
- 基于归一化权重的分散-凝聚机制，给出熵差下界
- 在有限样本下高概率成立，不受生命周期绝对尺度影响
- 适用于神经网络、同步模型等多类系统的相变检测
持久熵是用于检测系统状态变化的标量统计量，但尚无理论说明何时条形图的结构变化必然导致熵值可检测的变化。本文建立了一个无需依赖模型的定理，提供此类条件。将持久图视为随控制参数变化的随机对象，识别出归一化持久权重的分散-凝聚机制，并推导出两个阶段间熵差的显式下界，该下界在有限样本下以高概率成立，且对条形寿命的绝对尺度不敏感。同时提出验证经验条形图假设的流程。应用于卷积网络时，该判据表明Gabrielsson和Carlsson报道的滤波器环状组织通过尖锐的拓扑相变形成，其起点在MNIST上约数百次迭代内出现，而在CIFAR-10上则晚一个数量级；同一判据还可检测库朗托同步与维克赛克有序-无序相变。
原文摘要 · Abstract (English)
Persistent entropy is a scalar summary of persistence barcodes widely used to detect regime changes, yet there is no account of when a structural change in a barcode must produce a detectable change in entropy. We establish a model-agnostic theorem supplying such conditions. Treating persistence diagrams as random objects indexed by a control parameter, we identify a dispersion-condensation mechanism in the normalized persistence weights and derive an explicit lower bound on the entropy difference between the two regimes, valid with high probability at finite sample size and insensitive to the absolute scale of bar lifetimes. We also give a procedure for verifying the hypotheses on empirical barcodes. Applied to convolutional networks, the criterion shows that the circular organization of learned filters reported by Gabrielsson and Carlsson emerges through a sharp topological phase transition, and locates its onset: within a few hundred iterations on MNIST, but an order of magnitude later on CIFAR-10. The same criterion detects the Kuramoto synchronization and Vicsek order-disorder transitions.
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