持久熵变换：一种用于拓扑数据分析的基于熵的描述符
Persistent Entropy Transform: An entropy-based descriptor for topological data analysis
浏览论文内容
中文总结 AI 辅助
研究针对持久熵丢弃数据几何信息的局限，引入持久熵变换（PET）。通过理论证明其具有平移、尺度和正交等变性，利用合成形状和真实数据集测试其性能，验证PET可作为紧凑且易计算的拓扑数据分析描述符。
中文摘要 AI 辅助
持久熵为持久图提供了紧凑的摘要，但丢弃了数据固有的几何信息。这种局限性在计算高效但几何粗糙的标量摘要与表达性强但高维且计算要求高的方向拓扑变换之间造成了差距。在这项工作中，我们引入了持久熵变换（PET），这是一种新颖的方向拓扑描述符，可解释为基于熵的方向拓扑变换压缩。我们建立了PET的基本理论性质，证明了平移不变性、正均匀缩放时的尺度不变性和正交等变性。通过合成形状进行实证评估，包括方向敏感性、与旋转等变性的一致性、受控扰动下的鲁棒性以及对方向采样密度的依赖性。此外，我们在两个实时序列基准数据集上测试了该新工具，证明PET嵌入可成功用作实时序列基准上的紧凑特征向量。这些实验支持并验证了PET作为紧凑且计算易处理的描述符。
英文摘要
Persistent entropy provides a compact summary of persistence diagrams, but discards geometric information inherent to the data. This limitation creates a gap between scalar summaries, which are computationally efficient but geometrically coarse, and directional topological transforms, which are expressive but high-dimensional and computationally demanding. In this work, we introduce the Persistent Entropy Transform (PET), a novel directional topological descriptor which can be interpreted as an entropy-based compression of directional topological transforms. We establish basic theoretical properties of PET. In particular, we prove translation invariance, scale invariance under positive uniform scalings, and orthogonal equivariance. Empirically, we use synthetic shapes to assess directional sensitivity, consistency with rotational equivariance, robustness under controlled perturbations, and dependence on directional sampling density. In addition, we test the novel tool on two real time-series benchmark datasets, the TwoLead- ECG and the MIT-BIH, to provide a proof of concept showing that PET embeddings can be used succesfully as compact feature vectors on real time-series benchmarks. These experiments support and validate PET as a compact and computationally tractable descriptor.
