持久Fisher核：持久图的一个黎曼流形核
Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
摘要：
代数拓扑方法近年来在复杂几何结构数据（如形状、链接扭曲图和材料数据）的统计分析中发挥了重要作用。其中，\textit{persistent homology}是一个众所周知的工具，以提取鲁棒的拓扑特征，并输出为\textit{persistence diagrams}（PD）。然而，PD是点多集，不能用于矢量数据的机器学习算法。一种新兴的方法是使用核方法，而合适的几何形状是衡量PD相似性的重要因素。PD的常用几何形状是\textit{Wasserstein metric}。然而，瓦瑟斯坦距离不是\textit{negative definite}。因此，在瓦瑟斯坦距离\textit{without approximation}上构建正定核是有限的。在这项工作中，我们依赖于替代\textit{Fisher information geometry}提出一个正定核PD \textit{without approximation}，即持久性Fisher（PF）内核。然后，我们分析了核机器的积分算子的特征系统。在此基础上，我们通过覆盖数和Rademacher平均值推导出具有PF核的核机器的泛化误差界。此外，我们展示了一些很好的性质，如稳定性和无限整除建议的内核。此外，我们还提出了一个线性的时间复杂度在PD的点的数量近似我们提出的内核有界误差。在各种基准数据集上进行的许多不同任务的实验中，我们说明了PF内核与PD的其他基线内核相比毫不逊色。
英文摘要：
Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, \textit{persistent homology} is a well-known tool to extract robust topological features, and outputs as \textit{persistence diagrams} (PDs). However, PDs are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for PDs is an important factor to measure the similarity of PDs. A popular geometry for PDs is the \textit{Wasserstein metric}. However, Wasserstein distance is not \textit{negative definite}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance \textit{without approximation}. In this work, we rely upon the alternative \textit{Fisher information geometry} to propose a positive definite kernel for PDs \textit{without approximation}, namely the Persistence Fisher (PF) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the PF kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in PDs for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the PF kernel compares favorably with other baseline kernels for PDs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
