Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
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发表时间：
2018-02
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通讯作者：
Tam Le;M. Yamada
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作者：
Tam Le;M. Yamada
近年来，代数拓扑学方法在形状、关联扭曲图和材料数据等复杂几何结构数据的统计分析中发挥了重要作用。其中，Texttit{Persistent Homology}是一种著名的提取稳健拓扑特征的工具，并以PDS输出。然而，PD是点多集，不能用于矢量数据的机器学习算法。为了解决这一问题，一种新兴的方法是使用核方法，而合适的PDS几何是衡量PDS相似性的一个重要因素。PDS的一个流行的几何是\textit{Wasserstein度量}。然而，Wasserstein距离不是负定的。因此，它仅限于在Wasserstein距离上构造正定核。在这项工作中，我们利用另一种选择-正定核然后，我们分析了由所提出的核诱导的核机器的积分算子的特征系统。在此基础上，通过覆盖数和Rademacher平均得到了PF核机器的泛化误差界。此外，我们还证明了所提出的核的稳定性和无限整除性等性质。此外，我们还提出了在具有有界误差的情况下，对于我们所提出的核的近似，在PD中的点数上的线性时间复杂度。通过在不同基准数据集上的许多不同任务的实验，我们说明了PF内核在PD方面优于其他基准内核。
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.