[Paper Review] Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams

Tam Le, Makoto Yamada|arXiv (Cornell University)|Feb 10, 2018

Topological and Geometric Data AnalysisComputer Science51 citations

TL;DR

The paper introduces the Persistence Fisher (PF) kernel for persistence diagrams, based on Fisher information geometry, providing a positive definite, non-approximate kernel with theoretical guarantees and competitive empirical performance.

ABSTRACT

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, extit{persistent homology} is a well-known tool to extract robust topological features, and outputs as extit{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 extit{Wasserstein metric}. However, Wasserstein distance is not extit{negative definite}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance extit{without approximation}. In this work, we rely upon the alternative extit{Fisher information geometry} to propose a positive definite kernel for PDs extit{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.

Motivation & Objective

• Motivate robust statistical analysis of persistence diagrams (PDs) via kernels that respect their geometry.
• Propose a positive definite PF kernel computed directly from the Fisher information metric without approximation.
• Establish theoretical guarantees including eigenstructure, generalization bounds, and stability properties.
• Demonstrate empirical performance of PF across multiple PD-based learning tasks against baselines.

Proposed method

• Represent each PD as a smoothed, normalized measure on a finite set via Gaussian smoothing.
• Define the Fisher information metric between two PDs using the smoothed measures and the probability simplex.
• Construct the PF kernel as k_PF(Dg_i, Dg_j) = exp(-t d_FIM(Dg_i, Dg_j)) with t > 0 and show d_FIM is negative definite up to a shift.
• Analyze the eigenstructure of the integral operator induced by k_PF to derive covering-number and Rademacher-average generalization bounds.
• Propose a linear-time approximation using the Fast Gauss Transform to reduce cost while preserving bounded error.
• Demonstrate infinite divisibility of the PF kernel and discuss stability with respect to the underlying Fisher information geometry.

Experimental results

Research questions

• RQ1 How to define a geometry-aware, positive definite kernel for persistence diagrams without approximating the underlying metric?
• RQ2 What are the theoretical properties (eigenstructure, generalization bounds, stability) of a kernel based on the Fisher information metric on PDs?
• RQ3 How does the PF kernel perform empirically against existing PD kernels across classification and change-point tasks?

Key findings

• The PF kernel is positive definite and built directly from the Fisher information metric, without approximation.
• The integral-operator eigensystem derived for PF shows nonnegative Legendre expansion coefficients, enabling kernel-learning bounds.
• PF kernel achieves competitive or superior performance compared to baseline PD kernels on benchmarks (e.g., MPEG7 and Orbit datasets).
• PF-based SVM results: MPEG7 accuracy 80.00 ± 4.08; Orbit accuracy 85.87 ± 0.77, outperforming PSS, PWG, and SW baselines.
• The PF kernel admits linear-time approximation via Fast Gauss Transform, and is infinitely divisible with favorable stability properties.