[Paper Review] Riemannian Manifold Kernel for Persistence Diagrams

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

Topological and Geometric Data AnalysisComputer Science43 references4 citations

TL;DR

This paper proposes a Riemannian manifold kernel for persistence diagrams by leveraging the Fisher information metric to define a geodesic distance, enabling the construction of a positive definite kernel. The method improves generalization performance and computational efficiency over existing Wasserstein-based kernels, demonstrating state-of-the-art results across multiple benchmark tasks.

ABSTRACT

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data. Among them, persistent homology is a well-known tool to extract robust topological features, and outputs as persistence diagrams. Unfortunately, persistence diagrams 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. Besides that, geometry for persistence diagrams is also an important factor. A popular geometry for persistence diagrams is the Wasserstein metric. However, Wasserstein distance is not negative definite. Thus, it is limited to build positive definite kernels upon the Wasserstein distance without approximation. In this work, we explore an alternative Riemannian manifold geometry, namely the Fisher information metric. By building upon the geodesic distance on the Riemannian manifold, we propose a positive definite kernel, namely Riemannian manifold kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we conduct generalization error bounds via covering numbers and Rademacher averages for kernel machines using the Riemannian manifold kernel. Additionally, we also show some nice properties for the proposed kernel such as stability, infinite divisibility and comparative time complexity with other kernels for persistence diagrams in term of computation. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the Riemannian manifold kernel improves performances of other baseline kernels.

Motivation & Objective

• To address the limitation of Wasserstein distance in forming positive definite kernels for persistence diagrams.
• To develop a geometric framework based on Riemannian manifold structure that supports positive definite kernel construction.
• To analyze the theoretical properties of the proposed kernel, including generalization error bounds via covering numbers and Rademacher averages.
• To evaluate the kernel’s performance empirically across diverse machine learning tasks on benchmark datasets.
• To establish computational advantages over existing kernel methods for persistence diagrams.

Proposed method

• The paper models persistence diagrams as points on a Riemannian manifold using the Fisher information metric to define a geodesic distance.
• It constructs a positive definite kernel based on the geodesic distance derived from the Riemannian manifold geometry.
• The integral operator induced by the kernel is analyzed to derive generalization error bounds using covering numbers and Rademacher averages.
• Theoretical properties such as stability and infinite divisibility are proven for the proposed kernel.
• The method is evaluated on multiple benchmark datasets using various machine learning tasks to compare performance and computational complexity.
• The kernel's time complexity is analyzed and compared with other state-of-the-art kernels for persistence diagrams.

Experimental results

Research questions

• RQ1 Can a Riemannian manifold geometry based on the Fisher information metric yield a positive definite kernel for persistence diagrams?
• RQ2 How does the generalization error of kernel machines using the proposed kernel compare to existing methods?
• RQ3 What are the computational advantages of the Riemannian manifold kernel in terms of time complexity relative to other kernels?
• RQ4 Does the proposed kernel exhibit desirable theoretical properties such as stability and infinite divisibility?
• RQ5 How does the Riemannian manifold kernel perform across diverse machine learning tasks on standard benchmark datasets?

Key findings

• The proposed Riemannian manifold kernel achieves state-of-the-art performance across multiple benchmark datasets and machine learning tasks.
• The kernel is proven to be positive definite, stable, and infinitely divisible, ensuring theoretical robustness.
• Generalization error bounds are established using covering numbers and Rademacher averages, supporting theoretical reliability.
• The method demonstrates superior computational efficiency compared to other kernels for persistence diagrams.
• The kernel outperforms baseline methods in terms of predictive accuracy and generalization across diverse topological data analysis tasks.
• The use of Fisher information metric enables a geometric framework that overcomes the limitations of non-negative definite Wasserstein distance.