[Paper Review] Persistence weighted Gaussian kernel for topological data analysis

Genki Kusano, Kenji Fukumizu+1 more|arXiv (Cornell University)|Jan 8, 2016

Topological and Geometric Data AnalysisComputer Science40 references89 citations

TL;DR

This paper introduces the persistence weighted Gaussian kernel (PWGK), a novel kernel method for topological data analysis that embeds persistence diagrams into a reproducing kernel Hilbert space (RKHS) while explicitly controlling for persistence. The method enhances stability, reduces noise influence by downweighting low-persistence features, and enables fast approximation; it outperforms existing methods on protein and oxide glass datasets by providing more robust and accurate topological descriptors.

ABSTRACT

Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy topological properties. This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA. The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method allows a fast approximation technique. The method is applied into practical data on proteins and oxide glasses, and the results show the advantage of our method compared to other relevant methods on persistence diagrams.

Motivation & Objective

• To develop a stable, statistically sound kernel method for analyzing persistence diagrams in topological data analysis.
• To explicitly control the influence of topological features based on their persistence, reducing noise impact.
• To enable efficient computation via fast approximation techniques for large-scale applications.
• To provide a vectorized representation of persistence diagrams suitable for standard kernel methods in machine learning.
• To demonstrate superior performance on real-world datasets such as proteins and oxide glasses compared to existing approaches.

Proposed method

• Proposes the persistence weighted Gaussian kernel (PWGK), a positive definite kernel that weights points in a persistence diagram by their persistence.
• Uses a weight function based on persistence to reduce the contribution of noisy, low-persistence features near the diagonal.
• Employs kernel embedding of measures into an RKHS via the Bochner integral, enabling vectorization of persistence diagrams.
• Introduces a fast approximation scheme based on random Fourier features to scale the method to large datasets.
• Derives theoretical stability bounds showing the kernel’s robustness to perturbations in the input data.
• Applies the kernel to statistical learning tasks such as classification and regression using standard kernel methods.

Experimental results

Research questions

• RQ1 Can a kernel method be designed for persistence diagrams that explicitly controls the influence of persistence to enhance robustness to noise?
• RQ2 How can persistence diagrams be embedded into a Hilbert space in a way that preserves topological stability and enables efficient computation?
• RQ3 Does the proposed kernel outperform existing kernel methods on persistence diagrams in practical classification tasks?
• RQ4 To what extent can the PWGK achieve fast approximation without sacrificing accuracy or stability?
• RQ5 How does the method perform on real-world datasets such as protein structures and oxide glass data compared to baseline approaches?

Key findings

• The persistence weighted Gaussian kernel achieves theoretical stability under perturbations, with a bound proportional to the Wasserstein distance between persistence diagrams.
• The method effectively downweights low-persistence features (noise) while preserving high-persistence topological structures, improving statistical robustness.
• The PWGK enables fast approximation using random Fourier features, significantly reducing computational cost while maintaining accuracy.
• On protein and oxide glass datasets, the proposed method outperformed existing kernel methods on persistence diagrams in classification tasks.
• Theoretical analysis confirms that the kernel induces a stable RKHS norm, ensuring reliable statistical inference from persistence diagrams.