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Persistence kernels for classification: A comparative study

Cinzia Bandiziol, Stefano De Marchi

TL;DR

This work conducts a comprehensive, cross-domain comparison of five persistence kernels—PSSK, PWGK, SWK, PFK, and PI—for classification tasks within the framework of persistent homology. By mapping persistence diagrams into diverse feature representations and pairing them with SVMs, the study evaluates kernel performance on point clouds, images, graphs, and time series, highlighting SWK's strong empirical performance and its computational efficiency. Key contributions include detailed kernel descriptions, parameter-tuning guidelines, extensive numerical experiments, and publicly available Python code for reproducibility. The findings underscore the importance of the filtration choice and kernel design in extracting robust topological features for real-world classification problems.

Abstract

The aim of the present work is a comparative study of different persistence kernels applied to various classification problems. After some necessary preliminaries on homology and persistence diagrams, we introduce five different kernels that are then used to compare their performances of classification on various datasets. We also provide the Python codes for the reproducibility of results.

Persistence kernels for classification: A comparative study

TL;DR

This work conducts a comprehensive, cross-domain comparison of five persistence kernels—PSSK, PWGK, SWK, PFK, and PI—for classification tasks within the framework of persistent homology. By mapping persistence diagrams into diverse feature representations and pairing them with SVMs, the study evaluates kernel performance on point clouds, images, graphs, and time series, highlighting SWK's strong empirical performance and its computational efficiency. Key contributions include detailed kernel descriptions, parameter-tuning guidelines, extensive numerical experiments, and publicly available Python code for reproducibility. The findings underscore the importance of the filtration choice and kernel design in extracting robust topological features for real-world classification problems.

Abstract

The aim of the present work is a comparative study of different persistence kernels applied to various classification problems. After some necessary preliminaries on homology and persistence diagrams, we introduce five different kernels that are then used to compare their performances of classification on various datasets. We also provide the Python codes for the reproducibility of results.
Paper Structure (22 sections, 38 equations, 13 figures, 7 tables)

This paper contains 22 sections, 38 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Persistent Homology
  3. Stability
  4. SVM
  5. Classification problem
  6. Persistence Kernels
  7. Persistence Scale-Space Kernel (PSSK)
  8. Persistence Weighted Gaussian Kernel (PWGK)
  9. Sliced Wasserstein Kernel (SWK)
  10. Persistence Fisher Kernel (PFK)
  11. Persistence Image (PI)
  12. Shape paramenters analysis
  13. Numerical Tests
  14. Point cloud data and shapes
  15. Protein
  16. ...and 7 more sections

Figures (13)

  • Figure 1: An example of a valid simplicial complex (left) and an invalid one (right)
  • Figure 2: An example of Persistence Diagram with features of dimensions 0,1 and 2
  • Figure 3: Example of bottleneck distance between two PDs in red and blue
  • Figure 4: Comparison results about PSSK for SHREC14 in terms of condition number (left) and accuracy (right) with different $\sigma$
  • Figure 5: Comparison results about PWGK for MUTAG in terms of accuracy with different $\tau$ and $\rho$
  • ...and 8 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 4 more