A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

Authors

  • Rolando Kindelan Nuñez Universidad de Chile
  • Mircea Petrache UC Chile
  • Mauricio Cerda Universidad de Chile
  • Nancy Hitschfeld Universidad de Chile

DOI:

https://doi.org/10.1609/aaai.v38i12.29220

Keywords:

ML: Evaluation and Analysis, DMKM: Other Foundations of Data Mining & Knowledge Management, ML: Classification and Regression, ML: Clustering, ML: Kernel Methods, ML: Transparent, Interpretable, Explainable ML

Abstract

Persistence diagrams (PD)s play a central role in topological data analysis, and are used in an ever increasing variety of applications. The comparison of PD data requires computing distances among large sets of PDs, with metrics which are accurate, theoretically sound, and fast to compute. Especially for denser multi-dimensional PDs, such comparison metrics are lacking. While on the one hand, Wasserstein-type distances have high accuracy and theoretical guarantees, they incur high computational cost. On the other hand, distances between vectorizations such as Persistence Statistics (PS)s have lower computational cost, but lack the accuracy guarantees and theoretical properties of a true distance over PD space. In this work we introduce a class of pseudodistances called Extended Topological Pseudodistances (ETD)s, which have tunable complexity, and can approximate Sliced and classical Wasserstein distances at the high-complexity extreme, while being computationally lighter and close to Persistence Statistics at the lower complexity extreme, and thus allow users to interpolate between the two metrics. We build theoretical comparisons to show how to fit our new distances at an intermediate level between persistence vectorizations and Wasserstein distances. We also experimentally verify that ETDs outperform PSs in terms of accuracy and outperform Wasserstein and Sliced Wasserstein distances in terms of computational complexity.

Published

2024-03-24

How to Cite

Kindelan Nuñez, R., Petrache, M., Cerda, M., & Hitschfeld, N. (2024). A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams. Proceedings of the AAAI Conference on Artificial Intelligence, 38(12), 13202-13210. https://doi.org/10.1609/aaai.v38i12.29220

Issue

Section

AAAI Technical Track on Machine Learning III