Fuzzy c-Means Clustering for Persistence Diagrams

TL;DR

This paper introduces a fuzzy c-means clustering algorithm adapted for persistence diagrams, enabling unsupervised topological data analysis with theoretical guarantees and practical applications in machine learning model selection and materials science.

Contribution

It extends fuzzy c-means clustering to the space of persistence diagrams, providing convergence guarantees and demonstrating practical utility in diverse datasets.

Findings

Successfully captures topological structures in data

Enables model selection based on topological features

Classifies lattice structures in materials science

Abstract

Persistence diagrams concisely represent the topology of a point cloud whilst having strong theoretical guarantees, but the question of how to best integrate this information into machine learning workflows remains open. In this paper we extend the ubiquitous Fuzzy c-Means (FCM) clustering algorithm to the space of persistence diagrams, enabling unsupervised learning that automatically captures the topological structure of data without the topological prior knowledge or additional processing of persistence diagrams that many other techniques require. We give theoretical convergence guarantees that correspond to the Euclidean case, and empirically demonstrate the capability of our algorithm to capture topological information via the fuzzy RAND index. We end with experiments on two datasets that utilise both the topological and fuzzy nature of our algorithm: pre-trained model selection in machine learning and lattices structures from materials science. As pre-trained models can perform well on multiple tasks, selecting the best model is a naturally fuzzy problem; we show that fuzzy clustering persistence diagrams allows for model selection using the topology of decision boundaries. In materials science, we classify transformed lattice structure datasets for the first time, whilst the probabilistic membership values let us rank candidate lattices in a scenario where further investigation requires expensive laboratory time and expertise.