# A Stable Cardinality Distance for Topological Classification

## TL;DR

This paper introduces a new stable distance metric for persistence diagrams that improves topological classification of point cloud data, especially in noisy, sparse materials science applications.

## Contribution

It proposes a novel distance measure on persistence diagrams that accounts for cardinality differences and proves its stability, enhancing topological data analysis for material classification.

## Key findings

- Effective classification of crystal structures from noisy data
- The new distance is stable under data perturbations
- Successful application to synthetic atom probe tomography data

## Abstract

This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on the space of persistence diagrams generates relevant input features for a classification algorithm for materials science data. This distance measures the similarity of persistence diagrams using the cost of matching points and a regularization term corresponding to cardinality differences between diagrams. Establishing stability properties of this distance provides theoretical justification for the use of the distance in comparisons of such diagrams. The classification scheme succeeds in determining the crystal structure of materials on noisy and sparse data retrieved from synthetic atom probe tomography experiments.