Tropical Coordinates on the Space of Persistence Barcodes

TL;DR

This paper introduces tropical coordinates for the space of persistence barcodes, providing a stable numerical representation that facilitates their use in machine learning applications.

Contribution

The paper identifies and proves the stability of tropical coordinates on the space of barcodes, enabling their effective use in applied topology and data analysis.

Findings

Tropical coordinates are stable under bottleneck and Wasserstein distances.

They provide a numerical encoding of barcodes for machine learning.

The approach bridges topological invariants and practical data analysis.

Abstract

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well-adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes and these outputs can then be used as input to standard algorithms. It is the purpose of this paper to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.