Metrics for Learning in Topological Persistence

TL;DR

This paper introduces metrics based on persistent homology for analyzing data connectivity, focusing on stable ranks as robust topological descriptors, and demonstrates their effectiveness in classifying physical activities and atmospheric cloud patterns.

Contribution

It proposes a new framework for defining and optimizing metrics via contour functions to stabilize topological invariants for data analysis.

Findings

Stable ranks serve as robust data fingerprints.

Optimizing contours improves classification accuracy.

Application to atmospheric cloud patterns shows practical effectiveness.

Abstract

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects. We outline how so called contour functions induce relevant metrics for stabilizing the rank invariant. On the practical level, the stable ranks are used as fingerprints for data. Different choices of contour lead to different stable ranks and the topological learning is then the question of finding the optimal contour. We outline our analysis pipeline and show how it can enhance classification of physical activities data. As our main application we study how stable ranks and contours provide robust descriptors of spatial patterns of atmospheric cloud fields.