Through the Grapevine: Vineyard Distance as a Measure of Topological Dissimilarity

TL;DR

This paper introduces vineyard distance, a new topological data analysis measure that captures dataset dissimilarity by quantifying topological changes along data interpolations, offering a balance between existing distance metrics.

Contribution

The paper presents vineyard distance, a novel topological measure that is less sensitive than $L^p$ but more sensitive than Wasserstein, with theoretical bounds and real-world applications.

Findings

Vineyard distance effectively distinguishes datasets with subtle topological differences.

It provides theoretical bounds relating to existing metrics.

Demonstrated usefulness in geospatial and neural network data analysis.

Abstract

We introduce a new measure of distance between datasets, based on vineyards from topological data analysis, which we call the vineyard distance. Vineyard distance measures the extent of topological change along an interpolation from one dataset to another, either along a pre-computed trajectory or via a straight-line homotopy. We demonstrate through theoretical results and experiments that vineyard distance is less sensitive than $L^p$ distance (which considers every single data value), but more sensitive than Wasserstein distance between persistence diagrams (which accounts only for shape and not location). This allows vineyard distance to reveal distinctions that the other two distance measures cannot. In our paper, we establish theoretical results for vineyard distance including as upper and lower bounds. We then demonstrate the usefulness of vineyard distance on real-world data through applications to geospatial data and to neural network training dynamics.