The Convex Matching Distance in Multiparameter Persistence

TL;DR

This paper introduces the convex matching distance, a new, computationally efficient metric for comparing functions with two real components, which can be more discriminative and stable than traditional methods, validated on various datasets.

Contribution

The paper proposes the convex matching distance, a novel metric for multiparameter persistence that improves computational efficiency and discriminative power over traditional matching distances.

Findings

The convex matching distance is more discriminative in certain cases.

It is stable under perturbations of the functions.

Experimental results show it is faster and reliable on diverse datasets.

Abstract

We introduce the convex matching distance, a novel metric for comparing functions with values in the real plane. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations of the two function components. Similarly to the traditional matching distance, the convex matching distance aggregates the information provided by two real-valued components. However, whereas the matching distance depends on two parameters, the convex matching distance depends on only one, offering improved computational efficiency. We further show that the convex matching distance can be more discriminative than the traditional matching distance in certain cases, although the two metrics are generally not comparable. Moreover, we prove that the convex matching distance is stable and characterize the coefficients of the convex combination at which it is attained. Finally, we demonstrate that this new aggregation framework benefits from the computational advantages provided by the Pareto grid, a collection of curves in the plane whose points lie in the image of the Pareto critical set associated with functions assuming values on the real plane. Experimental validation on MNIST digits, synthetic shapes, and chaotic attractors suggests that the convex matching distance provides a reliable and efficient alternative to the matching distance, at a significantly lower computational cost.