The Convex Matching Distance in Multiparameter Persistence

TL;DR

This paper tackles the challenge of comparing vector-valued functions in multiparameter persistence by introducing the convex matching distance cmd_k, a single-parameter, smooth aggregation of component filtrations. cmd_k computes the maximal bottleneck distance between persistence diagrams across convex combinations of the two components, reducing parameter complexity and enabling efficient Pareto-grid based computation. It establishes stability, a Position Theorem linking diagram coordinates to Pareto contours, and a main result showing the maximum is attained at a special set of t-values, with practical computation via the Pareto grid and GENEOs. The work offers both theoretical insight and practical benefits for TDA applications involving two real-valued coordinates.

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.

Figures (3)

• Figure 1: The sets $K$ and $C$ cited in the Examples \ref{['exKC1']}, \ref{['exKC2']}.
• Figure 2: The Pareto grid of the function $\boldsymbol{{\varphi}}:\mathbb S^2\to{\mathbb R}^2$ defined by setting $\boldsymbol{{\varphi}}(x,y,z):=(x,z)$.
• Figure 3: Osculating circles $\hat{\boldsymbol{\alpha}}$ and $\hat{\boldsymbol{\alpha}'}$ of centres $(x,y),(x',y')$ and radii $\rho,\rho'$ to a negative and positive parabola denoted by $\boldsymbol{\alpha}$ and $\boldsymbol{\alpha}'$, respectively, at a non-critical point. The signed radius $\ell$ from Definition \ref{['def:signed_radius']} is positive on the leftmost case and negative on the rightmost one.

Theorems & Definitions (26)

• Remark 1
• Definition 1
• Remark 2
• Remark 3
• Remark 4
• Proposition 1
• proof
• Remark 5
• Proposition 2: Stability of $\mathop{\mathrm{\mathbf{cmd}}}\nolimits_k$
• proof