On the choice of weight functions for linear representations of persistence diagrams

TL;DR

This paper investigates how to select weight functions for linear feature maps of persistence diagrams, ensuring stability and convergence, and provides practical heuristics for tuning based on data dimensionality.

Contribution

It extends stability results to general weight functions and characterizes conditions for convergence of feature maps in an asymptotic setting.

Findings

Stability of linear feature maps is improved and extended to general weight functions.

Conditions for convergence of feature maps are characterized for data sampled from a density.

A heuristic for choosing weight functions based on data manifold dimension is proposed.

Abstract

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations) are commonly used for the analysis. A large class of feature maps, which we call linear, depends on some weight functions, the choice of which is a critical issue. An important criterion to choose a weight function is to ensure stability of the feature maps with respect to Wasserstein distances on diagrams. We improve known results on the stability of such maps, and extend it to general weight functions. We also address the choice of the weight function by considering an asymptotic setting; assume that $\mathbb{X}_n$ is an i.i.d. sample from a density on $[0,1]^d$. For the \v{C}ech and Rips filtrations, we characterize the weight functions for which the corresponding feature maps converge as $n$ approaches infinity, and by doing so, we prove laws of large numbers for the total persistences of such diagrams. Those two approaches (stability and convergence) lead to the same simple heuristic for tuning weight functions: if the data lies near a $d$-dimensional manifold, then a sensible choice of weight function is the persistence to the power $\alpha$ with $\alpha \geq d$.