Skip to Main content Skip to Navigation
Journal articles

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

Vincent Divol 1 Wolfgang Polonik 2
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
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 X_n is an i.i.d. sample from a density on [0,1]^d. For the Čech 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. Both approaches 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 α with α \geq d.
Complete list of metadata

Cited literature [40 references]  Display  Hide  Download
Contributor : Vincent Divol Connect in order to contact the contributor
Submitted on : Wednesday, November 11, 2020 - 1:37:07 PM
Last modification on : Wednesday, November 3, 2021 - 10:01:33 AM


Files produced by the author(s)



Vincent Divol, Wolfgang Polonik. On the choice of weight functions for linear representations of persistence diagrams. Journal of Applied and Computational Topology, Springer, 2019, 3 (3), pp.249-283. ⟨10.1007/s41468-019-00032-z⟩. ⟨hal-01833660v3⟩



Les métriques sont temporairement indisponibles