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

Abstract

A. 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. 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 α with α ≥ d.

Functional central limit theorems for persistent Betti numbers on cylindrical networks

“…First, we recall the definition of the filtration time as the first time r > 0 where a simplex is contained in the complex . In (Divol & Polonik, 2020, Lemma 6.10) it is shown that both in the Č‐ and the VR‐filtration, the filtration times of iid uniform points in a ball admit a bounded and continuous density on , which is bounded above by some c 1, d > 0.…”

Topology Applied to Machine Learning: From Global to Local

“…Another approach to hypothesis testing is given by Robinson and Turner (2017). Other statistical approaches include Fasy et al (2014), which describes confidence intervals and a statistical approach to distinguishing important features from noise, Divol and Polonik (2019) and Maroulas et al (2019), which consider probability density functions for persistence diagrams, and Maroulas et al (2020), which describes a Bayesian framework. See Wasserman (2018) for a review of statistical techniques in the context of topological data analysis.…”

“…Other techniques for vectorizing persistence barcodes involve heat kernels (Carrière et al, 2015), entropy (Merelli et al, 2015;Atienza et al, 2020), rings of algebraic functions (Adcock et al, 2016), tropical coordinates (Kališnik, 2019), complex polynomials (Di Fabio andFerri, 2015), and optimal transport (Carrière et al, 2017), among others. Some of these techniques, including those by Zhao and Wang (2019) and Divol and Polonik (2019), allow one to learn the vectorization parameters that are best suited for a machine learning task on a given dataset. Others allow one to plug persistent homology information directly into a neural network (Hofer et al, 2017).…”

Topological Distances Between Brain Networks

“…This loss of information can be detrimental to the analysis of the given dataset as important features may be over smoothed or unintentionally grouped together. One method of combatting this potential loss of information is to utilize weighting function w(ξ, τ) on the PI representation to emphasize certain features (Divol and Polonik, 2019;Adams et al, 2017). A popular method used is to assign linear or exponential weight such as w(ξ, τ) = (τ − ξ) for each point in the PD that increases in correlation with the distance from the diagonal of the persistence diagram before diffusion via the Gaussian kernel.…”