DeepAI AI Chat
Log In Sign Up

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

by   Divol Vincent, et al.
University of California-Davis

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≥ d.


page 1

page 2

page 3

page 4


Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric

Persistence diagrams do not admit an inner product structure compatible ...

Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport

We consider a generalization of persistence diagrams, namely Radon measu...

Asynchronously Trained Distributed Topographic Maps

Topographic feature maps are low dimensional representations of data, th...

On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces

Persistence diagrams are important descriptors in Topological Data Analy...

Fuzzy c-Means Clustering for Persistence Diagrams

Persistence diagrams, a key tool in the field of Topological Data Analys...

Estimation and Quantization of Expected Persistence Diagrams

Persistence diagrams (PDs) are the most common descriptors used to encod...

Steady and ranging sets in graph persistence

Generalised persistence functions (gp-functions) are defined on (ℝ, ≤)-i...