DeepAI AI Chat
Log In Sign Up

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

by   Mathieu Carrière, et al.
Technische Universität München
Columbia University

Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their diagram distances, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of persistence diagrams into Reproducing Kernel Hilbert Spaces, in which all computations can be performed easily. Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the metric properties of the feature map, i.e., the relationship between the Hilbert distance and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding. In this article, we study the possibility of embedding persistence diagrams into separable Hilbert spaces, with bi-Lipschitz maps. In particular, we show that for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature, any lower bound must depend on the cardinalities of the persistence diagrams, and that when the Hilbert space is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.


page 1

page 2

page 3

page 4


On the Metric Distortion of Embedding Persistence Diagrams into Reproducing Kernel Hilbert Spaces

Persistence diagrams are important feature descriptors in Topological Da...

Embeddings of Persistence Diagrams into Hilbert Spaces

Since persistence diagrams do not admit an inner product structure, a ma...

Geometry and clustering with metrics derived from separable Bregman divergences

Separable Bregman divergences induce Riemannian metric spaces that are i...

A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering

Persistence diagrams have been widely used to quantify the underlying fe...

Edit Distance and Persistence Diagrams Over Lattices

We build a functorial pipeline for persistent homology. The input to thi...

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

Persistence diagrams are efficient descriptors of the topology of a poin...

Learning Hyperbolic Representations of Topological Features

Learning task-specific representations of persistence diagrams is an imp...