Paper Info
Title
On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces.
Abstract
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {em 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 {em 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.
Year
DOI
Venue
2019
10.4230/LIPIcs.SoCG.2019.21
symposium on computational geometry
Field
DocType
ISSN
Hilbert space,Topological data analysis,Mathematical optimization,Embedding,Pure mathematics,Separable space,Cardinality,Diagram,Kernel method,Mathematics,Bounded function
Conference
35th International Symposium on Computational Geometry (SoCG 2019), 21:1-15
Citations 
PageRank 
References 
0
0.34
0
Authors
2
Name
Order
Citations
PageRank
Mathieu Carrière1253.39
Ulrich Bauer210210.84