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

Persistence diagrams are important feature 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 (RKHS), in which all computations can be performed easily. Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the {\em metric properties} of a kernel $k$, i.e., the relationship between the RKHS distance $d_k$ 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 RKHS with bi-Lipschitz maps. In particular, we show that when the RKHS is infinite dimensional, any lower bound must depend on the cardinalities of the persistence diagrams, and that when the RKHS is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.