On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces
This addresses a theoretical limitation in applying persistence diagrams to machine learning, showing that metric guarantees do not fully carry over, which is incremental for topological data analysis.
The paper investigates whether persistence diagrams can be embedded into separable Hilbert spaces with bi-Lipschitz maps, finding that for infinite-dimensional spaces, lower bounds depend on diagram cardinalities, and for finite-dimensional spaces, such embeddings are impossible even with bounded cardinalities.
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.