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.
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.
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feature
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Machine Learning
machine learning
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(RKHS)
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a kernel k
the feature map
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RKHS distance d_k
Hilbert distance
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RKHS
separable Hilbert spaces,
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when the RKHS is infinite dimensional
for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature
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RKHS
Hilbert space