Embeddings of Persistence Diagrams into Hilbert Spaces

05/11/2019

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by   Peter Bubenik, et al.

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Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence diagrams with the bottleneck distance do not admit a coarse embedding into a Hilbert space. As part of our proof, we show that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance. As corollaries, we also calculate the generalized roundness, negative type, and asymptotic dimension of this space.