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Persistence diagrams do not admit an inner product structure compatible withany Wasserstein metric. Hence, when applying kernel methods to persistencediagrams, the underlying feature map necessarily causes distortion. We provepersistence diagrams with the p-Wasserstein metric do not admit a coarseembedding into a Hilbert space when p > 2.

Nonembeddability of Persistence Diagrams with $p>2$ Wasserstein Metric