Nonembeddability of Persistence Diagrams with $p>2$ Wasserstein Metric
This addresses a theoretical limitation for researchers in topological data analysis, specifically those using kernel methods with persistence diagrams, and is incremental as it extends prior nonembeddability results to p > 2.
The paper tackled the problem of applying kernel methods to persistence diagrams by proving that persistence diagrams with the p-Wasserstein metric do not admit a coarse embedding into a Hilbert space when p > 2, showing inherent distortion in such embeddings.
Persistence diagrams do not admit an inner product structure compatible with any Wasserstein metric. Hence, when applying kernel methods to persistence diagrams, the underlying feature map necessarily causes distortion. We prove persistence diagrams with the p-Wasserstein metric do not admit a coarse embedding into a Hilbert space when p > 2.