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

TL;DR

This paper proves that persistence diagrams with the p-Wasserstein metric for p > 2 cannot be embedded into Hilbert spaces, highlighting limitations for kernel methods in topological data analysis.

Contribution

It establishes the nonembeddability of persistence diagrams with p > 2 Wasserstein metric into Hilbert spaces, revealing fundamental geometric constraints.

Findings

Persistence diagrams with p > 2 cannot be coarsely embedded into Hilbert spaces.

Kernel methods based on these diagrams may introduce distortions.

The result clarifies limitations of applying kernel techniques to certain topological summaries.

Abstract

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.