Coarse embeddability of Wasserstein space and the space of persistence diagrams

Abstract

We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e.~Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the $p$-Wasserstein distance for $1\leq p\leq 2$ remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams implies embeddability for Wasserstein space on $\mathbb{R}^2$, with the converse holding when $p > 1$. To prove this, we show that finite subsets of Wasserstein space uniformly coarsely embed into the space of persistence diagrams, and vice versa (when $p>1$).Comment: 11 pages, 1 figur