Nonembeddability of persistence diagrams with 𝑝>2 Wasserstein metric

Wagner, Alexander

doi:10.1090/proc/15451

Nonembeddability of persistence diagrams with $p>2$ Wasserstein metric
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Proc. Amer. Math. Soc. 149 (2021), 2673-2677 Request permission

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 that persistence diagrams with the $p$-Wasserstein metric do not admit a coarse embedding into a Hilbert space when $p > 2$.

References

Peter Bubenik and Alexander Wagner, Embeddings of persistence diagrams into Hilbert spaces, J. Appl. Comput. Topol. 4 (2020), no. 3, 339–351. MR 4130975, DOI 10.1007/s41468-020-00056-w
Mathieu Carrière and Ulrich Bauer, On the metric distortion of embedding persistence diagrams into separable Hilbert spaces, 35th International Symposium on Computational Geometry, LIPIcs. Leibniz Int. Proc. Inform., vol. 129, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019, pp. Art. No. 21, 15. MR 3968607
William B. Johnson and N. Lovasoa Randrianarivony, $l_p\ (p>2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1045–1050. MR 2196037, DOI 10.1090/S0002-9939-05-08415-7
Piotr W. Nowak, Coarse embeddings of metric spaces into Banach spaces, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2589–2596. MR 2146202, DOI 10.1090/S0002-9939-05-08150-5
Katharine Turner and Gard Spreemann, Same but different: Distance correlations between topological summaries, Topological Data Analysis, Abel Symposia, Vol. 15, Springer, 2020, DOI 10.1007/978-3-030-43408-3_18.