| Abstract: |
| Persistence diagrams may be thought of as countable multisets of points in a metric space $X$, considered relative to a distinguished nonempty closed subset $A\subset X$. Their comparison is usually based on the bottleneck distance or on $p$-Wasserstein distances. In this talk, I will present a common framework in which these distances arise as instances of a more general construction involving a normalized permutation-invariant Banach sequence ideal $E$, with the norm of $E$ used as the cost function in the corresponding optimal transport problem between persistence diagrams. This yields a family of spaces of persistence diagrams parametrized by the Banach sequence ideal $E$, extending earlier definitions of diagram metrics. I will explain natural assumptions on $E$ under which the resulting matching formula defines a metric and the associated spaces of persistence diagrams are complete, separable, and geodesic. In this way, the framework enlarges the class of metrics available for applications while preserving the main structural properties of the classical diagram spaces. The bottleneck and $p$-Wasserstein distances are recovered as the special cases $E=\ell^\infty$ and $E=\ell^p$. |
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