A Coarse Embedding of n-Point Persistence Diagram Spaces into Hilbert Spaces

Husham Sabhan Khalaf

Authors

Husham Sabhan Khalaf College of Education for Pure Sciences, University of Maragheh, Maragheh, Iran

Keywords:

Persistence diagrams, Coarse embedding, Hilbert spaces, Asymptotic dimension

Abstract

We establish that the metric space of persistence diagrams consisting of exactly n points admits a coarse embedding into a Hilbert space when equipped with either the bottleneck distance or a Wasserstein distance. This result is obtained by demonstrating that such spaces possess asymptotic dimension equal to 2n. As a consequence, a broad range of analytical and geometric tools native to Hilbert spaces become applicable to the study of persistence diagrams with bounded cardinality. In contrast, we show that if the number of points in persistence diagrams is allowed to grow without bound, the associated metric spaces fail to have finite asymptotic dimension. Moreover, for the bottleneck metric, we prove that the corresponding unbounded persistence diagram space does not admit a coarse embedding into any Hilbert space.

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