Injective and coarse embeddings of persistence diagrams and Wasserstein space

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Christopher Neil Pritchard (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Thomas Weighill

Abstract: In this dissertation we will examine questions related to two fields of mathematics, topological data analysis (TDA) and optimal transport (OT). Both of these fields center on complex data types to which one often needs to apply standard machine learning or statistical methods. Such application will typically mandate that these data types are embedded into a vector space. It has been shown that for many natural metrics such embeddings necessarily have high distortion, i.e. are not even coarse embeddings. Whether coarse embeddings exist with respect to the p-Wasserstein distance for 1 = p = 2 remains an open question, however, both for persistence diagrams (from TDA) and planar distributions (from OT). In this first part of this dissertation, we use coarse geometric techniques to show that the TDA and OT sides of this open question are equivalent for p > 1. In the second, we study an embedding of persistence diagrams, and show that under mild conditions it is injective, i.e. distinguishes between distinct diagrams.

Additional Information

Language: English
Date: 2023
Coarse Geometry, Embeddings, Persistence Diagrams
Embeddings (Mathematics)
Mathematical analysis
Metric spaces

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