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Explore the metric properties of embedding Persistence Diagrams (PDs) into Reproducing Kernel Hilbert Spaces (RKHS) in this 55-minute lecture from the Applied Algebraic Topology Network. Delve into the challenges of using PDs in Machine Learning through kernel methods and understand the importance of maintaining stability guarantees when embedding PDs into RKHS. Examine the relationship between RKHS distance and diagram distances, and investigate the possibility of creating bi-Lipschitz maps for PD embeddings. Learn about the limitations of embedding PDs into both infinite-dimensional and finite-dimensional RKHS, including the dependence on PD cardinalities and the impossibility of finding bi-Lipschitz embeddings in finite-dimensional spaces, even with bounded cardinality restrictions.
