On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

Abstract

Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector spaces or compared through kernels. Although many persistence diagram vectorizations and persistence kernels have been proposed, existing methods are typically developed as individual constructions, and existing work does not provide a canonical persistence diagram vectorization from which many existing vectorizations and kernels can be derived. In this paper, we develop such a canonical persistence diagram vectorization for certain classes of persistence diagrams using Fourier analysis on groups of virtual persistence diagrams. Spectral synthesis asks whether other persistence diagram vectorizations can be represented through this canonical vectorization. We prove this result for a class of Lipschitz persistence diagram vectorizations. We extend this result from uniformly discrete metric pairs to separable metric pairs.