On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

[2607.16957] On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2607.16957 (math) [Submitted on 18 Jul 2026 ( v1 ), last revised 12 Oct 2026 (this version, v4)] Title: On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams Authors: Charles Fanning , Mehmet Emin Aktas View PDF HTML (experimental) 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, there is no general framework for determining which persistence diagram vectorizations are synthesizable. Spectral synthesis asks whether the exponential monomials contained in the translation-invariant variety generated by a function span a dense subspace of that variety. We introduce uniformly discrete spectral synthesis for persistence diagram vectorizations using Fourier analysis on groups of virtual persistence diagrams. When the space of persistence intervals is separable, we prove sufficient conditions for uniformly discrete spectral synthesis and derive approximation results for virtual persistence diagrams with bounded total multiplicity. We then classify many persistence diagram vectorizations and persistence kernels from the literature by uniformly discrete spectral synthesis. Comments: 73 pages, 25 figures Subjects: Functional Analysis (math.FA) ; Algebraic Topology (math.AT) MSC classes: 43A45, 46E40, 55N31 Cite as: arXiv:2607.16957 [math.FA] (or arXiv:2607.16957v4 [math.FA] for this version) https://doi.org/10.48550/arXiv.2607.16957 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Charles Fanning [ view email ] [v1] Sat, 18 Jul 2026 20:22:28 UTC (111 KB) [v2] Sat, 1 Aug 2026 17:36:31 UTC (111 KB) [v3] Mon, 24 Aug 2026 23:47:27 UTC (83 KB) [v4] Mon, 12 Oct 2026 23:13:31 UTC (15,834 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-07 Change to browse by: math math.AT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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