On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams

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.

Persistence diagrams of finite topological spaces
Akhilesh Badra, Hemant Kumar Singh
arXiv12 Oct 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv
A result of continuous stability for generalized persistence diagrams
Yanga Bavuma, Martin Mijangos, Francesco G. Russo
arXiv1 Nov 2026PreprintCC BY 4.0Full textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Higher-order Persistence Diagrams
Charles Fanning, Mehmet Aktas
arXiv25 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEs Geometry & topologyarXiv
Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation
Pramita Bagchi, Sushovan Majhi, Atish Mitra +1
arXiv7 Sep 2026PreprintarXiv licenceFull textStatisticsarXiv
Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget
Pramita Bagchi, Edward Bae, Atish Mitra +3
arXiv6 Oct 2026PreprintarXiv licenceFull textStatisticsarXiv

Cite

BibTeX RIS

BibTeX

@misc{fanning2026spectral,
  title = {{On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams}},
  author = {Charles Fanning and Mehmet Emin Aktas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.16957},
  eprint = {2607.16957},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.16957},
}

RIS

TY  - UNPB
TI  - On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams
AU  - Charles Fanning
AU  - Mehmet Emin Aktas
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2607.16957
UR  - https://arxiv.org/abs/2607.16957
AB  - 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.
ER  -

Abstract

Related

Cite