Persistent Laplacian Diagrams
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915656292630528 |
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| author | Jung, Inkee Kang, Wonwoo Park, Heehyun |
| author_facet | Jung, Inkee Kang, Wonwoo Park, Heehyun |
| contents | Vectorization methods for \emph{Persistent Homology} (PH), such as the \emph{Persistence Image} (PI), encode persistence diagrams into finite dimensional vector spaces while preserving stability. In parallel, the \emph{Persistent Laplacian} (PL) has been proposed, whose spectra contain the information of PH as well as richer geometric and combinatorial features. In this work, we develop an analogous vectorization for PL. We introduce \emph{signatures} that map PL to real values and assemble these into a \emph{Persistent Laplacian Diagram} (PLD) and a \emph{Persistent Laplacian Image} (PLI). We prove the stability of PLI under the noise on PD. Furthermore, we illustrate the resulting framework on explicit graph examples that are indistinguishable by both PH and a signature of the combinatorial Laplacian but are separated by the signature of PL. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05463 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Persistent Laplacian Diagrams Jung, Inkee Kang, Wonwoo Park, Heehyun Algebraic Topology Computational Geometry 55N31, 05C62, 68T09, 62R40 G.2.2 Vectorization methods for \emph{Persistent Homology} (PH), such as the \emph{Persistence Image} (PI), encode persistence diagrams into finite dimensional vector spaces while preserving stability. In parallel, the \emph{Persistent Laplacian} (PL) has been proposed, whose spectra contain the information of PH as well as richer geometric and combinatorial features. In this work, we develop an analogous vectorization for PL. We introduce \emph{signatures} that map PL to real values and assemble these into a \emph{Persistent Laplacian Diagram} (PLD) and a \emph{Persistent Laplacian Image} (PLI). We prove the stability of PLI under the noise on PD. Furthermore, we illustrate the resulting framework on explicit graph examples that are indistinguishable by both PH and a signature of the combinatorial Laplacian but are separated by the signature of PL. |
| title | Persistent Laplacian Diagrams |
| topic | Algebraic Topology Computational Geometry 55N31, 05C62, 68T09, 62R40 G.2.2 |
| url | https://arxiv.org/abs/2512.05463 |