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Persistent Laplacian Diagrams

Inkee Jung, Wonwoo Kang, Heehyun Park

TL;DR

The paper develops a vectorization framework for the Persistent Laplacian by introducing signatures that yield a Persistent Laplacian Diagram (PLD) and a Persistent Laplacian Image (PLI). It proves stability of PLI under perturbations of persistence diagrams and demonstrates that PL-based signatures can distinguish graphs that PH cannot, including cospectral examples when using geometry-based signatures. The approach enables stable, fixed-dimensional embeddings suitable for integration with Graph Neural Networks, providing both global descriptors and local spectral encodings that capture higher-order topological and geometric information. This work bridges topological inference with geometric deep learning, expanding the toolkit for multiscale graph analysis and beyond.

Abstract

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.

Persistent Laplacian Diagrams

TL;DR

The paper develops a vectorization framework for the Persistent Laplacian by introducing signatures that yield a Persistent Laplacian Diagram (PLD) and a Persistent Laplacian Image (PLI). It proves stability of PLI under perturbations of persistence diagrams and demonstrates that PL-based signatures can distinguish graphs that PH cannot, including cospectral examples when using geometry-based signatures. The approach enables stable, fixed-dimensional embeddings suitable for integration with Graph Neural Networks, providing both global descriptors and local spectral encodings that capture higher-order topological and geometric information. This work bridges topological inference with geometric deep learning, expanding the toolkit for multiscale graph analysis and beyond.

Abstract

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.

Paper Structure

This paper contains 27 sections, 13 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistent Homology(PH) and Persistent Laplacian(PL)
  4. Persistence Diagrams and Persistence Image
  5. Filtrations
  6. Examples of filtrations
  7. Vietoris–Rips filtration
  8. Graph data
  9. Algorithms for the Persistent Laplacian
  10. Schur complement and Kron reduction
  11. Algorithmic construction
  12. Time complexity
  13. Signatures
  14. Definition and Properties
  15. Persistent Laplacian Diagram
  16. ...and 12 more sections

Key Result

Lemma 5

Let $U$ be the subset of extended plane $\overline{\mathbb{R}}^2$ such that $U = \{(x,y) \in \overline\mathbb{R}^2:y \geq x\}$. Let $B_1$ and $B_2$ be two multisets in $U$. $C_{B_{i}}$ denotes the induced set defined as where $I_i = \Pi_1 |B_i| \bigcup \Pi_2 |B_i|$ with the projections $\Pi_1, \Pi_2$ and the underlying set $|B_i|$ of $B_i$. Then $W_p(C_{B_1}, C_{B_2}) \leq 4n W_p(B_1, B_2)$, wher

Figures (4)

  • Figure 1: Non-isomorphic codegree pair of graphs $G_1$ and $G_2$. They are indistinguishable by PH in degree filtration.
  • Figure 2: Non-isomorphic codegree pair of graphs $G$ and $H$. For $k=3, 4, 5$, the graphs $G_k$ and $H_k$ denote the subgraphs obtained from $G$ and $H$, respectively, by degree filtration at level $k$. They coincide at $k=3$, but evolve differently for higher levels. The full graphs $G$ and $H$ share the same second smallest eigenvalue. Note that $H_1(G_4) = H_1(H_4) = 4$, and $H_1(G_5)=H_1(H_5)=8$.
  • Figure 3: Persistent Laplacian diagrams of $G$ and $H$ in Figure \ref{['fig:eigenval_dist2']} with signature being their second smallest eigenvalues
  • Figure 4: Shrikhande graph $G_S$ (left) and $4\times 4$ rook's graph $G_R$(right).

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Signatures
  • Definition 5: Persistent Laplacian surface and image
  • Remark
  • Remark
  • Lemma 5
  • Lemma 5
  • Theorem 6
  • ...and 16 more