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Local Laplacian: theory and models for data analysis

Jian Liu, Hongsong Feng, Kefeng Liu

TL;DR

The persistent local Laplacian formalism is introduced, which is designed to extract fine-grained local topological and geometric signatures while enabling a highly efficient, parallelizable computational workflow.

Abstract

While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations and suffer from prohibitive computational costs on large-scale datasets. To bridge this gap, we introduce the persistent local Laplacian formalism, which is designed to extract fine-grained local topological and geometric signatures while enabling a highly efficient, parallelizable computational workflow. On the theoretical front, we prove a generalized persistent Hodge isomorphism, establishing that the harmonic space of the persistent local Laplacian is isomorphic to the persistent local homology. Furthermore, we derive a unitary equivalence between the persistent local Laplacian and the persistent Laplacian of its corresponding link complex at a shifted dimension. This spectral conjugacy establishes the mathematical foundation for developing efficient computational schemes to resolve persistent local spectral invariants. We further extend this construction to point clouds and graph-structured data, characterizing their persistent local spectral properties through combinatorial filtrations. The resulting architecture is inherently decoupled, facilitating massive parallelization and rendering it uniquely scalable for large-scale network analysis and distributed computational environments.

Local Laplacian: theory and models for data analysis

TL;DR

The persistent local Laplacian formalism is introduced, which is designed to extract fine-grained local topological and geometric signatures while enabling a highly efficient, parallelizable computational workflow.

Abstract

While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations and suffer from prohibitive computational costs on large-scale datasets. To bridge this gap, we introduce the persistent local Laplacian formalism, which is designed to extract fine-grained local topological and geometric signatures while enabling a highly efficient, parallelizable computational workflow. On the theoretical front, we prove a generalized persistent Hodge isomorphism, establishing that the harmonic space of the persistent local Laplacian is isomorphic to the persistent local homology. Furthermore, we derive a unitary equivalence between the persistent local Laplacian and the persistent Laplacian of its corresponding link complex at a shifted dimension. This spectral conjugacy establishes the mathematical foundation for developing efficient computational schemes to resolve persistent local spectral invariants. We further extend this construction to point clouds and graph-structured data, characterizing their persistent local spectral properties through combinatorial filtrations. The resulting architecture is inherently decoupled, facilitating massive parallelization and rendering it uniquely scalable for large-scale network analysis and distributed computational environments.
Paper Structure (44 sections, 25 theorems, 161 equations, 6 figures)

This paper contains 44 sections, 25 theorems, 161 equations, 6 figures.

Table of Contents

  1. Abstract
  2. Keywords
  3. Introduction
  4. Local Laplacians on simplicial complexes
  5. Local homology of simplicial complexes
  6. Relative homology
  7. Local homology
  8. Closed star, link, and open star
  9. Computing local homology
  10. Local homology and interaction homology
  11. Local Laplacian
  12. Inner product structure and Hilbert space
  13. Combinatorial Laplacian
  14. Inner product on the relative chain complex
  15. The relative combinatorial Laplacian
  16. ...and 29 more sections

Key Result

Theorem 2.5

For $K=L$, the interaction Betti numbers $\beta_{n} = \mathrm{rank}(H_n(K, K))$ provide a categorical refinement of the Wu characteristic, yielding the identity

Figures (6)

  • Figure 1: Schematic of the Laplacian family: from classical graph Laplacian to persistent local Laplacian.
  • Figure 2: Construction of the link complex from a simplicial complex $K$.
  • Figure 3: The local structure of a $3$-simplex at vertex $0$. The link $\mathrm{Lk}_K(0)$ is a $2$-simplex (triangle), whose combinatorial Laplacian $\Delta_0^{\mathrm{Lk}_K(0)}$ defines the local Laplacian $\Delta_1^{K,0}$ of the original complex at vertex $0$.
  • Figure 4: Illustration of the simplicial map $\phi\colon K \to L$.
  • Figure 5: Relationship between a graph and its local persistent structure. (b) The clique complex $\mathop{\mathrm{Clq}}\nolimits(G)$ highlights connected substructures in blue. (c) The link complex focuses on the relational structure between neighbors by removing vertex $v$.
  • ...and 1 more figures

Theorems & Definitions (77)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: knill2018cohomologyliu2023interaction
  • Theorem 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8: Simplicial Hodge theorem
  • Definition 2.9
  • ...and 67 more