Local Laplacian: theory and models for data analysis

TL;DR

This paper introduces the persistent local Laplacian, a new tool for analyzing local topological features in data that is computationally efficient and scalable, with theoretical foundations linking it to local homology.

Contribution

The paper develops the persistent local Laplacian formalism, proves a generalized persistent Hodge isomorphism, and extends the approach to point clouds and graphs for scalable data analysis.

Findings

Establishes a theoretical link between harmonic space and local homology.

Provides a spectral conjugacy between local and global Laplacians.

Demonstrates scalability for large-scale network analysis.

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.