Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Persistent Laplacians and their Spectral Properties

Arne Wolf, Jiyu Fan, Anthea Monod

TL;DR

The paper develops a unifying operator-theoretic framework for generalized Laplacians that covers both discrete combinatorial settings and de Rham complexes on manifolds. It introduces generalized persistent chain Laplacians built from Hilbert complexes and proves that the up- and down-persistent components are monotone and stable, and that their spectra determine the full persistent Laplacian spectrum. The results extend to finite and infinite-dimensional settings and provide a sufficient condition for full monotonicity, enabling robust spectral invariants beyond persistent homology. This work lays a foundational, spectral-theoretic perspective on Laplacian-based invariants with potential broad impact across topology, geometry, and data analysis.

Abstract

Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in modern computational and data science fields such as applied and computational topology and application areas such as machine learning and network science. In this paper, we introduce a unifying operator-theoretic framework of generalized Laplacians as invariants that encompasses and extends all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, we introduce and study a generalized notion of persistent Laplacians. While the classical persistent Laplacian fails to satisfy the desirable properties of monotonicity and stability -- both crucial for robustness and interpretability -- our framework allows us to isolate and analyze these properties systematically. We demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, we prove that the spectra of these separate components fully determine the spectra of the full Laplacians, making them not only preferable but sufficient for analysis. We study these questions comprehensively, in both the finite and infinite dimensional settings. Our work expands and strengthens the theoretical foundation of generalized Laplacian-based methods in pure, applied, and computational mathematics.

Generalized Persistent Laplacians and their Spectral Properties

TL;DR

The paper develops a unifying operator-theoretic framework for generalized Laplacians that covers both discrete combinatorial settings and de Rham complexes on manifolds. It introduces generalized persistent chain Laplacians built from Hilbert complexes and proves that the up- and down-persistent components are monotone and stable, and that their spectra determine the full persistent Laplacian spectrum. The results extend to finite and infinite-dimensional settings and provide a sufficient condition for full monotonicity, enabling robust spectral invariants beyond persistent homology. This work lays a foundational, spectral-theoretic perspective on Laplacian-based invariants with potential broad impact across topology, geometry, and data analysis.

Abstract

Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in modern computational and data science fields such as applied and computational topology and application areas such as machine learning and network science. In this paper, we introduce a unifying operator-theoretic framework of generalized Laplacians as invariants that encompasses and extends all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, we introduce and study a generalized notion of persistent Laplacians. While the classical persistent Laplacian fails to satisfy the desirable properties of monotonicity and stability -- both crucial for robustness and interpretability -- our framework allows us to isolate and analyze these properties systematically. We demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, we prove that the spectra of these separate components fully determine the spectra of the full Laplacians, making them not only preferable but sufficient for analysis. We study these questions comprehensively, in both the finite and infinite dimensional settings. Our work expands and strengthens the theoretical foundation of generalized Laplacian-based methods in pure, applied, and computational mathematics.

Paper Structure

This paper contains 31 sections, 40 theorems, 194 equations, 19 figures.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Contributions.
  4. Outline.
  5. Related Work
  6. Background and Preliminaries
  7. Simplicial Homology and The Combinatorial Laplacian
  8. The Persistent Combinatorial Laplacian
  9. Operator Theory: Main Tools and Supporting Results
  10. Adjoint and Closed Operators
  11. Self-Adjoint Operators and Quadratic Forms
  12. Spectra of Self-Adjoint Operators
  13. The Polar Decomposition of the Operators $T^*T$ and $TT^*$
  14. The Chain Laplacians
  15. Hilbert Complexes
  16. ...and 16 more sections

Key Result

Theorem 2.3

lim$\ker(\Delta^K_k) \cong H_k(K)$ and the isomorphism is given by $v \mapsto [v]$.

Figures (19)

  • Figure 1: Examples of a Čech complex (left) and a Vietoris--Rips complex (right) for the same $\varepsilon$sctypes.
  • Figure 2: Setting in which the persistent Laplacian is defined.
  • Figure 3: Above: The simplicial complexes studied in \ref{['ex:algperslap']}, $\textcolor{blue}{K} \subset L$. Below: The $k\in \{0,1,2\}$ component of the diagram from \ref{['fig:perslapsc']} for the chosen purple bases.
  • Figure 4: The chain Laplacian. The colored parts represent the respective up and down chain Laplacian: $\textcolor{red}{\Delta^{P}_{+, k}}$ and $\textcolor{blue}{\Delta^{P}_{-, k}}$.
  • Figure 5: Defining the generalized persistent chain Laplacian.
  • ...and 14 more figures

Theorems & Definitions (129)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7: Persistent Combinatorial Hodge Theorem, mww
  • Definition 3.1: Adjoint Operator, schroedingerop
  • Proposition 3.2: Closed Operators, schroedingerop
  • proof
  • ...and 119 more