[Paper Review] The algebraic stability for persistent Laplacians

Jian Liu, Jingyan Li+1 more|arXiv (Cornell University)|Feb 8, 2023

Topological and Geometric Data Analysis11 citations

TL;DR

The paper develops a categorical framework for persistent Laplacians, introduces Laplacian trees, proves an algebraic stability theorem for these trees, and applies the results to real-valued functions on simplicial complexes and digraphs.

ABSTRACT

The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of persistent Laplacians has emerged as a novel approach to topological persistence, attracting significant attention and finding applications in various fields. In this paper, we investigate the stability of persistent Laplacians. We introduce the notion of ``Laplacian trees'', which captures the collection of persistent Laplacians that persist from a given parameter. To formalize our study, we construct the category of Laplacian trees and establish an algebraic stability theorem for persistent Laplacian trees. Notably, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.

Motivation & Objective

• Motivate stability questions for topological persistence beyond standard persistence diagrams by focusing on persistent Laplacians.
• Introduce Laplacian trees as a structured, categorical collection of persistent Laplacians.
• Develop an algebraic stability theorem within the category of differential graded inner product spaces and their Laplacian trees.
• Show how the theory recovers and relates persistent homology and persistent harmonic spaces via functorial constructions.
• Provide concrete applications to real-valued functions on simplicial complexes and digraphs to illustrate stability results.

Proposed method

• Define the persistence objects in the category (R,≤) mapped to differential graded inner product spaces (DGI).
• Establish the persistent Laplacian Δ^{a,b}_{S} using the morphisms between DGIs and inclusions, and derive a persistent Hodge decomposition (Theorem 3.10).
• Construct Laplacian trees as a categorical object (V,A) encoding a family of Laplacians parameterized by morphisms, and study their properties.
• Prove an algebraic stability theorem showing ε-interleaving equivalence between persistence Laplacian trees corresponds to ε-interleaving of the underlying persistence DGIs (Theorem 1.3).
• Derive corollaries linking interleaving distance of Laplacian trees to that of the underlying persistence objects (Corollary 1.4).
• Apply the framework to real-valued functions on simplicial complexes and digraphs to illustrate stability (Theorems 1.5–1.6).

Experimental results

Research questions

• RQ1 How can persistent Laplacians be categorically organized to support stability analysis?
• RQ2 Can the interleaving framework used for persistence diagrams be extended to persistence Laplacian trees?
• RQ3 What is the precise relationship between persistent harmonic spaces and persistent homology within this algebraic setting?
• RQ4 How do stability bounds for persistence Laplacian trees translate to stability results for real-valued functions on simplicial complexes and digraphs?

Key findings

• There exists a natural isomorphism between persistent harmonic spaces and persistent homology as persistence modules (Theorem 1.1).
• The persistence Laplacian trees are ε-interleaved if and only if the underlying persistence DGIs are ε-interleaved (Theorem 1.3).
• The interleaving distance of Laplacian trees matches the interleaving distance of the underlying persistence objects (Corollary 1.4).
• For non-decreasing real-valued functions on a simplicial complex, the interleaving distance between their persistence Laplacian trees is bounded by the infinity-norm distance between the functions (Theorem 1.5).
• In the digraph setting, an algebraic stability theorem relates persistent harmonic spaces and persistent homology with respect to interleaving distance of Laplacians (Theorem 1.6).
• Corollaries provide explicit bounds in terms of function differences and infinity-norm for digraphs (Corollary 1.7).