Persistent Laplacian Diagrams

TL;DR

This paper introduces a new vectorization method for the Persistent Laplacian, creating Persistent Laplacian Diagrams and Images that capture richer geometric features and are stable under noise, enhancing topological data analysis.

Contribution

It develops a novel vectorization framework for Persistent Laplacian, including signatures, diagrams, and images, with proven stability and improved discrimination over existing methods.

Findings

Persistent Laplacian signatures distinguish graphs indistinguishable by PH.

The framework produces stable Persistent Laplacian Images under noise.

Experimental results demonstrate enhanced geometric feature capture.

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.