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PETLS: PErsistent Topological Laplacian Software

Benjamin Jones, Guo-Wei Wei

TL;DR

This work implements existing and new persistent Laplacian algorithms in an efficient and flexible C++ library with Python bindings, titled PETLS: PErsistent Topological Laplacian Software and provides recommendations on how to use algorithms and complexes for data analysis in machine learning.

Abstract

Persistent topological Laplacians are operators that provide persistent Betti numbers and additional multiscale geometric information through the eigenvalues of the persistent topological Laplacian matrix. We introduce a framework and novel algorithm to aid in the computation of persistent topological Laplacians. We implement existing and new persistent Laplacian algorithms in an efficient and flexible C++ library with Python bindings, titled PETLS: PErsistent Topological Laplacian Software. As part of this library, we interface with several complexes commonly used in topological data analysis (TDA), such as simplicial, alpha, directed flag, Dowker, and cellular Sheaf. Because increased efficiency broadens the set of computationally feasible applications, we provide recommendations on how to use algorithms and complexes for data analysis in machine learning.

PETLS: PErsistent Topological Laplacian Software

TL;DR

This work implements existing and new persistent Laplacian algorithms in an efficient and flexible C++ library with Python bindings, titled PETLS: PErsistent Topological Laplacian Software and provides recommendations on how to use algorithms and complexes for data analysis in machine learning.

Abstract

Persistent topological Laplacians are operators that provide persistent Betti numbers and additional multiscale geometric information through the eigenvalues of the persistent topological Laplacian matrix. We introduce a framework and novel algorithm to aid in the computation of persistent topological Laplacians. We implement existing and new persistent Laplacian algorithms in an efficient and flexible C++ library with Python bindings, titled PETLS: PErsistent Topological Laplacian Software. As part of this library, we interface with several complexes commonly used in topological data analysis (TDA), such as simplicial, alpha, directed flag, Dowker, and cellular Sheaf. Because increased efficiency broadens the set of computationally feasible applications, we provide recommendations on how to use algorithms and complexes for data analysis in machine learning.

Paper Structure

This paper contains 26 sections, 5 theorems, 20 equations, 15 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Abstract
  2. Keywords
  3. Introduction
  4. Background
  5. Combinatorial Laplacian
  6. Persistent Laplacian
  7. Computation of persistent up-Laplacian
  8. Software design
  9. Overview
  10. Persistent sheaf Laplacian
  11. Computation
  12. The default pipeline
  13. Comparison with other methods
  14. Eigenvalue algorithms
  15. Top-dimensional Laplacian eigenvalues
  16. ...and 11 more sections

Key Result

Theorem 1

For a simplicial complex $K$, $\ker\Delta_n \cong H_n(K;\mathbb{R})$.

Figures (15)

  • Figure 1: The process for creating and using persistent topological Laplacians (PTLs). Pictured is an alpha filtration of a torus with maximum PTL dimension $N=2$, with summary function $\Theta$ as the maximum eigenvalue function.
  • Figure 2: Simplices and a simplicial complex
  • Figure 3: Simplicial complex for Example \ref{['ex:homology']} and Example \ref{['ex:combinatorial']}.
  • Figure 4: A filtration step of simplicial complexes where restricting $d_n^b$ to $C_n^{a,b}$ is not as simple as removing rows or columns.
  • Figure 5: The least nonzero eigenvalues of persistent Laplacian matrices of a Rips complex from $30$ points sampled from the unit sphere. Values are shown from one individual point sample and are not averaged.
  • ...and 10 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Definition 5
  • Theorem 1
  • Theorem 2
  • Example 2
  • Definition 6
  • ...and 7 more