This is an implementation of the notion Persistent Laplacian first invented by André Lieutier in https://project.inria.fr/gudhi/files/2014/10/Persistent-Harmonic-Forms.pdf and later rediscovered by Rui Wang, Duc Duy Nguyen, Guo-Wei Wei in the paper https://arxiv.org/abs/1912.04135.
The algorithm implemented in this repo was developed by Facundo Memoli, Zhengchao Wan and Yusu Wang in the paper https://arxiv.org/abs/2012.02808. Our algorithm crucially exploits the notion of Kron reduction in network theory (Dorfler and Bullo, 2013).
persistLap.m is a Matlab implementation of the q-th persistent Laplacian of a simplicial pair via the Schur complement formulation.
Syntax:
% persistLap - computes the q-th (up) persistent Laplacian of a simplicial pair K in L
%
% [pL,upL] = persistLap(B1,B2,Gind)
%
% B1 - the (q+1)-th boundary matrix of the large complex L
% B2 - the q-th boundary matrix of the small complex K
% Gind - index set of q-simplices of K in the set of q-simplices of L
%
% Returns:
% pL - q-th persistent Laplacian
% upL - q-th up persistent Laplacian
boundary.py is a python code based on dionysus generating boundary matrices required by persistLap.m. It takes as input a point cloud and extracts two Vietoris-Rips complexes determined by two given thresholds as a simplicial pair K in L. The output consists of q1Boundary.mat: a (q+1)-boundary matrix of L, and qBoundary.mat: a q-boundary matrix of K, where q is the dimension specified by the user.
Inside the example folder, we include a point cloud file Point.txt, which represents a 35-point point cloud randomly generated from a circle illustrated in the figure below. We include two precomputed boundary matrices q1Boundary.mat and qBoundary.mat when q=2 for a Vietoris-Rips simplicial pair K in L with thresholds 1 and 1.5. VR-persistLap takes q1Boundary.mat and qBoundary.mat as input and compute the 2-persistent Laplacian and the corresponding eigenvalues.
