Convergence Rate of the Symmetrically Normalized Graph Laplacian

TL;DR

This paper proves that the symmetrically normalized graph Laplacian converges to the continuous manifold Laplacian at a rate of O(1/N) as the sample size increases, providing theoretical guarantees for graph-based manifold learning.

Contribution

It provides a rigorous proof of the convergence rate of the symmetrically normalized graph Laplacian to the manifold Laplacian as sampling density increases.

Findings

Convergence rate of O(1/N) established

The normalized graph Laplacian approximates the manifold Laplacian

Theoretical validation for graph-based manifold learning methods

Abstract

This short note aims at (re)proving that the symmetrically normalized graph Laplacian $L=\Id - D^{-1/2}WD^{-1/2}$ (from a graph defined from a Gaussian weighting kernel on a sampled smooth manifold) converges towards the continuous Manifold Laplacian when the sampling become infinitely dense. The convergence rate with respect to the number of samples $N$ is $O(1/N)$.