Spectral Convergence Rate of Graph Laplacian

TL;DR

This paper establishes the spectral convergence rate of the graph Laplacian constructed from data sampled on a manifold, ensuring the consistency of spectral clustering algorithms and highlighting the importance of denoising.

Contribution

It provides the first rigorous proof of the spectral convergence rate for graph Laplacians on manifolds, connecting spectral convergence with clustering consistency.

Findings

Spectral convergence rate is established for graph Laplacians on manifolds.

Consistency of spectral clustering is demonstrated via perturbation analysis.

Numerical results suggest denoising improves spectral algorithm performance.

Abstract

Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional compact submanifold $M$ in $\mathbb{R}^D$, we establish the spectral convergence rate of the graph Laplacian. It implies the consistency of the spectral clustering algorithm via a standard perturbation argument. A simple numerical study indicates the necessity of a denoising step before applying spectral algorithms.