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A kernel-based analysis of Laplacian Eigenmaps

Martin Wahl

TL;DR

The main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of $\mathcal{M}$.

Abstract

Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of $\mathcal{M}$. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of $\mathcal{M}$ as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.

A kernel-based analysis of Laplacian Eigenmaps

TL;DR

The main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of .

Abstract

Given i.i.d. observations uniformly distributed on a closed manifold , we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of . In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.
Paper Structure (30 sections, 26 theorems, 194 equations)

This paper contains 30 sections, 26 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Laplacian Eigenmaps and Diffusion Maps
  4. Laplace-Beltrami operator, heat semigroup and heat kernel
  5. Two main consequences
  6. Previous and new proof map
  7. Further basic notation.
  8. Heat kernel principal component analysis
  9. The heat kernel as reproducing kernel
  10. Heat kernel PCA
  11. Perturbation bounds for heat kernel PCA
  12. The approximation error
  13. Different graph Laplacians
  14. Analysis of the approximation error
  15. Additional proofs
  16. ...and 15 more sections

Key Result

Corollary 1

Grant Assumption ass:manifold:hypothesis. Then there are constants $c,C>0$ depending only on the constants in (C1), (C3), (C4) and (C5) such that the following holds. Let $n\geq 2$ and let $t\in(0,1]$ be such that $\frac{\log n}{nt^{d/2}}\leq c$. Then with high probability.

Theorems & Definitions (53)

  • Corollary 1
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 43 more