[Paper Review] Spectral Convergence of the connection Laplacian from random samples

Amit Singer, Hau‐Tieng Wu|arXiv (Cornell University)|Jun 7, 2013

Topological and Geometric Data Analysis31 references26 citations

TL;DR

This paper establishes spectral convergence of the graph connection Laplacian (GCL) to the connection Laplacian on vector bundles over manifolds when data points are sampled independently from a general (non-uniform) distribution, including manifolds with boundary. The authors generalize Belkin and Niyogi's spectral convergence results by proving that eigenvectors and eigenvalues of the GCL converge to those of the connection Laplacian, leveraging a unified framework based on principal bundle structure and contraction mapping principles for perturbation analysis.

ABSTRACT

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

Motivation & Objective

• To establish spectral convergence of the graph connection Laplacian (GCL) to the connection Laplacian on vector bundles over smooth manifolds.
• To generalize prior spectral convergence results—originally for the Laplace-Beltrami operator under uniform sampling—to non-uniform sampling and manifolds with boundary.
• To unify the approximation of connection Laplacians via a principal bundle framework, enabling application to vector bundles arising from group actions.
• To extend the theoretical foundation of Vector Diffusion Maps (VDM) by rigorously proving convergence of eigenspaces and eigenvalues from random samples.

Proposed method

• Formalizes the connection Laplacian on vector bundles induced by group actions (e.g., orthogonal or unitary groups), ensuring symmetry and isometry.
• Defines an invariant metric $ d_G $ on the orbit space $ \mathcal{X}/G $ using optimal alignment, replacing Euclidean distance to factor out nuisance parameters.
• Constructs the graph connection Laplacian (GCL) using Gaussian kernel weights based on $ d_G $, encoding both data affinities and group-invariant structure.
• Applies a contraction mapping principle to solve a system of PDEs arising from perturbation of local embeddings, ensuring existence of symmetric isometric embeddings.
• Uses Schauder estimates and Sobolev embedding theorems to control regularity and convergence of solutions in $ C^{2,\alpha} $ spaces.
• Extends convergence results to non-uniform sampling and manifolds with boundary by adapting the analysis of graph Laplacian convergence beyond the uniform i.i.d. case.

Experimental results

Research questions

• RQ1 Does the graph connection Laplacian (GCL) converge spectrally to the connection Laplacian on the tangent bundle of a manifold under non-uniform sampling?
• RQ2 Can spectral convergence be established for connection Laplacians on general vector bundles, not just the tangent bundle, using a unified principal bundle framework?
• RQ3 How does the convergence of eigenvalues and eigenvectors of the GCL behave when the underlying manifold has a non-empty boundary?
• RQ4 What conditions ensure that the GCL approximates the connection Laplacian in the limit of infinitely many independent random samples, even when sampling is non-uniform?
• RQ5 Can the theoretical framework of Vector Diffusion Maps (VDM) be rigorously justified via spectral convergence of the GCL to the connection Laplacian?

Key findings

• The eigenvectors and eigenvalues of the graph connection Laplacian (GCL) converge to those of the connection Laplacian on the vector bundle as the number of i.i.d. samples tends to infinity.
• Spectral convergence is proven for non-uniform sampling measures and for manifolds with non-empty boundary, extending Belkin and Niyogi's original result.
• The convergence is established via a contraction mapping argument applied to a system of PDEs arising from local isometric embedding perturbations.
• The method ensures symmetric isometric embeddings of the manifold and its double cover in $ \mathbb{R}^p $, preserving group action symmetries.
• A unified framework based on principal bundle structure enables the approximation of various connection Laplacians, generalizing previous results on the Laplace-Beltrami operator.
• The existence of a symmetric isometric embedding into Euclidean space is proven for any smooth, closed, non-orientable manifold, using the double cover and the contraction method.