Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

Trillos, Nicolás García; Gerlach, Moritz; Hein, Matthias; Slepčev, Dejan

doi:10.1007/s10208-019-09436-w

Cited by 111 publications

(125 citation statements)

References 20 publications

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“…The spectrum of the normalized graph Laplacian of the latter is known to behave like the eigensystem of the Laplace-Beltrami operator over the submanifold corresponding to the k-th connected subset M k . In particular, we will use existing results [4,34] on error estimates by using the spectrum of a random geometric graph to approximate the eigensystem of the Laplace-Beltrami operator in the numerical analysis literature.…”

Section: 5mentioning

confidence: 99%

Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations

Chen

Yang

2021

Applied and Computational Harmonic Analysis

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We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion K-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion K-means. In addition, we also propose a nuclear norm (i.e., trace norm) regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion K-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the localized diffusion K-means by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion K-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds.

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Section: 5mentioning

confidence: 99%

Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations

Chen

Yang

2021

Applied and Computational Harmonic Analysis

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“…This truncation has the added benefit of denoising the diffusion distances, since the eigenvectors associated with eigenvalues away from 1 in modulus (in some sense the high frequency eigenvectors) correspond not to intrinsic geometric structures in the data, but to random fluctuations produced by sampling [28]. The embedding…”

Section: Background On Diffusion Geometrymentioning

confidence: 99%

Learning by active nonlinear diffusion

Maggioni¹,

Murphy²

2019

Foundations of Data Science

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This article proposes an active learning method for high dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. Diffusion distances are used to parametrize low-dimensional structures on the dataset, which allow for high-accuracy labelings of the dataset with only a small number of carefully chosen labels. The geometric structure of the data suggests regions that have homogeneous labels, as well as regions with high label complexity that should be queried for labels. The proposed method enjoys theoretical performance guarantees on a general geometric data model, in which clusters corresponding to semantically meaningful classes are permitted to have nonlinear geometries, high ambient dimensionality, and suffer from significant noise and outlier corruption. The proposed algorithm is implemented in a manner that is quasilinear in the number of unlabeled data points, and exhibits competitive empirical performance on synthetic datasets and real hyperspectral remote sensing images.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

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“…One would only need to adjust the statements using manifold analogues of the weighted Dirichlet energy and the Laplacian. The convergence of graph Laplacian in the manifold setting has already been established in the standard setting [19]. In the manifold setting the dimension d in the results above should be replaced by the dimension of the data manifold.…”

Section: Discrete To Continuum Convergencementioning

confidence: 99%

“…Remark 1.4. We remark that the discrete functional (13) can be rewritten as (19) GE n,ε,ζ puq " 1 2n 2 ε 2 ÿ x,yPXn pγ ζ pxq`γ ζ pyqqη ε px´yq|upxq´upyq| 2 , and so the problem has a symmetric weight matrix.…”

Section: Introductionmentioning

confidence: 99%

Properly-Weighted Graph Laplacian for Semi-supervised Learning

Calder

Slepčev

2019

Appl Math Optim

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The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit.In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement.

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Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

Cited by 111 publications

References 20 publications

Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations

Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations

Learning by active nonlinear diffusion

Properly-Weighted Graph Laplacian for Semi-supervised Learning

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