Spectral convergence of the connection Laplacian from random samples

Singer, Amit; Wu, Hau‐Tieng

doi:10.1093/imaiai/iaw016

Cited by 84 publications

(147 citation statements)

References 36 publications

(66 reference statements)

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“…In fact, near the boundary, graph Laplacian is dominated by the first-order derivative and thus fails to be a true Laplacian [7,27]. Recently, Singer and Wu [45] showed the spectral convergence of the graph Laplacian in the presence of the Neumann boundary. the convergence analysis in both [6] and [45] is based on the connection between the graph Laplacian and the heat operator.…”

Section: R(r)mentioning

confidence: 99%

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Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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The Laplace-Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace-Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228-258, 2017). In this paper, we analyze the convergence of PIM for Poisson equation with Neumann boundary condition on submanifolds that are isometrically embedded in Euclidean spaces. Keywords: Laplace-Beltrami operator, Neumann boundary, Point cloud, Point integral method, Convergence analysisMathematics Subject Classification: 65N12, 65N25, 65N75 BackgroundThe partial differential equations on manifolds arise in a wide variety of applications. In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space. One fundamental assumption in the manifold model is that the point cloud samples a smooth manifold. Thus, the information of the manifold is very useful to understand the data or images. PDEs on the manifold, particularly the Laplace-Beltrami equation, encode several intrinsic information of the manifold, thus helping reveal the underlying structures in the data or images. To get the information encoded in PDEs, we need to solve them in the unstructured point cloud. Given that the point cloud is embedded in a high-dimensional space, the traditional methods for PDEs on 2D surfaces do not work.

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Section: R(r)mentioning

confidence: 99%

“…Recently, Singer and Wu [45] showed the spectral convergence of the graph Laplacian in the presence of the Neumann boundary. the convergence analysis in both [6] and [45] is based on the connection between the graph Laplacian and the heat operator. Therefore, Gaussian weights are essential.…”

Section: R(r)mentioning

confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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“…However, there are several recent works on point wise estimates between graph Laplacians and continuum operators or their spectral convergence, e.g. [SW17,GS18] and the references therein. Currently, however, only consistency results for eigenvectors and eigenprojections have been obtained.…”

Section: Fokker-planckmentioning

confidence: 99%

A Geometrical Method for Low-Dimensional Representations of Simulations

Iza-Teran¹,

Garcke²

2019

SIAM/ASA J. Uncertainty Quantification

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We propose a new data analysis approach for the efficient post-processing of bundles of finite element data from numerical simulations. The approach is based on the mathematical principles of symmetry. We consider the case where simulations of an industrial product are contained in the space of surface meshes embedded in R 3 . Furthermore, we assume that distance preserving transformations exist, albeit unknown, which map simulation to simulation. In this setting, a discrete Laplace-Beltrami operator can be constructed on the mesh, which is invariant to isometric transformations and therefore valid for all simulations. The eigenfunctions of such an operator are used as a common basis for all (isometric) simulations. One can use the projection coefficients instead of the full simulations for further analysis. To extend the idea of invariance, we employ a discrete Fokker-Planck operator, that in the continuous limit converges to an operator invariant to a nonlinear transformation, and use its eigendecomposition accordingly.The data analysis approach is applied to time-dependent datasets from numerical car crash simulations. One observes that only a few spectral coefficients are necessary to describe the data variability, and low dimensional structures are obtained. The eigenvectors are seen to recover different independent variation modes such as translation, rotation, or global and local deformations. An effective analysis of the data from bundles of numerical simulations is made possible, in particular an analysis for many simulations in time. *

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“…Pointwise estimates between graph Laplacians and the continuum operators were studied by Belkin and Niyogi [8], Coifman and Lafon [11], Giné and Koltchinskii [18], Hein, Audibert and von Luxburg [23], and Singer [37]. Spectral convergence was studied in the works of Ting, Huang, and Jordan [45], Belkin and Niyogi [7] on the convergence of Laplacian eigenmaps, von Luxburg, Belkin and Bousquet on graph Laplacians, and of Singer and Wu [38] on connection graph Laplacian. The convergence of the eigenvalues and eigenvectors these works obtain is of great relevance to machine learning.…”

Section: Introductionmentioning

confidence: 99%

A variational approach to the consistency of spectral clustering

Trillos

Slepčev

2018

Applied and Computational Harmonic Analysis

142

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Abstract. This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.

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Spectral convergence of the connection Laplacian from random samples

Cited by 84 publications

References 36 publications

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

A Geometrical Method for Low-Dimensional Representations of Simulations

A variational approach to the consistency of spectral clustering

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