Graph approximations to the Laplacian spectra

Lu, Jingzhou

doi:10.1142/s1793525320500442

Cited by 7 publications

(14 citation statements)

References 11 publications

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“…Re-arranging the terms gives that µ k < λ k + 0.8γ K . This can be verified for all 2 ≤ k ≤ K, and note that the good event E (1) is w.r.t X, and E U B is constructed for fixed k max , and none is for specific k ≤ K. The positive integer kmax is fixed, and the constant γ K is half of the minimum first-K eigen-gaps, defined as in (22). Eigenvalue UB and initial LB are proved for k ≤ K, which guarantees (41).…”

Section: Un-normalized Graph Laplacianmentioning

confidence: 83%

“…The current result may be extended in several directions. First, for manifold with smooth boundary, the random-walk graph Laplacian recovers the Neumann Laplacian [10], and one can expect to prove the spectral convergence as well, such as in [22]. Second, extension to kernel with variable or adaptive bandwidth [5,9], and other normalization schemes, e.g., bi-stochastic normalization [23,20,36], would be important to improve the robustness against low sampling density and noise in data, and even the spectral convergence as well.…”

Section: Discussionmentioning

confidence: 99%

“…Proposition 5.2. Under Assumptions 1(A1), p being uniform on M, and h is Gaussian, for fixed k max ∈ N, K = k max + 1, assume that the eigenvalues µ k for k ≤ K are all single multiplicity, and γ K > 0 as defined in (22), the constant c K as in Proposition 4.…”

Section: Then We Use the Relation That For Eachmentioning

confidence: 99%

“…Figure 1: Population eigenvalues µ k of −∆, and empirical eigenvalues λ k of graph Laplacian matrix L N , L N can be Lun or Lrw.The positive integer kmax is fixed, and the constant γ K is half of the minimum first-K eigen-gaps, defined as in(22). Eigenvalue UB and initial LB are proved for k ≤ K, which guarantees (41).…”

mentioning

confidence: 91%

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Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

Cheng¹,

Wu²

2021

Preprint

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This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from N random samples on a d-dimensional manifold embedded in a possibly high dimensional space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter ∼ (log N/N ) 1/(d/2+2) such that the eigenvalue convergence rate is N −1/(d/2+2) and the eigenvector convergence in 2-norm has rate N −1/(d+4) ; When ∼ N −1/(d/2+3) , both eigenvalue and eigenvector rates are N −1/(d/2+3) . These rates are up to a log N factor and proved for finitely many low-lying eigenvalues. The result holds for un-normalized and random-walk graph Laplacians when data are uniformly sampled on the manifold, as well as the density-corrected graph Laplacian (where the affinity matrix is normalized by the degree matrix from both sides) with non-uniformly sampled data. As an intermediate result, we prove new point-wise and Dirichlet form convergence rates for the density-corrected graph Laplacian. Numerical results are provided to verify the theory.

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Section: Un-normalized Graph Laplacianmentioning

confidence: 83%

Section: Discussionmentioning

confidence: 99%

Section: Then We Use the Relation That For Eachmentioning

confidence: 99%

mentioning

confidence: 91%

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Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

Cheng¹,

Wu²

2021

Preprint

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“…However, theoretical estimates for spectral data in the literature have been much weaker than this. The standard bound on the bias error in the spectral data has been the naive estimate of O(ε 1/2 ), corresponding to the L p → L p operator error (Hein et al 2005, Shi 2015, Lu 2020. While the decay of the variance error as M → ∞ with ε fixed has been long known as a result of the theory of Glivenko-Cantelli function classes (von Luxburg et al 2004, 2008, Belkin & Niyogi 2007, this approach has yielded only weak quantitative bounds of O(M −1/2 ε −d−3 ) on the variance error (Shi 2015).…”

Section: Introductionmentioning

confidence: 99%

Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation

Wormell,

Reich

2020

Preprint

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Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are however generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localised compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE, and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretisation.We also introduce an alternative normalisation for diffusion maps based on Sinkhorn weights. This normalisation approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalisation on flat domains, and present a highly efficient algorithm to compute the Sinkhorn weights.

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Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

Jiang

Harlim

2021

Comm Pure Appl Math

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In this paper, we extend the class of kernel methods, the so‐called diffusion maps (DM) and its local kernel variants to approximate second‐order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite‐difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well‐posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh‐free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. © 2021 Wiley Periodicals LLC.

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Graph approximations to the Laplacian spectra

Cited by 7 publications

References 11 publications

Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

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