Approximation of eigenfunctions in kernel-based spaces

Santin, Gabriele; Schaback, Robert

doi:10.1007/s10444-015-9449-5

Cited by 35 publications

(39 citation statements)

References 28 publications

(37 reference statements)

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“…In some problems, the prior knowledge of a good separable approximation of the kernel function may inform the choice of a low rank model. An initial subsampling procedure can also be combined with an eigendecomposition (Williams and Seeger, 2001;Santin and Schaback, 2016) to improve the accuracy of the low rank approximation. This approach is in some sense similar to that of Halko et al (2011), where a random projection followed by an SVD is used to obtain low rank approximations of matrices.…”

Section: Existing Methodsmentioning

confidence: 99%

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

Schäfer¹,

Sullivan²,

Owhadi³

2021

Multiscale Model. Simul.

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Dense kernel matrices Θ ∈ R N ×N obtained from point evaluations of a covariance function G at locations {x i } 1≤i≤N arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions elliptic boundary value problems and approximately equally spaced sampling points, we show how to identify a subset S ⊂ {1, . . . , N } × {1, . . . , N }, with #S = O N log(N ) log d (N/ ) , such that the zero fill-in block-incomplete Cholesky decomposition of Θ i,j 1 (i,j)∈S is an -approximation of Θ. This block-factorisation can provably be obtained in O N log 2 (N ) log (1/ ) + log 2 (N ) 4d+1 complexity in time. Numerical evidence further suggests that element-wise Cholesky decomposition with the same ordering constitutes an O N log 2 (N ) log 2d (N/ ) solver. The algorithm only needs to know the spatial configuration of the x i and does not require an analytic representation of G. Furthermore, an approximate PCA with optimal rate of convergence in the operator norm can be easily read off from this decomposition. Hence, by using only subsampling and the incomplete Cholesky decomposition, we obtain at nearly linear complexity the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky decomposition we also obtain a near-linear-time solver for elliptic PDEs.

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Section: Existing Methodsmentioning

confidence: 99%

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

Schäfer¹,

Sullivan²,

Owhadi³

2021

Multiscale Model. Simul.

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show abstract

“…• Meshless techniques based on collocation schemes (strong forms), such as the meshless collocation method based on radial basis functions (RBFs) (Parand et al 2011;Kansa 1990;Jakobsson et al 2009;Abbasbandy et al 2012Abbasbandy et al , 2013Kamranian et al 2016;Moradipour and Yousefi 2018). • Meshless techniques based on the combination of weak forms and collocation method (Santin and Schaback 2016;Schaback 2015;Young et al 2008;Assari and Dehghan 2018;Shirzadi 2010, 2011;Dehghan and Ghesmati 2010;Liu et al 2002;Liu and Gu 2001;Shivanian 2013Shivanian , 2014Khodabandehlo 2014, 2016;Hosseini et al 2015Hosseini et al , 2016.…”

Section: Preliminariesmentioning

confidence: 99%

Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

Shivanian

Abbasbandy

2020

Comp. Appl. Math.

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In this study, we develop an approximate formulation for a generalization form of biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as Cahn-Hilliard type boundary conditions with application to spinodal decomposition. Since the rigorous steps to analyze such a problem is of high-order derivatives, implementing multiple boundary conditions and especially when the geometry of domain of the problem is complex. In PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high-order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by an erudite application of PSMRPI on nodal points near the boundaries of the domain. The main results of generalized biharmonic problem are demonstrated by some examples to show validity and trustworthy of PSMRPI technique.

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“…Since one can add another basis function to the low-rank factorization without recomputing the others, they call the obtained basis the "Newton" basis, in analogy to Newton interpolation. This kernelbased approach has been extended in [34] to compute a Karhunen-Loève expansion if radial basis functions are used for the spatial discretization.…”

Section: Related Workmentioning

confidence: 99%

Error-Controlled Model Approximation for Gaussian Process Morphable Models

et al. 2018

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Gaussian Process Morphable Models (GP-MMs) unify a variety of non-rigid deformation models for surface and image registration. Deformation models, such as B-splines, radial basis functions, and PCA models are defined as a probability distribution using a Gaussian process. The method depends heavily on the low-rank approximation of the Gaussian process, which is mandatory to obtain a parametric representation of the model. In this article, we propose the use of the pivoted Cholesky decomposition for this task, which has the following advantages: 1) Compared to the current state of the art used in GPMMs, it provides a fully controllable approximation error. The algorithm greedily computes new basis functions until the userdefined approximation accuracy is reached. 2) Unlike the currently used approach, this method can be used in a black-box-like scenario, whereas the method automatically chooses the amount of basis functions for a given model and accuracy. 3) We propose the Newton basis as an alternative basis for GPMMs. The proposed basis does not need an SVD computation and can be iteratively refined. We show that the proposed basis functions achieve competitive registration results, while providing the mentioned advantages for its computation. † Author names in alphabetical order.

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Approximation of eigenfunctions in kernel-based spaces

Cited by 35 publications

References 28 publications

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

Error-Controlled Model Approximation for Gaussian Process Morphable Models

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