On Convergence of Flat Multivariate Interpolation by Translation Kernels with Finite Smoothness

Lee, Yeon Ju; Micchelli, Charles A.; Yoon, Jungho

doi:10.1007/s00365-014-9233-7

Cited by 15 publications

(13 citation statements)

References 19 publications

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“…The results we present in detail below appear in Schaback (2005) and Lee et al (2007); see also (Schaback, 2008). Finitely smooth kernels are considered in Song et al (2012) and Lee et al (2014), with further generalisations appearing in Lee et al (2015).…”

Section: Abbreviationsmentioning

confidence: 99%

Classical quadrature rules via Gaussian processes

Karvonen

Särkkä

2017

2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP)

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all of whom have been kind enough to host me at their home institutions, some of them several times, during the past four years. The probabilistic numerics research community is still compact enough for one to meet almost everybody during the short span of a few years. The various conferences, workshops, and visits have been made much more enjoyable and productive by the presence, often recurring, of in particular Dr.

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Section: Abbreviationsmentioning

confidence: 99%

Classical quadrature rules via Gaussian processes

Karvonen

Särkkä

2017

2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP)

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“…Consider then the case → ∞. This scenario is called the flat limit in scattered data approximation literature where it has been proved 1 that the kernel interpolant associated to an isotropic kernel with increasing length-scale converges to (i) the unique polynomial interpolant of degree N − 1 to the data if the kernel is infinitely smooth [22,34,24] or (ii) to a polyharmonic spline interpolant if the kernel is of finite smoothness [23]. In our case, → ∞ results in…”

Section: Effect Of the Length-scalementioning

confidence: 88%

Gaussian kernel quadrature at scaled Gauss–Hermite nodes

Karvonen

Särkkä

2019

Bit Numer Math

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This article derives an accurate, explicit, and numerically stable approximation to the kernel quadrature weights in one dimension and on tensor product grids when the kernel and integration measure are Gaussian. The approximation is based on use of scaled Gauss-Hermite nodes and truncation of the Mercer eigendecomposition of the Gaussian kernel. Numerical evidence indicates that both the kernel quadrature and the approximate weights at these nodes are positive. An exponential rate of convergence for functions in the reproducing kernel Hilbert space induced by the Gaussian kernel is proved under an assumption on growth of the sum of absolute values of the approximate weights.

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“…In particular, the assumptions hold for instance for the α-exponential kernels with α ∈ (0, 2) (see [6]) and the popular Matern RBF kernels with ν < 2. The proof of the theorem can be found in Appendix D.1 and is based on the flat limit literature on RBFs [13,22,25,38]. Finally, we remark that Theorem 3.3 applies only for the interpolating estimator f0 .…”

Section: Inconsistency Of Kernel Interpolation For τ D Effmentioning

confidence: 99%

“…In addition, denote with d α Z the vector with entries (d α Z ) i = z i − Z α 2 and with d α the function d α (z, z ) = z − z α 2 . First, although the limit lim τ →∞ K −1 does not exists, we can apply Theorem 3.12 in [25] to show that the flat limit interpolator f FL := lim τ →∞ f0 of any kernel satisfying the assumption in Theorem 3.3 exists and has the form…”

Section: D1 Proof Of Theorem 33mentioning

confidence: 99%

How rotational invariance of common kernels prevents generalization in high dimensions

Donhauser,

Wu,

Yang

2021

Preprint

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Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for kernel regression under certain assumptions on the ground truth function and the distribution of the input data. In this paper, we show that the rotational invariance property of commonly studied kernels (such as RBF, inner product kernels and fully-connected NTK of any depth) induces a bias towards low-degree polynomials in high dimensions. Our result implies a lower bound on the generalization error for a wide range of distributions and various choices of the scaling for kernels with different eigenvalue decays. This lower bound suggests that general consistency results for kernel ridge regression in high dimensions require a more refined analysis that depends on the structure of the kernel beyond its eigenvalue decay.

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On Convergence of Flat Multivariate Interpolation by Translation Kernels with Finite Smoothness

Cited by 15 publications

References 19 publications

Classical quadrature rules via Gaussian processes

Classical quadrature rules via Gaussian processes

Gaussian kernel quadrature at scaled Gauss–Hermite nodes

How rotational invariance of common kernels prevents generalization in high dimensions

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