A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines

Kimeldorf, George; Wahba, Grace

doi:10.1214/aoms/1177697089

Cited by 798 publications

(504 citation statements)

References 4 publications

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“…Note that the regularized risk approach can also be dealt with in a reproducing kernel Hilbert space (RKHS) approach which may lead to sometimes more elegant exposition of the subject, see Kimeldorf and Wahba (1971);Micchelli (1986); Wahba (1990); Girosi (1997);Schölkopf (1997).…”

Section: Discussionmentioning

confidence: 99%

“…For instance by choosing a suitable operator that penalizes large variations of f one can reduce the well-known overfitting effect. Another possible setting also might be an operatorP mapping from L 2 (R n ) into some reproducing kernel Hilbert space (Kimeldorf and Wahba, 1971;Girosi, 1997). In Appendix A, we provide a worked through example (mainly taken from Girosi et al, 1993) for a simple regularization operator to illustrate our reasoning.…”

Section: Regularization Networkmentioning

confidence: 99%

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The connection between regularization operators and support vector kernels

Smola¹,

Schölkopf²,

Müller³

1998

Neural Networks

553

316

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In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels. ᭧

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Section: Discussionmentioning

confidence: 99%

Section: Regularization Networkmentioning

confidence: 99%

The connection between regularization operators and support vector kernels

Smola¹,

Schölkopf²,

Müller³

1998

Neural Networks

553

316

View full text Add to dashboard Cite

show abstract

“…Several authors including [41,50,10,56] have already pointed out this connection, a comprehensive overview over both approaches is given by [4]. While the authors of [4] also establish the link between stochastic processes and reproducing kernel Hilbert spaces (RKHS), the equivalence of kernel interpolation and kriging is shown by the usual algebraic arguments.…”

Section: Introductionmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

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“…The resulting learned decision function, implied by the representer theorem (Kimeldorf and Wahba, 1970), is the solution…”

Section: Model Constructionmentioning

confidence: 99%

A signal theory approach to support vector classification: The sinc kernel

et al. 2009

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Fourier-based regularisation is considered for the support vector machine classification problem over absolutely integrable loss functions. By invoking the modest assumption that the decision function belongs to a Paley-Wiener space, it is shown that the classification problem can be developed in the context of signal theory. Furthermore, by employing the Paley-Wiener reproducing kernel, namely the sinc function, it is shown that a principled and finite kernel hyper-parameter search space can be discerned, a priori.Subsequent simulations performed on a commonly-available hyperspectral image data set reveal that the approach yields results that surpass state-of-the-art benchmarks.

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A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines

Cited by 798 publications

References 4 publications

The connection between regularization operators and support vector kernels

The connection between regularization operators and support vector kernels

Interpolation of spatial data – A stochastic or a deterministic problem?

A signal theory approach to support vector classification: The sinc kernel

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