Journal article
Differentiable positive definite kernels and Lipschitz continuity
Mathematical Proceedings of the Cambridge Philosophical Society, Vol.104(02)
1988
Abstract
Reade[ll] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order a on a bounded region have eigenvalues which are asymptotically O(l/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) ≡ ∆2rK(x, t)/∆xr∆tr is in Lipα and establish λn(K) = O(l/n2r+1+α). If ∆Krr/∆t is in Lipα, the anticipated asymptotic estimate is also derived. The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.
Details
- Title
- Differentiable positive definite kernels and Lipschitz continuity
- Authors/Creators
- J.A. Cochran (Author/Creator)M.A. Lukas (Author/Creator)
- Publication Details
- Mathematical Proceedings of the Cambridge Philosophical Society, Vol.104(02)
- Publisher
- Cambridge University Press
- Identifiers
- 991005540511407891
- Copyright
- © 1988, Cambridge Philosophical Society.
- Murdoch Affiliation
- School of Mathematical and Physical Sciences
- Language
- English
- Resource Type
- Journal article
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- 9 Mathematics
- 9.92 Statistical Methods
- 9.92.2786 Kernel Methods
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- Mathematics
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