Derivative reproducing properties for kernel methods in learning theory

By Ding-Xuan Zhou. 2008

174 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Mathematical Physics, Modeling and Simulation

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Citations in the corpus

A Kernelized Stein Discrepancy for Goodness-of-fit Tests (2016, methodology): “We ﬁrst prove (12) by applying the reproducing property k ( x, x (cid:48) ) = (cid:104) k ( x, · ) , k ( x (cid:48) , · ) (cid:105) H on (8): where we used the fact that β where we used the fact that ∇ (Zhou, 2008; Steinwart & Christmann, 2008).”
Optimal Rates for Random Fourier Features (2015, background): “As motivated in Section 1, the question of approximating the derivatives of the kernel through finite dimensioal random feature map is also important as it enables to speed up several interesting machine learning tasks that involve the derivatives of the kernel (e.g., [32, 21, 17, 18, 29, 23]).”
Density Estimation in Infinite Dimensional Exponential Families (2013, neutral): “ntiable if ∂α,αk: X × X → R exists and is continuous for all α∈ Nd 0 with |α| ≤ mwhere ∂α,α := ∂α1 1 ...∂ αd d ∂ α1 1+d...∂ αd 2d. Corollary 4.36 in Steinwart and Christmann (2008) and Theorem 1 8 in Zhou (2008) state that if ∂α,αkexists and is continuous, then ∂ αk(x,·) = ∂ 1 1 ...∂ αd d k(x,·) = ∂”
Nonparametric sparsity and regularization (2012, background): “Then, from Theorem 1 in [59] we have that, if k is at least a C2(X × X ), (∂ak)x belongs to H for all x ∈ X and most importantly ∂f(x) ∂xa = 〈f, (∂ak)x〉H, for a = 1, .”
A Regularization Approach to Nonlinear Variable Selection (2010, background): “In fact if the kernel is sufficiently smooth it is possible to prove (Zhou, 2008) that zj,x ∈ H for all x ∈ X where zj,x is defined in (4), and the following reproducing property for derivatives, that will be the main tool towards deriving Algorithm 1, holds ∂jf(x0) = 〈f, zj,x0〉H .”
Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels (2008, methodology): “The purpose of this article is to continue the study in [19, 21 ] and investigate real analyticity of functions from HK associated with translation-invariant Mercer kernels of the type K(x,y) = ϕ(|x − y| 2 ) with ϕ : R → R. Then we shall apply our results to two mathematical problems concerning lea”