Capacity of reproducing kernel spaces in learning theory

By Ding-Xuan Zhou. 2003

251 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

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Citations in the corpus

Variance-based Regularization with Convex Objectives (2016, methodology): “t space generated by some suﬃciently nice kernel K; even for the Gaussian kernel K(x,z) = exp(−1 2 kx−zk 2), log covering numbers for such function spaces grow at least exponentially in the dimension [51, 26]. Example 6(Reproducing kernels and least-absolute-deviation regression): We now give an exa”
Shannon sampling and function reconstruction from point values (2004, background): “This assumption is true if X is compact and the inclusion map of HK,t into the space of Lipschitz functions on X is bounded (this is the case when K is a C(2) Mercer kernel; see [33]).”
Learning Theory Estimates via Integral Operators and Their Approximations (2007, background): “It was proved in [17] that when K ∈ C (X × X) with 0 < < 2 and X is the closure of a domain in IR, the inclusion HK ⊂ C (X) is well defined and bounded.”
Distributed Learning with Regularized Least Squares (2016, methodology): “An example of a C ∞ Mercer kernel was presented in (Zhou, 2002, 2003) to show that smoothness of the Mercer kernel does not guarantee the uniform boundedness of the eigenfunctions.”
Application of integral operator for regularized least-square regression (2009, background): “For example, the covering numbers of RKHS is only well estimated for certain very smooth kernels such as polynomial and Gaussians and the underlying input space X is finite dimensional compact set [12,16,17].”
Regularized Policy Iteration with Nonparametric Function Spaces (2016, background): “Assumption A4 is suitable for large function spaces and is indeed satisfied for the Sobolev spaces and various RKHS. Refer to van de Geer (2000); Zhou (2002, 2003); Steinwart and Christmann (2008) for many examples.”
Theory of classification : a survey of some recent advances (2005, unknown): “The study of universal approximation properties of kernels and statistical consistency of Support Vector Machines is due to Steinwart [203–205], Lin [140,141], Zhou [243], and Blanchard, Bousquet, and Massart [39].”
Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels (2008, methodology): “This problem was initiated in [ 19 ] and further studied recently in [21].”
Statistical performance of support vector machines (2008, background): “ntropy is that, by deﬁnition, it is distribution independent. Furthermore, some relatively general results are known on this entropy depending on the regularity properties of the kernel function; see [41]. 4. A model selection theorem and its application. 4.1. An abstract model selection theorem. Th”
Iterative Value-Aware Model Learning (2018, result): “…the behaviour of the metric entropy is p log( 1 ε ) , which can be seen as having α → 0 with a certain rate, For many examples of the covering number and metric entropy results, refer to van de Geer [2000], Györﬁ et al. [2002], Zhou [2003], Steinwart and Christmann [2008], Giné and Nickl [2015].”