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[|| ||.sub.[upsilon]] is the norm in the RKHS. [eta] is set to be a positive tunable parameter, which is used to control the trade-off between the fitness error and the solution complexity as measured by the norm in the RKHS.
With the Stein kernel, we can map the SPD manifold into RKHS:
The RKHS of [mathematical expression not reproducible] is crucial for our results.
Following the previous works, Kernel mean matching (KMM)  is introduced to weight the training data by minimizing the difference between the means of weighted-training and test data distribution in RKHS. Different from the previous works, the sample reweighting procedure and matching feature selection are modeled in a unified optimization problem.
Following Nordgren and Rosenthal , we say that RKHS H(Q) is standard if the underlying set [OMEGA] is a subset of a topological space and the boundary [partial derivative][OMEGA] is non empty and has the property that the normalized reproducing kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges weakly to 0 whenever [([[lambda].sub.n]).sub.k[greater than or equal to]1] [member of] [OMEGA] converges to any point in [partial derivative][OMEGA].
In order to handle nonlinear classification, the kernelization trick  is used to map the n-dimensional date points into an arbitrary reproducing kernel Hilbert space (RKHS)  via a mapping function [phi] : [R.sup.n] [??] H; that is, [x.sub.i] [??] [phi]([x.sub.i]).
Recently, a lot of research work has been devoted to the applications of RKHS method for wide classes of stochastic and deterministic problems involving operator equations, differential equations, integral equations, and integrodifferential equations.
A Hilbert space H consisting of complex valued functions on a set E is called a reproducing kernel Hilbert space (RKHS) if there is a function q(s, t) on E x E, called the reproducing kernel of H, satisfying
The hypothesis space H is defined as a reproducing kernel Hilbert space (RKHS) whose elements are real continuous functions defined on X with a kernel K.
Organic revenue growth for the first half was 0.4%, an increase of 1.9%, reportedly as a result of the integration of Score (France), Zehnacker (Germany), Comfort Keepers (US) and RKHS (India).
Interestingly, it turns out that these complexity issues can be overcome by using some duality techniques from optimization theory (Bertsekas, Nedic, and Ozdaglar 2003) and reproducing kernel Hilbert spaces (RKHS) (Kimeldorf and Wahba 1971).
The key of kernel method is that if kernel function [kappa] is positive definite, there exists a mapping [phi] into the reproducing kernel Hilbert spaces (RKHS), such that
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