Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

Abstract

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We rst give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l 2 -norm for some general Mercer kernel matrices are provided. As an example, we give the l 2 -norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l 2 -norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

Adaptive Kernel Learning Kalman Filtering With Application to Model-Free Maneuvering Target Tracking

“…where 𝜑 is the feature projection operator from input vector space ℛ to feature vector space ℱ. When 𝜅 satisfies Mercer condition [30][31], i.e., 𝜅 is continuous, symmetric and positive definite, it has 1) ∀ 𝑥 ∈ ℛ, 𝜅(𝑥, 𝑦) ∈ ℱ;…”

“…Equations ( 26) to (31) complete the process of one-step estimation in a kernel Kalman filter, where 𝒂 𝑖 , 𝑷 ̃𝑖 − , 𝑮 ̃𝑖, 𝑷 ̃𝑖, and 𝒃 𝑖 all can be expressed with inner product of embedded variables so that they can be calculated conveniently by kernel trick with given kernel function.…”

“…For 𝜇̂𝑖 in (31) which is defined in RKHS, it needs to revive the state estimation 𝒙 ̂𝑖 in the original input space. This kind of inverse kernel projection can be realized as the following process [34].…”

Error analysis of multicategory support vector machine classifiers

“…Putting(21) and(25)into(20) with f = η m (f M ), we get E(f z,λ ) − E(f M ) ≤ λ kC {E(f z,λ ) − E(f M )} + c(m),where c(m) is defined in(25), we have with confidence at least 1 − 2δ , thatE(f z,λ ) − E(f M ) ≤ 2λ kC 2 1…”