Moduli of smoothness, \(K\)-functionals and Jackson-type inequalities associated with Kernel function approximation in learning theory (Q6587592)

The authors give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. The paper is organized as follows: Section 1 contains the motivations for the tasks under consideration.\N\NIn Section 2, the authors provide some general notions and propositions for kernel function approximations, for example, \(K\)-functionals and moduli of smoothness and a Jackson-type inequality.\N\NIn Section 3, the authors apply the general theory and results of Section 2 to some concrete kernel function spaces, which include the kernel function spaces defined on the unit sphere, the kernel function spaces defined on the unit ball, the kernel function spaces defined on a mixed domain by both the unit sphere and the unit ball, and the kernel function spaces defined on a simplex.\N\NIn Section 4, the authors apply the Jackson inequalities established in Sections 2 and 3 to learning theory and express the learning rates.\N\NIn Section 5, the authors give some analysis and explanations of this paper, which contain five sections. The relationship between a modulus of smoothness and the heat kernel is stated in Section 5.1. The authors show a view point that, in some concrete kernel function spaces, the RKHS approximation may be sum up to kernel convolutional approximation. In Section 5.2, the authors provide some properties of Jackson inequality. The authors show that there is a family of radial kernel functions whose best approximation error cannot be described with the classical moduli of smoothness. In Section 5.3, the authors provide the problem of deep RKHS approximation. In Section 5.4, the authors give some comments on the advantages of this paper. In Section 5.5, the authors give explicit expressions for some moduli of smoothness and reproducing kernels with convolutional operators.