reproducing kernel hilbert spaces
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Author(s):  
Osval Antonio Montesinos López ◽  
Abelardo Montesinos López ◽  
Jose Crossa

AbstractThe fundamentals for Reproducing Kernel Hilbert Spaces (RKHS) regression methods are described in this chapter. We first point out the virtues of RKHS regression methods and why these methods are gaining a lot of acceptance in statistical machine learning. Key elements for the construction of RKHS regression methods are provided, the kernel trick is explained in some detail, and the main kernel functions for building kernels are provided. This chapter explains some loss functions under a fixed model framework with examples of Gaussian, binary, and categorical response variables. We illustrate the use of mixed models with kernels by providing examples for continuous response variables. Practical issues for tuning the kernels are illustrated. We expand the RKHS regression methods under a Bayesian framework with practical examples applied to continuous and categorical response variables and by including in the predictor the main effects of environments, genotypes, and the genotype ×environment interaction. We show examples of multi-trait RKHS regression methods for continuous response variables. Finally, some practical issues of kernel compression methods are provided which are important for reducing the computation cost of implementing conventional RKHS methods.


2021 ◽  
Vol 15 (5) ◽  
Author(s):  
Monika Drewnik ◽  
Tomasz Miller ◽  
Zbigniew Pasternak-Winiarski

AbstractThe aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.


2021 ◽  
Vol 15 (4) ◽  
Author(s):  
Franciszek Hugon Szafraniec

AbstractThis short note has to make alive the overlooked work of Pedrick. It is rather a tour guide than exhausted examination of the content and intends to serve potential explorers of diverse kinds of reproducing kernel (Hilbert) spaces, the topic mushrooming nowadays.


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