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Random Measures in Infinite-Dimensional Dynamics
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Palle E.T. Jorgensen, Feng Tian
A reproducing kernel Hilbert space (RKHS) is a Hilbert space H of functions on a prescribed set, say V, with the property that point-evaluation for functions f ∈ H is continuous with respect to the H-norm. They are called kernel spaces, because, for every x ∈ V, the point-evaluation for functions f ∈ H, f (x) must then be given as a H-inner product of f and a vector kx in H; called the kernel, i.e., f(x)=〈kx,f〉H, ∀f∈H, x ∈ V.
Regularization and Kernel Methods
Published in Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman, Data Science and Machine Learning, 2019
Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman
As can be seen from the proof of Theorem 6.2, the RKHS of functions corresponding to the linear kernel is the space of linear functions on ℝp. This space is isomorphic to ℝp itself, as discussed in the introduction (see also Exercise 12).
Fast and Exact Leave-One-Out Analysis of Large-Margin Classifiers
Published in Technometrics, 2022
Among many existing classification methods, the kernel support vector machine (Cortes and Vapnik SVM, 1995; Vapnik SVM, 1995, 1999) is widely recognized as one of the most competitive classifiers. With extensive numerical studies, Fernández-Delgado et al. (2014) declared that the kernel SVM is one of the best among hundreds of popular classifiers, in the same league as random forest, boosting ensemble, and neural nets. The statistical view of the SVM reveals its connection to nonparametric function estimation in a reproducing kernel Hilbert space (Hastie, Tibshirani, and Friedman RKHS, 2009), which also suggests a unified derivation of many kernel classifiers based on a penalized loss formulation. Let denote the class label in a binary classification problem. Given a random sample , the kernel SVM can be defined as a function estimation problemwhere is the so-called hinge loss and f is found within an RKHS with reproducing kernel K. The classification rule for x is . One can replace the hinge loss with other classification calibrated margin-based loss functions (Bartlett, Jordan, and McAuliffe 2006) in problem (1), and the resulting classifier is
Comment: From Ridge Regression to Methods of Regularization
Published in Technometrics, 2020
Although the use of RKHS in statistics can be traced back at least to the early 1960s, it gained tremendous popularity in the late 1990s and early 2000s with the rise of kernel methods in machine learning (see, e.g., Scholkopf and Smola 2001 and references therein). While the concept of RKHS may be abstract to many, kernels on the other hand are more intuitive—they are generalizations of the inner product in Euclidean spaces to a generic domain and can be viewed as a similarity measure between a pair of objects from the same domain. Fundamentally, any sensible prediction is based on the idea that similar inputs result in similar outcomes. While the similarity between outcomes is determined by the choice of a loss function, the similarity between inputs now rests on the choice of a kernel or similarity measure.
Monitoring the mean vector with Mahalanobis kernels
Published in Quality Technology & Quantitative Management, 2018
Edgard M. Maboudou-Tchao, Ivair R. Silva, Norou Diawara
If we can find a Mercer kernel K that computes a dot product in the feature space we are interested in, we can use the kernel evaluations to replace the dot products in the SVDD algorithm. The advantage is that in certain cases, one can find a Mercer kernel that allows one to efficiently simulate dot products in the high-dimensional feature space using computations in only the original, lower-dimensional space. The RKHS theory (Aronszajn, 1944) precisely states which kernel functions correspond to a dot product and which linear spaces are implicitly induced by these kernel functions. Although the corresponding feature space has infinite dimension, all computations can be performed without ever computing a feature vector.