Theory of Reproducing Kernels.

By N. Aronszajn. 1950

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In Statistical Learning Theory

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “The positive definiteness of the kernel function guarantees the existence of a dot product space H and a mapping φ : X → H such that k(x,y) = 〈φ(x), φ(y)〉H (Aronszajn 1950) without needing to compute φ explicitly (Boser et al. 1992, Cortes and Vapnik 1995, Vapnik 2000, Schölkopf and Smola 2002).”
Minimax Estimation of Kernel Mean Embeddings (2016, background): “However, note that Kh need not be positive definite (and therefore need not be the reproducing kernel of some RKHS).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, background): “…applied to various problems, such as density estimations (Smola et al., 2007; Song et al., 2008; McCalman et al., 2013), hypothesis tests (two sample test, Gretton et al. 2012, indepen- c©2014 Yu Nishiyama and Kenji Fukumizu. dence test, Gretton and Györfi 2010, conditional independence test,…”
Filtering with State-Observation Examples via Kernel Monte Carlo Filter (2013, background): “1 Any positive-definite kernel is uniquely associated with a reproducing kernel Hilbert space (RKHS) (Aronszajn, 1950).”
Density Estimation in Infinite Dimensional Exponential Families (2013, background): “…generalization [7, 14] of (1) as where the function space F is deﬁned as A ( f ) is the cumulant generating function and ( H , h· , ·i H ) is a reproducing kernel Hilbert space (RKHS) [1] with k as its reproducing kernel (see Section 2 for deﬁnitions, and a more detailed introduction to the RKHS).”
On the optimal estimation of probability measures in weak and strong topologies (2013, background): “Here H k denotes a reproducing kernel Hilbert space (RKHS) [2] (also see [5] and [35, Chapter 4] for a nice introduction to RKHS and its applications in probability, statistics and learning theory) with k : R d × R d → R as the reproducing kernel (and therefore positive deﬁnite) and K is a cone of…”
Kernels for Vector-Valued Functions: a Review (2011, background): “This latter concept plays a key role, so we review a few essential concepts (see [87, 6, 105, 25]).”
Kernel Belief Propagation (2011, methodology): “We begin with a description of the properties of a message mut(xt), given it is in the reproducing kernel Hilbert space (RKHS) F of functions on the separable metric space X (Aronszajn, 1950; Schölkopf & Smola, 2002).”
Kernel Bayes' Rule (2010, methodology): “We begin with a review of some basic concepts and tools concerning statistics on RKHS [1, 3, 6, 7].”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “y between f (which is learned from the training data) and f⋆goes to zero (in some sense) as the sample size goes to inﬁnity. Since X j∈Nn cjk(·,xj) : n∈ N,{cj} ⊂ R,{xj} ⊂ X is dense in H (Aronszajn, 1950), and assuming that the kernel-based algorithm makes f “converge to an appropriate functio”
InfoMax-GAN: Improved Adversarial Image Generation via Information Maximization and Contrastive Learning (2020, neutral): “RKHS dimensions.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, methodology): “Keywords: Probability metrics, Homogeneity tests, Independence tests, Kernel methods, Universal kernels, Characteristic kernels, Hilbertian metric, Weak topology.”
On the mathematical foundations of learning (2001, neutral): “Aronszajn [2].”
Kernel dimension reduction in regression (2009, methodology): “e Borel σ-ﬁeld is always assumed. Let (HX,kX) and (HY,kY) be RKHSs of functions on X and Y, respectively, with measurable positive deﬁnite kernels 6 K. FUKUMIZU, F. R. BACH AND M. I. JORDAN kX and kY [1]. We consider a random vector (X,Y):Ω → X ×Y with the law P XY . The marginal distribution of Xan”
Testing for Homogeneity with Kernel Fisher Discriminant Analysis (2007, background): “To each point x ∈ X , there corresponds an element Φ( x ) ∈ H such that Φ( x ) , f i H = f ( x ) for all f ∈ H , and Φ( x ) , Φ( y ) i H = k ( x, y ), where k : X × X → R is a positive deﬁnite kernel (Aronszajn, 1950).”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, background): “Since K̂ is symmetric and positive definite, let Ĥ be the corresponding reproducing kernel Hilbert space, whose elements are real functions on X̂ [2].”
Kernel methods in machine learning (2007, neutral): “RKHSs from many L 2 Hilbert spaces]. From the point evaluation functional, one can then construct the reproducing kernel using the Riesz representation theorem. The Moore–Aronszajn theorem (Aronszajn [7]) states that, for every positive deﬁnite kernel on X×X, there exists a unique RKHS and vice vers”
Hidden physics models: Machine learning of nonlinear partial differential equations (2017, background): “, covariance function) gives rise to a Reproducing Kernel Hilbert Space (RKHS) [14, 15, 16] that defines a class of functions that can be represented by this kernel.”