Reproducing kernel Hilbert spaces in probability and statistics

By Alain F. Berlinet, Christine Thomas-Agnan. 2004

1,893 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics, Statistical Learning Theory

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Citations in the corpus

Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering (2015, methodology): “ 2Halready. If 1 2H= , we dene H0to be the Hilbert sum of the RKHS Hand the one generated by the constant kernel 1 (and thus H 0is a RKHS with kernel k = 1 + kwhere kis the original kernel for H; seeBerlinet and Thomas-Agnan (2004, Thm. 5)). We can show that running the Frank-Wolfe algorithm using t”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Detailed exposition on RKHSs can be found in Schölkopf and Smola (2002) and Berlinet and Thomas-Agnan (2004), for example.”
Prototypal Analysis and Prototypal Regression (2017, background): “Probability distributions or samples thereof can also be mapped to a reproducing kernel Hilbert space via kernel embedding (see Berlinet and Thomas-Agnan, 2004; Gretton et al., 2006; Smola et al., 2007; Sriperumbudur et al., 2010; Sejdinovic et al., 2012; Muandet et al., 2017).”
Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families (2015, background): “Consequently, the stationary distribution of the chain remains correct, given that we take care when adapting the surrogate.”
On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, background): “This RKHS has two important characteristic properties (see, e.g., Berlinet and Thomas-Agnan, 2004): (a) Membership of kernel evaluations : for any x ∈ X , the function k ( , x ) : y 7→ k ( y, x ) is an element of F .”
K2-ABC: Approximate Bayesian Computation with Infinite Dimensional Summary Statistics via Kernel Embeddings (2015, methodology): “The problem is equivalent to learning a regression function in the RKHSs of s(y) and θ induced by their respective kernels (Grünewälder et al., 2012).”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “…(2008, 2009); an example thereof is the Gaussian kernel (26) The kernel map of a distribution p can be deﬁned as the expectation of the feature map (Berlinet and Agnan 2004; Smola et al. 2007; Gretton et al. 2012), where we overload the symbol μ and assume, here and below, that p is a Borel…”
Learning Theory for Distribution Regression (2014, background): “It is possible, however, to view set kernels in a distribution setting, as they represent valid kernels between (mean) embeddings of empirical probability measures into a reproducing kernel Hilbert space (RKHS; Berlinet and Thomas-Agnan, 2004).”
Kernel Mean Estimation via Spectral Filtering (2014, background): “) from P, the most common way to estimate the kernel mean is by replacing P by the empirical measure, Pn := 1 n ∑n i=1 δXi whereδx is a Dirac measure at x [5, 6].”
Control functionals for Monte Carlo integration (2014, background): “ ; 2H 0. In addition, equip H + with the norm kfk H+ := kck C+ k k H 0 , again, well-dened by uniqueness of representation. It can be shown that H + is a RKHS with kernel k +(x;x0) := 1 + k 0(x;x0) (Berlinet and Thomas-Agnan, 2004, Thm. 5, p24). For the analysis we assume a basic well-posedness cond”
A Wild Bootstrap for Degenerate Kernel Tests (2014, background): “The kernel embedding of a probability measure P on X is an element μk(P ) ∈ Hk, given by μk(P ) = ∫ k(·, x) dP (x) [3, 29].”
Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels (2014, background): “Under a natural choice of inner product, these spaces are calledPaley-Wiener spacesand they constitute a RKHS. Proposition 8 (Kernel of Paley-Wiener RKHS [Berlinet and Thomas-Agnan, 2004, Yao, 1967, Peloso, 2011]).”
Kernel Mean Shrinkage Estimators (2014, background): “The kernel mean has recently gained attention in the machine learning community, thanks to the introduction of Hilbert space embedding for distributions (Berlinet and Thomas-Agnan, 2004; Smola et al., 2007).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, background): “ss of the pdfs of P and mP . A scalar multiplication (c > 0) changes as follows: ¯mP := EX∼P [k¯(·,X)] = cEX∼P[k(·,X)] = cmP and hm¯P ,m¯QiH¯ = chmP ,mQiH, where hf,giH¯ = 1 c hf,giH, ∀f,g ∈ H,H¯ (Berlinet and Thomas-Agnan, 2004, p.37). 7 Nishiyamaand Fukumizu Conversely, for any γ ∈ Rd, symmetri”
A Kernel Independence Test for Random Processes (2014, background): “We associate to the random variable X a mean embedding µX(x) := EXk(X,x), such that ∀f ∈ HX , 〈f, µX〉HX = EX(f(X)) (Berlinet & Thomas-Agnan, 2004; Smola et al., 2007).”
On the optimal estimation of probability measures in weak and strong topologies (2013, methodology): “We start with the deﬁnition of an RKHS, which we quote from [5].”