On the Influence of the Kernel on the Consistency of Support Vector Machines

By Ingo Steinwart. 2001

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In Statistical Learning Theory

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Assuming that X is compact, the RKHS H induced by the kernel k is dense in F if k is universal in the sense of Steinwart (2002).”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “If, on the other hand, our RKHS is sufﬁciently rich [e.g., associated with a universal kernel (Steinwart 2002)], we can use (19) to approximate, for instance, the probability of any interval ( a , b ) on a bounded domain, by approximating the indicator function I ( a , b ) as a kernel expansion…”
Learning from Distributions via Support Measure Machines (2012, neutral): “Then, for every function f ∈ C(X ) and every ε > 0 there exists a function g ∈ H induced by k with supx∈X |f(x)−g(x)| ≤ ε [13].”
Kernel Belief Propagation (2011, methodology): “If we can estimate this quantity directly from data, we can perform message updates via a simple inner product, avoiding a two-step procedure where the conditional distribution is first estimated and the expectation then taken.”
Kernel Bayes' Rule (2010, background): “Gaussian kernels, however, are known to be able to approximate any continuous function on a compact subset with arbitrary accuracy [23].”
Generative and Latent Mean Map Kernels (2010, background): “The operator μ maps distributions to elements of the RKHS and is injective for RKHSs induced by universal kernels [21], such as the Gaussian radial basis function (RBF) kernel k(x, y) = exp(−λ‖x − y‖(2)) and the Laplace kernel k(x, y) = exp(−β ∑n i=1 |xi − yi|) for λ, β ∈ R.”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “the uniform norm, that is, for every function g ∈C(X) and all ε > 0, there exists an f ∈H such that ‖ f −g‖u ≤ ε (Steinwart, 2001).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “, B2n+1splines (Steinwart, 2001), which can be shown to be characteristic.”
A Kernel Method for the Two-Sample-Problem (2006, methodology): “We shall use the unit balls in universal reproducing kernel Hilbert spaces [22] as our function class, since these will be shown to satisfy both of the foregoing properties.”
Kernel dimension reduction in regression (2009, background): “A positive denite kernel on a compact set is called universal if the corresponding RKHS is dense in the Banach space of continuous functions with sup norm [ 23 ].”
Kernel methods in machine learning (2007, neutral): “This can be made precise using the concept of universal kernels (Steinwart [128], cf.”
Domain Adaptive Neural Networks for Object Recognition (2014, methodology): “Unlabeled images from both source and target domains are considered as inputs to the DAE pretraining.”
A kernel-based causal learning algorithm (2007, background): “Furthermore, it is known that ΣY X =0 ⇔ X⊥ Y for universal kernels, in the sense of (Steinwart, 2001), or for Gaussian kernels on the entire IR, proved by Bach and Jordan (2002).”
Deep Model Based Domain Adaptation for Fault Diagnosis (2017, background): “The MMD definition is written as follows: MMD(Xs ,Xt) = ∥ ∥ ∥ ∥ ∥ ∥ 1 ns ns∑ i=1 φ (xsi ) − 1 nt nt∑ j=1 φ ( xtj ) ∥ ∥ ∥ ∥ ∥ ∥ H (2) where H is a universal RKHS [30], φ : X,Y → H.”
Density-ratio matching under the Bregman divergence: a unified framework of density-ratio estimation (2012, methodology): “The basic idea of KMM is essentially the same as the finite-order approach, but a universal reproducing kernel K(x,x′) (Steinwart, 2001) is used as a non-linear transformation.”
Direct importance estimation for covariate shift adaptation (2008, background): “The basic idea of KMM is to find ŵ(x) such that the mean discrepancy between nonlinearly transformed samples drawn from pte(x) and ptr(x) is minimized in a universal reproducing kernel Hilbert space (Steinwart, 2001).”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “The notion of c-universality was introduced by [42], who showed that many widely-used kernels are c-universal on compact sets.”
Nonparametric Scoring Rules (2015, background): “A kernel K is universal if its RKHS is dense in C(Ω), the set of continuous real functions on Ω (Steinwart 2002).”
Multicategory Support Vector Machines , Theory , and Application to the Classification of Microarray Data and Satellite Radiance Data (2004, background): “Theoretical justifications of the SVM in Vapnik’s structural risk minimization approach can be found in Vapnik (1995), and Vapnik (1998). These arguments are based on upper bounds of its generalization error in terms of VapnikChervonenkis dimensions, which are often too pessimistic to explain the su”