Universal Kernels

By Charles A. Micchelli, Yuesheng Xu, Haizhang Zhang. 2006

688 citations recorded by the source corpus.

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Citations in the corpus

Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “Micchelli et al. (2006, Proposition 1) have shown that for any compact set Z ⊂ X , the embedding in (16) is injective if and only if the set K ( Z ) = span k ( , y ) : y ∈ Z is dense in C ( Z ) w.r.t. the uniform norm.”
Deep Kernel Learning (2015, background): “…with RBF kernels correspond to models which have an inﬁnite basis expansion in a dual space, and have compelling theoretical properties: these models are universal approxi-mators, and have prior support to within an arbitrarily small epsilon band of any continuous function (Micchelli et al., 2006).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “This means, if a translation invariant kernel on Rd is characteristic, then it is also universal in the sense of Micchelli et al. (2006), while the converse is not true.”
A Kernel Method for the Two-Sample-Problem (2006, background): “ our tests are able to employ universal kernels that cannot be written as inner products between Parzen windows, normalized or otherwise: several examples are given by Steinwart (2001, Section 3) and Micchelli et al. (2006, Section 3). We may further generalize to kernels on structured objects such ”
Adversarial Feature Matching for Text Generation (2017, background): “If a space X has a continuous bijective mapping λ : X → F , the composed kernel ˜ k : Micchelli et al. (2006), k is continuous and its RKHS H is dense in C ( F ) (cid:44) { g : F → R | g continuous } , i.e. , for any n points { f i } i ∈ N n ∈ F , for ∀ g ∈ C ( F ) , there exists a i : i ∈ N n that…”
Reproducing kernel Hilbert spaces (2019, background): “1 General Results Everything in this section is derived from [3].”
Scatter Component Analysis: A Unified Framework for Domain Adaptation and Domain Generalization (2015, background): “If we assume that we have a universal kernel [69], [70], i.”
Radial kernels and their reproducing kernel Hilbert spaces (2010, methodology): “[21] have used the integral representation (1) to show that all nonconstant radial kernels are universal for all compact subsets, in the sense that their RKHSs are dense in the Banach space C(X) of continuous functions.”
Learning to Score Behaviors for Guided Policy Optimization (2019, methodology): “First, we replace X and Y with the functions from a RKHS corresponding to universal kernels (Micchelli et al., 2006).”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “ work with a kernel kthat is c-universal. The notion of c-universality was introduced by Steinwart (2001), who showed that many widely-used kernels are c-universal on compact XRd. Indeed, Prop. 1 of Micchelli et al. (2006) proves that an reproducing kernel Hilbert space with kernel kis c-universal i”
Risk-Averse Bayes-Adaptive Reinforcement Learning (2021, background): “…Ξ × Ξ → R , where Ξ is the compact set of admissible adversary actions according to Equation 11 Because the Gaussian kernel is a “universal” kernel [23], the reproducing kernel Hilbert space (RKHS), H k , corresponding to kernel k is dense in C (Ξ) , the set of real-valued continuous functions on…”
Vector valued reproducing kernel Hilbert spaces and universality (2008, methodology): “Moreover, we present several examples of vector valued RKHS, most of them considered in [22, 5].”
Deep Convolutional Networks are Hierarchical Kernel Machines (2015, background): “In fact, under the hypothesis of normalization of the vectors x,t we have that 2 which is a universal kernel (see also th 17 of [21]).”
Spectral Algorithms for Supervised Learning (2008, methodology): “Finally, we assume that the kernel K is bounded by 1 and is universal (see Micchelli, Xu, & Zhang, 2006, and references therein), that is, the set of functions is dense in L 2 ( X ), the Hilbert space of functions that are square-integrable with respect to ρ X .”