Vector valued reproducing kernel Hilbert spaces and universality

By Claudio Carmeli, Ernesto De Vito, Alessandro Toigo, Veronica Umanità. 2008

236 citations recorded by the source corpus.

This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

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In Applied Mathematics

In Computational Mechanics

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Citations in the corpus

A Kernel Test of Goodness of Fit (2016, background): “We now prove the claims made in the previous section.”
Learning Theory for Distribution Regression (2014, background): “For a K : X × X → L(Y ) operator-valued kernel, the same properties hold (Carmeli et al., 2010, Proposition 3) if (i) K is a Mercer kernel (H = H(K) ⊆ C(X,Y ) = {X → Y continuous functions}), (ii) K is bounded (∃BK ∞ such that ‖K(x, x)‖L(Y ) ≤ BK), and (iii) K(x, x) is a compact operator for all…”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, methodology): “Recently, this has been answered by Carmeli et al. (2009, Deﬁnition 2, Theorem 1) and Sriperumbudur et al. (2010), wherein they deﬁned k to be c 0 -universal if k is bounded, k ( , x ) ∈ C 0 ( X ) , ∀ x ∈ X and its corresponding RKHS, H is dense in C 0 ( X ) w.r.t. the uniform norm, where X is a…”
A Consistent Regularization Approach for Structured Prediction (2016, background): “[8] Ingo Steinwart, Andreas Christmann, et al.”
Operator-valued Kernels for Learning from Functional Response Data (2015, methodology): “…matrix-valued kernels for infinite-dimensional output spaces, commonly known as operator-valued kernels, have been considered in previous studies (Micchelli and Pontil, 2005a; Caponnetto et al., 2008; Carmeli et al., 2006, 2010); however, they have been only studied in a theoretical perspective.”
Measuring Sample Quality with Kernels (2017, background): “3) established that if k is C0universal (Carmeli et al., 2010, Defn. 4.1) or integrally strictly positive definite (ISPD, Stewart, 1976, Sec. 6) and Eμ[k0(X,X) + ‖∇ log p(X) r(X)‖ 2 2] < ∞ for k0 , ∑d j=1 k j 0, then S(μ, TP ,Gk) = 0 only if μ = P . However, this property is insufficient to conclude”
Vector Field Learning via Spectral Filtering (2010, background): “In particular, note that the results in [31] allow to work on a non compact domain, at least if the kernel is well-behaved.”
Learning Sets with Separating Kernels (2012, background): “Note that in a supervised learning framework a similar role is played by the so called universal kernels [15, 62].”
Kernel Implicit Variational Inference (2017, methodology): “This is due to the accurate estimation guaranteed by the global optimum in the RKHS, which is a universal family when RBF kernels are used (Carmeli et al., 2010).”
Input Output Kernel Regression (2016, background): “…regression problems, operator-valued kernels (OVK) have then been studied under the angle of their universality (Caponnetto et al. (2008); Carmeli et al. (2010)) and developed in di ↵ erent contexts such as structured classiﬁcation (Dinuzzo et al., 2011), functional regression (Kadri et…”
Matrix-valued kernels for shape deformation analysis (2013, background): “Much more recently, Carmeli et al. [15] derived regularity results of vector-valued RKHS and analyzed some of their properties from the point of view of the machine learning community.”
Multiple Operator-valued Kernel Learning (2012, background): “Recently, there has been considerable interest in estimating vector-valued functions [21, 5, 7].”
A Tutorial on Environment-Aware Communications via Channel Knowledge Map for 6G (2023, background): “Next, different from UAVs that can fly freely over the 3D space, vehicles/robots normally have constrained moving paths and trajectories, thus making the trajectory planning more challenging.”
Calibration tests in multi-class classification: A unifying framework (2019, background): “A more detailed general introduction to RKHSs of vector-valued functions can be found in the publications by Caponnetto et al. (2008); Carmeli et al. (2010); Micchelli and Pontil (2005).”