Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (Noyaux reproduisants)

By Laurent Schwartz. 1964

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Citations in the corpus

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]).”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, methodology): “However, to our knowledge, a thorough error analysis for regularized least-squares algorithm when the output space is a general Hilbert space had never been given before.”
VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM (2006, methodology): “Our results hold for RKH spaces of functions f : X → K where X is a measurable set and K is a Hilbert space, following the general setting of vector valued RKH spaces outlined in [7, 21, 26].”
Stochastic integration of functions with values in a Banach space (2005, background): “For more information on this topic we refer to [19, 21].”
Multi-output learning via spectral filtering (2012, methodology): “Similarly to the scalar case, it can be shown (Schwartz 1964), that for any given reproducing kernel Γ , a unique RKHS can be defined with Γ as its reproducing kernel.”
Kernel methods and the exponential family (2006, background): “The following corollary arises from [Schwartz, 1964, Proposition 23] and [Wahba, 1990, Theorem 1.1.1] Proposition 3 (Bijection between RKHS and Kernel) There is a bi-jection between the set of all possible RKHS and the set of all positive kernels.”
Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality (2012, background): “The theory and formalism of vector-valued RKHS can be traced as far back as the work of Laurent Schwarz in 1964 [32].”
Some Properties of Regularized Kernel Methods (2004, methodology): “Let K be the closed subspace of It is known (Schwartz, 1964) that, given s ∈ S , there is a unique From Eq.”
γ-Radonifying Operators: A Survey (2010, unknown): “The more general problem of radonification of cylindrical semimartingales has been covered by Badrikian and Üstünel [5], Schwartz [109] and Jakubowski, Kwapień, Raynaud de Fitte, Rosiński [55].”
Learning from Examples as an Inverse Problem (2005, unknown): “In particular it allowed to cast a large class of algorithms in a common framework, namely regularization networks or regularized kernel methods (Evgeniou et al., 2000; Schölkopf and Smola, 2002).”
Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels (2008, background): “The derivative reproducing properties of HK , stated in (2.1), (2.2), and (2.3) below, can be found in [ 13 ] and [11] where analytic kernels are also considered.”
On regularization algorithms in learning theory (2007, methodology): “Before recalling some basic facts on such spaces we discuss some implication of considering approximation schemes in a fixed hypothesis space and in particular in RKHSs.”
On the Capacity of Channels with Gaussian and Non-Gaussian Noise (1981, background): “Schwartz (1964) showed that if E is quasi-complete then each linear operator, R: E ' ~ E , which is symmetric and non-negative, has a unique Hilbert space H, which is a vector subspace of E, such that the injection j of H into E is continuous and R = j j * .”
Vector valued reproducing kernel Hilbert spaces and universality (2008, background): “Most of the results are well known for scalar kernels, whereas for vector valued kernels they are consequences of the theory developed in [27] in a more general context.”