VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM

By Claudio Carmeli, Ernesto De Vito, Alessandro Toigo. 2006

274 citations recorded by the source corpus.

We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.

Subfields: Applied Mathematics, Mathematical Physics, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Second, it improves the performance analysis of conditional mean embedding as one has access to a rich theory of vector-valued regression (Micchelli and Pontil 2005, Carmeli et al. 2006, Caponnetto and De Vito 2007, Caponnetto et al. 2008).”
Learning Theory for Distribution Regression (2014, background): “By this property with the strong continuity of K(·), the separability of H follows (Carmeli et al., 2006, Proposition 12: 2 ⇒ 1; Corollary 3).”
Conditional mean embeddings as regressors (2012, background): “In our case we have (Carmeli et al., 2006)[Prop.”
Kernels for Vector-Valued Functions: a Review (2011, background): “The deﬁnition of RKHS for vector valued functions parallels the one in the scalar, with the main difference that the reproducing kernel is now matrix valued, see for example [65, 19] .”
Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions (2016, background): “The machine learning community gradually introduced more and more variants of universal kernels (Steinwart, 2001; Micchelli et al., 2006; Carmeli et al., 2006; Caponnetto et al., 2008), but instead of also introducing variants of characteristic kernels, it stuck to the original deﬁnition given by…”
Elastic-net regularization in learning theory (2008, background): “4 of [33] (withK = Y, Ĥ = `2, γ (x) = Φ x and A = Φ) gives thatΦ is a partial isometry from `2 onto the reproducing kernel Hilbert space H , with reproducing kernel K(x, t).”
Learning Output Kernels with Block Coordinate Descent (2011, background): “See Micchelli & Pontil (2005); Carmeli et al. (2006) for further details.”
Kernel methods in system identification, machine learning and function estimation: A survey (2014, methodology): “Some theoretical results which corroborate its robustness, by finding connections with MSE minimization, are described in Aravkin, Burke, Chiuso, and Pillonetto (2012) and Carli, Chen, Chiuso, Ljung, and Pillonetto (2012). Once η is determined, following the Empirical Bayes paradigm (Berger, 1985; M”
Learning Sets with Separating Kernels (2012, background): “Denote by H the reproducing kernel Hilbert space with kernel K (cid:48) , and deﬁne the feature map Ψ : is a unitary operator with K (cid:48) x = Ψ ∗ (Ψ( x )) [14].”
Operator-valued Kernels for Learning from Functional Response Data (2015, background): “More recently, interest has grown in exploring reproducing Hilbert spaces of vector functions for learning vector-valued functions (Micchelli and Pontil, 2005a; Carmeli et al., 2006; Caponnetto et al., 2008; Carmeli et al., 2010; Zhang et al., 2012), even though the idea of extending the theory of R”
Cross-Domain Object Recognition Via Input-Output Kernel Analysis (2013, methodology): “Different from previous works that model the cla ss label as scalar and use a scalar function as the predictive function, the proposed algorithm represents the first attem p to investigate domain adaptation with the analysis of the RKHS for vector-valued functions, inspired by [16], [17], [30].”
Robust Point Matching via Vector Field Consensus (2014, background): “We review the basic theory of vector-valued reproducing kernel Hilbert space, and for further details and references we refer to [37] and [10].”
Multiple Operator-valued Kernel Learning (2012, background): “K F is positive deﬁnite on G x if it is Hermitian and for every natural number r and all { ( w i , u i ) i =1 ,...,r } ∈ G x × G y , i,j K F ( w i , w j ) u j , u i i G y ≥ 0 .”
Vector Field Learning via Spectral Filtering (2010, result): “The development of the theory of RKHS in the vector case is essentially the same as in the scalar case and we refer to [1,19] for further details and references.”
Multi-output learning via spectral filtering (2012, methodology): “A general class of matrix-valued kernels (Evgeniou et al. 2005), that can be related to specific regularizers, is composed of kernels of the form: Γ ( x, x ′ ) = K(x, x ′)A (19) where K is a scalar kernel and A a positive semidefinite d × d matrix that encodes how the outputs are related.”
Non-Rigid Point Set Registration by Preserving Global and Local Structures (2015, unknown): “…maps evx : H → Y (i.e., evx(f) = f(x)) are bounded, i.e., if ∀x ∈ X there exists a positive constant Cx such that ‖evx(f)‖ = ‖f(x)‖ ≤ Cx‖f‖H, ∀f ∈ H. (15) A reproducing kernel : X × X → B(Y) is then defined as: (x, x′) := evxev∗x′ , where B(Y) is the Banach space of bounded linear operators (i.e.,…”