Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operatorvalued kernels. We propose a general principle for Operator-valued Random Four...

In this paper, some sharp inequalities for certain multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained. As an application, we obtain the (L p , L q)-norm inequalities and Morrey spaces boundedness for the multilinear operators.

We consider the problem of learning a vector-valued function f in an online learning setting. The function f is assumed to lie in a reproducing Hilbert space of operator-valued kernels. We describe two online algorithms for learning f while taking into account the output structure. A first contribution is an algorithm, ONORMA, that extends the standard kernel-based online learning algorithm NOR...

Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multipliers theorems, we obtain necessary and sufficient conditions to guarantee existence and uniqueness of periodic solutions to the equation d dt (Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our res...

In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, ...

Let T be the Calderón-Zygmund singular integral operator, a well-known result of Coifman et al. (see [6]) states that the commutator [b,T]( f ) = T(b f )− bT( f ) (where b ∈ BMO) is bounded on Lp(Rn) (1 < p <∞); Chanillo (see [1]) proves a similar result when T is replaced by the fractional integral operator; in [8, 9], these results on the TriebelLizorkin spaces and the case b ∈ Lipβ (where Li...

Let T be the singular integral operator, a well-known result of Coifman, Rochberg and Weiss [6] which states that the commutator [b,T ] = T (b f )− bT f (where b ∈ BMO) is bounded on Lp(Rn)(1 < p < ∞). Chanillo [1] proves a similar result when T is replaced by the fractional integral operator. In [9,11], these results on the Triebel–Lizorkin spaces and the case b∈Lipβ (where Lipβ is the homogen...