Operator-Valued Kernels for Nonparametric Operator Estimation

We consider the supervised learning problem when both covariates and responses are real functions rather than scalars or finite dimensional vectors. In this setting, we aim at developing a sound and effective nonparametric operator estimation approach based on optimal approximation in reproducing kernel Hilbert spaces of function-valued functions. In a first step, we exhibit a class of operator-valued kernels that perform the mapping between two spaces of functions: this is the first contribution of this paper. Then, we show how to solve the problem of minimizing a regularized functional without discretizing covariate and target functions. Finally, we apply this framework to a standard functional regression problem