  In this paper we consider the problems of supervised classification and
regression in the case where attributes and labels are functions: a data is
represented by a set of functions, and the label is also a function. We focus
on the use of reproducing kernel Hilbert space theory to learn from such
functional data. Basic concepts and properties of kernel-based learning are
extended to include the estimation of function-valued functions. In this
setting, the representer theorem is restated, a set of rigorously defined
infinite-dimensional operator-valued kernels that can be valuably applied when
the data are functions is described, and a learning algorithm for nonlinear
functional data analysis is introduced. The methodology is illustrated through
speech and audio signal processing experiments.
