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            <funder>This work was supported by Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council and FEDER through the ‘Contrat de Projets Etat Region (CPER) 2007-2013’. H.K. acknowledges the support of a Junior Researcher Contract No. 4297 from the Nord-Pas-de-Calais region. H.K. and P.P. acknowledge the support of the French National Research Agency (ANR-09-EMER-007 project LAMPADA). H.K. was also supported in part by French grants from the CNRS-LAGIS (ANR KernSig Project). A.R. was supported by the PASCAL2 Network of Excellence, ICT-216886, ANR Project ASAP ANR-09-EMER-001 and the INRIA ARC MABI. P.P. also acknowledges the support of INRIA.</funder>
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              <p>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.</p>
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