Chapter

Learning Theory

Volume 3120 of the series Lecture Notes in Computer Science pp 255-269

A Function Representation for Learning in Banach Spaces

  • Charles A. MicchelliAffiliated withCarnegie Mellon UniversityDepartment of Mathematics and Statistics, State University of New York, The University at Albany
  • , Massimiliano PontilAffiliated withCarnegie Mellon UniversityDepartment of Computer Sciences, University College London

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Abstract

Kernel–based methods are powerful for high dimensional function representation. The theory of such methods rests upon their attractive mathematical properties whose setting is in Hilbert spaces of functions. It is natural to consider what the corresponding circumstances would be in Banach spaces. Led by this question we provide theoretical justifications to enhance kernel–based methods with function composition. We explore regularization in Banach spaces and show how this function representation naturally arises in that problem. Furthermore, we provide circumstances in which these representations are dense relative to the uniform norm and discuss how the parameters in such representations may be used to fit data.