Generalized support vector regression: duality and tensor-kernel representation
This work addresses the challenge of learning in Banach spaces for researchers in machine learning and functional analysis, offering a novel computational framework, but it appears incremental as it builds on existing support vector regression methods.
The paper tackles the variational problem of support vector regression in Banach function spaces by using Fenchel-Rockafellar duality to derive explicit dual formulations and optimality conditions, and introduces a tensor-kernel representation for computation, overcoming typical learning difficulties in such spaces, with applications to power series tensor kernels like generalizations of exponential and polynomial kernels.
In this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving the problem which relies on a tensor-kernel representation. This analysis overcomes the typical difficulties connected to learning in Banach spaces. We finally present a large class of tensor-kernels to which our theory fully applies: power series tensor kernels. This type of kernels describe Banach spaces of analytic functions and include generalizations of the exponential and polynomial kernels as well as, in the complex case, generalizations of the Szegö and Bergman kernels.