Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

Abstract

Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semi-discrete inverse problem, in certain functional spaces. The choice of an appropriate space is crucial for solutions of these problems. Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems in certain Banach spaces in order to obtain their sparse solutions, which is a feasible approach to overcome challenges coming from the big data nature of most practical applications. It is the goal of this paper to provide a systematic study of the representer theorems for these problems in Banach spaces. There are a few existing results for these problems in a Banach space, with all of them regarding implicit representer theorems. We aim at obtaining explicit representer theorems based on which convenient solution methods will then be developed. For the minimum norm interpolation, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is $\ell_1(\mathbb{N})$.