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

Constructing or learning a function from a finite number of sampled data\npoints (measurements) is a fundamental problem in science and engineering. This\nis often formulated as a minimum norm interpolation problem, regularized\nlearning problem or, in general, a semi-discrete inverse problem, in certain\nfunctional spaces. The choice of an appropriate space is crucial for solutions\nof these problems. Motivated by sparse representations of the reconstructed\nfunctions such as compressed sensing and sparse learning, much of the recent\nresearch interest has been directed to considering these problems in certain\nBanach spaces in order to obtain their sparse solutions, which is a feasible\napproach to overcome challenges coming from the big data nature of most\npractical applications. It is the goal of this paper to provide a systematic\nstudy of the representer theorems for these problems in Banach spaces. There\nare a few existing results for these problems in a Banach space, with all of\nthem regarding implicit representer theorems. We aim at obtaining explicit\nrepresenter theorems based on which convenient solution methods will then be\ndeveloped. For the minimum norm interpolation, the explicit representer\ntheorems enable us to express the infimum in terms of the norm of the linear\ncombination of the interpolation functionals. For the purpose of developing\nefficient computational algorithms, we establish the fixed-point equation\nformulation of solutions of these problems. We reveal that unlike in a Hilbert\nspace, in general, solutions of these problems in a Banach space may not be\nable to be reduced to truly finite dimensional problems (with certain infinite\ndimensional components hidden). We demonstrate how this obstacle can be\nremoved, reducing the original problem to a truly finite dimensional one, in\nthe special case when the Banach space is $\\ell_1(\\mathbb{N})$.\n