Approximate Representer Theorems in Non-reflexive Banach Spaces

The representer theorem is one of the most important mathematical foundations for regularised learning and kernel methods. Classical formulations of the theorem state sufficient conditions under which a regularisation problem on a Hilbert space admits a solution in the subspace spanned by the representers of the data points. This turns the problem into an equivalent optimisation problem in a finite dimensional space, making it computationally tractable. Moreover, Banach space methods for learning have been receiving more and more attention. Considering the representer theorem in Banach spaces is hence of increasing importance. Recently the question of the necessary condition for a representer theorem to hold in Hilbert spaces and certain Banach spaces has been considered. It has been shown that a classical representer theorem cannot exist in general in non-reflexive Banach spaces. In this paper we propose a notion of approximate solutions and approximate representer theorem to overcome this problem. We show that for these notions we can indeed extend the previous results to obtain a unified theory for the existence of representer theorems in any general Banach spaces, in particular including $l_1$-type spaces. We give a precise characterisation when a regulariser admits a classical representer theorem and when only an approximate representer theorem is possible.