When is there a Representer Theorem? Reflexive Banach spaces

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes the problem computationally tractable. Most literature deals only with sufficient conditions for representer theorems in Hilbert spaces. We prove necessary and sufficient conditions for the existence of representer theorems in reflexive Banach spaces and illustrate why in a sense reflexivity is the minimal requirement on the function space. We further show that if the learning relies on the linear representer theorem, then the solution is independent of the regulariser and in fact determined by the function space alone. This in particular shows the value of generalising Hilbert space learning theory to Banach spaces.Comment: 25 pages, 2 figures. arXiv admin note: text overlap with arXiv:1709.00084, arXiv:1804.0960

When is there a Representer Theorem? Nondifferentiable Regularisers and Banach spaces

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is the core of kernel methods in machine learning as it makes the problem computationally tractable. Necessary and sufficient conditions for differentiable regularisers on Hilbert spaces to admit a representer theorem have been proved. We extend those results to nondifferentiable regularisers on uniformly convex and uniformly smooth Banach spaces. This gives a (more) complete answer to the question when there is a representer theorem. We then note that for regularised interpolation in fact the solution is determined by the function space alone and independent of the regulariser, making the extension to Banach spaces even more valuable.Comment: 20 pages, 3 figure

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.Comment: 18 pages, 1 figur