Approximate Representer Theorems in Non-reflexive Banach Spaces

by   Kevin Schlegel, et al.

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.


page 1

page 2

page 3

page 4


When is there a Representer Theorem? Reflexive Banach spaces

We consider a general regularised interpolation problem for learning a p...

Representer Theorems for Metric and Preference Learning: A Geometric Perspective

We explore the metric and preference learning problem in Hilbert spaces....

The (abc,pqr)-problem for Approximate Schauder Frames for Banach Spaces

Motivated from the complete solution of important abc-problem for Gabor ...

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

We consider a general regularised interpolation problem for learning a p...

A Generalized Representer Theorem for Hilbert Space - Valued Functions

The necessary and sufficient conditions for existence of a generalized r...

Galerkin Approximation In Banach and Hilbert Spaces

In this paper we study the conforming Galerkin approximation of the prob...

A Generalization of the Borkar-Meyn Theorem for Stochastic Recursive Inclusions

In this paper the stability theorem of Borkar and Meyn is extended to in...

Please sign up or login with your details

Forgot password? Click here to reset