When is there a representer theorem?

Kevin Schlegel ()
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Kevin Schlegel: University of Oxford

Journal of Global Optimization, 2019, vol. 74, issue 2, No 9, 415 pages

Abstract: Abstract 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. 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.

Keywords: Representer theorem; Regularised interpolation; Regularisation; Semi-inner product spaces; Kernel methods; 68T05 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10898-019-00767-0

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