The representer theorem for Hilbert spaces: a necessary and sufficient condition
By Francesco Dinuzzo, Bernhard Schölkopf. 2012
98 citations recorded by the source corpus.
The representer theorem is a property that lies at the foundation of regularization theory and kernel methods. A class of regularization functionals is said to admit a linear representer theorem if every member of the class admits minimizers that lie in the finite dimensional subspace spanned by the representers of the data. A recent characterization states that certain classes of regularization functionals with differentiable regularization term admit a linear representer theorem for any choice of the data if and only if the regularization term is a radial nondecreasing function. In this paper, we extend such result by weakening the assumptions on the regularization term. In particular, the main result of this paper implies that, for a sufficiently large family of regularization functionals, radial nondecreasing functions are the only lower semicontinuous regularization terms that guarantee existence of a representer theorem for any choice of the data.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis
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In Computational Theory and Mathematics
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Citations in the corpus
- On Decomposing the Proximal Map (2013, background): “3 that when the subdifferential of one function is scaling invariant, then the prox-decomposition always holds if and only if another function is radial—which is, quite unexpectedly, exactly the same condition proven recently for the validity of the representer theorem in the context of kernel metho”