The representer theorem for Hilbert spaces: a necessary and sufficient condition

A family of regularization functionals is said to admit a linear representer theorem if every member of the family admits minimizers that lie in a fixed finite dimensional subspace. In this paper, we show that a general class of extended real-valued regularization functionals admits a linear representer theorem if and only if the regularizer is a non-decreasing function of the norm. In this way, we extend a recent characterization stating that such condition is necessary and sufficient for differentiable regularizers.