The representer theorem for Hilbert spaces: a necessary and sufficient condition
This work provides a more general theoretical foundation for representer theorems in Hilbert spaces, which is incremental but important for machine learning practitioners using regularization methods.
The authors tackled the problem of characterizing when regularization functionals admit a linear representer theorem, improving a prior result by removing the differentiability assumption and making the proof independent of dimensionality.
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. A recent characterization states that a general class of regularization functionals with differentiable regularizer admits a linear representer theorem if and only if the regularization term is a non-decreasing function of the norm. In this report, we improve over such result by replacing the differentiability assumption with lower semi-continuity and deriving a proof that is independent of the dimensionality of the space.