Kernel methods for learning, regularized interpolation, and inverse problems: A unified variational framework

We present a unified variational framework connecting kernel learning, regularized interpolation from noisy data, and linear inverse problems within a common infinite-dimensional approximation problem in a reproducing kernel Hilbert space (RKHS). The central object is a risk functional defined through a loss function, a family of linear measurement functionals, and a probability distribution on the data space. When the range of a compact linear operator carries a natural RKHS structure, we prove that minimizers of the approximation problem are related to normal minimizers and Tikhonov-regularized solutions of the associated inverse problem via the first isomorphism theorem. As a consequence, the representer theorem extends to classes of linear inverse problems whose ranges are identified with RKHSs. The statistical foundations rest on the interplay between the choice of loss function and the probabilistic assumptions on the data, connecting proper scoring rules and maximum likelihood within the same variational structure, and accommodating heteroscedastic noise models including Gaussian and Poisson data. Convergence of discrete approximations is established via the continuous argmax theorem in the statistical setting and via 𝛤 -convergence in the deterministic setting.