Representation by Integrating Reproducing Kernels
Provides a theoretical unification for kernel methods, benefiting researchers in machine learning and approximation theory by offering a general perspective on kernel construction and analysis.
The paper develops a framework for integrating parametrized families of reproducing kernels, unifying several known kernel constructions (Mercer kernels, integral transforms, mixtures, scale-mixtures) and generalizing results like the Kramer sampling theorem and representer theorem applications.
Based on direct integrals, a framework allowing to integrate a parametrised family of reproducing kernels with respect to some measure on the parameter space is developed. By pointwise integration, one obtains again a reproducing kernel whose corresponding Hilbert space is given as the image of the direct integral of the individual Hilbert spaces under the summation operator. This generalises the well-known results for finite sums of reproducing kernels; however, many more special cases are subsumed under this approach: so-called Mercer kernels obtained through series expansions; kernels generated by integral transforms; mixtures of positive definite functions; and in particular scale-mixtures of radial basis functions. This opens new vistas into known results, e.g. generalising the Kramer sampling theorem; it also offers interesting connections between measurements and integral transforms, e.g. allowing to apply the representer theorem in certain inverse problems, or bounding the pointwise error in the image domain when observing the pre-image under an integral transform.