Embeddings of Reproducing Kernel Hilbert Spaces with General Weights
Provides theoretical embedding results for a class of RKHS that may be useful for practitioners working with high-dimensional function approximation, but the contribution is incremental and theoretical.
The paper studies embeddings between reproducing kernel Hilbert spaces with kernels formed by weighted tensor products of a univariate kernel, showing that a change of kernel can be compensated by transforming the weights. The results are applied to computational problems like numerical integration and function recovery.
We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic idea for the embeddings is to compensate a change of the univariate kernel by a suitable transformation of the weights. For the proofs we employ ($d \in \mathbb{N}$) and develop ($d = \infty$) a discrete calculus on the cone of all weights, where completely monotone weights play a particular role. We sketch how to apply the embedding results to computational problems, as, e.g., numerical integration or function recovery.