Kernel regression, minimax rates and effective dimensionality: beyond the regular case
This work addresses theoretical limitations in kernel regression for researchers in statistical learning, but appears incremental as it extends prior analyses under more general conditions.
The paper investigates whether kernel regularization methods can achieve minimax convergence rates under weaker assumptions on eigenvalue decay, moving beyond polynomial decay to allow for more complex data structures, but does not report specific numerical results.
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.