Effect of dimensionality on convergence rates of kernel ridge regression estimator

Despite the curse of dimensionality, kernel ridge regression often exhibits good performance in practical applications, even when the dimension is moderately large. However, it has been shown that kernel ridge regression cannot be free from the curse of dimensionality. Until now, the literature on kernel ridge regression has suggested that the gap between theory and practice in relation to dimensionality has not narrowed. In this study, we first investigate when the influence of dimensionality does not significantly affect the convergence rate of the kernel ridge regression. Specifically, we study the convergence rate of and risks for the kernel ridge estimator, with a focus on reproducing kernel Hilbert space (RKHS) generated by a product kernel. We show that the univariate optimal convergence rate up to a logarithmic factor in and risks can be achieved by controlling the size of the RKHS. The result of a numerical study confirms our theoretical findings.