  Kernel ridge regression (KRR) is a well-known and popular nonparametric
regression approach with many desirable properties, including minimax
rate-optimality in estimating functions that belong to common reproducing
kernel Hilbert spaces (RKHS). The approach, however, is computationally
intensive for large data sets, due to the need to operate on a dense $n \times
n$ kernel matrix, where $n$ is the sample size. Recently, various approximation
schemes for solving KRR have been considered, and some analyzed. Some
approaches such as Nystr\"{o}m approximation and sketching have been shown to
preserve the rate optimality of KRR. In this paper, we consider the simplest
approximation, namely, spectrally truncating the kernel matrix to its largest
$r < n$ eigenvalues. We derive an exact expression for the maximum risk of this
truncated KRR, over the unit ball of the RKHS. This result can be used to study
the exact trade-off between the level of spectral truncation and the
regularization parameter. We show that, as long as the RKHS is
infinite-dimensional, there is a threshold on $r$, above which, the
spectrally-truncated KRR surprisingly outperforms the full KRR in terms of the
minimax risk, where the minimum is taken over the regularization parameter.
This strengthens the existing results on approximation schemes, by showing that
not only one does not lose in terms of the rates, truncation can in fact
improve the performance, for all finite samples (above the threshold).
Moreover, we show that the implicit regularization achieved by spectral
truncation is not a substitute for Hilbert norm regularization. Both are needed
to achieve the best performance.
