  It is well known that kernel ridge regression (KRR) is a popular
nonparametric regression estimator. Nonetheless, in the presence of a large
data set with size $n\gg 1,$ the KRR estimator has the drawback to require an
intensive computational load. Recently, scalable KRR approaches have been
proposed with the aims to reduce the computational complexity of the KRR, while
maintaining its superb convergence rate. In this work, we study a new scalable
KRR based
  approach for nonparametric regression. Our truncated kernel ridge regression
(TKRR) approach is simple. It is based on substituting the full $n\times n$
random kernel or Gram matrix $B_n,$ associated with a Mercer's kernel $\mathbb
K,$ by its main $n\times N$ sub-matrix $A_N,$ where usually $N \ll n.$ Also, we
show that the TKRR works with $d-$dimensional random sampling data following an
unknown probability law. To do so, we give a spectral analysis for the compact
kernel integral operator, associated with a probability measure, different from
its usual probability measure. This decay estimate is then extended to the
decay of the tail of the trace of the associated random Gram matrix. A special
interest is devoted to develop rules for the optimal choices of the involved
truncation order $N$ and the value for regularization parameter $\lambda >0.$
The proposed rules are based on the behavior and the decay rate of the spectrum
of the positive integral operator, associated with the kernel $\mathbb K.$
These optimal values of the parameters ensure that in terms of the empirical
risk error, the TKRR and the full KRR estimators have the same optimal
convergence rate. Finally, we provide the reader with some numerical
simulations that illustrate the performance of our proposed TKRR estimator.
