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It is well known that kernel ridge regression (KRR) is a popularnonparametric regression estimator. Nonetheless, in the presence of a largedata set with size $n\gg 1,$ the KRR estimator has the drawback to require anintensive computational load. Recently, scalable KRR approaches have beenproposed with the aims to reduce the computational complexity of the KRR, whilemaintaining its superb convergence rate. In this work, we study a new scalableKRR 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 $\mathbbK,$ by its main $n\times N$ sub-matrix $A_N,$ where usually $N \ll n.$ Also, weshow that the TKRR works with $d-$dimensional random sampling data following anunknown probability law. To do so, we give a spectral analysis for the compactkernel integral operator, associated with a probability measure, different fromits usual probability measure. This decay estimate is then extended to thedecay of the tail of the trace of the associated random Gram matrix. A specialinterest is devoted to develop rules for the optimal choices of the involvedtruncation order $N$ and the value for regularization parameter $\lambda >0.$The proposed rules are based on the behavior and the decay rate of the spectrumof the positive integral operator, associated with the kernel $\mathbb K.$These optimal values of the parameters ensure that in terms of the empiricalrisk error, the TKRR and the full KRR estimators have the same optimalconvergence rate. Finally, we provide the reader with some numericalsimulations that illustrate the performance of our proposed TKRR estimator.

A Distribution Free Truncated Kernel Ridge Regression Estimator and Related Spectral Analyses