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Cross-validation is a widely used technique for assessing the performance ofpredictive models on unseen data. Many predictive models, such as Kernel-BasedPartial Least-Squares (PLS) models, require the computation of$\mathbf{X}^{\mathbf{T}}\mathbf{X}$ and $\mathbf{X}^{\mathbf{T}}\mathbf{Y}$using only training set samples from the input and output matrices,$\mathbf{X}$ and $\mathbf{Y}$, respectively. In this work, we present threealgorithms that efficiently compute these matrices. The first one allows nocolumn-wise preprocessing. The second one allows column-wise centering aroundthe training set means. The third one allows column-wise centering andcolumn-wise scaling around the training set means and standard deviations.Demonstrating correctness and superior computational complexity, they offersignificant cross-validation speedup compared with straight-forwardcross-validation and previous work on fast cross-validation - all without dataleakage. Their suitability for parallelization is highlighted with anopen-source Python implementation combining our algorithms with Improved KernelPLS.

Shortcutting Cross-Validation: Efficiently Deriving Column-Wise Centered  and Scaled Training Set $\mathbf{X}^\mathbf{T}\mathbf{X}$ and  $\mathbf{X}^\mathbf{T}\mathbf{Y}$ Without Full Recomputation of Matrix  Products or Statistical Moments