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

Cross-validation is a widely used technique for assessing the performance of predictive models on unseen data. Many predictive models, such as Kernel-Based Partial 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 three algorithms that efficiently compute these matrices. The first one allows no column-wise preprocessing. The second one allows column-wise centering around the training set means. The third one allows column-wise centering and column-wise scaling around the training set means and standard deviations. Demonstrating correctness and superior computational complexity, they offer significant cross-validation speedup compared with straight-forward cross-validation and previous work on fast cross-validation - all without data leakage. Their suitability for parallelization is highlighted with an open-source Python implementation combining our algorithms with Improved Kernel PLS.