Improving Improved Kernel PLS

Improved Kernel Partial Least Squares (IKPLS) algorithms 1 and 2 are among the fastest PLS calibration algorithms. This article focuses on two shared steps, the computation of the X rotations, R, and the Y loadings, Q, and accelerates both. For R, term-by-term accumulation is replaced by a direct evaluation strategy that requires the same number of multiplications but parallelizes better on modern hardware. For Q, I identify - to the best of my knowledge, for the first time - equivalences showing that each Y loading is obtainable, up to explicitly derived constants, from quantities already computed earlier in the same iteration, and I exploit them in IKPLS to reduce the cost of each loading from Thetaleft(KMright) to Thetaleft(Mright) operations whenever M = 1 or 2 ≤ M < K, with K predictor variables (number of columns in X) and M response variables (number of columns in Y). Both improvements provably yield exactly the same W, P, Q, R, and T as the original algorithms. Benchmarks with NumPy (CPU) and JAX (GPU) show speedups of up to two orders of magnitude for the isolated steps and of approximately 2× (CPU) and 6× (GPU) for entire fits. Both improvements are implemented in the free, open-source Python package textttikpls.