发表机构
FOSS Analytical A/S（FOSS分析有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究聚焦改进的核偏最小二乘法中X旋转矩阵R和Y载荷矩阵Q的计算，通过新策略加速R计算，利用等价关系降低Q计算成本，经基准测试验证改进可显著加速，且已在开源包中实现。
AI 中文摘要
改进的核偏最小二乘法（IKPLS）算法1和2是最快的偏最小二乘校准算法之一。本文聚焦于两个共享步骤，即X旋转矩阵R和Y载荷矩阵Q的计算，并对两者进行加速。对于R，逐元素累加被直接评估策略取代，该策略乘法次数相同但在现代硬件上并行性更好。对于Q，首次发现等价关系，表明每个Y载荷在同一迭代中可由先前计算的量得到，利用此关系将每次载荷计算成本从Θ(KM)降至Θ(M)操作（当M = 1或2 ≤ M < K时）。两种改进证明能产生与原始算法相同的W、P、Q、R和T。NumPy（CPU）和JAX（GPU）基准测试显示，孤立步骤加速高达两个数量级，整体拟合加速约2倍（CPU）和6倍（GPU）。两种改进均在免费开源的Python包ikpls中实现。
英文摘要
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 $\mathbf{X}$ rotations, $\mathbf{R}$, and the $\mathbf{Y}$ loadings, $\mathbf{Q}$, and accelerates both. For $\mathbf{R}$, 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 $\mathbf{Q}$, I identify - to the best of my knowledge, for the first time - equivalences showing that each $\mathbf{Y}$ 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 $Θ\left(KM\right)$ to $Θ\left(M\right)$ operations whenever $M = 1$ or $2 \leq M < K$, with $K$ predictor variables (number of columns in $\mathbf{X}$) and $M$ response variables (number of columns in $\mathbf{Y}$). Both improvements provably yield exactly the same $\mathbf{W}$, $\mathbf{P}$, $\mathbf{Q}$, $\mathbf{R}$, and $\mathbf{T}$ 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 $2\times$ (CPU) and $6\times$ (GPU) for entire fits. Both improvements are implemented in the free, open-source Python package \texttt{ikpls}.
