CORAL：带锚定载荷的约束斜旋转，用于保真度约束的去相关
CORAL: Constrained Oblique Rotation with Anchored Loadings for Fidelity-Constrained Decorrelation
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中文总结 AI 辅助
CORAL是带锚定载荷的约束斜旋转算法，可在最小化残余互相关的同时保证变换变量与指定源的最低相关性，其在多变量系统去相关中能大幅提升源变量保真度，优于PCA。
中文摘要 AI 辅助
对多变量系统进行去相关无需破坏源变量的同一性。我们引入带锚定载荷的约束斜旋转（Constrained Oblique Rotation with Anchored Loadings，CORAL），该方法在最小化残余互相关的同时，保证每个变换后的变量与其指定源之间的相关性不低于声明的最小值。对于p变量相关矩阵R，我们证明每个精确去相关器均可表示为R^{-1/2}Q（其中Q为正交矩阵），并将ρ_⋆(R)定义为与精确去相关兼容的最大公共源保真度。在p=6、18、50的模拟中，ρ_⋆分别被构造性下界和严格分析上界紧密约束在[0.972,0.976]、[0.959,0.962]、[0.949,0.950]区间，而PCA在不同主成分与匹配源变量间可实现的最大最小相关性仅为0.358、0.329、0.172。对应区间在世界发展指标数据集上为[0.751,0.758]，在葡萄酒化学数据集上为[0.826,0.835]。因此，源变量同一性的损失并非精确去相关的固有属性，而是取决于所选去相关器。CORAL采用约束黎曼优化，且可扩展至精确支持限制。
英文摘要
Decorrelating a multivariate system need not destroy source-variable identity. We introduce Constrained Oblique Rotation with Anchored Loadings (CORAL), which minimizes residual cross-correlation while guaranteeing a declared minimum correlation between each transformed variable and its designated source. For a p-variable correlation matrix $R$, we show that every exact decorrelator can be written as $R^{-1/2}Q$ for some orthogonal matrix $Q$, and define $ρ_\star(R)$ as the maximum common source fidelity compatible with exact decorrelation. Constructive lower bounds and rigorous analytical upper bounds tightly bracket $ρ_\star$ at [0.972,0.976], [0.959,0.962], and [0.949,0.950] in simulations with p={6,18,50}, respectively, compared with PCA's largest achievable minimum correlation between distinct principal components and matched source variables of 0.358, 0.329, and 0.172. Corresponding intervals are [0.751,0.758] for World Development Indicators and [0.826,0.835] for wine chemistry data sets. Thus, loss of source-variable identity is not inherent to exact decorrelation but depends on the decorrelator selected. CORAL uses constrained Riemannian optimization and extends to exact support restrictions.
