CORAL: Constrained Oblique Rotation with Anchored Loadings for Fidelity-Constrained Decorrelation

Abstract

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 ${\rho }_{\star }(R)$ as the maximum common source fidelity compatible with exact decorrelation. Constructive lower bounds and rigorous analytical upper bounds tightly bracket ${\rho }_{\star }$ 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.