Generalized continuum regression (GCR): An advanced multivariate method for precise dimensionality reduction and efficient regression modeling

The collinearity inherent in high-dimensional, low-sample-size (HDLSS) data critically undermines the accuracy of chemometric regression. We have proven that when predictor matrix $X$ has full row rank, the optimal dimensionality of X-block latent variables (LVs) in multivariate linear regression equals $rank\left(Y\right)$. Based on this theoretical foundation, we develop generalized continuum regression (GCR), an advanced multivariate regression method rooted in continuum canonical correlation (CCC). GCR's core innovation lies in the extension of CCC's scalar parameter $\alpha$ to a vector form for precise dimension reduction in multivariate regression. We also develop an efficient numeric algorithm for computational speed. Real-world implementations on two spectroscopic datasets confirm that GCR adopts LVs with dimensionality equal to the rank of $Y$, validating its precise dimensionality reduction. When compared to CCC regression (CCCR), GCR exhibits superior performance with: (1) a 7.28 %–43.70 % reduction in mean-squared error for validation (MSEV) when utilizing two or three latent variables (LVs); and (2) a 30 to 55-fold increase in solution speed. These findings highlight GCR's potential as a valuable tool for dimensionality reduction and regression modeling in chemometrics, specifically for HDLSS analysis.