Directionally Paired Principal Component Analysis for Bivariate Estimation Problems.

We propose Directionally Paired Principal Component Analysis (DP-PCA), a novel linear dimension-reduction model for estimating coupled yet partially observable variable sets. Unlike partial least squares methods (e.g., partial least squares regression and canonical correlation analysis) that maximize correlation/covariance between the two datasets, our DP-PCA directly minimizes, either conditionally or unconditionally, the reconstruction and prediction errors for the observable and unobservable part, respectively. We demonstrate the optimality of the proposed DP-PCA approach, we compare and evaluate relevant linear cross-decomposition methods with data reconstruction and prediction experiments on synthetic Gaussian data, multi-target regression datasets, and a single-channel image dataset. Results show that when only a single pair of bases is allowed, the conditional DP-PCA achieves the lowest reconstruction error on the observable part and the total variable sets as a whole; meanwhile, the unconditional DP-PCA reaches the lowest prediction errors on the unobservable part. When an extra budget is allowed for the observable part's PCA basis, one can reach an optimal solution using a combined method: standard PCA for the observable part and unconditional DP-PCA for the unobservable part.