Principal Component Analysis (PCA) is a widely used technique for dimensionality reduction in various problem domains including data compression, image processing, visualization, exploratory data analysis, pattern recognition, time series prediction and machine learning. Often, data is presented in a correlated paired manner such there exists observable and correlated unobservable measurements. Unfortunately, traditional PCA techniques generally fail to optimally capture the leverageable correlations between such paired data as it does not yield a maximally correlated basis between the observable and unobservable counterparts. This instead is the objective of Canonical Correlation Analysis (and the more general Partial Least Squares methods); however, such techniques are still symmetric in maximizing correlation (covariance for PLSR) over all choices of basis for both datasets without differentiating between observable and unobservable variables (except for the regression phase of PLSR). Further, these methods deviate from PCA's formulation objective to minimize approximation error, seeking instead to maximize correlation or covariance. While these are sensible optimization objectives, they are not equivalent to error minimization. We therefore introduce a new method of leveraging PCA between paired datasets in an asymmetric manner which is optimal with respect to approximation error during training. We generate an asymmetrically paired basis for which we relax orthogonality constraints on the orthogonality in decomposing unreliable unobservable measurements. In doing so, this allows us to optimally capture the variations of the observable data while conditionally minimizing the expected prediction error for the unobservable component. We show preliminary results that demonstrate improved learning of our proposed method compared to that of traditional techniques.

We propose Asymmetrically-Paired Principal Component Analysis (APPCA), a novel linear dimension-reduction model for estimating coupled yet partially available variable sets. Unlike partial least square methods (e.g., partial least square regression and canonical correlation analysis) which maximize correlation/covariance between the two datasets, our APPCA 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 APPCA 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 single-channel image datasets. Results show that when only a single pair of bases is allowed, the conditional APPCA achieves lowest reconstruction error on the observable part and the total variable sets as a whole, meanwhile the unconditional APPCA reaches lowest prediction errors on the unobservable part. When extra budget is allowed for the PCA basis of the observable part, one can reach optimal solution using a combine method: standard PCA for the observable part and unconditional APPCA for the unobservable part.