Covariance Operator Based Dimensionality Reduction with Extension to Semi-Supervised Settings

We consider the task of dimensionality reduction for regression (DRR) informed by realvalued multivariate labels. The problem is often treated as a regression task where the goal is to find a low dimensional representation of the input data that preserves the statistical correlation with the targets. Recently, Covariance Operator Inverse Regression (COIR) was proposed as an effective solution that exploits the covariance structures of both input and output. COIR addresses known limitations of recent DRR techniques and allows a closed-form solution without resorting to explicit output space slicing often required by existing IR-based methods. In this work we provide a unifying view of COIR and other DRR techniques and relate them to the popular supervised dimensionality reduction methods including the canonical correlation analysis (CCA) and the linear discriminant analysis (LDA). We then show that COIR can be effectively extended to a semi-supervised learning setting where many of the input points lack their corresponding multivariate targets. A study of benefits of proposed approaches is presented on several important regression problems in both fullysupervised and semi-supervised settings.