Dimensionality Reduction for Nonlinear Regression with Two Predictor Vectors

Many variables that we would like to predict depend nonlinearly on two types of attributes. For example, prices are influenced by supply and demand. Movie ratings are determined by demographic attributes and genre attributes. This paper addresses the dimensionality reduction problem in such regression problems with two predictor vectors. In particular, we assume a discriminative model where low-dimensional linear embeddings of the two predictor vectors are sufficient statistics for predicting a dependent variable. We show that a simple algorithm involving singular value decomposition can accurately estimate the embeddings provided that certain sample complexities are satisfied, surprisingly, without specifying the nonlinear regression model. These embeddings improve the efficiency and robustness of subsequent training, and can serve as a pre-training algorithm for neural networks. The main results establish sample complexities under multiple settings. Sample complexities for different regression models only differ by constant factors.