  Many machine learning problems, especially multi-modal learning problems,
have two sets of distinct features (e.g., image and text features in news story
classification, or neuroimaging data and neurocognitive data in cognitive
science research). This paper addresses the joint dimensionality reduction of
two feature vectors in supervised learning problems. In particular, we assume a
discriminative model where low-dimensional linear embeddings of the two feature
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, without specifying the nonlinear link function (regressor or
classifier). The main results establish sample complexities under multiple
settings. Sample complexities for different link functions only differ by
constant factors.
