Nonparametric Canonical Correlation Analysis
Authors: Tomer Michaeli, Weiran Wang, Karen Livescu

Abstract: Canonical correlation analysis (CCA) is a fundamental technique in multi-view data analysis and representation learning. Several nonlinear extensions of the classical linear CCA method have been proposed, including kernel and deep neural network methods. These approaches restrict attention to certain families of nonlinear projections, which the user must specify (by choosing a kernel or a neural network architecture), and are computationally demanding. Interestingly, the theory of nonlinear CCA without any functional restrictions, has been studied in the population setting by Lancaster already in the 50’s. However, these results, have not inspired practical algorithms. In this paper, we revisit Lancaster’s theory, and use it to devise a practical algorithm for nonparametric CCA (NCCA). Specifically, we show that the most correlated nonlinear projections of two random vectors can be expressed in terms of the singular value decomposition of a certain operator associated with their joint density. Thus, by estimating the population density from data, NCCA reduces to solving an eigenvalue system, superficially like kernel CCA but, importantly, without having to compute the inverse of any kernel matrix. We also derive a partially linear CCA (PLCCA) variant in which one of the views undergoes a linear projection while the other is nonparametric. PLCCA turns out to have a similar form to the classical linear CCA, but with a nonparametric regression term replacing the linear regression in CCA. Using a kernel density estimate based on a small number of nearest neighbors, our NCCA and PLCCA algorithms are memory-efficient, often run much faster, and achieve better performance than kernel CCA and comparable performance to deep CCA.