Canonical Correlation Analysis with Implicit Distributions

Canonical Correlation Analysis (CCA) is a ubiquitous technique that shows promising performance in multi-view learning problems. Due to the conjugacy of the prior and the likelihood, probabilistic CCA (PCCA) presents the posterior with an analytic solution, which provides probabilistic interpretation for classic linear CCA. As the multi-view data are usually complex in practice, nonlinear mappings are adopted to capture nonlinear dependency among the views. However, the interpretation provided in PCCA cannot be generalized to this nonlinear setting, as the distribution assumptions on the prior and the likelihood makes it restrictive to capture nonlinear dependency. To overcome this bottleneck, in this paper, we provide a novel perspective for CCA based on implicit distributions. Specifically, we present minimum Conditional Mutual Information (CMI) as a new criteria to capture nonlinear dependency for multi-view learning problem. To eliminate the explicit distribution requirement in direct estimation of CMI, we derive an objective whose minimization implicitly leads to the proposed criteria. Based on this objective, we present an implicit probabilistic formulation for CCA, named Implicit CCA (ICCA), which provides a flexible framework to design CCA extensions with implicit distributions. As an instantiation, we present adversarial CCA (ACCA), a nonlinear CCA variant which benefits from consistent encoding achieved by adversarial learning. Quantitative correlation analysis and superior performance on cross-view generation task demonstrate the superiority of the proposed ACCA.