  Canonical correlation analysis (CCA) is a classic statistical method for
discovering latent co-variation that underpins two or more observed random
vectors. Several extensions and variations of CCA have been proposed that have
strengthened our capabilities in terms of revealing common random factors from
multiview datasets. In this work, we first revisit the most recent
deterministic extensions of deep CCA and highlight the strengths and
limitations of these state-of-the-art methods. Some methods allow trivial
solutions, while others can miss weak common factors. Others overload the
problem by also seeking to reveal what is not common among the views -- i.e.,
the private components that are needed to fully reconstruct each view. The
latter tends to overload the problem and its computational and sample
complexities. Aiming to improve upon these limitations, we design a novel and
efficient formulation that alleviates some of the current restrictions. The
main idea is to model the private components as conditionally independent given
the common ones, which enables the proposed compact formulation. In addition,
we also provide a sufficient condition for identifying the common random
factors. Judicious experiments with synthetic and real datasets showcase the
validity of our claims and the effectiveness of the proposed approach.
