  We introduce three novel semi-parametric extensions of probabilistic
canonical correlation analysis with identifiability guarantees. We consider
moment matching techniques for estimation in these models. For that, by drawing
explicit links between the new models and a discrete version of independent
component analysis (DICA), we first extend the DICA cumulant tensors to the new
discrete version of CCA. By further using a close connection with independent
component analysis, we introduce generalized covariance matrices, which can
replace the cumulant tensors in the moment matching framework, and, therefore,
improve sample complexity and simplify derivations and algorithms
significantly. As the tensor power method or orthogonal joint diagonalization
are not applicable in the new setting, we use non-orthogonal joint
diagonalization techniques for matching the cumulants. We demonstrate
performance of the proposed models and estimation techniques on experiments
with both synthetic and real datasets.
