A Comprehensive Guide to Multiset Canonical Correlation Analysis and its Application to Joint Blind Source Separation

Multiset Canonical Correlation Analysis (mCCA), also called Generalized Canonical Correlation Analysis (GCCA), is a technique to identify correlated variables across multiple datasets, which can be used for feature extraction in fields like neuroscience, cross-language information retrieval, and recommendation systems, among others. Besides its wide use, there is still a lack of comprehensive understanding of its theory and implementation with different objective functions all under one umbrella. In this paper, we review the five commonly used mCCA methods sumcor, maxvar, minvar, genvar, and ssqcor. We provide a concise overview of their optimization problems along with their solutions and pseudocodes. After this, we discuss the application of mCCA for estimating underlying latent components in the Joint Blind Source Separation (JBSS) problem and propose the source identification conditions of the different mCCA methods, i.e., the conditions under which they are able to achieve JBSS. We substantiate the proposed theoretical conditions with numerical results and test the statistical efficiency of the methods for finite samples. We observe in our experiments that genvar appears to have the least restrictive source identification conditions and to be more statistically efficient that the other methods. This suggests that genvar is generally the best-performing mCCA method for JBSS except for special cases, which is an important finding, as the most commonly used mCCA methods are maxvar and sumcor.