Canonical correlation analysis (CCA) is used to discover relations between two or more multivariate sets of variables, called views. Data to be processed are collected for a population of individuals, and for one individual its state is described by groups of values of associated variables, every group being part of the corresponding view. Not only linear relations can be extracted, but also nonlinear relations may provide enhanced knowledge on the subject under investigation.

This valuable paper succeeds in covering the most important aspects of CCA. A unified framework is used to present linear models in both over- and under-determined settings. Regularization techniques are used for under-determined cases. The robustness of CCA is improved in a Bayesian framework. Therefore, the Bayesian CCA approach is obtained. The nonlinear relations can be extracted using kernel methods, or deep CCA based on neural networks. Solving sparse versions of CCA (enforcing some components of position vectors to null values) with appropriate algorithms is also covered.

The reader will appreciate a well-written introduction, a valuable section dedicated to a discussion of the presented topic, and an excellent list of references.

The second section is dedicated to the over-determined case. Solved as an optimization problem, the unknown parameters can be computed by solving the linear CCA through one of the following approaches: (1) solving a standard eigenvalue problem; (2) solving a generalized eigenvalue problem; and (3) using the single-value decomposition method. Not only are the computational procedures well described, but the included examples and interpretations for each case bring value. The models are evaluated for statistical significance and tested for generalizability.

The third section contains the rest of approaches, where three algorithms are described: (1) application of repeated k-fold cross-validation for regularized CCA; (2) computation of a 1-rank approximation of the covariance matrix; and (3) solving the convex sparse least square problem. Also, inspired examples illustrate the applicability of the described methods.

The fourth section is dedicated to discussion. Every section ends with a short summary.

The tutorial is well structured and clearly written (minor typographical errors can be easily identified). I highly recommend this paper as a good resource for researchers, master’s students, or undergraduates enrolled in advanced data analysis courses.