From Common Information to Canonical Correlation Analysis and Beyond: Bridging Information Theory, Signal Processing, and Machine Learning

Many problems in signal processing, machine learning, and neuroimaging provide several jointly observed measurements of one process, each mixing a component common to all of them with a component private to itself, and require the common component to be recovered without supervision. This problem has been studied in several communities, largely independently. Statisticians proposed canonical correlation analysis (CCA), introduced by Hotelling in 1936. Information theorists proposed two notions of common information, due to Gács and Körner and to Wyner. Engineers developed subspace methods for sensor arrays and multiuser communications to design multiview receivers. This tutorial presents the main results and tools each community developed for this problem in a single exposition, and explains the connections that emerge where they meet, distinguishing exact equivalences from special cases and analogies. We develop the information-theoretic foundation from first principles. We then introduce CCA and its extension to several views, followed by the engineering thread from blind multiuser detection to CCA as a blind receiver. Next, we turn to nonlinear CCA, covering kernel, deep, and probabilistic formulations, the conditions under which the common factor is identifiable, and connections to self-supervised learning. We conclude with a discussion of further extensions and open directions. The exposition is self-contained, so that readers from each of these communities can follow the parts developed by the others.