A Local Characterization for Wyner Common Information

While the Hirschfeld-Gebelein-Renyi (HGR) maximal correlation and the Wyner common information share similar information processing purposes of extracting common knowledge structures between random variables, the relationships between these approaches are generally unclear. In this paper, we demonstrate such relationships by considering the Wyner common information in the weakly dependent regime, called ϵ-common information. We show that the HGR maximal correlation functions coincide with the relative likelihood functions of estimating the auxiliary random variables in ϵ-common information, which establishes the fundamental connections these approaches. Moreover, we extend the ϵ-common information to multiple random variables, and derive a novel algorithm for extracting feature functions of data variables regarding their common information. Our approach is validated by the MNIST problem, and can potentially be useful in multi-modal data analyses.