Minimum HGR correlation principle: From marginals to joint distribution

摘要: Given low order moment information over the random variables X = (X 1 , 2 …, p ) and Y, what distribution minimizes Hirschfeld-Gebelein-Renyi (HGR) maximal correlation coefficient between while remains faithful to given moments? The answer this question is important especially in fit models (X, Y) with minimum dependence among Y. In paper, we investigate first continuous setting by showing that jointly Gaussian achieves HGR distributions second moments. Then, pose a similar discrete scenario fixing pairwise marginals of Subsequently, derive lower bound for class fixed marginals. Then show tight if there exists certain additive structure satisfying Moreover, coefficient. Finally, conclude event obtaining containing an structured has positive Lebesgue measure probability simplex.