# Minimum Rényi Correlation Principle: From Marginals to Joint Distribution

Given low order moment information over the random variables $\bX = (X_1,X_2,\ldots,X_p)$ and $Y$, what distribution maximizes the Rényi maximal correlation coefficient between $\bX$ and $Y$, while remains faithful to the given moments? The answer to this question is important especially in order to fit models over $(\bX,Y)$ with minimum dependence among the random variables $\bX$ and $Y$. In this talk, we investigate this question first in the continuous setting by showing that the jointly Gaussian distribution achieves the minimum Rényi correlation coefficient among distributions with the same first and second order moments. Then, we pose a similar question in the discrete scenario by fixing the pairwise marginals of the random variables $\bX$ and $Y$. We first derive a lower bound for the Rényi correlation coefficient over the class of distributions with fixed pairwise marginals. Then we show that this lower bound is tight if there exists a distribution with certain additive model structure satisfying the given pairwise marginals. Moreover, the distribution with the additive structure achieves the minimum Rényi maximal correlation coefficient.

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