  Given low order moment information over the random variables $\mathbf{X} =
(X_1,X_2,\ldots,X_p)$ and $Y$, what distribution minimizes the
Hirschfeld-Gebelein-R\'{e}nyi (HGR) maximal correlation coefficient between
$\mathbf{X}$ and $Y$, while remains faithful to the given moments? The answer
to this question is important especially in order to fit models over
$(\mathbf{X},Y)$ with minimum dependence among the random variables
$\mathbf{X}$ and $Y$. In this paper, we investigate this question first in the
continuous setting by showing that the jointly Gaussian distribution achieves
the minimum HGR correlation coefficient among distributions with the given
first and second order moments. Then, we pose a similar question in the
discrete scenario by fixing the pairwise marginals of the random variables
$\mathbf{X}$ and $Y$. To answer this question in the discrete setting, we first
derive a lower bound for the HGR 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 {\it additive} structure
satisfying the given pairwise marginals. Moreover, the distribution with the
additive structure achieves the minimum HGR correlation coefficient. Finally,
we conclude by showing that the event of obtaining pairwise marginals
containing an additive structured distribution has a positive Lebesgue measure
over the probability simplex.
