Minimum HGR correlation principle: From marginals to joint distribution
Minimum HGR correlation principle: From marginals to joint distribution
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最小HGR相关性原理：从边际分布到联合分布
DOI：
10.1109/isit.2015.7282681
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
David Tse
中科院分区：
文献类型：
--
作者：
Farzan Farnia;Meisam Razaviyayn;Sreeram Kannan;David Tse
给定随机变量 X = (X1, X2, ..., Xp) 和 Y 的低阶矩信息，哪种分布可以最小化 X 和 Y 之间的 Hirschfeld-Gebelein-Rényi (HGR) 最大相关系数，同时保持忠实于给定矩？这个问题的答案非常重要，尤其是为了在随机变量 X 和 Y 之间以最小依赖性拟合 (X, Y) 上的模型。在本文中，我们首先在连续设置中研究这个问题，通过证明联合高斯分布在给定一阶和二阶矩的分布中实现了最小 HGR 相关系数。然后，我们通过固定随机变量 X 和 Y 的成对边际，在离散场景中提出类似的问题。随后，我们在具有固定成对边际的分布类别上推导出 HGR 相关系数的下界。然后我们证明，如果存在满足给定成对边际的特定加性结构的分布，则该下界是紧的。此外，具有加性结构的分布实现了最小的HGR相关系数。最后，我们得出结论，获得包含加性结构化分布的成对边缘的事件对于概率单纯形具有正的勒贝格测度。
Given low order moment information over the random variables X = (X1, X2, ..., Xp) and Y, what distribution minimizes the Hirschfeld-Gebelein-Rényi (HGR) maximal correlation coefficient between X and Y, while remains faithful to the given moments? The answer to this question is important especially in order to fit models over (X, Y) with minimum dependence among the random variables 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 X and Y. Subsequently, we 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 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.