单模李群上的信息论不等式
Information-Theoretic Inequalities on Unimodular Lie Groups
摘要：
在信息论中使用的经典不等式，如de Bruijn、Fisher和Kullback的不等式，是从概率论在欧氏空间中的设置延续到单模李群的设置。这些群包含在左移和右移下不变的积分测度，这意味着即使在非交换的情况下，它们也具有欧几里得空间的许多有用特征。在实际工程中，旋转群和欧几里得运动群是最令人感兴趣的单模李群，适用于这些李群的信息论的发展开辟了研究与不规则或随机投影方向的图像重建、移动机器人的信息收集、卫星姿态控制、细菌趋化性和信息处理有关的问题的潜力。几个定义从欧几里德的情况下，李群，包括费舍尔信息矩阵，和类似的不平等，在经典的信息论推导和陈述的形式，15个小定理。在所有这样的不等式中，随机变量的加法被群积所取代，并且采用了概率密度卷积的适当推广。
英文摘要：
Classical inequalities used in information theory such as those of de Bruijn, Fisher, and Kullback carry over from the setting of probability theory on Euclidean space to that of unimodular Lie groups. These are groups that posses integration measures that are invariant under left and right shifts, which means that even in noncommutative cases they share many of the useful features of Euclidean space. In practical engineering terms the rotation group and Euclidean motion group are the unimodular Lie groups of most interest, and the development of information theory applicable to these Lie groups opens up the potential to study problems relating to image reconstruction from irregular or random projection directions, information gathering in mobile robotics, satellite attitude control, and bacterial chemotaxis and information processing. Several definitions are extended from the Euclidean case to that of Lie groups including the Fisher information matrix, and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed.
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