标题： 基于Bregman散度的逆em问题及其在经典和量子信息理论中的应用 显示英文标题

标题： Reverse em-problem based on Bregman divergence and its application to classical and quantum information theory

摘要： 最近的论文（IEEE Trans. IT 69, 1680）介绍了一种用于计算信道容量的解析方法，而无需迭代。这种方法存在一些限制，限制了其适用性。此外，该论文并未解释为何在此特定情况下可以解析地求解信道容量。为了扩大该方法的适用范围并解决其局限性，我们将注意力转向丰田（Information Geometry, 3, 1355 (2020)) 提出的反向EM问题。这个反向EM问题涉及迭代应用EM迭代的逆映射来计算信道容量，这代表最大互信息。然而，在丰田的工作中仍存在几个未解决的开放问题。为了解决这些挑战，我们基于Bregman散度构建反向EM问题，并提供这些问题的解决方案。在这些结果的基础上，我们将反向EM问题转化为EM问题，并推导出反向EM问题的非迭代公式。此公式可视为上述解析计算方法的推广。重要的是，这一推导揭示了该特殊情况下的信息几何结构。通过有效解决之前解析方法的局限性并提供对底层信息几何结构的更深入理解，我们的工作显著扩展了该方法在无迭代计算信道容量中的适用性。 显示英文摘要

摘要： The recent paper (IEEE Trans. IT 69, 1680) introduced an analytical method for calculating the channel capacity without the need for iteration. This method has certain limitations that restrict its applicability. Furthermore, the paper does not provide an explanation as to why the channel capacity can be solved analytically in this particular case. In order to broaden the scope of this method and address its limitations, we turn our attention to the reverse em-problem, proposed by Toyota (Information Geometry, 3, 1355 (2020)). This reverse em-problem involves iteratively applying the inverse map of the em iteration to calculate the channel capacity, which represents the maximum mutual information. However, several open problems remained unresolved in Toyota's work. To overcome these challenges, we formulate the reverse em-problem based on Bregman divergence and provide solutions to these open problems. Building upon these results, we transform the reverse em-problem into em-problems and derive a non-iterative formula for the reverse em-problem. This formula can be viewed as a generalization of the aforementioned analytical calculation method. Importantly, this derivation sheds light on the information geometrical structure underlying this special case. By effectively addressing the limitations of the previous analytical method and providing a deeper understanding of the underlying information geometrical structure, our work significantly expands the applicability of the proposed method for calculating the channel capacity without iteration.