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.

翻译：近期论文（IEEE Trans. IT 69, 1680）提出了一种无需迭代即可计算信道容量的解析方法，但该方法存在若干限制其适用性的局限。此外，该论文并未解释为何在此特定情形下信道容量能够被解析求解。为拓展该方法的应用范围并克服其局限，我们将研究焦点转向Toyota提出的逆向EM问题（Information Geometry, 3, 1355 (2020)）。该逆向EM问题通过迭代应用EM迭代的逆映射来计算表征最大互信息的信道容量，然而Toyota的工作中仍存在若干未解决的开放问题。为应对这些挑战，我们基于布雷格曼散度构建了逆向EM问题，并为这些开放问题提供了解决方案。基于上述成果，我们将逆向EM问题转化为EM问题，并推导出逆向EM问题的非迭代公式。该公式可视为前述解析计算方法的推广，更重要的是，这一推导揭示了该特例所蕴含的信息几何结构。通过有效解决原有解析方法的局限并深化对底层信息几何结构的理解，本研究显著拓展了所提无迭代信道容量计算方法的应用范围。