The mutual information is analyzed as a function of the input distribution using an identity due to Tops\o{e} for channels with (possibly multiple) linear cost constraints and finite input and output sets. The mutual information is bounded above by a function decreasing quadratically with the distance to the set of all capacity-achieving input distributions for the case when the distance is less than a certain threshold. The closed-form expressions for the threshold and the coefficient of the quadratic decrease are derived. A counter-example demonstrating the non-existence of such a quadratic bound in the case of infinitely many linear cost constraints is provided. Implications of these observations for the channel coding problem and applications of the proof technique to related problems are discussed.

翻译：本文利用Topsøe恒等式，针对具有（可能多个）线性代价约束以及有限输入输出集的信道，分析了互信息作为输入分布函数的性质。当输入分布到所有容量逼近输入分布集合的距离小于某阈值时，互信息的上界由一个与该距离呈二次递减的函数给出。本文推导了该阈值及二次递减系数的闭式表达式，并给出了一个反例，证明在存在无穷多个线性代价约束的情况下，此类二次界限不存在。最后，讨论了这些观察结果对信道编码问题的启示，以及该证明技术在相关问题中的应用。