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$H$-mutual information ($H$-MI) is a wide class of information leakage measures, where $H=(\eta, F)$ is a pair of monotonically increasing function $\eta$ and a concave function $F$, which is a generalization of Shannon entropy. $H$-MI is defined as the difference between the generalized entropy $H$ and its conditional version, including Shannon mutual information (MI), Arimoto MI of order $\alpha$, $g$-leakage, and expected value of sample information. This study presents a variational characterization of $H$-MI via statistical decision theory. Based on the characterization, we propose an alternating optimization algorithm for computing $H$-capacity.

The problem of computing $\alpha$-capacity for $\alpha>1$ is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu--Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the $\alpha$-capacity for $\alpha>1$ based on a variational characterization of the Augustin--Csisz{\'a}r mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.

The Augustin--Csisz{\' a}r mutual information (MI) and Lapidoth--Pfister MI are well-known generalizations of the Shannon MI, but do not have known closed-form expressions, so they need to be calculated by solving optimization problems. In this study, we propose alternating optimization algorithms for computing these types of MI and present proofs of their global convergence properties. We also provide a novel variational characterization of the Augustin--Csisz{\' a}r MI that is similar to that of the Sibson MI.

The Sibson and Arimoto capacity, which are based on the Sibson and Arimoto mutual information (MI) of order {\alpha}, respectively, are well-known generalizations of the channel capacity C. In this study, we derive novel alternating optimization algorithms for computing these capacities by providing new variational characterizations of the Sibson and Arimoto MI. Moreover, we prove that all iterative algorithms for computing these capacities are equivalent under appropriate conditions imposed on their initial distributions.