We formulate em algorithm in the framework of Bregman divergence, which is a general problem setting of information geometry. That is, we address the minimization problem of the Bregman divergence between an exponential subfamily and a mixture subfamily in a Bregman divergence system. Then, we show the convergence and its speed under several conditions. We apply this algorithm to rate distortion and its variants including the quantum setting, and show the usefulness of our general algorithm. In fact, existing applications of Arimoto-Blahut algorithm to rate distortion theory make the optimization of the weighted sum of the mutual information and the cost function by using the Lagrange multiplier. However, in the rate distortion theory, it is needed to minimize the mutual information under the constant constraint for the cost function. Our algorithm directly solves this minimization. In addition, we have numerically checked the convergence speed of our algorithm in the classical case of rate distortion problem.
翻译：我们在布雷格曼差异的框架内制定电子算法,这是信息几何学上的一般问题设置。也就是说,我们解决了布雷格曼指数子家庭与混合子家庭在布雷格曼差异系统中的差异最小化问题。然后,我们在若干条件下展示了趋同及其速度。我们将这种算法应用于计算扭曲率及其变体,包括量子设定,并展示了我们一般算法的有用性。事实上,阿里莫托-布拉赫特算法在评估扭曲理论方面的现有应用,通过使用拉格兰奇乘数优化了相互信息的加权和成本功能。然而,在率扭曲理论中,我们需要在成本函数的经常限制下将相互信息最小化。我们的算法直接解决了这种最小化问题。此外,我们在典型的汇率扭曲问题中用数字检查了我们算法的趋同速度。
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