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We formulate em algorithm in the framework of Bregman divergence, which is ageneral problem setting of information geometry. That is, we address theminimization problem of the Bregman divergence between an exponential subfamilyand a mixture subfamily in a Bregman divergence system. Then, we show theconvergence and its speed under several conditions. We apply this algorithm torate distortion and its variants including the quantum setting, and show theusefulness of our general algorithm. In fact, existing applications ofArimoto-Blahut algorithm to rate distortion theory make the optimization of theweighted sum of the mutual information and the cost function by using theLagrange multiplier. However, in the rate distortion theory, it is needed tominimize the mutual information under the constant constraint for the costfunction. Our algorithm directly solves this minimization. In addition, we havenumerically checked the convergence speed of our algorithm in the classicalcase of rate distortion problem.

Bregman divergence based em algorithm and its application to classical and quantum rate distortion theory