  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.
