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

TL;DR

This paper introduces a Bregman divergence-based EM algorithm within information geometry, demonstrating its convergence and applying it to classical and quantum rate distortion problems, offering a direct optimization approach.

Contribution

It formulates a novel EM algorithm using Bregman divergence, applicable to classical and quantum rate distortion theory, improving optimization methods.

Findings

Proves convergence and speed of the proposed algorithm.

Successfully applies the algorithm to quantum rate distortion.

Numerical validation confirms efficiency in classical cases.

Abstract

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.