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

TL;DR

This work develops an EM algorithm within a general Bregman divergence framework, enabling direct minimization of divergences between exponential and mixture subfamilies and providing convergence guarantees and speed analyses. The method is then applied to rate-distortion theory in both classical and quantum settings, including scenarios with side information and multiple distortions, avoiding the traditional Lagrange-multiplier approach. By unifying classical and quantum RD under information-geometric projections, the paper delivers a versatile and theoretically grounded optimization tool with numerical demonstrations of classical RD convergence. The broad formulation promises applicability beyond RD, with potential extensions to memory channels and Rényi-type divergences, highlighting the method’s generality and practicality for information-theoretic optimization problems.

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.

Figures (7)

• Figure 1: Figure for Step 3 of the proof of Lemma \ref{['LMGO']}.
• Figure 2: Algorithms \ref{['protocol1-0']} and \ref{['protocol1']}: This figure shows the topological relation among $\theta_*$, $\theta^*$, $\theta_{(t+1 )}$, $\theta^{(t+1 )}$, and $\theta_{(t)}$, which is used in the application of Phythagorean theorem (Proposition \ref{['MNL']}). $\mathcal{M}_{\theta_*\to \mathcal{E}} =\mathcal{E}_{\theta^*\to \mathcal{M}}$ and $\mathcal{M}_{\theta^{(t+1)} \to \mathcal{E}}$ are the mixture subfamilies to project $\theta(\epsilon_1)$ and $\theta^{(t+1)}$ to the exponential subfamily $\mathcal{E}$, respectively. $\mathcal{E}_{\theta_{(t)}\to \mathcal{M}}$ is the exponential subfamily to project $\theta_{(t)}$ to the mixture subfamily $\mathcal{M}$.
• Figure 3: Behavior of the error $I(X;Y)_{P_{X,Y}^{(t)}}-I(X;Y)_{P_{X,Y}^{*}}$ of the minimum mutual information. Red points show the value of the $I(X;Y)_{P_{X,Y}^{(t)}}-I(X;Y)_{P_{X,Y}^{*}}$ depending on the number of iteration $t$. The blue points show its upper bound given in \ref{['mma2BYY2']}.
• Figure 4: The behavior of the parameter $\tau$ depending on the number of iteration $t$. The green points show the parameter $\tau$ in algorithm \ref{['protocol1-2']}.
• Figure 5: Algorithms \ref{['protocol1-0']} and \ref{['protocol1']}: This figure shows the topological relation among $\theta(\epsilon_1 )_*$, $\theta(\epsilon_1 )$, $\theta_{(t+1 )}$, $\theta^{(t+1 )}$, and $\theta_{(t)}$, which is used in the application of Phythagorean theorem (Proposition \ref{['MNL']}). $\mathcal{M}_{\theta(\epsilon_1)\to \mathcal{E}}$ and $\mathcal{M}_{\theta^{(t+1)} \to \mathcal{E}}$ are the mixture subfamilies to project $\theta(\epsilon_1)$ and $\theta^{(t+1)}$ to the exponential subfamily $\mathcal{E}$, respectively. $\mathcal{E}_{\theta_{(t)}\to \mathcal{M}}$ is the exponential subfamily to project $\theta_{(t)}$ to the mixture subfamily $\mathcal{M}$.

Theorems & Definitions (47)

• Definition 1: Bregman Divergence
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof