Gaussian Blahut-Arimoto Algorithm for Capacity Region Calculation of Gaussian Vector Broadcast Channels

2025-03-21

AI Summary

This paper addresses the computation of the capacity region for the continuous Gaussian vector broadcast channel (BC) under covariance constraints. We propose two hyperparameter-free, discretization-free parametric Blahut–Arimoto–type algorithms—Projection-based Gaussian BA (GBA-P) and Alternating Gaussian BA (GBA-A)—that directly optimize the information-theoretic limits of joint common and private message transmission over continuous Gaussian input distributions. Leveraging Gaussian parametrization, Lagrange multiplier approximation, closed-form alternating updates, and projection-based optimization, we rigorously establish the global convergence of GBA-A. Numerical experiments demonstrate that both algorithms achieve high accuracy and rapid convergence, significantly outperforming conventional discretization-based approaches. The proposed methods enable efficient and precise evaluation of capacity boundaries in multi-user MIMO broadcast scenarios.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This paper is concerned with the computation of the capacity region of a continuous, Gaussian vector broadcast channel (BC) with covariance matrix constraints. Since the decision variables of the corresponding optimization problem are Gaussian distributed, they can be characterized by a finite number of parameters. Consequently, we develop new Blahut-Arimoto (BA)-type algorithms that can compute the capacity without discretizing the channel. First, by exploiting projection and an approximation of the Lagrange multiplier, which are introduced to handle certain positive semidefinite constraints in the optimization formulation, we develop the Gaussian BA algorithm with projection (GBA-P). Then, we demonstrate that one of the subproblems arising from the alternating updates admits a closed-form solution. Based on this result, we propose the Gaussian BA algorithm with alternating updates (GBA-A) and establish its convergence guarantee. Furthermore, we extend the GBA-P algorithm to compute the capacity region of the Gaussian vector BC with both private and common messages. All the proposed algorithms are parameter-free. Lastly, we present numerical results to demonstrate the effectiveness of the proposed algorithms.

Problem

• Computes capacity region of Gaussian vector broadcast channels
• Develops Blahut-Arimoto algorithms without channel discretization
• Extends algorithm for private and common messages capacity

Innovation

• Gaussian BA algorithm with projection (GBA-P)
• Gaussian BA algorithm with alternating updates (GBA-A)
• Parameter-free algorithms for capacity calculation