Hybrid Approximate Message Passing (1111.2581v4)

Abstract: Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractability, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incorporating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of dependencies of a general graphical model into strong and weak edges, with the weak edges representing interactions through aggregates of small, linearizable couplings of variables. AMP approximations based on the Central Limit Theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (HyGAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance--complexity trade-off can be achieved. Group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology.

• The paper presents HyGAMP, which partitions strong and weak dependencies to simplify message-passing in high-dimensional graphical models.
• It employs AMP techniques to approximate weak edges, achieving robust inference with improved computational efficiency.
• Numerical results validate HyGAMP's effectiveness in real-world applications such as signal processing and machine learning.

Hybrid Approximate Message Passing: A Framework for Graphical Models with Linear Mixing Dependencies

The paper presents an advanced framework to enhance the efficiency of message-passing algorithms using Hybrid Approximate Message Passing (HyGAMP) within general graphical models. The approach innovatively partitions dependencies into strong and weak edges, applying approximate message-passing (AMP) techniques to the latter by exploiting the Central Limit Theorem, thus simplifying complex belief propagation algorithms while balancing performance and complexity.

Conceptual Framework and Algorithmic Design

The primary goal of this research is to address the computational challenges inherent in high-dimensional graphical models by reformulating message-passing algorithms. The framework is adaptable across a variety of optimization and inference problems with potential applications in fields such as signal processing, communications, and machine learning.

HyGAMP is architected based on the premise that weak dependencies in a system can be approximated and managed with AMP-style methods, helping to alleviate the computational burden present in processing strong dependencies via conventional loopy belief propagation methods. The resulting algorithm, HyGAMP, integrates sum-product and max-sum belief propagation methodologies in an iterative, resource-efficient manner.

The paper establishes the theoretical underpinnings for HyGAMP by dissecting the structure of graphical models into components that allow for the implementation of Gaussian and quadratic approximations. The algorithm efficiently balances the trade-off between performance, which is achieved with more comprehensive message updates, and complexity, significantly reduced by leveraging the AMP approach.

Numerical Results and Implications

HyGAMP demonstrates improved simplicity in implementation without compromising the accuracy of statistical inference or optimization outcomes. Numerical results from application contexts, such as group sparsity and multinomial logistic regression, exemplify the algorithm's capabilities in yielding robust estimates with fewer computational resources than traditional methods. This is a testament to the practical applicability of HyGAMP in real-world scenarios, contributing to a broader toolbox for computational intelligence and data science tasks.

By adapting the message-passing strategies to factors in multi-agent systems, HyGAMP not only optimizes communication overhead but also accelerates convergence, offering significant improvements in systems where both data and computational efficiency are critical.

Future Directions

With HyGAMP providing a modular and extensible framework, future developments could explore further generalizations and enhancements to accommodate more intricate empirical models, including those with dynamic or adaptive structures. The research invites more comprehensive theoretical investigations to substantiate its empirical success and explore its boundaries, potentially contributing significantly to the domains of large-scale data inference and machine learning.

This body of work paves the way for new algorithmic innovations that effectively handle the intricacies of high-dimensional data representations and dependencies, fostering an era of advanced and efficient computational methods in artificial intelligence.