Approximating Univariate Factored Distributions via Message-Passing Algorithms

Gaussian Mixture Models (GMMs) commonly arise in communication systems, particularly in bilinear joint estimation and detection problems. Although the product of GMMs is still a GMM, as the number of factors increases, the number of components in the resulting product GMM grows exponentially. To obtain a tractable approximation for a univariate factored probability density function (PDF), such as a product of GMMs, we investigate iterative message-passing algorithms. Based on Belief Propagation (BP), we propose a Variable Duplication and Gaussian Belief Propagation (VDBP)-based algorithm. The key idea of VDBP is to construct a multivariate measurement model whose marginal posterior is equal to the given univariate factored PDF. We then apply Gaussian BP (GaBP) to transform the global inference problem into local ones. Expectation propagation (EP) is another branch of message passing algorithms. In addition to converting the global approximation problem into local ones, it features a projection operation that ensures the intermediate functions (messages) belong to a desired family. Due to this projection, EP can be used to approximate the factored PDF directly. However, even if every factor is integrable, the division operation in EP may still cause the algorithm to fail when the mean and variance of a non-integrable belief are required. Therefore, this paper proposes two methods that combine EP with our previously proposed techniques for handling non-integrable beliefs to approximate univariate factored distributions.

Expectations in Expectation Propagation

Expectation Propagation (EP) is a widely used message-passing algorithm that decomposes a global inference problem into multiple local ones. It approximates marginal distributions (beliefs) using intermediate functions (messages). While beliefs must be proper probability distributions that integrate to one, messages may have infinite integral values. In Gaussian-projected EP, such messages take a Gaussian form and appear as if they have "negative" variances. Although allowed within the EP framework, these negative-variance messages can impede algorithmic progress. In this paper, we investigate EP in linear models and analyze the relationship between the corresponding beliefs. Based on the analysis, we propose both non-persistent and persistent approaches that prevent the algorithm from being blocked by messages with infinite integral values. Furthermore, by examining the relationship between the EP messages in linear models, we develop an additional approach that avoids the occurrence of messages with infinite integral values.

Decentralized Expectation Propagation for Semi-Blind Channel Estimation in Cell-Free Networks

This paper serves as a correction to the conference version. In this work, we explore uplink communication in cell-free (CF) massive multiple-input multiple-output (MaMIMO) systems, employing semi-blind transmission structures to mitigate pilot contamination. We propose a simplified, decentralized method based on Expectation Propagation (EP) for semi-blind channel estimation. By utilizing orthogonal pilots, we preprocess the received signals to establish a simplified equivalent factorization scheme for the transmission process. Moreover, this study integrates Central Limit Theory (CLT) with EP, eliminating the need to introduce new auxiliary variables in the factorization scheme. We also refine the algorithm by assessing the variable scales involved. Finally, a decentralized approach is proposed to significantly reduce the computational demands on the Central Processing Unit (CPU).