Low complexity sparse Bayesian learning using combined belief propagation and mean field with a stretched factor graph

Abstract

This paper concerns message passing based approaches to sparse Bayesian learning (SBL) with a linear model corrupted by additive white Gaussian noise with unknown variance. With the conventional factor graph, mean field (MF) message passing based algorithms have been proposed in the literature. In this work, instead of using the conventional factor graph, we modify the factor graph by adding some extra hard constraints (the graph looks like being 'stretched'), which enables the use of combined belief propagation (BP) and MF message passing. We then propose a low complexity BP-MF SBL algorithm based on which an approximate BP-MF SBL algorithm is also developed to further reduce the complexity. Thanks to the use of BP, the BP-MF SBL algorithms show their merits compared with state-of-the-art MF SBL algorithms: they deliver even better performance with much lower complexity compared with the vector-form MF SBL algorithm and they significantly outperform the scalar-form MF SBL algorithm with similar complexity.Index Terms-sparse Bayesian learning, message passing, BP-MF.

Variational Message Passing and Local Constraint Manipulation in Factor Graphs

“…As a result, the variational posterior also loses flexibility. To improve inference performance for sparse Bayesian learning, the authors of [ 40 ] proposes a hybrid mechanism by augmenting naive mean-field VMP with sum-product updates. This hybrid scheme reduces the complexity of the sum-product algorithm, while improving the accuracy of the naive VMP approach.…”

Joint spare channel estimation and decoding for orthogonal frequency division multiplexing using combined message passing

“…However, the complexity of BP applied to such continuous and discrete variable coexisting factors is prohibitive, which motivates [5] the use of the EP rule to obtain Gaussian messages. To efficiently deal with such complex observation nodes, we employ the stretching method [26] to further decompose it into several sub‐nodes (factors).…”

“…With the message (26), the m-type message m f h i → α l (α l ) from factor node f h i to variable node α l is computed by the BP rule, which yields…”

“…The joint channel estimation and decoding iteration is presented in lines (18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36), denoted by global iteration, which includes the detection-decoding (lines [19][20], channel estimation (lines [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and noise precision estimation (line 36). Note that the channel estimation and noise precision estimation can be arranged as an inner iteration, i.e.…”

“…However, the observation nodes are of a complex form, which may not be handled efficiently by solo MF or BP. Inspired by our recent work [8,26], we tackle the observation nodes by using combined MF and BP message passing. This can be achieved by breaking each of the observation nodes into several sub-nodes, enabling us to choose proper message passing rules to best handle each of the sub-nodes.…”