Belief Propagation with Deep Unfolding for High-dimensional Inference in Communication Systems

High-dimensional probability distributions are important objects in a wide variety of applications for example, most prediction and inference applications focus on computing the posterior marginal of a subset of variables conditioned on observations of another subset of variables. In practice, this is untractable due to the curse of dimensionality. In some problems, high-dimensional joint probability distributions can be represented by factor graphs. For such problems, belief propagation (BP) is a polynomial-time algorithm that provides an efficient approximation of the posterior marginals, and it is exact if the factor graph does not contain cycles. With rapid improvements in machine learning over the past decade, using machine learning techniques to optimize system parameters is an emerging field in communication research.

This thesis considers applying BP for communication systems, and focuses on incorporating domain knowledge into machine learning models. For compressive sensing, two variants of relaxed belief propagation (RBP) algorithm are proposed. One improves the stability over a larger class of measurement matrices and the other reduces the computational complexity when measurement matrix is in the product of several sparse matrices. For optical communication, the non-linear Schrodinger equation is solved by modeling the signal in each step of split-step Fourier method as a multivariate complex Gaussian distribution. Then, the parameters of the Gaussian are tracked through in digital back-propagation. For recursive decoding for Reed–Muller codes, the algebraic structure of the code is utilized and a recursive BP approach for redundant factor graphs is developed for near-optimal decoding. Finally, we use deep unfolding to unroll BP decoding as a recursive neural network and introduce the idea of a the parameter adaptive network to learn the relation between channel SNR and optimal BP weight factors.