Non-parametric Belief Propagation Applications

The canonical problem of solving a system of linear equations arises in numerous contexts in information theory, communication theory, and related fields. Gaussian belief propagation (GaBP) has been shown to solve this problem in a manner that does not involve direct matrix inversion. The iterative nature of GaBP allows for a distributed message-passing implementation of the solution algorithm. Non-parametric Belief Propagation (NBP) is an extension of GaBP that allows the prior distributions to be Gaussian mixtures instead of single Gaussian, which enables approximating many complex distributions. The importance of these prior distribution is in solving systems of linear equations with constraints on the values of x, the prior approximation is a very intuitive yet accurate approximation. We first address the problem of Low Density Lattice Codes (LDLC) decoding algorithm, with bipolar constraints on x, and equal probability for x = 1,−1. In general the LDLC problem has integer contraints on x, and the NBP solver supports this as well. We show that an NBP based solver allows for low Symbol Error Rate (SER), especially when working close to channel capacity, compared to present solvers. We also give some general convergence conditions borrowed from the world of GaBP, and while no proof of convergence has been found to date we report preliminary results which show good experimental convergence. The Fault Detection problem further expands our formulation to allow for different probability of x = 1, x = −1, resulting in a closely related problem. We again show the NBP based solver has better results than current state of the art algorithms from several related fields. We also compare a solution via NBP based on 3 different prior distributions, and show that best results are achieved when we incorporate our full knowledge of the constraints on the solution into the prior distribution.