Survey Propagation Revisited

Survey propagation (SP) is an exciting new technique that has been remarkably successful at solving very large hard combinatorial problems, such as determining the satisfiability of Boolean formulas. In a promising attempt at understanding the success of SP, it was recently shown that SP can be viewed as a form of belief propagation, computing marginal probabilities over certain objects called covers of a formula. This explanation was, however, shortly dismissed by experiments suggesting that non-trivial covers simply do not exist for large formulas. In this paper, we show that these experiments were misleading: not only do covers exist for large hard random formulas, SP is surprisingly accurate at computing marginals over these covers despite the existence of many cycles in the formulas. This re-opens a potentially simpler line of reasoning for understanding SP, in contrast to some alternative lines of explanation that have been proposed assuming covers do not exist.

Belief Propagation and Beyond for Particle Tracking

We describe a novel approach to statistical learning from particles tracked while moving in a random environment. The problem consists in inferring properties of the environment from recorded snapshots. We consider here the case of a fluid seeded with identical passive particles that diffuse and are advected by a flow. Our approach rests on efficient algorithms to estimate the weighted number of possible matchings among particles in two consecutive snapshots, the partition function of the underlying graphical model. The partition function is then maximized over the model parameters, namely diffusivity and velocity gradient. A Belief Propagation (BP) scheme is the backbone of our algorithm, providing accurate results for the flow parameters we want to learn. The BP estimate is additionally improved by incorporating Loop Series (LS) contributions. For the weighted matching problem, LS is compactly expressed as a Cauchy integral, accurately estimated by a saddle point approximation. Numerical experiments show that the quality of our improved BP algorithm is comparable to the one of a fully polynomial randomized approximation scheme, based on the Markov Chain Monte Carlo (MCMC) method, while the BP-based scheme is substantially faster than the MCMC scheme.