In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements DMM . The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.
According to the standard prescription, messages used in the the sum-product algorithm should be probability measures over the real line @math , cf. Eqs. ), ). This is impractical from a computational point of view. (A low complexity message-passing algorithm for a related problem was used in @cite_11 ).
@cite_11
2137813581 2290712804 1479971791 1982531027
Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of "local" functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph, In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes-either exactly or approximately-various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's (1988) belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math . Probabilistic inference problems arise naturally in distributed systems such as sensor networks and teams of mobile robots. Inference algorithms that use message passing are a natural fit for distributed systems, but they must be robust to the failure situations that arise in real-world settings, such as unreliable communication and node failures. Unfortunately, the popular sum--product algorithm can yield very poor estimates in these settings because the nodes' beliefs before convergence can be arbitrarily different from the correct posteriors. In this paper, we present a new message passing algorithm for probabilistic inference which provides several crucial guarantees that the standard sum--product algorithm does not. Not only does it converge to the correct posteriors, but it is also guaranteed to yield a principled approximation at any point before convergence. In addition, the computational complexity of the message passing updates depends only upon the model, and is independent of the network topology of the distributed system. We demonstrate the approach with detailed experimental results on a distributed sensor calibration task using data from an actual sensor network deployment. We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a message-passing procedure which generalizes the usual methods like Sum-Product or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional heuristic. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005