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Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 792 (27 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Messagepassing for graphstructured linear programs: Proximal methods and rounding schemes
, 2008
"... The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly convergen ..."
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Cited by 60 (0 self)
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The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly convergent algorithms for solving these LPs, based on proximal minimization schemes using Bregman divergences. As with standard messagepassing on graphs, the algorithms are distributed and exploit the underlying graphical structure, and so scale well to large problems. Our algorithms have a doubleloop character, with the outer loop corresponding to the proximal sequence, and an inner loop of cyclic Bregman divergences used to compute each proximal update. Different choices of the Bregman divergence lead to conceptually related but distinct LPsolving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graphstructured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a “reparameterization ” property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LPsolver converges to the optimum. We also propose a graphstructured randomized rounding scheme that applies to iterative LP solving algorithms in general. We analyze the performance of our rounding schemes, giving bounds on the number of iterations required, when the LP is integral, for the rounding schemes to obtain the MAP solution. These bounds are expressed in terms of the strength of the potential functions, and the energy gap, which measures how well the integral MAP solution is separated from other integral configurations. We also report simulations comparing these rounding schemes. 1
Probabilistic Analysis of Linear Programming Decoding
, 2008
"... We initiate the probabilistic analysis of linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probabilit ..."
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Cited by 25 (6 self)
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We initiate the probabilistic analysis of linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finitelength result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bitflipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting byproduct of our analysis is to establish the existence of “probabilistic expansion ” in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.
Guessing Facets: Polytope Structure and Improved LP Decoder
, 2009
"... We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We then prove that for expan ..."
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Cited by 22 (0 self)
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We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword. We further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as LP decoding that provably performs better. The method is very simple: it first applies ordinary LP decoding, and when it fails, it proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that the proposed algorithm, when applied to suitable expander codes, succeeds unless there exists a certain number of pseudocodewords, all adjacent to the ML codeword on the LP decoding polytope, and with higher likelihood than the ML codeword. We then describe an extended algorithm, still with polynomial complexity, that succeeds as long as there are at most polynomially many pseudocodewords above the ML codeword.
Variational inference in graphical models: The view from the marginal polytope
, 2003
"... Underlying a variety of techniques for approximate inferenceamong them mean field, sumproduct, and cluster variational methodsis a classical variational principle from statistical physics, which involves a "free energy" optimization problem over the set of all distributions. Work ..."
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Cited by 19 (1 self)
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Underlying a variety of techniques for approximate inferenceamong them mean field, sumproduct, and cluster variational methodsis a classical variational principle from statistical physics, which involves a "free energy" optimization problem over the set of all distributions. Working within the framework of exponential families, we describe an alternative view, in which the optimization takes place over the (typically) much lowerdimensional space of mean parameters. The associated constraint set consists of all mean parameters that are globally realizable; for discrete random variables, we refer to this set as a marginal polytope. As opposed to the classical formulation, the representation given here clarifies that there are two distinct components to variational inference algorithms: (a) an approximation to the entropy function; and (b) an approximation to the marginal polytope.
Robust messagepassing for statistical inference in sensor networks
 IN: PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON INFORMATION PROCESSING IN SENSOR NETWORKS IPSN’07
, 2007
"... Largescale sensor network applications require innetwork processing and data fusion to compute statistically relevant summaries of the sensed measurements. This paper studies distributed messagepassing algorithms, in which neighboring nodes in the network pass local information relevant to a glob ..."
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Cited by 12 (1 self)
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Largescale sensor network applications require innetwork processing and data fusion to compute statistically relevant summaries of the sensed measurements. This paper studies distributed messagepassing algorithms, in which neighboring nodes in the network pass local information relevant to a global computation, for performing statistical inference. We focus on the class of reweighted belief propagation (RBP) algorithms, which includes as special cases the standard sumproduct and maxproduct algorithms for general networks with cycles, but in contrast to standard algorithms has attractive theoretical properties (uniqueness of fixed points, convergence, and robustness). Our main contribution is to design and implement a practical and modular architecture for implementing RBP algorithms in real networks. In addition, we show how intelligent scheduling of RBP messages can be used to minimize communication between motes and prolong the lifetime of the network. Our simulation and Mica2 mote deployment indicate that the proposed algorithms achieve accurate results despite realworld problems such as dying motes, dead and asymmetric links, and dropped messages. Overall, the class of RBP provides provides an ideal fit for sensor networks due to their distributed nature, requiring only local knowledge and coordination, and little requirements on other services such as reliable transmission.
Messagepassing in stochastic processing networks
, 2011
"... Simple, distributed and iterative algorithms, popularly known as messagepassing, have become the architecture of choice for emerging infrastructure networks and the canonical behavioral model for natural networks. Therefore designing, as well as understanding, messagepassing algorithms has become i ..."
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Simple, distributed and iterative algorithms, popularly known as messagepassing, have become the architecture of choice for emerging infrastructure networks and the canonical behavioral model for natural networks. Therefore designing, as well as understanding, messagepassing algorithms has become important. The purpose of this survey is to describe the stateofart of messagepassing algorithms in the context of dynamic resource allocation in the presence of uncertainty, a problem that is central to operations research (OR) and management science (MS). Various directions for future research are described in this context as well as connections beyond OR and MS are explained. Through this survey, we hope to convey the opportunity presented to the OR and MS community to benefit from and contribute to the growing interdisciplinary area of messagepassing algorithms.