This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.
Fourth, the Lazy Flipper guarantees that the best approximation found with a search depth @math is optimal within a Hamming distance @math . A similar guarantee known as the Single Loop Tree (SLT) neighborhood @cite_17 is given by BP in case of convergence. The SLT condition states that in any alteration of an assignment of values to the variables that leads to a lower energy, the altered variables form a subgraph in the graphical model that has at least two loops. The fact that Hamming optimality and SLT optimality differ can be exploited in practice. We show in one experiment in that BP approximations can be further improved by means of lazy flipping.
@cite_17
2098387242
Graphical models, such as Bayesian networks and Markov random fields (MRFs), represent statistical dependencies of variables by a graph. The max-product "belief propagation" algorithm is a local-message-passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the fixed point yields the most probable values of the unobserved variables given the observed ones. Good empirical performance has been obtained by running the max-product algorithm (or the equivalent min-sum algorithm) on graphs with loops, for applications including the decoding of "turbo" codes. Except for two simple graphs (cycle codes and single-loop graphs) there has been little theoretical understanding of the max-product algorithm on graphs with loops. Here we prove a result on the fixed points of max-product on a graph with arbitrary topology and with arbitrary probability distributions (discrete- or continuous-valued nodes). We show that the assignment based on a fixed point is a "neighborhood maximum" of the posterior probability: the posterior probability of the max-product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment. The region includes all assignments that differ from the max-product assignment in any subset of nodes that form no more than a single loop in the graph. In some graphs, this neighborhood is exponentially large. We illustrate the analysis with examples. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze and generalize the Belief Propagation (BP) approach to computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. We introduce a fractional free energy functional parameterized by a scalar parameter @math , where @math corresponds to the BP limit and @math corresponds to the exclusion principle Mean-Field (MF) limit, and show monotonicity and continuity of the functional with @math . We observe that the optimal value of @math , where the @math -parameterized functional is equal to the exact free energy (defined as the minus log of the permanent), lies in the @math range, with the low and high values from the range producing provable low and upper bounds for the permanent. Our experimental analysis suggests that the optimal @math varies for different ensembles considered but it always lies in the @math interval. Besides, for all ensembles considered the behavior of the optimal @math is highly distinctive, thus offering a lot of practical potential for estimating permanents of non-negative matrices via the fractional free energy functional approach. Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate. We prove an optimal @math lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive @math -bit strings @math and @math , respectively. They are promised that the Hamming distance between @math and @math is either at least @math or at most @math , and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283--289], it had been conjectured that the naive protocol, which uses @math bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is de...