map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

message passing for maximum weight independent set in this paper we investigate the use of message passing algorithms for the problem of finding the max weight independent set mwis in a graph first we study the performance of the classical loopy max product belief propagation we show that each fixed point estimate of max product can be mapped in a natural way to an extreme point of the linear programming lp polytope associated with the mwis problem however this extreme point may not be the one that maximizes the value of node weights the particular extreme point at final convergence depends on the initialization of max product we then show that if max product is started from the natural initialization of uninformative messages it always solves the correct lp if it converges this result is obtained via a direct analysis of the iterative algorithm and cannot be obtained by looking only at fixed points the tightness of the lp relaxation is thus necessary for max product optimality but it is not sufficient motivated by this observation we show that a simple modification of max product becomes gradient descent on a smoothed version of the dual of the lp and converges to the dual optimum we also develop a message passing algorithm that recovers the primal mwis solution from the output of the descent algorithm we show that the mwis estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the mwis is unique finally we show that any problem of maximum a posteriori map estimation for probability distributions over finite domains can be reduced to an mwis problem we believe this reduction will yield new insights and algorithms for map estimation

a local algorithm for the computation of image velocity via constructive interference of global fourier components a novel fourier based technique for local motion detection from image sequences is proposed in this method the instantaneous velocities of local image points are inferred directly from the global 3d fourier components of the image sequence this is done by selecting those velocities for which the superposition of the corresponding fourier gratings leads to constructive interference at the image point hence image velocities can be assigned locally even though position is computed from the phases and amplitudes of global fourier components spanning the whole image sequence that have been filtered based on the motion constraint equation reducing certain aperture effects typically arising from windowing in other methods regularization is introduced for sequences having smooth flow fields aperture effects and their effect on optic flow regularization are investigated in this context the algorithm is tested on both synthetic and real image sequences and the results are compared to those of other local methods finally we show that other motion features i e motion direction can be computed using the same algorithmic framework without requiring an intermediate representation of local velocity which is an important characteristic of the proposed method

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

fourier integral operators on lie groupoids abstract as announced in 36 we develop a calculus of fourier integral g operators on any lie groupoid g for that purpose we study lagrangian conic submanifolds of the symplectic groupoid t g this includes their product transposition and inversion we also study the relationship between these lagrangian submanifolds and the equivariant families of lagrangian submanifolds of t g x t g x parametrized by the units x g 0 of g this allows us to select a subclass of lagrangian distributions on any lie groupoid g that deserve the name of fourier integral g operators g fios by construction the class of g fios contains the class of equivariant families of ordinary fourier integral operators on the manifolds g x x g 0 we then develop for g fios the first stages of the calculus in the spirit of hormander s work finally we illustrate this calculus in the case of manifolds with boundary

the bethe permanent of a nonnegative matrix it has recently been observed that the permanent of a nonnegative square matrix i e of a square matrix containing only nonnegative real entries can very well be approximated by solving a certain bethe free energy function minimization problem with the help of the sum product algorithm we call the resulting approximation of the permanent the bethe permanent in this paper we give reasons why this approach to approximating the permanent works well namely we show that the bethe free energy function is convex and that the sum product algorithm finds its minimum efficiently we then discuss the fact that the permanent is lower bounded by the bethe permanent and we comment on potential upper bounds on the permanent based on the bethe permanent we also present a combinatorial characterization of the bethe permanent in terms of permanents of so called lifted versions of the matrix under consideration moreover we comment on possibilities to modify the bethe permanent so that it approximates the permanent even better and we conclude the paper with some observations and conjectures about permanent based pseudocodewords and permanent based kernels

groupoids and pseudodifferential calculus on manifolds with corners we associate to any manifold with corners even with non embedded hyperfaces a non hausdorff longitudinally smooth lie groupoid on which we define a pseudodifferential calculus this calculus generalizes the b calculus of melrose defined for manifolds with embedded corners the groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension using tools from the theory of c algebras of groupoids we also obtain new proofs for the study of b calculus

scale and rotation invariant matching using linearly augmented trees we propose a novel linearly augmented tree method for efficient scale and rotation invariant object matching the proposed method enforces pairwise matching consistency defined on trees and high order constraints on all the sites of a template the pairwise constraints admit arbitrary metrics while the high order constraints use l1 norms and therefore can be linearized such a linearly augmented tree formulation introduces hyperedges and loops into the basic tree structure but different from a general loopy graph its special structure allows us to relax and decompose the optimization into a sequence of tree matching problems that are efficiently solvable by dynamic programming the proposed method also works on continuous scale and rotation parameters we can match with a scale up to any large value with the same efficiency our experiments on ground truth data and a variety of real images and videos show that the proposed method is efficient accurate and reliable

inference methods for crfs with co occurrence statistics the markov and conditional random fields crfs used in computer vision typically model only local interactions between variables as this is generally thought to be the only case that is computationally tractable in this paper we consider a class of global potentials defined over all variables in the crf we show how they can be readily optimised using standard graph cut algorithms at little extra expense compared to a standard pairwise field this result can be directly used for the problem of class based image segmentation which has seen increasing recent interest within computer vision here the aim is to assign a label to each pixel of a given image from a set of possible object classes typically these methods use random fields to model local interactions between pixels or super pixels one of the cues that helps recognition is global object co occurrence statistics a measure of which classes such as chair or motorbike are likely to occur in the same image together there have been several approaches proposed to exploit this property but all of them suffer from different limitations and typically carry a high computational cost preventing their application on large images we find that the new model we propose produces a significant improvement in the labelling compared to just using a pairwise model and that this improvement increases as the number of labels increases

unbounded pseudodifferential calculus on lie groupoids abstract we develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids also called lie groupoids with compact basis we analyze these operators as unbounded operators acting on hilbert modules over c g and we show in particular that elliptic operators are regular we construct a scale of sobolev modules which are the abstract analogues of the ordinary sobolev spaces and analyze their properties furthermore we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense

improved belief propagation algorithm finds many bethe states in the random field ising model on random graphs we first present an empirical study of the belief propagation bp algorithm when run on the random field ising model defined on random regular graphs in the zero temperature limit we introduce the notion of extremal solutions for the bp equations and we use them to fix a fraction of spins in their ground state configuration at the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size this in turn makes the associated optimization problem highly non trivial in the critical region using the bounds on the bp messages provided by the extremal solutions we design a new and very easy to implement bp scheme which is able to output a large number of stable fixed points on one hand this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time on the other hand we found that the number of fixed points of the bp algorithm grows with the system size in the critical region this unexpected feature poses new relevant questions about the physics of this class of models

message passing algorithms for inference and optimization belief propagation and divide and concur message passing algorithms can solve a wide variety of optimization inference and constraint satisfaction problems the algorithms operate on factor graphs that visually represent and specify the structure of the problems after describing some of their applications i survey the family of belief propagation bp algorithms beginning with a detailed description of the min sum algorithm and its exactness on tree factor graphs and then turning to a variety of more sophisticated bp algorithms including free energy based bp algorithms splitting bp algorithms that generalize tree reweighted bp and the various bp algorithms that have been proposed to deal with problems with continuous variables

theta lifts for lorentzian lattices and coefficients of mock theta functions we evaluate regularized theta lifts for lorentzian lattices in three different ways in particular we obtain formulas for their values at special points involving coefficients of mock theta functions by comparing the different evaluations we derive recurrences for the coefficients of mock theta functions such as hurwitz class numbers andrews spt function and ramanujan s mock theta functions

belief propagation and loop calculus for the permanent of a non negative matrix we consider computation of the permanent of a positive n n non negative matrix p pji i j 1 n or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph kn n the problem is known to be of likely exponential complexity stated as the partition function z of a graphical model the problem allows for exact loop calculus representation chertkov m and chernyak v 2006 phys rev e 72 065102 in terms of an interior minimum of the bethe free energy functional over non integer doubly stochastic matrix of marginal beliefs ji i j 1 n also correspondent to a fixed point of the iterative message passing algorithm of the belief propagation bp type our main result is an explicit expression of the exact partition function permanent in terms of the matrix of bp marginals as z perm p zbp perm ji 1 ji i j 1 ji where zbp is the bp expression for the permanent stated explicitly in terms of we give two derivations of the formula a direct one based on the bethe free energy and an alternative one combining the ihara graph function and the loop calculus approaches assuming that the matrix of the bp marginals is calculated we provide two lower bounds and one upper bound to estimate the multiplicative term two complementary lower bounds are based on the gurvits van der waerden theorem and on a relation between the modified permanent and determinant respectively

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

fixed points of generalized approximate message passing with arbitrary matrices the estimation of a random vector with independent components passed through a linear transform followed by a componentwise possibly nonlinear output map arises in a range of applications approximate message passing amp methods based on gaussian approximations of loopy belief propagation have recently attracted considerable attention for such problems for large random transforms these methods exhibit fast convergence and admit precise analytic characterizations with testable conditions for optimality even for certain non convex problem instances however the behavior of amp under general transforms is not fully understood in this paper we consider the generalized amp gamp algorithm and relate the method to more common optimization techniques this analysis enables a precise characterization of the gamp algorithm fixed points that applies to arbitrary transforms in particular we show that the fixed points of the so called max sum gamp algorithm for map estimation are critical points of a constrained maximization of the posterior density the fixed points of the sum product gamp algorithm for estimation of the posterior marginals can be interpreted as critical points of a certain free energy

high dimensional gaussian graphical model selection walk summability and local separation criterion we consider the problem of high dimensional gaussian graphical model selection we identify a set of graphs for which an efficient estimation algorithm exists and this algorithm is based on thresholding of empirical conditional covariances under a set of transparent conditions we establish structural consistency or sparsistency for the proposed algorithm when the number of samples n jmin 2 log p where p is the number of variables and jmin is the minimum absolute edge potential of the graphical model the sufficient conditions for sparsistency are based on the notion of walk summability of the model and the presence of sparse local vertex separators in the underlying graph we also derive novel non asymptotic necessary conditions on the number of samples required for sparsistency

shimura lifts of weakly holomorphic modular forms we show how to realize the shimura lift of arbitrary level and character using the vector valued theta lifts of borcherds using the regularization of borcherds lift we extend the shimura lift to take weakly holomorphic modular forms of half integral weight to meromorphic modular forms of even integral weight having poles at cm points

image segmentation via probabilistic graph matching this paper presents an unsupervised and semi automatic image segmentation approach where we formulate the segmentation as an inference problem based on unary and pairwise assignment probabilities computed using low level image cues the inference is solved via a probabilistic graph matching scheme which allows rigorous incorporation of low level image cues and automatic tuning of parameters the proposed scheme is experimentally shown to compare favorably with contemporary semi supervised and unsupervised image segmentation schemes when applied to contemporary state of the art image sets

matching by linear programming and successive convexification we present a novel convex programming scheme to solve matching problems focusing on the challenging problem of matching in a large search range and with cluttered background matching is formulated as metric labeling with l1 regularization terms for which we propose a novel linear programming relaxation method and an efficient successive convexification implementation the unique feature of the proposed relaxation scheme is that a much smaller set of basis labels is used to represent the original label space this greatly reduces the size of the searching space a successive convexification scheme solves the labeling problem in a coarse to fine manner importantly the original cost function is reconvexified at each stage in the new focus region only and the focus region is updated so as to refine the searching result this makes the method well suited for large label set matching experiments demonstrate successful applications of the proposed matching scheme in object detection motion estimation and tracking

multi periodic neural coding for adaptive information transfer information processing in the presence of noise has been a key challenge in multiple disciplines including computer science communications and neuroscience among such noise reduction mechanisms the shift map code represents an analog variable by its residues with respect to distinct moduli that are chosen as geometric scalings of an integer motivated by the multi periodic neural code in the entorhinal cortex i e the coding mechanism of grid cells this work extends the shift map code by generalizing the choices of moduli in particular it is shown that using similarly sized moduli for instance evenly and closely spaced integers which tend to have large co prime factors results in a code whose codewords are separated in an interleaving way such that when the decoder has side information regarding the source then error control is significantly improved compared to the original shift map code this novel structure allows the system to dynamically adapt to the side information at the decoder even if the encoder is not privy to the side information a geometrical interpretation of the proposed coding scheme and a method to find such codes are detailed as an extension it is shown that this novel code also adapts to scenarios when only a fraction of codeword symbols is available at the decoder

stereo matching with color weighted correlation hierarchical belief propagation and occlusion handling in this paper we formulate a stereo matching algorithm with careful handling of disparity discontinuity and occlusion the algorithm works with a global matching stereo model based on an energy minimization framework the global energy contains two terms the data term and the smoothness term the data term is first approximated by a color weighted correlation then refined in occluded and low texture areas in a repeated application of a hierarchical loopy belief propagation algorithm the experimental results are evaluated on the middlebury data sets showing that our algorithm is the top performer among all the algorithms listed there

the atiyah singer index formula for subelliptic operators on contact manifolds part i the atiyah singer index theorem gives a topological formula for the index of an elliptic differential operator the topological index depends on a cohomology class that is constructed from the principal symbol of the operator on contact manifolds the important fredholm operators are not elliptic but hypoelliptic their symbolic calculus is noncommutative and is closely related to analysis on the heisenberg group for a hypoelliptic differential operator in the heisenberg calculus on a contact manifold we construct a symbol class in the k theory of a noncommutative c algebra that is associated to the algebra of symbols there is a canonical map from this analytic k theory group to the ordinary cohomology of the manifold which gives a de rham class to which the atiyah singer formula can be applied we prove that the index formula holds for these hypoelliptic operators our methods derive from connes tangent groupoid proof of the index theorem

nonbacktracking expansion of finite graphs message passing equations yield a sharp percolation transition in finite graphs as an artifact of the locally treelike approximation for an arbitrary finite connected undirected graph we construct an infinite tree having the same local structural properties as this finite graph when observed by a nonbacktracking walker formally excluding the boundary this infinite tree is a generalization of the bethe lattice we indicate an infinite locally treelike random network whose local structure is exactly given by this infinite tree message passing equations for various cooperative models on this construction are the same as for the original finite graph but here they provide the exact solutions of the corresponding cooperative problems these solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated we show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix as representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction the nonbacktracking expansion we analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations notably in a range of important cases simulations turn out to be more efficient computationally than the message passing

on resolving simultaneous congruences using belief propagation graphical models and related algorithmic tools such as belief propagation have proven to be useful tools in approximately solving combinatorial optimization problems across many application domains a particularly combinatorially challenging problem is that of determining solutions to a set of simultaneous congruences specifically a continuous source is encoded into multiple residues with respect to distinct moduli and the goal is to recover the source efficiently from noisy measurements of these residues this problem is of interest in multiple disciplines including neural codes decentralized compression in sensor networks and distributed consensus in information and social networks this letter reformulates the recovery problem as an optimization over binary latent variables then we present a belief propagation algorithm a layered variant of affinity propagation to solve the problem the underlying encoding structure of multiple congruences naturally results in a layered graphical model for the problem over which the algorithms are deployed resulting in a layered affinity propagation lap solution first the convergence of lap to an approximation of the maximum likelihood ml estimate is shown second numerical simulations show that lap converges within a few iterations and that the mean square error of lap approaches that of the ml estimation at high signal to noise ratios

efficient semidefinite branch and cut for map mrf inference we propose a branch and cut b c method for solving general map mrf inference problems the core of our method is a very efficient bounding procedure which combines scalable semidefinite programming sdp and a cutting plane method for seeking violated constraints in order to further speed up the computation several strategies have been exploited including model reduction warm start and removal of inactive constraints we analyze the performance of the proposed method under different settings and demonstrate that our method either outperforms or performs on par with state of the art approaches especially when the connectivities are dense or when the relative magnitudes of the unary costs are low we achieve the best reported results experiments show that the proposed algorithm achieves better approximation than the state of the art methods within a variety of time budgets on challenging non submodular map mrf inference problems

a max product em algorithm for reconstructing markov tree sparse signals from compressive samples we propose a bayesian expectation maximization em algorithm for reconstructing markov tree sparse signals via belief propagation the measurements follow an underdetermined linear model where the regression coefficient vector is the sum of an unknown approximately sparse signal and a zero mean white gaussian noise with an unknown variance the signal is composed of large and small magnitude components identified by binary state variables whose probabilistic dependence structure is described by a markov tree gaussian priors are assigned to the signal coefficients given their state variables and the jeffreys noninformative prior is assigned to the noise variance our signal reconstruction scheme is based on an em iteration that aims at maximizing the posterior distribution of the signal and its state variables given the noise variance we construct the missing data for the em iteration so that the complete data posterior distribution corresponds to a hidden marcov tree hmt probabilistic graphical model that contains no loops and implement its maximization m step via a max product algorithm this em algorithm estimates the vector of state variables as well as solves iteratively a linear system of equations to obtain the corresponding signal estimate we select the noise variance so that the corresponding estimated signal and state variables obtained upon convergence of the em iteration have the largest marginal posterior distribution we compare the proposed and existing state of the art reconstruction methods via signal and image reconstruction experiments

message passing algorithms for quadratic minimization gaussian belief propagation gabp is an iterative algorithm for computing the mean and variances of a multivariate gaussian distribution or equivalently the minimum of a multivariate positive definite quadratic function sufficient conditions such as walk summability that guarantee the convergence and correctness of gabp are known but gabp may fail to converge to the correct solution given an arbitrary positive definite covariance matrix as was observed by malioutov et al 2006 the gabp algorithm fails to converge if the computation trees produced by the algorithm are not positive definite in this work we will show that the failure modes of the gabp algorithm can be understood via graph covers and we prove that a parameterized generalization of the min sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite we demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the gauss seidel and jacobi algorithms finally we observe empirically that there always exists a choice of parameters such that the above generalization of the gabp algorithm converges

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

maximum weight matching using odd sized cycles max product belief propagation and half integrality we study the maximum weight matching mwm problem for general graphs through the max product belief propagation bp and related linear programming lp the bp approach provides distributed heuristics for finding the maximum a posteriori map assignment in a joint probability distribution represented by a graphical model gm and respective lps can be considered as continuous relaxations of the discrete map problem it was recently shown that a bp algorithm converges to the correct map mwm assignment under a simple gm formulation of mwm as long as the corresponding lp relaxation is tight first under the motivation for forcing the tightness condition we consider a new gm formulation of mwm say c gm using non intersecting odd sized cycles in the graph the new corresponding lp relaxation say c lp becomes tight for more mwm instances however the tightness of c lp now does not guarantee such convergence and correctness of the new bp on c gm to address the issue we introduce a novel graph transformation applied to c gm which results in another gm formulation of mwm and prove that the respective bp on it converges to the correct map mwm assignment as long as c lp is tight finally we also show that c lp always has half integral solutions which leads to an efficient bp based mwm heuristic consisting of making sequential cutting plane modifications to the underlying gm our experiments show that this bp based cutting plane heuristic performs as well as that based on traditional lp solvers

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

linear scale and rotation invariant matching matching visual patterns that appear scaled rotated and deformed with respect to each other is a challenging problem we propose a linear formulation that simultaneously matches feature points and estimates global geometrical transformation in a constrained linear space the linear scheme enables search space reduction based on the lower convex hull property so that the problem size is largely decoupled from the original hard combinatorial problem our method therefore can be used to solve large scale problems that involve a very large number of candidate feature points without using prepruning in the search this method is more robust in dealing with weak features and clutter we apply the proposed method to action detection and image matching our results on a variety of images and videos demonstrate that our method is accurate efficient and robust

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

the holonomy groupoid of a singular foliation we construct the holonomy groupoid of any singular foliation in the regular case this groupoid coincides with the usual holonomy groupoid of winkelnkemper h e winkelnkemper the graph of a foliation ann glob anal geom 1 3 1983 51 75 the same holds in the singular cases of j pradines how to define the differentiable graph of a singular foliation c top geom diff cat xxvi 4 1985 339 381 b bigonnet j pradines graphe d un feuilletage singulier c r acad sci paris 300 13 1985 439 442 c debord local integration of lie algebroids banach center publ 54 2001 21 33 c debord holonomy groupoids of singular foliations j diff geom 58 2001 467 500 which from our point of view can be thought of as being almost regular in the general case the holonomy groupoid can be quite an ill behaved geometric object on the other hand it often has a nice longitudinal smooth structure nonetheless we use this groupoid to generalize to the singular case connes construction of the c algebra of the foliation we also outline the construction of a longitudinal pseudo differential calculus the analytic index of a longitudinally elliptic operator takes place in the k theory of our c algebra in our construction the key notion is that of a bi submersion which plays the role of a local lie groupoid defining the foliation our groupoid is the quotient of germs of these bi submersions with respect to an appropriate equivalence relation

linear response algorithms for approximate inference in graphical models belief propagation bp on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph however it does not prescribe a way to compute joint distributions over pairs of distant nodes in the graph in this article we propose two new algorithms for approximating these pairwise probabilities based on the linear response theorem the first is a propagation algorithm that is shown to converge if bp converges to a stable fixed point the second algorithm is based on matrix inversion applying these ideas to gaussian random fields we derive a propagation algorithm for computing the inverse of a matrix

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

creating large scale city models from 3d point clouds a robust approach with hybrid representation we present a novel and robust method for modeling cities from 3d point data our algorithm provides a more complete description than existing approaches by reconstructing simultaneously buildings trees and topologically complex grounds a major contribution of our work is the original way of modeling buildings which guarantees a high generalization level while having semantized and compact representations geometric 3d primitives such as planes cylinders spheres or cones describe regular roof sections and are combined with mesh patches that represent irregular roof components the various urban components interact through a non convex energy minimization problem in which they are propagated under arrangement constraints over a planimetric map our approach is experimentally validated on complex buildings and large urban scenes of millions of points and is compared to state of the art methods

graph rigidity cyclic belief propagation and point pattern matching a recent paper caetano et al 2006 proposed a provably optimal polynomial time method for performing near isometric point pattern matching by means of exact probabilistic inference in a chordal graphical model its fundamental result is that the chordal graph in question is shown to be globally rigid implying that exact inference provides the same matching solution as exact inference in a complete graphical model this implies that the algorithm is optimal when there is no noise in the point patterns in this paper we present a new graph which is also globally rigid but has an advantage over the graph its maximal clique size is smaller rendering inference significantly more efficient however this graph is not chordal and thus standard junction tree algorithms cannot be directly applied nevertheless we show that loopy belief propagation in such a graph converges to the optimal solution this allows us to retain the optimality guarantee in the noiseless case while substantially reducing both memory requirements and processing time our experimental results show that the accuracy of the proposed solution is indistinguishable when there is noise in the point patterns

context aware patch based image inpainting using markov random field modeling in this paper we first introduce a general approach for context aware patch based image inpainting where textural descriptors are used to guide and accelerate the search for well matching candidate patches a novel top down splitting procedure divides the image into variable size blocks according to their context constraining thereby the search for candidate patches to nonlocal image regions with matching context this approach can be employed to improve the speed and performance of virtually any patch based inpainting method we apply this approach to the so called global image inpainting with the markov random field mrf prior where mrf encodes a priori knowledge about consistency of neighboring image patches we solve the resulting optimization problem with an efficient low complexity inference method experimental results demonstrate the potential of the proposed approach in inpainting applications like scratch text and object removal improvement and significant acceleration of a related global mrf based inpainting method is also evident

finiteness of 2 reflective lattices of signature 2 n a modular form for an even lattice l of signature 2 n is said to be 2 reflective if its zero divisor is set theoretically contained in the heegner divisor defined by the 2 vectors in l we prove that there are only finitely many even lattices with n geq7 which admit 2 reflective modular forms in particular there is no such lattice in n geq26 except the even unimodular lattice of signature 2 26 this proves a conjecture of gritsenko and nikulin in the range n geq7

on the optimality of solutions of the max product belief propagation algorithm in arbitrary graphs 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

finding maps for belief networks is np hard given a probabilistic world model an important problem is to find the maximum a posteriori probability map instantiation of all the random variables given the evidence numerous researchers using such models employ some graph representation for the distributions such as a bayesian belief network this representation simplifies the complexity of specifying the distributions from exponential in n the number of variables in the model to linear in n in many interesting cases we show however that finding the map is np hard in the general case when these representations are used even if the size of the representation happens to be linear in n furthermore minor modifications to the proof show that the problem remains np hard for various restrictions of the topology of the graphs the same technique can be applied to the results of a related paper by cooper to further restrict belief network topology in the proof that probabilistic inference is np hard

analytic k theory on manifolds with corners let y be a compact manifold in 1 atiyah proposed an analytic delinition of the k homology groups k y defined abstractly as the dual theory to the atiyah hirzebruch k cohomology theory these ideas were developed by kasparov 15 and brown douglas and fillmore lo cycles for this analytic k homology theory are abstract elliptic operators i e bounded linear operators between hilbert spaces satisfying additional conditions derived from the properties of elliptic pseudo differential operators in particular elliptic pseudodifferential operators between sections of vector bundles define such cycles the homology class of the cycle depends only on the principal symbol and indeed only on the k cohomology class it represents on the cotangent bundle in fact it had already been shown in essence by atiyah that the resulting map

correctness of local probability propagation in graphical models with loops graphical models such as bayesian networks and markov networks represent joint distributions over a set of variables by means of a graph when the graph is singly connected local propagation rules of the sort proposed by pearl 1988 are guaranteed to converge to the correct posterior probabilities recently a number of researchers have empirically demonstrated good performance of these same local propagation schemes on graphs with loops but a theoretical understanding of this performance has yet to be achieved for graphical models with a single loop we derive an analytical relationship between the probabilities computed using local propagation and the correct marginals using this relationship we show a category of graphical models with loops for which local propagation gives rise to provably optimal maximum a posteriori assignments although the computed marginals will be incorrect we also show how nodes can use local information in the messages they receive in order to correct their computed marginals we discuss how these results can be extended to graphical models with multiple loops and show simulation results suggesting that some properties of propagation on single loop graphs may hold for a larger class of graphs specifically we discuss the implication of our results for understanding a class of recently proposed error correcting codes known as turbo codes

approximating the permanent with fractional belief propagation we discuss schemes for exact and approximate computations of permanents and compare them with each other specifically we analyze the belief propagation bp approach and its fractional belief propagation fbp generalization for 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 the fractional free energy ffe function is parameterized by a scalar parameter 1 1 where 1 corresponds to the bp limit and 1 corresponds to the exclusion principle but ignoring perfect matching constraints mean field mf limit ffe shows monotonicity and continuity with respect to for every non negative matrix we define its special value 1 0 to be the for which the minimum of the parameterized ffe function is equal to the permanent of the matrix where the lower and upper bounds of the interval corresponds to respective bounds for the permanent our experimental analysis suggests that the distribution of varies for different ensembles but always lies within the 1 1 2 interval moreover for all ensembles considered the behavior of is highly distinctive offering an empirical practical guidance for estimating permanents of non negative matrices via the ffe approach

on the converse theorem for borcherds products abstract we prove a new converse theorem for borcherds multiplicative theta lift which improves the previously known results to this end we develop a newform theory for vector valued modular forms for the weil representation which might be of independent interest we also derive lower bounds for the ranks of the picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups

fourier integral operators and the index of symplectomorphisms on manifolds with boundary given two compact manifolds with boundary x y and a boundary preserving symplectomorphism t y 0 t x 0 which is one homogeneous in the fibers and satisfies the transmission condition we introduce fourier integral operators of boutet de monvel type associated with we study their mapping properties between sobolev spaces develop a calculus and prove a egorov type theorem we also introduce a notion of ellipticity which implies the fredholm property finally we show how in the spirit of a classical construction by a weinstein a fredholm operator of this type can be associated with and a section of the maslov bundle if dim y 2 or the maslov bundle is trivial the index is independent of the section and thus an invariant of the symplectomorphism

holonomy groupoids of singular foliations we give a new construction of lie groupoids which is particularly well adapted to the generalization of holonomy groupoids to singular foliations given a family of local lie groupoids on open sets of a smooth manifold m satisfying some hypothesis we construct a lie groupoid which contains the whole family this construction involves a new way of considering local morita equivalences not only as equivalence relations but also as generalized isomorphisms in particular we prove that almost injective lie algebroids are integrable

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

bounded approximate decentralised coordination via the max sum algorithm in this paper we propose a novel approach to decentralised coordination that is able to efficiently compute solutions with a guaranteed approximation ratio our approach is based on a factor graph representation of the constraint network it builds a tree structure by eliminating dependencies between the functions and variables within the factor graph that have the least impact on solution quality it then uses the max sum algorithm to optimally solve the resulting tree structured constraint network and provides a bounded approximation specific to the particular problem instance in addition we present two generic pruning techniques to reduce the amount of computation that agents must perform when using the max sum algorithm when this is combined with the above mentioned approximation algorithm the agents are able to solve decentralised coordination problems that have very large action spaces with a low computation and communication overhead we empirically evaluate our approach in a mobile sensor domain where mobile agents are used to monitor and predict the state of spatial phenomena e g temperature or gas concentration such sensors need to coordinate their movements with their direct neighbours to maximise the collective information gain while predicting measurements at unobserved locations when applied in this domain our approach is able to provide solutions which are guaranteed to be within 2 of the optimal solution moreover the two pruning techniques are extremely effective in decreasing the computational effort of each agent by reducing the size of the search space by up to 92

pseudodifferential operators on manifolds with a lie structure at infinity we define and study an algebra 0 v m0 of pseudodifferential operators canonically associated to a noncompact riemannian manifold m0 whose geometry at infinity is described by a lie algebra of vector fields v on a compactification m of m0 to a compact manifold with corners we show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to 0 v m0 we also consider the algebra diff m0 of differential operators on m0 generated by v and c m and show that 0 v m0 is a microlocalization of diff m0 our construction solves a problem posed by melrose in 1990 finally we introduce and study semi classical and suspended versions of the algebra 0 v m0

on decoding irregular tanner codes with local optimality guarantees we consider decoding of binary linear tanner codes using message passing iterative decoding and linear programming lp decoding in memoryless binary input output symmetric mbios channels we present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular tanner codes with respect to any mbios channel this characterization is a generalization of arora proc acm symp theory of computing 2009 and vontobel proc inf theory and appl workshop 2010 and is based on a conical combination of normalized weighted subtrees in the computation trees of the tanner graph these subtrees may have any finite height h even equal or greater than half of the girth of the tanner graph in addition the degrees of local code nodes in these subtrees are not restricted to two i e these subtrees are not restricted to skinny trees we prove that local optimality in this new characterization implies maximum likelihood ml optimality and lp optimality and show that a certificate can be computed efficiently we also present a new message passing iterative decoding algorithm called normalized weighted min sum nwms nwms decoding is a belief propagation bp type algorithm that applies to any irregular binary tanner code with single parity check local codes e g low density and high density parity check codes we prove that if a locally optimal codeword with respect to height parameter h exists whereby notably h is not limited by the girth of the tanner graph then nwms decoding finds this codeword in h iterations the decoding guarantee of the nwms decoding algorithm applies whenever there exists a locally optimal codeword because local optimality of a codeword implies that it is the unique ml codeword the decoding guarantee also provides an ml certificate for this codeword finally we apply the new local optimality characterization to regular tanner codes and prove lower bounds on the noise thresholds of lp decoding in mbios channels when the noise is below these lower bounds the probability that lp decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the tanner graph

a rigorous analysis of the cavity equations for the minimum spanning tree we analyze a new general representation for the minimum weight steiner tree problem that translates the topological connectivity constraint into a set of local conditions which can be analyzed by the so called cavity equation techniques for the limit case of the spanning tree we prove that the fixed point of the algorithm arising from the cavity equations leads to the global optimum

an orientation inference framework for surface reconstruction from unorganized point clouds in this paper we present an orientation inference framework for reconstructing implicit surfaces from unoriented point clouds the proposed method starts from building a surface approximation hierarchy comprising of a set of unoriented local surfaces which are represented as a weighted combination of radial basis functions we formulate the determination of the globally consistent orientation as a graph optimization problem by treating the local implicit patches as nodes an energy function is defined to penalize inconsistent orientation changes by checking the sign consistency between neighboring local surfaces an optimal labeling of the graph nodes indicating the orientation of each local surface can thus be obtained by minimizing the total energy defined on the graph the local inference results are propagated over the model in a front propagation fashion to obtain the global solution the reconstructed surfaces are consolidated by a simple and effective inspection procedure to locate the erroneously fitted local surfaces a progressive reconstruction algorithm that iteratively includes more oriented points to improve the fitting accuracy and efficiently updates the rbf coefficients is proposed we demonstrate the performance of the proposed method by showing the surface reconstruction results on some real world 3 d data sets with comparison to those by using the previous methods

mathematical programming decoding of binary linear codes theory and algorithms mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches challenging established but often heuristic algorithms based on iterative message passing concepts from mathematical programming used in the context of decoding include linear integer and nonlinear programming network flows notions of duality as well as matroid and polyhedral theory this paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary input memoryless symmetric channels

edge preserving image denoising with a closed form solution this paper addresses the problem of image denoising which is still a valid challenge at the crossing of functional analysis and statistics we herein propose a novel pixel based algorithm which formulates the image denoising problem as the maximum a posterior map estimation problem using markov random fields mrfs such an map estimation problem is equivalent to a maximum likelihood ml estimation constrained on spatial homogeneity and is np hard in discrete domain to make it tractable we convert it to a continuous label assignment problem based on a gaussian mrf model and then obtain a closed form globally optimal solution since the gaussian mrfs tend to over smooth images and blur edges our algorithm incorporates the pre estimated image edge information into the energy function construction and therefore better preserves the image structures in the algorithm patch similarity based pairwise interaction is also involved to better preserve image details and make the algorithm more robust to noise based on the theoretical analysis on the deviation caused by the discretization from obtained continuous global optimum to discrete output we demonstrate the guaranteed optimal property of our algorithm both quantitative and qualitative comparative experimental results are given to demonstrate the better performance of our algorithm over several existing state of the art related algorithms

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks

a comparative study of energy minimization methods for markov random fields with smoothness based priors among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel labeling tasks such as depth or texture computation it has been known for decades that such problems can be elegantly expressed as markov random fields yet the resulting energy minimization problems have been widely viewed as intractable algorithms such as graph cuts and loopy belief propagation lbp have proven to be very powerful for example such methods form the basis for almost all the top performing stereo methods however the trade offs among different energy minimization algorithms are still not well understood in this paper we describe a set of energy minimization benchmarks and use them to compare the solution quality and runtime of several common energy minimization algorithms we investigate three promising methods graph cuts lbp and tree reweighted message passing in addition to the well known older iterated conditional mode icm algorithm our benchmark problems are drawn from published energy functions used for stereo image stitching interactive segmentation and denoising we also provide a general purpose software interface that allows vision researchers to easily switch between optimization methods the benchmarks code images and results are available at http vision middlebury edu mrf

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

estimating the wrong graphical model benefits in the computation limited setting consider the problem of joint parameter estimation and prediction in a markov random field that is the model parameters are estimated on the basis of an initial set of data and then the fitted model is used to perform prediction e g smoothing denoising interpolation on a new noisy observation working under the restriction of limited computation we analyze a joint method in which the same convex variational relaxation is used to construct an m estimator for fitting parameters and to perform approximate marginalization for the prediction step the key result of this paper is that in the computation limited setting using an inconsistent parameter estimator i e an estimator that returns the wrong model even in the infinite data limit is provably beneficial since the resulting errors can partially compensate for errors made by using an approximate prediction technique en route to this result we analyze the asymptotic properties of m estimators based on convex variational relaxations and establish a lipschitz stability property that holds for a broad class of convex variational methods this stability result provides additional incentive apart from the obvious benefit of unique global optima for using message passing methods based on convex variational relaxations we show that joint estimation prediction based on the reweighted sum product algorithm substantially outperforms a commonly used heuristic based on ordinary sum product

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

belief propagation an asymptotically optimal algorithm for the random assignment problem the random assignment problem asks for the minimum cost perfect matching in the complete n n bipartite graph knn with i i d edge weights say uniform on 0 1 in a remarkable work by aldous aldous d 2001 the 2 limit in the random assignment problem rsa18 381 418 the optimal cost was shown to converge to 2 as n as conjectured by mezard and parisi mezard m g parisi 1987 on the solution of the random link matching problem j phys 48 1451 1459 through the so called cavity method the latter also suggested a nonrigorous decentralized strategy for finding the optimum which turned out to be an instance of the belief propagation bp heuristic discussed by pearl pearl j 1988 probabilistic reasoning in intelligent systems networks of plausible inference morgan kaufmann san francisco in this paper we use the objective method to analyze the performance of bp as the size of the underlying graph becomes large specifically we establish that the dynamic of bp on knn converges in distribution as n to an appropriately defined dynamic on the poisson weighted infinite tree and we then prove correlation decay for this limiting dynamic as a consequence we obtain that bp finds an asymptotically correct assignment in o n2 time only this contrasts with both the worst case upper bound for convergence of bp derived by bayati et al bayati m d shah m sharma 2008 max product for maximum weight matching convergence correctness and lp duality ieee trans inform theory54 3 1241 1251 and the best known computational cost of n3 achieved by edmonds and karp s algorithm edmonds j r karp 1972 theoretical improvements in algorithmic efficiency for network flow problems j acm19 248 264

convergence of weighted min sum decoding via dynamic programming on trees applying the max product and sum product algorithms to loopy graphs is now quite popular for best assignment problems this is largely due to their low computational complexity and impressive performance in practice still there is no general understanding of the conditions required for convergence or optimality of converged solutions or both this paper presents an analysis of both attenuated max product decoding and weighted min sum decoding for low density parity check ldpc codes which guarantees convergence to a fixed point when a weight factor is sufficiently small it also shows that if the fixed point satisfies some consistency conditions then it must be both a linear programming lp and maximum likelihood ml decoding solution for dv dc regular ldpc codes the weight factor must satisfy dv 1 1 are given finally connections are explored with recent work on the threshold of lp decoding

message passing for graph structured linear programs proximal methods and rounding schemes the problem of computing a maximum a posteriori map configuration is a central computational challenge associated with markov random fields there has been some focus on tree based linear programming lp relaxations for the map problem this paper develops a family of super linearly convergent algorithms for solving these lps based on proximal minimization schemes using bregman divergences as with standard message passing on graphs the algorithms are distributed and exploit the underlying graphical structure and so scale well to large problems our algorithms have a double loop character with the outer loop corresponding to the proximal sequence and an inner loop of cyclic bregman projections used to compute each proximal update we establish convergence guarantees for our algorithms and illustrate their performance via some simulations we also develop two classes of rounding schemes deterministic and randomized for obtaining integral configurations from the lp solutions our deterministic rounding schemes use a re parameterization property of our algorithms so that when the lp solution is integral the map solution can be obtained even before the lp solver converges to the optimum we also propose graph structured randomized rounding schemes applicable to iterative lp solving algorithms in general we analyze the performance of and report simulations comparing these rounding schemes

boutet de monvel operators on lie manifolds with boundary abstract we introduce and study a general pseudodifferential calculus for boundary value problems on a class of non compact manifolds with boundary so called lie manifolds with boundary this is accomplished by constructing a suitable generalization of the boutet de monvel calculus for boundary value problems the data consists of a compact manifold with corners m that is endowed with a lie structure of vector fields 2 v a so called lie manifold the manifold m is split into two equal parts x and x which intersect in an embedded hypersurface y x our goal is to describe a transmission boutet de monvel calculus for boundary value problems compatible with the structure of lie manifolds starting with the example of b vector fields we show that there are two groupoids integrating the lie structures on m and on y respectively these two groupoids form a bibundle or a groupoid correspondence and under some mild assumptions these groupoids are morita equivalent with the help of the bibundle structure and canonically defined manifolds with corners which are blow ups in particular cases we define a class of boutet de monvel type operators we then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem finally we consider appropriate fredholm conditions and construct the parametrices for elliptic operators in the calculus

sufficient conditions for convergence of the sum product algorithm novel conditions are derived that guarantee convergence of the sum product algorithm also known as loopy belief propagation or simply belief propagation bp to a unique fixed point irrespective of the initial messages for parallel synchronous updates the computational complexity of the conditions is polynomial in the number of variables in contrast with previously existing conditions our results are directly applicable to arbitrary factor graphs with discrete variables and are shown to be valid also in the case of factors containing zeros under some additional conditions the conditions are compared with existing ones numerically and if possible analytically for binary variables with pairwise interactions sufficient conditions are derived that take into account local evidence i e single variable factors and the type of pair interactions attractive or repulsive it is shown empirically that this bound outperforms existing bounds

correlation decay in random decision networks we consider a decision network on an undirected graph in which each node corresponds to a decision variable and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes the goal is to construct a decision vector that maximizes the total reward this decision problem encompasses a variety of models including maximum likelihood inference in graphical models markov random fields combinatorial optimization on graphs economic team theory and statistical physics the network is endowed with a probabilistic structure in which rewards are sampled from a distribution our aim is to identify sufficient conditions on the network structure and rewards distributions to guarantee average case polynomiality of the underlying optimization problem additionally we wish to characterize the efficiency of a decentralized solution generated on the basis of local information we construct a new decentralized algorithm called cavity expans

residue traces for certain algebras of fourier integral operators abstract in this paper a trace functional is defined for certain algebras of fourier integral operators including for instance the algebra of toeplitz operators on the boundary of a strictly pseudoconvex domain and it is shown that vanishing of this trace characterizes commutators in each of these algebras

iterative decoding using eigenmessages the eigenmessage approach to iterative decoding introduces a degree of nonlocality into a belief propagation decoder by representing an entire set of messages around a cycle of the tanner graph as a linear operator the eigenvector for the operator represents a fixed point of the belief propagation algorithm around a cycle with incident messages fixed a multiple eigenmessage approach is also presented in which messages around several cycles are simultaneously expressed the eigenmessage approach may be applied to any graph with cycles but we demonstrate its use on ldpc decoding in this setting computational results compare eigenmessage methods with conventional belief propagation decoding showing using simulation and exit charts that the eigenmessage approaches slightly reduce the number of decoder iterations compared to belief propagation decoding while preserving the probability of error of conventional decoding