linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

improving variational methods via pairwise linear response identities inference methods are often formulated as variational approximations these approximations allow easy evaluation of statistics by marginalization or linear response but these estimates can be inconsistent we show that by introducing constraints on covariance one can ensure consistency of linear response with the variational parameters and in so doing inference of marginal probability distributions is improved for the bethe approximation and its generalizations improvements are achieved with simple choices of the constraints the approximations are presented as variational frameworks iterative procedures related to message passing are provided for finding the minima

cccp algorithms to minimize the bethe and kikuchi free energies convergent alternatives to belief propagation this article introduces a class of discrete iterative algorithms that are provably convergent alternatives to belief propagation bp and generalized belief propagation gbp our work builds on recent results by yedidia freeman and weiss 2000 who showed that the fixed points of bp and gbp algorithms correspond to extrema of the bethe and kikuchi free energies respectively we obtain two algorithms by applying cccp to the bethe and kikuchi free energies respectively cccp is a procedure introduced here for obtaining discrete iterative algorithms by decomposing a cost function into a concave and a convex part we implement our cccp algorithms on two and three dimensional spin glasses and compare their results to bp and gbp our simulations show that the cccp algorithms are stable and converge very quickly the speed of cccp is similar to that of bp and gbp unlike cccp bp will often not converge for these problems gbp usually but not always converges the results found by cccp applied to the bethe or kikuchi free energies are equivalent or slightly better than those found by bp or gbp respectively when bp and gbp converge note that for these and other problems bp and gbp give very accurate results see yedidia et al 2000 and failure to converge is their major error mode finally we point out that our algorithms have a large range of inference and learning applications

code aided maximum likelihood ambiguity resolution through free energy minimization in digital communication receivers ambiguities in terms of timing and phase need to be resolved prior to data detection in the presence of powerful error correcting codes which operate in low signal to noise ratios snr long training sequences are needed to achieve good performance in this contribution we develop a new class of code aided ambiguity resolution algorithms which require no training sequence and achieve good performance with reasonable complexity in particular we focus on algorithms that compute the maximum likelihood ml solution exactly or in good approximation with a tractable complexity using a factor graph representation the complexity of the proposed algorithm is discussed and reduced complexity variations including stopping criteria and sequential implementation are developed

projective curves of degree codimension 2 in this article we study nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay note that x pi p widetilde x for a rational normal curve widetilde x subset mathbb p d and a point p in mathbb p d setminus widetilde x 2 our main result is about the relation between the geometric properties of x and the position of p with respect to widetilde x we show that the graded betti numbers of x are uniquely determined by the rank hbox rk widetilde x p of p with respect to widetilde x in particular x satisfies property n 2 p if and only if p leq quad rk widetilde x p 3 therefore property n 2 p of x is controlled by quad rk widetilde x p and conversely quad rk widetilde x p can be read off from the minimal free resolution of x this result provides a non linearly normal example for which the converse to theorem 1 1 in eisenbud et al compositio math 141 1460 1478 2005 holds also our result implies that for nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay there are exactly lfloor frac d 2 2 rfloor distinct betti tables

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

norm product belief propagation primal dual message passing for approximate inference inference problems in graphical models can be represented as a constrained optimization of a free energy function in this paper we treat both forms of probabilistic inference estimating marginal probabilities of the joint distribution and finding the most probable assignment through a unified message passing algorithm architecture in particular we generalize the belief propagation bp algorithms of sum product and max product and tree reweighted trw sum and max product algorithms trbp and introduce a new set of convergent algorithms based on convex free energy and linear programming lp relaxation as a zero temperature of a convex free energy the main idea of this work arises from taking a general perspective on the existing bp and trbp algorithms while observing that they all are reductions from the basic optimization formula of f ihi where the function f is an extended valued strictly convex but nonsmooth and the functions hi are extended valued functions not necessarily convex we use tools from convex duality to present the primal dual ascent algorithm which is an extension of the bregman successive projection scheme and is designed to handle optimization of the general type f ihi we then map the fractional free energy variational principle for approximate inference onto the optimization formula above and introduce the norm product message passing algorithm special cases of the norm product include sum product and max product bp algorithms trbp and nmplp algorithms when the fractional free energy is set to be convex convex free energy the norm product is globally convergent for the estimation of marginal probabilities and for approximating the lp relaxation we also introduce another branch of the norm product which arises as the zero temperature of the convex free energy which we refer to as the convex max product the convex max product is convergent unlike max product and aims at solving the lp relaxation

the complexity of approximating a bethe equilibrium this paper resolves a common complexity issue in the bethe approximation of statistical physics and the belief propagation bp algorithm of artificial intelligence the bethe approximation and the bp algorithm are heuristic methods for estimating the partition function and marginal probabilities in graphical models respectively the computational complexity of the bethe approximation is decided by the number of operations required to solve a set of nonlinear equations the so called bethe equation although the bp algorithm was inspired and developed independently yedidia freeman and weiss showed that the bp algorithm solves the bethe equation if it converges however it often does not this naturally motivates the following question to understand limitations and empirical successes of the bethe and bp methods is the bethe equation computationally easy to solve we present a message passing algorithm solving the bethe equation in a polynomial number of operations for general binary graphical models of n variables where the maximum degree in the underlying graph is o log n equivalently it finds a stationary point of the bethe free energy function our algorithm can be used as an alternative to bp fixing its convergence issue and is the first fully polynomial time approximation scheme for the bp fixed point computation in such a large class of graphical models whereas the approximate fixed point computation is known to be polynomial parity arguments on directed graphs ppad hard in general we believe that our technique is of broader interest to understand the computational complexity of the cavity method in statistical physics

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

cortical circuitry implementing graphical models in this letter we develop and simulate a large scale network of spiking neurons that approximates the inference computations performed by graphical models unlike previous related schemes which used sum and product operations in either the log or linear domains the current model uses an inference scheme based on the sum and maximization operations in the log domain simulations show that using these operations a large scale circuit which combines populations of spiking neurons as basic building blocks is capable of finding close approximations to the full mathematical computations performed by graphical models within a few hundred milliseconds the circuit is general in the sense that it can be wired for any graph structure it supports multistate variables and it uses standard leaky integrate and fire neuronal units following previous work which proposed relations between graphical models and the large scale cortical anatomy we focus on the cortical microcircuitry and propose how anatomical and physiological aspects of the local circuitry may map onto elements of the graphical model implementation we discuss in particular the roles of three major types of inhibitory neurons small fast spiking basket cells large layer 2 3 basket cells and double bouquet neurons subpopulations of strongly interconnected neurons with their unique connectivity patterns in different cortical layers and the possible role of minicolumns in the realization of the population based maximum operation

reduced arithmetically gorenstein schemes and simplicial polytopes with maximal betti numbers abstract an si sequence is a finite sequence of positive integers which is symmetric unimodal and satisfies a certain growth condition these are known to correspond precisely to the possible hilbert functions of graded artinian gorenstein algebras with the weak lefschetz property a property shared by a nonempty open set of the family of all graded artinian gorenstein algebras having a fixed hilbert function that is an si sequence starting with an arbitrary si sequence we construct a reduced arithmetically gorenstein configuration g of linear varieties of arbitrary dimension whose artinian reduction has the given si sequence as hilbert function and has the weak lefschetz property furthermore we show that g has maximal graded betti numbers among all arithmetically gorenstein subschemes of projective space whose artinian reduction has the weak lefschetz property and the given hilbert function as an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h vector contains a polytope with maximal graded betti numbers

stable propagation algorithm for the minimization of the bethe free energy we propose a new propagation algorithm for the minimization of the bethe free energy for a generic lattice model with pair interactions the algorithm turns out to be more stable than belief propagation as it reaches a fixed point also for highly frustrated systems such as spin glasses and faster than the provably convergent double loop algorithms

computing loop corrections by message passing any spanning tree in a loopy interaction graph can be used for communicating the effect of the loopy interactions by introducing messages that are passed along the edges in the spanning tree this defines an exact mapping of the problem on the loopy interaction graph onto an extended problem on a tree interaction graph where the thermodynamic quantities can be computed by a message passing algorithm based on the bethe equations we propose an approximation loop correction algorithm for the ising model relying on the above representation of the problem the algorithm deals at the same time with the short and long loops and can be used to obtain upper and lower bounds for the free energy

loop corrections for approximate inference on factor graphs we propose a method to improve approximate inference methods by correcting for the influence of loops in the graphical model the method is a generalization and alternative implementation of a recent idea from montanari and rizzo 2005 it is applicable to arbitrary factor graphs provided that the size of the markov blankets is not too large it consists of two steps i an approximate inference method for example belief propagation is used to approximate cavity distributions for each variable i e probability distributions on the markov blanket of a variable for a modified graphical model in which the factors involving that variable have been removed ii all cavity distributions are improved by a message passing algorithm that cancels out approximation errors by imposing certain consistency constraints this loop correction lc method usually gives significantly better results than the original uncorrected approximate inference algorithm that is used to estimate the effect of loops indeed we often observe that the loop corrected error is approximately the square of the error of the uncorrected approximate inference method in this article we compare different variants of the loop correction method with other approximate inference methods on a variety of graphical models including real world networks and conclude that the lc method generally obtains the most accurate results

a generalisation of the form method for accretive forms and operators abstract the form method as popularised by lions and kato is a successful device to associate m sectorial operators with suitable elliptic or sectorial forms m c intosh generalised the form method to an accretive setting thereby allowing to associate m accretive operators with suitable accretive forms classically the form domain is required to be densely embedded into the hilbert space recently this requirement was relaxed by arendt and ter elst in the setting of elliptic and sectorial forms here we study the prospects of a generalised form method for accretive forms to generate accretive operators in particular we work with the same relaxed condition on the form domain as used by arendt and ter elst we give a multitude of examples for many degenerate phenomena that can occur in the most general setting we characterise when the associated operator is m accretive and investigate the class of operators that can be generated for the case that the associated operator is m accretive we study form approximation and ouhabaz type invariance criteria

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

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

statistical physics of loopy interactions independent loop approximation and beyond we consider an interacting system of spin variables on a loopy interaction graph identified by a tree graph and a set of loopy interactions we start from a high temperature expansion for loopy interactions represented by a sum of non negative contributions from all the possible frustration free loop configurations we then compute the loop corrections using different approximations for the nonlocal loop interactions induced by the spin correlations in the tree graph for distant loopy interactions we can exploit the exponential decay of correlations in the tree interaction graph to compute loop corrections within an independent loop approximation higher orders of the approximation are obtained by considering the correlations between the nearby loopy interactions involving larger number of spin variables in particular the sum over the loop configurations can be computed exactly by the belief propagation algorithm in the low orders of the approximation as long as the loopy interactions have a tree structure these results might be useful in developing more accurate and convergent message passing algorithms exploiting the structure of loopy interactions

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

on equations defining arithmetically cohen macaulay schemes i in this paper we continue the study of the simplest arithmetically cohen macauly cm for short schemes cf 111 and 12 i we describe the hilbert function of an arbitrary cm scheme in p of given degree e lying on a hypersurface of given degree k and such that its general curvilinear section has minimal genus proposition 1 3 we present a geometric construction of all reduced irreducible codimension two cm varieties whose general curvilinear sections have minimal genus with respect to degree theorem 1 7 we also prove the following assertion essentially due to severi there exist smooth nonspecial cm curves of degree n and genus g in p iff r y1 g r d t y 1 2 proposition 1 6 any such curve has minimal genus with respect to degree in the last section we consider d dimensional cm schemes of degree e mvd j jed in p whose general curvilinear sections have minimal genus these are the most accessible cm schemes cf 12 i we prove that the corresponding part of the hilbert scheme is a connected smooth variety naturally birationally equivalent to nd symemvd p provided v d 2 theorem 2 6 this was known for m v d 2 3 1 cf 9 our proof relies on an explicit description of the infinitesimal deformations of a thick subspace of arbitrary codimension see 2 4 we work over an algebraically closed field of characteristic 0 by the genus of a cm curve we mean its arithmetic genus

truncating the loop series expansion for belief propagation recently chertkov and chernyak 2006b derived an exact expression for the partition sum normalization constant corresponding to a graphical model which is an expansion around the belief propagation bp solution by adding correction terms to the bp free energy one for each generalized loop in the factor graph the exact partition sum is obtained however the usually enormous number of generalized loops generally prohibits summation over all correction terms in this article we introduce truncated loop series bp tlsbp a particular way of truncating the loop series of chertkov chernyak by considering generalized loops as compositions of simple loops we analyze the performance of tlsbp in different scenarios including the ising model on square grids and regular random graphs and on promedas a large probabilistic medical diagnostic system we show that tlsbp often improves upon the accuracy of the bp solution at the expense of increased computation time we also show that the performance of tlsbp strongly depends on the degree of interaction between the variables for weak interactions truncating the series leads to significant improvements whereas for strong interactions it can be ineffective even if a high number of terms is considered

model reductions for inference generality of pairwise binary and planar factor graphs we offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model we investigate the expressive power of three classes of model those with binary variables with pairwise factors and with planar topology as well as their four intersections we formalize a notion of simple reduction for the problem of inferring marginal probabilities and consider whether it is possible to simply reduce marginal inference from general discrete factor graphs to factor graphs in each of these seven subclasses we characterize the reducibility of each class showing in particular that the class of binary pairwise factor graphs is able to simply reduce only positive models we also exhibit a continuous spectral reduction based on polynomial interpolation which overcomes this limitation experiments assess the performance of standard approximate inference algorithms on the outputs of our reductions

approximate inference in boltzmann machines inference in boltzmann machines is np hard in general as a result approximations are often necessary we discuss first order mean field and second order onsager truncations of the plefka expansion of the gibbs free energy the bethe free energy is introduced and rewritten as a gibbs free energy from there a convergent belief optimization algorithm is derived to minimize the bethe free energy an analytic expression for the linear response estimate of the covariances is found which is exact on boltzmann trees finally a number of theorems is proven concerning the plefka expansion relating the first order mean field and the second order onsager approximation to the bethe approximation experiments compare mean field approximation onsager approximation belief propagation and belief optimization

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

a new look at survey propagation and its generalizations this article provides a new conceptual perspective on survey propagation which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k sat problems even with densities close to the satisfiability threshold we first describe how any sat formula can be associated with a novel family of markov random fields mrfs parameterized by a real number 0 1 we then show that applying belief propagation a well known message passing technique for estimating marginal probabilities to this family of mrfs recovers a known family of algorithms ranging from pure survey propagation at one extreme e 1 to standard belief propagation on the uniform distribution over sat assignments at the other extreme e 0 configurations in these mrfs have a natural interpretation as partial satisfiability assignments on which a partial order can be defined we isolate cores as minimal elements in this partial ordering which are also fixed points of survey propagation and the only assignments with positive probability in the mrf for e 1 our experimental results for k e 3 suggest that solutions of random formulas typically do not possess non trivial cores this makes it necessary to study the structure of the space of partial assignments for 0 can be viewed as a smoothed version of the uniform distribution over satisfying assignments e 0 finally we isolate properties of gibbs sampling and message passing algorithms that are typical for an ensemble of k sat problems

on the extension problem for singular accretive differential operators on considere la description explicite des conditions aux limites pour tous les operateurs differentiels accretifs maximaux possibles deductibles d une expression differentielle formelle donnee

information geometry of turbo and low density parity check codes since the proposal of turbo codes in 1993 many studies have appeared on this simple and new type of codes which give a powerful and practical performance of error correction although experimental results strongly support the efficacy of turbo codes further theoretical analysis is necessary which is not straightforward it is pointed out that the iterative decoding algorithm of turbo codes shares essentially similar ideas with low density parity check ldpc codes with pearl s belief propagation algorithm applied to a cyclic belief diagram and with the bethe approximation in statistical physics therefore the analysis of the turbo decoding algorithm will reveal the mystery of those similar iterative methods in this paper we recapture and extend the geometrical framework initiated by richardson to the information geometrical framework of dual affine connections focusing on both of the turbo and ldpc decoding algorithms the framework helps our intuitive understanding of the algorithms and opens a new prospect of further analysis we reveal some properties of these codes in the proposed framework including the stability and error analysis based on the error analysis we finally propose a correction term for improving the approximation

nonlocal robin laplacians and some remarks on a paper by filonov on eigenvalue inequalities abstract the aim of this paper is twofold first we characterize an essentially optimal class of boundary operators which give rise to self adjoint laplacians in l 2 d n x with nonlocal and local robin type boundary conditions on bounded lipschitz domains r n n n n 2 second we extend friedlander s inequalities between neumann and dirichlet laplacian eigenvalues to those between nonlocal robin and dirichlet laplacian eigenvalues associated with bounded lipschitz domains following an approach introduced by filonov for this type of problems

a characterization of boundary conditions yielding maximal monotone operators abstract we provide a characterization for maximal monotone realizations for a certain class of nonlinear operators in terms of their corresponding boundary data spaces the operators under consideration naturally arise in the study of evolutionary problems in mathematical physics we apply our abstract characterization result to port hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco elasticity

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

estimation and marginalization using the kikuchi approximation methods in this letter we examine a general method of approximation known as the kikuchi approximation method for finding the marginals of a product distribution as well as the corresponding partition function the kikuchi approximation method defines a certain constrained optimization problem called the kikuchi problem and treats its stationary points as approximations to the desired marginals we show how to associate a graph to any kikuchi problem and describe a class of local message passing algorithms along the edges of any such graph which attempt to find the solutions to the problem implementation of these algorithms on graphs with fewer edges requires fewer operations in each iteration we therefore characterize minimal graphs for a kikuchi problem which are those with the minimum number of edges we show with empirical results that these simpler algorithms often offer significant savings in computational complexity without suffering a loss in the convergence rate we give conditions for the convexity of a given kikuchi problem and the exactness of the approximations in terms of the loops of the minimal graph more precisely we show that if the minimal graph is cycle free then the kikuchi approximation method is exact and the converse is also true generically together with the fact that in the cycle free case the iterative algorithms are equivalent to the well known belief propagation algorithm our results imply that generically the kikuchi approximation method can be exact if and only if traditional junction tree methods could also solve the problem exactly

inference algorithm for finite dimensional spin glasses belief propagation on the dual lattice starting from a cluster variational method and inspired by the correctness of the paramagnetic ansatz at high temperatures in general and at any temperature in the two dimensional 2d edwards anderson ea model we propose a message passing algorithm the dual algorithm to estimate the marginal probabilities of spin glasses on finite dimensional lattices we use the ea models in 2d and 3d as benchmarks the dual algorithm improves the bethe approximation and we show that in a wide range of temperatures compared to the bethe critical temperature our algorithm compares very well with monte carlo simulations with the double loop algorithm and with exact calculation of the ground state of 2d systems with bimodal and gaussian interactions moreover it is usually 100 times faster than other provably convergent methods as the double loop algorithm in 2d and 3d the quality of the inference deteriorates only where the correlation length becomes very large i e at low temperatures in 2d and close to the critical temperature in 3d

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

on the extension problem for accretive differential operators abstract let t0 denote the minimal operator corresponding to the formal differential expression y x w x 1 j 0n 1 j pn j x y j j x i in l2 i under the assumption that t0 is an accretive operator in l2 i a complete description of all of the maximal accretive extensions of t0 via explicit boundary conditions is given for a wide class of differential expressions

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

curves of degree two and ropes on a line their ideals and even liaison classes abstract in this paper we investigate degree two curves of arbitrary codimension this requires to study also ropes supported on a line after establishing several characterizations of ropes supported on a line we describe their homogeneous ideals and hartshorne rao modules in particular we characterize the arithmetically buchsbaum ropes then we describe the even gorenstein liaison classes of the non arithmetically buchsbaum non degenerate ropes generalizing a well know result of migliore on double lines of codimension two the result relies on the explicit description of certain arithmetically gorenstein curves with maximal tangent spaces at each point as a consequence we can decide if two curves of degree two belong to the same gorenstein liaison class finally we show as an application how ropes can be used in order to construct quasi extremal curves

on the uniqueness of loopy belief propagation fixed points we derive sufficient conditions for the uniqueness of loopy belief propagation fixed points these conditions depend on both the structure of the graph and the strength of the potentials and naturally extend those for convexity of the bethe free energy we compare them with a strengthened version of conditions derived elsewhere for pairwise potentials we discuss possible implications for convergent algorithms as well as for other approximate free energies

stochastic reasoning free energy and information geometry belief propagation bp is a universal method of stochastic reasoning it gives exact inference for stochastic models with tree interactions and works surprisingly well even if the models have loopy interactions its performance has been analyzed separately in many fields such as ai statistical physics information theory and information geometry this article gives a unified framework for understanding bp and related methods and summarizes the results obtained in many fields in particular bp and its variants including tree reparameterization and concave convex procedure are reformulated with information geometrical terms and their relations to the free energy function are elucidated from an information geometrical viewpoint we then propose a family of new algorithms the stabilities of the algorithms are analyzed and methods to accelerate them are investigated

a novel optimizing network architecture with applications we present a novel optimizing network architecture with applications in vision learning pattern recognition and combinatorial optimization this architecture is constructed by combining the following techniques 1 deterministic annealing 2 self amplification 3 algebraic transformations 4 clocked objectives and 5 softassign deterministic annealing in conjunction with self amplification avoids poor local minima and ensures that a vertex of the hypercube is reached algebraic transformations and clocked objectives help partition the relaxation into distinct phases the problems considered have doubly stochastic matrix constraints or minor variations thereof we introduce a new technique softassign which is used to satisfy this constraint experimental results on different problems are presented and discussed

projective curves with next to sharp bounds on castelnuovo mumford regularity this paper is devoted to the study of the next extremal case for a castelnuovo type bound regc degc 1 codimc max k c 1 for the castelnuovo mumford regularity for a nondegenerate projective curve c where k c is an invariant which measures the deficiency of the hartshorne rao module of c we show that a projective curve with next to the maximal regularity lies on either a hirzebruch surface or a normal del pezzo surface

convergence analysis of reweighted sum product algorithms markov random fields are designed to represent structured dependencies among large collections of random variables and are well suited to capture the structure of real world signals many fundamental tasks in signal processing e g smoothing denoising segmentation etc require efficient methods for computing approximate marginal probabilities over subsets of nodes in the graph the marginalization problem though solvable in linear time for graphs without cycles is computationally intractable for general graphs with cycles this intractability motivates the use of approximate ldquomessage passingrdquo algorithms this paper studies the convergence and stability properties of the family of reweighted sum product algorithms a generalization of the widely used sum product or belief propagation algorithm in which messages are adjusted with graph dependent weights for pairwise markov random fields we derive various conditions that are sufficient to ensure convergence and also provide bounds on the geometric convergence rates when specialized to the ordinary sum product algorithm these results provide strengthening of previous analyses we prove that some of our conditions are necessary and sufficient for subclasses of homogeneous models but not for general models the experimental simulations on various classes of graphs validate our theoretical results

mean field method with correlations determined by linear response we introduce a mean field approximation based on the reconciliation of maximum entropy and linear response for correlations in the cluster variation method within a general formalism that includes previous mean field methods we derive formulas improving on e g the bethe approximation and the sessak monasson result at high temperature applying the method to direct and inverse ising problems we find improvements over standard implementations

some linear syzygy conjectures in this paper we introduce four conjectures refining a very useful result of mark green s on the linear part of the free resolution of a graded module over a polynomial ring we prove a number of special cases of the conjectures including all cases pertaining to modules over a polynomial ring in at most four variables in a later paper we will give a complete classification of the nets of skew symmetric transformations on a sdimensional vector space over an algebraically closed field from which we deduce that our conjectures are true in five variables as well

on the convergence of kikuchi s natural iteration method in this article we investigate on the convergence of the natural iteration method a numerical procedure widely employed in the statistical mechanics of lattice systems to minimize kikuchi s cluster variational free energies we discuss a sufficient condition for the convergence based on the coefficients of the cluster entropy expansion depending on the lattice geometry we also show that such a condition is satisfied for many lattices usually studied in applications finally we consider a recently proposed class of methods for the minimization of kikuchi functionals showing that the natural iteration method turns out as a particular instance of that class

smooth varieties of almost minimal degree abstract in this article we study non linearly normal smooth projective varieties x p r of deg x codim x p r 2 we first give geometric characterizations for x theorem 1 1 indeed x is the image of an isomorphic projection of smooth varieties x p r 1 of minimal degree also if x is not the veronese surface then there exists a smooth rational normal scroll y p r which contains x as a divisor linearly equivalent to h 2 f where h is the hyperplane section of y and f is a fiber of the projection morphism y p 1 by using these characterizations 1 we determine all the possible types of y from the type of x theorem 1 2 and 2 we investigate the relation between the betti diagram of x and the type of y theorem 1 3 in particular we clarify the relation between the number of generators of the homogeneous ideal of x and the type of y as an application we construct non linearly normal examples where the converse to theorem 1 1 in d eisenbud m green k hulek s popescu restriction linear syzygies algebra and geometry compos math 141 2005 1460 1478 fails to hold remark 2

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed