the critical z invariant ising model via dimers the periodic case we study a large class of critical two dimensional ising models namely critical z invariant ising models on periodic graphs example of which are the classical mathbb z 2 triangular and honeycomb lattice at the critical temperature fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model we prove that the dimer characteristic polynomial is equal up to a constant to the critical laplacian characteristic polynomial and defines a harnack curve of genus 0 we prove an explicit expression for the free energy and for the gibbs measure obtained as weak limit of boltzmann measures

on meromorphic solutions of nonlinear partial differential equations of first order abstract we prove a uniqueness theorem in terms of value distribution for meromorphic solutions of a class of nonlinear partial differential equations of first order which shows that such solutions f are uniquely determined by the zeros and poles of f c j counting multiplicities for two distinct complex numbers c 1 and c 2

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

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

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

holomorphic curves with shift invariant hyperplane preimages if f c p n is a holomorphic curve of hyper order less than one for which 2n 1 hyperplanes in general position have forward invariant preimages with respect to the translation z z c then f is periodic with period c 2 c this result which can be described as a difference analogue of m green s picard type theorem for holomorphic curves follows from a more general result presented in this paper the proof relies on a new version of cartan s second main theorem for the casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives both of which are proved here finally an application to the uniqueness theory of meromorphic functions is given and the sharpness of the obtained results is demonstrated by examples

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

nonequidimensional value distribution theory and meromorphic connections this is a sequel to the paper si2 1 in whichwe introduced the use ofmeromorphic connections to handle nonequidimensional value distribution theory the meromorphic connection is chosen so that the divisor has zero second fundamental form with respect to the meromorphic connection when we have more than one divisor they must have zero second fundamental form with respect to the same meromorphic connection such a condition is so stringent that it is difficult to find examples to which the method can be applied and which cannot be handled by other means one example is a collection of fermat curves in p2 of the form aowg atw a2wg 0 for the same homogeneous coordinate system wo wt w2 shift man i sh2 told me that for such a collection of fermat curves defect relations can be obtained by using a technique of h cartan c2 see 9 whereas the general case of allowing fermat curves with respect to different homogeneous coordinates cannot be handled by any of the known methods the only known result on the defect of general divisors is obtained by using the veronese map and is given in shl in the following form let f c pn be a holomorphic curve such that f c is not contained in any algebraic hypersurface in pn let sj be a collection of algebraic hypersurfaces of degree d in p the sum of the defects of f for the

combinatorial resampling particle filter an effective and efficient method for articulated object tracking particle filter pf is a method dedicated to posterior density estimations using weighted samples whose elements are called particles in particular this approach can be applied to object tracking in video sequences in complex situations and in this paper we focus on articulated object tracking i e objects that can be decomposed as a set of subparts one of pf s crucial step is a resampling step in which particles are resampled to avoid degeneracy problems in this paper we propose to exploit mathematical properties of articulated objects to swap conditionally independent subparts of the particles in order to generate new particle sets we then introduce a new resampling method called combinatorial resampling that resamples over the particle set resulting from all the admissible swappings the so called combinatorial set in essence combinatorial resampling cr is quite similar to the combination of a crossover operator and a usual resampling but there exists a fundamental difference between cr and the use of crossover operators we prove that cr is sound i e in a bayesian framework it is guaranteed to represent without any bias the posterior densities of the states over time by construction the particle sets produced by cr better represent the density to estimate over the whole state space than the original set and therefore cr produces higher quality samples unfortunately the combinatorial set is generally of an exponential size and therefore to be scalable we show how it can be implicitly constructed and resampled from thus resulting in both an efficient and effective resampling scheme finally through experimentations both on challenging synthetic and real video sequences we also show that our resampling method outperforms all classical resampling methods both in terms of the quality of its results and in terms of computation times

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

finite alphabet iterative decoders part i decoding beyond belief propagation on the binary symmetric channel we introduce a new paradigm for finite precision iterative decoding on low density parity check codes over the binary symmetric channel the messages take values from a finite alphabet and unlike traditional quantized decoders which are quantized versions of the belief propagation bp decoder the proposed finite alphabet iterative decoders faids do not propagate quantized probabilities or log likelihoods and the variable node update functions do not mimic the bp decoder rather the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code thereby rendering these decoders capable of outperforming the bp in the error floor region on certain column weight three codes of practical interest we show that there exist faids that surpass the floating point bp decoder in the error floor region while requiring only three bits of precision for the representation of the messages hence faids are able to achieve a superior performance at much lower complexity we also provide a methodology for the selection of faids that is not code specific but gives a set of candidate faids containing potentially good decoders in the error floor region for any column weight three code we validate the code generality of our methodology by providing particularly good three bit precision faids for a variety of codes with different rates and lengths

defect relations for holomorphic maps between spaces of different dimensions there is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by nevanlinna stoll griffiths and many other people however for the case of maps between spaces of unequal dimension essentially there is only the theory developed by weyl weyl and ahlfors when the target space is the complex projective space and their theory is built on the projective linear structure of the projective space a partial list of references for these results is given at the end of this paper in this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold since the special case of maps from c to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case in order to avoid unnecessarily complicated notations and to make the arguments more easily understood we give complete details only for this special case and indicate the easy modifications needed for the general case besides the usual techniques in value distribution theory of using the poincar6 lelong formula the calculus lemma of estimating the integrand by its integral and the concavity of logarithm a new key point in our argument is to introduce a meromorphic connection so that the second fundamental form of a divisor with respect to the connection is zero more precisely for the case of a meromorphic map f from c to a compact kihler surface m to consider the defect for a nonsingular zero set of a holomorphic section s of a positive holomorphic line bundle l we introduce a meromorphic connection i v form with the following two properties i for some hermitian holomorphic line bundle f over m and some holomorphic section 0 of f

the abc theorem for higher dimensional function fields we generalize the abc theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non singular in codimension one we also obtain an upper bound for the minimal order sequence of wronskians over such function fields of positive characteristic

the critical z invariant ising model via dimers locality property we study a large class of critical two dimensional ising models namely critical z invariant ising models fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph our main result is an explicit local formula for the inverse kasteleyn matrix in the spirit of kenyon invent math 150 2 409 439 2002 as a contour integral of the discrete exponential function of mercat discrete period matrices and related topics 2002 and kenyon invent math 150 2 409 439 2002 multiplied by a local function using results of boutillier and de tiliere prob theor rel fields 147 3 4 379 413 2010 and techniques of de tiliere prob th rel fields 137 3 4 487 518 2007 and kenyon invent math 150 2 409 439 2002 this yields an explicit local formula for a natural gibbs measure and a local formula for the free energy as a corollary we recover baxter s formula for the free energy of the critical z invariant ising model baxter in exactly solved models in statistical mechanics academic press london 1982 and thus a new proof of it the latter is equal up to a constant to the logarithm of the normalized determinant of the laplacian obtained in kenyon invent math 150 2 409 439 2002

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

non archimedean nevanlinna theory in several variables and the non archimedean nevanlinna inverse problem cartan s method is used to prove a several variable non archimedean nevanlinna second main theorem for hyperplanes in projective space the corresponding defect relation is derived but unlike in the complex case we show that there can only be finitely many non zero non archimedean defects we then address the non archimedean nevanlinna inverse problem by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space there exists a non archimedean analytic function with the given defects at the specified hyperplanes and with no other defects 1 history and introduction nevanlinna theory broadly speaking studies to what extent something like the fundamental theorem of algebra holds for meromorphic functions unlike polynomials transcendental meromorphic functions in general have infinitely many zeros however they have only finitely many zeros inside a disc of radius r therefore in order to study the values of a meromorphic function nevanlinna theory associates to each meromorphic function f three functions of r the distance from the origin for their precise definitions see nevanlinna ne 2 the characteristic or height function tf r measures the growth of f and should be thought of as the analogue of the degree of a polynomial the counting function nf a r counts the number of times as a logarithmic average f takes on the value a in the disc of radius r finally the mean proximity function mf a r measures how often on average f stays close to the value a on the circle of radius r nevanlinna proved two main theorems about these functions the so called first main theorem states that tf r mf a r nf a r o 1 where the bounded term o 1 depends on f and a but not on r this should be thought of as a substitute for the fundamental theorem of algebra in the following sense the first main theorem says that mf a r nf a r is essentially independent of the value a and this is analogous to the fact that a polynomial takes on every finite value the same number of times counting multiplicity the first received by the editors october 14 1995 and in revised form june 17 1996 1991 mathematics subject classification primary 11j99 11s80 30d35 32h30 32p05

approximate learning algorithm in boltzmann machines boltzmann machines can be regarded as markov random fields for binary cases they are equivalent to the ising spin model in statistical mechanics learning systems in boltzmann machines are one of the np hard problems thus in general we have to use approximate methods to construct practical learning algorithms in this context in this letter we propose new and practical learning algorithms for boltzmann machines by using the belief propagation algorithm and the linear response approximation which are often referred as advanced mean field methods finally we show the validity of our algorithm using numerical experiments

a new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables abstract let c c m f c m p n c be a linearly nondegenerate meromorphic mapping over the field p c of c periodic meromorphic functions in c m and let h j 1 j q be q 2 n n 1 hyperplanes in n subgeneral position of p n c we prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the casorati determinant of f instead of its wronskian determinant as its applications we obtain a defect relation a uniqueness theorem and a difference analogue of generalized picard theorem

a variational inference framework for soft in soft out detection in multiple access channels we propose a unified framework for deriving and studying soft in soft out siso detection in multiple access channels using the concept of variational inference the proposed framework may be used in multiple access interference mai intersymbol interference isi and multiple input multiple output mimo channels without loss of generality we will focus our attention on turbo multiuser detection to facilitate a more concrete discussion it is shown that with some loss of optimality variational inference avoids the exponential complexity of a posteriori probability app detection by optimizing a closely related but much more manageable objective function called variational free energy in addition to its systematic appeal there are several other advantages to this viewpoint first of all it provides unified and rigorous justifications for numerous detectors that were proposed on radically different grounds and facilitates convenient joint detection and decoding utilizing the turbo principle when error control codes are incorporated second efficient joint parameter estimation and data detection is possible via the variational expectation maximization em algorithm such that the detrimental effect of inaccurate channel knowledge at the receiver may be dealt with systematically we are also able to extend bpsk based siso detection schemes to arbitrary square qam constellations in a rigorous manner using a variational argument

on the abc conjecture and diophantine approximation by rational points we show that an earlier conjecture of the author on diophantine approximation of rational points on varieties implies the abc conjecture of masser and oesterle in fact a weak form of the former conjecture is sufficient involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type analogues of this weaker conjecture are proved in the split function field case of characteristic zero and in the case of holomorphic curves nevanlinna theory the proof of the latter involves a geometric generalization of the classical lemma on the logarithmic derivative due to mcquillan this lemma may be of independent interest

iterative decoding of serially concatenated cpm in fading channels with noisy channel state information we use the sum product algorithm spa to propose a reduced complexity detector for serially concatenated continuous phase modulated sccpm signals with noisy channel state information csi at the receiver two channel effects phase noise pn and flat fading are tackled using the statistical model and associated factor graph fg of the problem although optimal performance is not assured due to the loopy nature of the fg as well as simplifying assumptions made in the algorithm development simulations show that very significant performance gains are possible and the gap between systems with noiseless and noisy csi can be reduced to about 1 db using the proposed methods the increase in complexity compared to the conventional decoder which treats csi estimates as if they were equal to their true values is small

logarithmic derivatives in annuli in this paper we define nevanlinna functions in annuli with two independent variables and use them to prove a version of the generalized logarithmic derivative lemma in annuli as a corollary we can conclude the logarithmic derivative lemma in a disk and the complex plane

second main theorem and uniqueness theorem with moving targets on parabolic manifolds abstract in this paper with the motivation from diophantine approximation a truncated second main theorem is established for meromorphic maps from m into p v with moving targets g j m p v 1 j q where m is a parabolic manifold and v is a hermitian vector space as an application of this second main theorem a uniqueness theorem without counting multiplicities is given

zeros of analytic functions with or without multiplicities the classical mason stothers theorem deals with nontrivial polynomial solutions to the equation a b c it provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a deg b and deg c we extend this to general analytic functions living on a reasonable bounded domain omega subset mathbb c rather than on the whole of mathbb c the estimates obtained are sharp for any and a generalization of the original result on polynomials can be recovered from them by a limiting argument

triangular approximation for ising model and its application to boltzmann machine abstract when we consider a problem in information processing it is convenient to formulate the problem by using a random ising model in statistical physics however a kind of computational difficulty arises in a case that the number of nodes becomes large hence approximation schemes such as a mean field approximation and a bethe approximation have been used extensively for overcoming the difficulty when frustration is essential in some problems the bethe approximation gives unfavorable results in those problems more advanced approximation schemes are needed beyond the bethe approximation in the present paper we present explicitly the triangular approximation which is the next approximation to the bethe approximation we apply the obtained approximation scheme to a boltzmann machine in order to investigate the validity of the triangular approximation

an analogue of continued fractions in number theory for nevanlinna theory we show an analogue of continued fractions in approximation to irrational numbers by rationals for nevanlinna theory the analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of nevanlinna theory

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

blending learning and inference in conditional random fields conditional random fields maximize the log likelihood of training labels given the training data e g objects given images in many cases the training labels are structures that consist of a set of variables and the computational complexity for estimating their likelihood is exponential in the number of the variables learning algorithms relax this computational burden using approximate inference that is nested as a sub procedure in this paper we describe the objective function for nested learning and inference in conditional random fields the devised objective maximizes the log beliefs probability distributions over subsets of training variables that agree on their marginal probabilities this objective is concave and consists of two types of variables that are related to the learning and inference tasks respectively importantly we afterwards show how to blend the learning and inference procedure and effectively get to the identical optimum much faster the proposed algorithm currently achieves the state of the art in various computer vision applications

on decoding of the 73 37 13 quadratic residue code in this paper a method to search the set of syndromes indices needed in computing the unknown syndromes for the 73 37 13 quadratic residue qr code is proposed according to the resulting index sets one computes the unknown syndromes and thus finds the corresponding error locator polynomial by using an inverse free berlekamp massey bm algorithm based on the modified chase ii algorithm the performance of soft decision decoding for the 73 37 13 qr code is given this result is new moreover the error rate performance of linear programming lp decoding for the 73 37 13 qr code is also investigated and lp based decoding is shown to be significantly superior in performance to the algebraic soft decision decoding while requiring almost the same computational complexity in fact the algebraic hard decision and soft decision decoding of the 89 45 17 qr code outperforms that of the 73 37 13 qr code because the former has a larger minimal distance however experimental results indicate that the 73 37 13 qr code outperforms the 89 45 17 qr code with much fewer arithmetic operations when using the lp based decoding algorithms the pseudocodewords analysis partially explains this seemingly strange phenomenon

a note on the abc conjecture in this note we have generalized the abc conjecture for integers and extended mason s theorem to polynomials in m by dint of nevanlinna s value distribution theory 2002 wiley periodicals inc

from cycle rooted spanning forests to the critical ising model an explicit construction fisher established an explicit correspondence between the 2 dimensional ising model defined on a graph g and the dimer model defined on a decorated version mathcal g of this graph fisher in j math phys 7 1776 1781 1966 in this paper we explicitly relate the dimer model associated to the critical ising model and critical cycle rooted spanning forests crsfs this relation is established through characteristic polynomials whose definition only depends on the respective fundamental domains and which encode the combinatorics of the model we first show a matrix tree type theorem establishing that the dimer characteristic polynomial counts crsfs of the decorated fundamental domain mathcal g 1 our main result consists in explicitly constructing crsfs of mathcal g 1 counted by the dimer characteristic polynomial from crsfs of g1 where edges are assigned kenyon s critical weight function kenyon in invent math 150 2 409 439 2002 thus proving a relation on the level of configurations between two well known 2 dimensional critical models

optimum estimation via gradients of partition functions and information measures a statistical mechanical perspective in continuation to a recent work on the statistical mechanical analysis of minimum mean square error mmse estimation in gaussian noise via its relation to the mutual information the i mmse relation here we propose a simple and more direct relationship between optimum estimation and certain information measures e g the information density and the fisher information which can be viewed as partition functions and hence are amenable to analysis using statistical mechanical techniques the proposed approach has several advantages most notably its applicability to general sources and channels as opposed to the i mmse relation and its variants which hold only for certain classes of channels e g additive white gaussian noise channels we then demonstrate the derivation of the conditional mean estimator and the mmse in a few examples two of these examples turn out to be generalizable to a fairly wide class of sources and channels for this class the proposed approach is shown to yield an approximate conditional mean estimator and an mmse formula that has the flavor of a single letter expression we also show how our approach can easily be generalized to situations of mismatched estimation

the second main theorem in the hyperbolic case we develop nevanlinna s theory for a class of holomorphic maps when the source is a disc such maps appear in the theory of foliations by riemann surfaces

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

on nevanlinna s second main theorem in projective space we first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables then we show the best nature of the second main theorem in nevanlinna theory under two different assumptions of non degeneracy of meromorphic mappingsf n n for arbitrary positive integersn andm moreover we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order finally we demonstrate the sharpness of all upper bounds in our main theorems

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

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

on the hyperbolicity of the complements of curves in algebraic surfaces the three component case the paper is a contribution to the conjecture of kobayashi that the complement of a generic curve in the projective plane is hyperbolic provided the degree is at least five previously the authors treated the cases of two quadrics and a line and three quadrics the main results are let c be the union of three curves in p 2 whose degrees are at least two one of which is at least three then for generic such configurations the complement of c is hyperbolic and hyperbolically embedded the same statement holds for complements of curves in generic hypersurfaces x of degree at least five and curves which are intersections of x with hypersurfaces of degree at least five furthermore results are shown for curves on surfaces with picard number one

schmidt s subspace theorem with moving targets in recent years due to work of vojta lang osgood etc people have started to realize that there is a close relationship between nevanlinna theory and diophantine approximation p vojta has compiled a dictionary see vojta v1 providing correspondences for the concepts fundamental to the two theories the second main theorem for meromorphic functions is the most important result in nevanlinna theory and its counterpart in number theory via vojta s dictionary is roth s theorem in 1929 r nevanlinna conjectured that the second main theorem remains correct if distinct points in p1 c are replaced by distinct holomorphic maps ai c p1 c with tai r o tf r where tf r is the nevanlinna characteristic function of the holomorphic map f c p1 c see rs 1 for the definition after many attempts by various people osgood proved the conjecture in 1985 which is now known as the second main theorem with moving targets in 1986 steinmetz gave another simple and elegant proof recently vojta v 2 and v 3 formulated and proved its counterpart in number theory theorem a roth s theorem with moving targets let k be a number field and let s be a finite set of places of k let q z 0 and let 0 then there does not exist an infinite sequence of distinct tuples

a linear homogeneous partial differential equation with entire solutions represented by bessel polynomials abstract we have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations specifically we deal with entire solutions of the equations that are represented in convergent series of bessel polynomials and determine orders and types of the solutions in terms of their taylor coefficients by establishing an analogue of lindelof pringsheim theorem as well as wiman valiron type theory for such functions finally by using value distribution theory of holomorphic functions we are able to exhibit some uniqueness theorems of the entire or meromorphic solutions

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

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

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

tree based reparameterization framework for analysis of sum product and related algorithms we present a tree based reparameterization trp framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles this class includes the belief propagation bp or sum product algorithm as well as variations and extensions of bp algorithms in this class can be formulated as a sequence of reparameterization updates each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph i e a tree or more generally a hypertree the ultimate goal is to obtain an alternative but equivalent factorization using functions that represent exact or approximate marginal distributions on cliques of the graph our framework highlights an important property of the sum product algorithm and the larger class of reparameterization algorithms the original distribution on the graph with cycles is not changed the perspective of tree based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency we develop interpretations of these results in terms of information geometry the invariance of the distribution in conjunction with the fixed point characterization enables us to derive an exact expression for the difference between the true marginals on an arbitrary graph with cycles and the approximations provided by belief propagation more broadly our analysis applies to any algorithm that minimizes the bethe free energy we also develop bounds on the approximation error which illuminate the conditions that govern their accuracy finally we show how the reparameterization perspective extends naturally to generalizations of bp e g kikuchi 1951 approximations and variants via the notion of hypertree reparameterization

a sharp form of nevanlinna s second fundamental theorem letf be meromorphic in the plane we find a sharp upper bound for the error term s r f m r f sum limits i 1 q m r a i f n 1 r f 2t r f in nevanlinna s second fundamental theorem for any positive increasing functions t t andp t with int limits 1 infty dt varphi t infty and int limits 1 infty dt p t infty we have s left r f right leqq log left frac varphi left t left r f right right p left r right right o left 1 right asr outside a sete with int limits e dr p r infty further if t t is positive and increasing and int limits 1 infty dt psi t infty then there is an entiref such thats r f log t r f outside a set of finite linear measure we also prove analogous results for functions meromorphic in a disk

learning bayesian network parameters under equivalence constraints we propose a principled approach for learning parameters in bayesian networks from incomplete datasets where the examples of a dataset are subject to equivalence constraints these equivalence constraints arise from datasets where examples are tied together in that we may not know the value of a particular variable but whatever that value is we know it must be the same across different examples we formalize the problem by defining the notion of a constrained dataset and a corresponding constrained likelihood that we seek to optimize we further propose a new learning algorithm that can effectively learn more accurate bayesian networks using equivalence constraints which we demonstrate empirically moreover we highlight how our general approach can be brought to bear on more specialized learning tasks such as those in semi supervised clustering and topic modeling where more domain specific approaches were previously developed

a new class of upper bounds on the log partition function we introduce a new class of upper bounds on the log partition function of a markov random field mrf this quantity plays an important role in various contexts including approximating marginal distributions parameter estimation combinatorial enumeration statistical decision theory and large deviations bounds our derivation is based on concepts from convex duality and information geometry in particular it exploits mixtures of distributions in the exponential domain and the legendre mapping between exponential and mean parameters in the special case of convex combinations of tree structured distributions we obtain a family of variational problems similar to the bethe variational problem but distinguished by the following desirable properties i they are convex and have a unique global optimum and ii the optimum gives an upper bound on the log partition function this optimum is defined by stationary conditions very similar to those defining fixed points of the sum product algorithm or more generally any local optimum of the bethe variational problem as with sum product fixed points the elements of the optimizing argument can be used as approximations to the marginals of the original model the analysis extends naturally to convex combinations of hypertree structured distributions thereby establishing links to kikuchi approximations and variants

on cartan s theorem and cartan s conjecture we give a mild generalization of cartan s theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes this generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author given hyperplanes in projective space in general position there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union even linearly degenerate curves satisfy the inequality of cartan s theorem except for the ramification term in addition it is shown how these methods can lead to a shorter proof of nochka s theorem on cartan s conjecture and in the number field case how nochka s theorem gives a short proof of wirsing s theorem on approximation of algebraic numbers by algebraic numbers of bounded degree