complements of graphs of meromorphic functions and complete vector fields given a meromorphic function s mathbb c rightarrow mathbb p 1 we obtain a family of fiber preserving dominating holomorphic maps from mathbb c 2 onto mathbb c 2 setminus text graph s defined in terms of the flows of complete vector fields of type mathbb c and of an entire function h mathbb c rightarrow mathbb c whose graph does not meet text graph s which was determined by buzzard and lu in particular we prove that the dominating map constructed by these authors to prove the dominability of mathbb c 2 setminus text graph s is in the above family we also study the complement of a double section in mathbb c times mathbb p 1 in terms of a complex flow moreover when s has at most one pole we prove that there are infinitely many complete vector fields tangent to text graph s describing explicit families of them with all their trajectories proper and of the same type mathbb c or mathbb c if text graph s does not contain zeros and families with almost all trajectories non proper and of type mathbb c or of type mathbb c if text graph s contains zeros we also study the dominability of mathbb c 2 setminus a when a subset mathbb c 2 is invariant by the flow of a complete holomorphic vector field

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

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

the τ function of the ablowitz segur family of solutions to painleve ii as a widom constant the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function

gap probability of the circular unitary ensemble with a fisher hartwig singularity and the coupled painleve v system we consider the circular unitary ensemble with a fisher hartwig singularity of both jump type and root type at z 1 a rescaling of the ensemble at the fisher hartwig singularity leads to the confluent hypergeometric kernel by studying the asymptotics of the toeplitz determinants we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the hamiltonian of the coupled painleve v system in dimension four this leads to a painleve type representation of the confluent hypergeometric kernel determinant moreover the large gap asymptotics including the constant terms are derived by evaluating the total integral of the hamiltonian in particular we reproduce the large gap asymptotics of the confluent hypergeometric kernel determinant obtained by deift krasovsky and vasilevska and the sine kernel determinant as a special case including the constant term conjectured earlier by dyson

structured prediction dual extragradient and bregman projections we present a simple and scalable algorithm for maximum margin estimation of structured output models including an important class of markov networks and combinatorial models we formulate the estimation problem as a convex concave saddle point problem that allows us to use simple projection methods based on the dual extragradient algorithm nesterov 2003 the projection step can be solved using dynamic programming or combinatorial algorithms for min cost convex flow depending on the structure of the problem we show that this approach provides a memory efficient alternative to formulations based on reductions to a quadratic program qp we analyze the convergence of the method and present experiments on two very different structured prediction tasks 3d image segmentation and word alignment illustrating the favorable scaling properties of our algorithm

random matrix theory and the sixth painleve equation a feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices study of such ensembles realized by matrices with gaussian entries leads to statistical quantities related to the eigenspectrum such as the distribution of the largest eigenvalue which can be expressed as multidimensional integrals or equivalently as determinants these distributions are well known to be functions for painleve systems allowing for the former to be characterized as the solution of certain nonlinear equations we consider the random matrix ensembles for which the nonlinear equation is the form of pvi known results are reviewed as is their implication by way of series expansions for the distributions new results are given for the boundary conditions in the neighbourhood of the fixed singularities at t 0 1 of pvi displayed by a generalization of the generating function for the distributions the structure of these expansions is related to jimbo s general expansions for the function of pvi in the neighbourhood of its fixed singularities and this theory is itself put in its context of the linear isomonodromy problem relating to pvi

z measures on partitions related to the infinite gelfand pair s 2 h abstract the paper deals with the z measures on partitions with the deformation jack parameters 2 or 1 2 we provide a detailed explanation of the representation theoretic origin of these measures and of their role in the harmonic analysis on the infinite symmetric group

the z measures on partitions pfaffian point processes and the matrix hypergeometric kernel abstract we consider a point process on one dimensional lattice originated from the harmonic analysis on the infinite symmetric group and defined by the z measures with the deformation jack parameter 2 we derive an exact pfaffian formula for the correlation function of this process namely we prove that the correlation function is given as a pfaffian with a 2 2 matrix kernel the kernel is given in terms of the gauss hypergeometric functions and can be considered as a matrix analogue of the hypergeometric kernel introduced by a borodin and g olshanski 2000 5 our result holds for all values of admissible complex parameters

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

deformation analysis of matrix models abstract in this paper we show that the non linear equations that play a role in tracy and widom s analysis of scaling limits for hermitian matrix models are special cases of the monodromy preserving deformation equations first discussed by sato miwa and jimbo in the early 1980 s

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

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

a note on the lax pairs for painleveequations for the classical painleveequations besides the method of similarity reduction of lax pairs for integrable partial differential equations two ways are known for lax pair generation the first is based on the confluence procedure in fuchs linear ode with four regular singularities isomonodromy deformation which is governed by the sixth painleveequation the second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of odes in whose non triangular extensions give rise to the lax pairs for the painleveequations the theory of integrable integral operators suggests a new way of lax pair generation for the classical painleveequations this method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions some of the lax pairs we introduce are known others are new the question of gauge equivalence of different lax pairs for the painleveequations is considered as well

how instanton combinatorics solves painleve vi v and iiis we elaborate on a recently conjectured relation of painlev transcendents and 2d conformal field theory general solutions of painlev vi v and iii are expressed in terms of c 1 conformal blocks and their irregular limits agt related to instanton partition functions in supersymmetric gauge theories with nf 0 1 2 3 4 the resulting combinatorial series representations of painlev functions provide an efficient tool for their numerical computation at finite values of the argument the series involves sums over bipartitions which in the simplest cases coincide with gessel expansions of certain toeplitz determinants considered applications include fredholm determinants of classical integrable kernels scaled gap probability in the bulk of the gaussian unitary ensemble and all order conformal perturbation theory expansions of correlation functions in the sine gordon field theory at the free fermion point

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

integrable fredholm operators and dual isomonodromic deformations the fredholm determinants of a special class of integrable integral operators k supported on the union of m curve segments in the complex plane are shown to be the functions of an isomonodromic family of meromorphic covariant derivative operators having regular singular points at the 2m endpoints of the curve segments and a singular point of poincare index 1 at infinity the rank r of the corresponding vector bundle over the riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel the matrix riemann hilbert problem method is used to deduce an identification of the fredholm determinant as a function in the sense of segal wilson and sato i e in terms of abelian group actions on the determinant line bundle over a loop space grassmannian an associated dual isomonodromic family of covariant derivative operators having rank n 2m and r finite regular singular points located at the values of the exponents defining the kernel of k is derived the deformation equations for this family are shown to follow from an associated dual set of riemann hilbert data in which the roles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged the operators are analogously associated to an integral operator whose fredholm determinant is equal to that of k

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

infinite random matrices and ergodic measures we introduce and study a 2 parameter family of unitarily invariant probability measures on the space of infinite hermitian matrices we show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line the correlation kernel for this process is explicitly computed

schlesinger transformations and quantum r matrices schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues we study schlesinger transformations for twisted valued connections on the torus a universal construction is presented which gives the elementary two point transformations in terms of belavin s elliptic quantum r matrix in particular the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted elementary one point transformations acting on the residue eigenvalues at a single pole are constructed in terms of the classical elliptic r matrix the action of these transformations on the function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection in the limit of a rational r matrix our construction and the quotients reduce to the classical results of jimbo and miwa in the complex plane

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

partition functions for matrix models and isomonodromic tau functions for one matrix models with polynomial potentials the explicit relationship between the partition function and the isomonodromic tau function for the 2 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived

isomonodromy transformations of linear systems of difference equations we introduce and study isomonodromy transformations of the matrix linear difference equation y z 1 a z y z with polynomial a z our main result is construction of an isomonodromy action of zm n 1 1 on the space of coefficients a z here m is the size of matrices and n is the degree of a z the birational action of certain rank n subgroups can be described by difference analogs of the classical schlesinger equations and we prove that for generic initial conditions these difference schlesinger equations have a unique solution we also show that both the classical schlesinger equations and the schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes similarly to the continuous case for m n 2 the difference schlesinger equations and their q analogs yield discrete painlev e equations examples include dpii dpiv dpv and q pvi

double sections dominating maps and the jacobian fibration we give two parametrized versions of the uniformization theorem of a noncompact non hyperbolic riemann surface using different but complementary methods the first constructs the uniformizing maps directly in terms of coordinates via classical complex analysis and provides a canonical form for the double sections of a conic bundle over a noncompact complex curve the second version which is coordinate independent works over any complex curve and is obtained by extending kodaira s theory of the jacobian fibration to a family of singular algebraic curves constructed via algebraic geometry then using the results obtained with the jacobian fibration we give two equivalent conditions for a complex analytic surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we show that this dominability is equivalent to the apparently weaker condition of the existence of a zariski dense image of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold euler characteristic we discuss also the sharpness of our result in various contexts as well as the lack of connection to the fundamental group

on the algorithmic implementation of multiclass kernel based vector machines in this paper we describe the algorithmic implementation of multiclass kernel based vector machines our starting point is a generalized notion of the margin to multiclass problems using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks our notion of margin yields a direct method for training multiclass predictors by using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size we describe an efficient fixed point algorithm for solving the reduced optimization problems and prove its convergence we then discuss technical details that yield significant running time improvements for large datasets finally we describe various experiments with our approach comparing it to previously studied kernel based methods our experiments indicate that for multiclass problems we attain state of the art accuracy

painleve vi and hankel determinants for the generalized jacobi weight we study the hankel determinant of the generalized jacobi weight x t x 1 x for x 0 1 with 0 t 0 and based on the ladder operators for the corresponding monic orthogonal polynomials pn x it is shown that the logarithmic derivative of the hankel determinant is characterized by a jimbo miwa okamoto form of the painleve vi system

a linear programming approach to max sum problem a review the max sum labeling problem defined as maximizing a sum of binary i e pairwise functions of discrete variables is a general np hard optimization problem with many applications such as computing the map configuration of a markov random field we review a not widely known approach to the problem developed by ukrainian researchers schlesinger et al in 1976 and show how it contributes to recent results most importantly those on the convex combination of trees and tree reweighted max product in particular we review schlesinger et al s upper bound on the max sum criterion its minimization by equivalent transformations its relation to the constraint satisfaction problem the fact that this minimization is dual to a linear programming relaxation of the original problem and the three kinds of consistency necessary for optimality of the upper bound we revisit problems with boolean variables and supermodular problems we describe two algorithms for decreasing the upper bound we present an example application for structural image analysis

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

the theory and practice of bayesian image labeling integrating disparate sources of information has been recognized as one of the keys to the success of general purpose vision systems image clues such as shading texture stereo disparities and image flows provide uncertain local and incomplete information about the three dimensional scene spatial a priori knowledge plays the role of filling in missing information and smoothing out noise this thesis proposes a solution to the longstanding open problem of visual integration it reports a framework based on bayesian probability theory for computing an intermediate representation of the scene from disparate sources of information the computation is formulated as a labeling problem local visual observations for each image entity are reported as label likelihoods they are combined consistently and coherently on hierarchically structured label trees with a new computationally simple procedure the pooled label likelihoods are fused with the a priori spatial knowledge encoded as markov random fields mrf s the a posteriori distribution of the labelings are thus derived in a bayesian formalism a new inference method highest confidence first hcf estimation is used to infer a unique labeling from the a posteriori distribution unlike previous inference methods based on the mrf formalism hcf is computationally efficient and predictable while meeting the principles of graceful degradation and least commitment the results of the inference process are consistent with both observable evidence and a priori knowledge the effectiveness of the approach is demonstrated with experiments on two image analysis problems intensity edge detection and surface reconstruction for edge detection likelihood outputs from a set of local edge operators are integrated with a priori knowledge represented as an mrf probability distribution for surface reconstruction intensity information is integrated with sparse depth measurements and a priori knowledge coupled mrf s provide a unified treatment of surface reconstruction and segmentation and an extension of hcf implements a solution method experiments using real image and depth data yield robust results the framework can also be generalized to higher level vision problems as well as to other domains

fredholm determinant and nekrasov sum representations of isomonodromic tau functions we derive fredholm determinant representation for isomonodromic tau functions of fuchsian systems with n regular singular points on the riemann sphere and generic monodromy in gl n mathbb c the corresponding operator acts in the direct sum of n n 3 copies of l2 s1 its kernel has a block integrable form and is expressed in terms of fundamental solutions of n 2 elementary 3 point fuchsian systems whose monodromy is determined by monodromy of the relevant n point system via a decomposition of the punctured sphere into pairs of pants for n 2 these building blocks have hypergeometric representations the kernel becomes completely explicit and has cauchy type in this case fredholm determinant expansion yields multivariate series representation for the tau function of the garnier system obtained earlier via its identification with fourier transform of liouville conformal block or a dual nekrasov okounkov partition function further specialization to n 4 gives a series representation of the general solution to painleve vi equation

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

on the properties of concept classes induced by multivalued bayesian networks the concept class c n induced by a bayesian network n can be embedded into some euclidean inner product space the vapnik chervonenkis vc dimension of the concept class and the minimum dimension of the inner product space are very important indicators for evaluating the classification capability of the bayesian network in this paper we investigate the properties of the concept class c n k induced by a multivalued bayesian network n k where each node x i of n k is a k valued variable we focus on the values of two dimensions i the vc dimension of the concept class c n k denoted as vcdim n k and ii the minimum dimension of the inner product space into which c n k can be embedded we show that the values of these two dimensions are k n 1 for fully connected k valued bayesian networks n f k with n variables for non fully connected k valued bayesian networks n k without v structure we prove that the two dimensional values are k 1 i 1 nk m i 1 where m i denotes the number of parents for the i t h variable we also derive the upper and lower bounds on the minimum dimension of the inner product space induced by non fully connected bayesian networks

the jacobi polynomial ensemble and the painleve vi equation we consider the jacobi polynomial ensemble of n n random matrices we show that the probability of finding no eigenvalues in the interval 1 z for a random matrix chosen from the ensemble viewed as a function of z satisfies a second order differential equation after a simple change of variable this equation can be reduced to the okamoto jimbo miwa form of the painleve vi equation the result is achieved by a comparison of the tracy widom and the virasoro approaches to the problem which both lead to different third order differential equations the virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments which drastically simplifies the computations

form factors of twist fields in the lattice dirac theory we study u 1 twist fields in a two dimensional lattice theory of massive dirac fermions factorized formulas for finite lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model elliptic determinant identities and theta functional interpolation we also investigate the thermodynamic and infinite volume scaling limit where the corresponding expressions reduce to form factors of the exponential fields of the sine gordon model at the free fermion point

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

discriminative learning of max sum classifiers the max sum classifier predicts n tuple of labels from n tuple of observable variables by maximizing a sum of quality functions defined over neighbouring pairs of labels and observable variables predicting labels as map assignments of a random markov field is a particular example of the max sum classifier learning parameters of the max sum classifier is a challenging problem because even computing the response of such classifier is np complete in general estimating parameters using the maximum likelihood approach is feasible only for a subclass of max sum classifiers with an acyclic structure of neighbouring pairs recently the discriminative methods represented by the perceptron and the support vector machines originally designed for binary linear classifiers have been extended for learning some subclasses of the max sum classifier besides the max sum classifiers with the acyclic neighbouring structure it has been shown that the discriminative learning is possible even with arbitrary neighbouring structure provided the quality functions fulfill some additional constraints in this article we extend the discriminative approach to other three classes of max sum classifiers with an arbitrary neighbourhood structure we derive learning algorithms for two subclasses of max sum classifiers whose response can be computed in polynomial time i the max sum classifiers with supermodular quality functions and ii the max sum classifiers whose response can be computed exactly by a linear programming relaxation moreover we show that the learning problem can be approximately solved even for a general max sum classifier

a framework for efficient structured max margin learning of high order mrf models we present a very general algorithm for structured prediction learning that is able to efficiently handle discrete mrfs crfs including both pairwise and higher order models so long as they can admit a decomposition into tractable subproblems at its core it relies on a dual decomposition principle that has been recently employed in the task of mrf optimization by properly combining such an approach with a max margin learning method the proposed framework manages to reduce the training of a complex high order mrf to the parallel training of a series of simple slave mrfs that are much easier to handle this leads to a very efficient and general learning scheme that relies on solid mathematical principles we thoroughly analyze its theoretical properties and also show that it can yield learning algorithms of increasing accuracy since it naturally allows a hierarchy of convex relaxations to be used for loss augmented map mrf inference within a max margin learning approach furthermore it can be easily adapted to take advantage of the special structure that may be present in a given class of mrfs we demonstrate the generality and flexibility of our approach by testing it on a variety of scenarios including training of pairwise and higher order mrfs training by using different types of regularizers and or different types of dissimilarity loss functions as well as by learning of appropriate models for a variety of vision tasks including high order models for compact pose invariant shape priors knowledge based segmentation image denoising stereo matching as well as high order potts mrfs

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

scaled limit and rate of convergence for the largest eigenvalue from the generalized cauchy random matrix ensemble in this paper we are interested in the asymptotic properties for the largest eigenvalue of the hermitian random matrix ensemble called the generalized cauchy ensemble gcye whose eigenvalues pdf is given by textrm const cdot prod 1 leq j k leq n x j x k 2 prod j 1 n 1 ix j s n 1 ix j overline s n dx j where s is a complex number such that s 1 2 and where n is the size of the matrix ensemble using results by borodin and olshanski commun math phys 223 1 87 123 2001 we first prove that for this ensemble the law of the largest eigenvalue divided by n converges to some probability distribution for all s such that s 1 2 using results by forrester and witte nagoya math j 174 29 114 2002 on the distribution of the largest eigenvalue for fixed n we also express the limiting probability distribution in terms of some non linear second order differential equation eventually we show that the convergence of the probability distribution function of the re scaled largest eigenvalue to the limiting one is at least of order 1 n