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

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

a family of tridiagonal pairs abstract let f denote a field and let v denote a vector space of finite positive dimension over f let a a denote a tridiagonal pair on v of diameter d 3 assume the eigenvalue and dual eigenvalue sequences of a a satisfy i q2i i q 2 d 2 i for some nonzero scalars q f where q is not a root of unity assume that not all eigenvalues of a and a have multiplicity one let m and m denote the subalgebras of end v generated by a and a respectively and assume that v mv m v for some eigenvectors v of a associated with 0 and v of a associated with d we find a nice basis for v and describe the action of a a on this basis in terms of six parameters

sharp tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 a conjecture of tatsuro ito and the second author states that if k is algebraically closed then a a is sharp in order to better understand and eventually prove the conjecture in this paper we begin a systematic study of the sharp tridiagonal pairs our results are summarized as follows assuming a a is sharp and using the data a vi i 0d a vi i 0d we define a finite sequence of scalars called the parameter array we display some equations that show the geometric significance of the parameter array we show how the parameter array is affected if is replaced by a vi i 0d a vi i 0d or a vd i i 0d a vi i 0d or a vi i 0d a vd i i 0d we prove that if the isomorphism class of is determined by the parameter array then there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v

solving matching problems with linear programming in this paper we describe an implementation of a cutting plane algorithm for the perfect matching problem which is based on the simplex method the algorithm has the following features it works on very sparse subgraphs ofkn which are determined heuristically global optimality is checked using the reduced cost criterion cutting plane recognition is usually accomplished by heuristics only if these fail the padberg rao procedure is invoked to guarantee finite convergence

a bilinear form for tridiagonal pairs of q serre type abstract let v denote a vector space of finite positive dimension and let v denote the vector space dual of v let a a be a tridiagonal pair on v and let a v v a v v be linear transformations such that for all v v and f v a f v f av and a f v f a v we show that a a is a tridiagonal pair on v we then show that if a a is of q serre type then a a is isomorphic to a a we also show that in this case there exist a unique bilinear form on v such that for all u v v au v u av and a u v u a v moreover this bilinear form is nondegenerate and symmetric

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

towards a classification of the tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension let end v denote the k algebra consisting of all k linear transformations from v to v we consider a pair a a end v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v let e0 denote the element of end v such that e0 i v0 0 and e0 vi 0 for 1 i d let d resp d denote the k subalgebra of end v generated by a resp a in this paper we prove that the span of e0 dd de0 equals the span of e0 de0 de0 and that the elements of e0 de0 mutually commute we relate these results to some conjectures of tatsuro ito and the second author that are expected to play a role in the classification of tridiagonal pairs

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

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

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

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

the cutting plane method is polynomial for perfect matchings the cutting plane approach to finding minimum cost perfect matchings has been discussed by several authors over past decades its convergence has been an open question we develop a cutting plane algorithm that converges in polynomial time using only edmonds blossom inequalities and which maintains half integral intermediate lp solutions supported by a disjoint union of odd cycles and edges our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values this allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solutions

tridiagonal pairs of krawtchouk type let f denote an algebraically closed field with characteristic 0 and let v denote a vector space over f with finite positive dimension let a a denote a tridiagonal pair on v with diameter d we say that a a has krawtchouk type whenever the sequence d 2i i 0d is a standard ordering of the eigenvalues of a and a standard ordering of the eigenvalues of a assume a a has krawtchouk type we show that there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for u v v we show that the following tridiagonal pairs are isomorphic i a a ii a a iii a a iv a a we give a number of related results and conjectures

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

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

two linear transformations each tridiagonal with respect to an eigenbasis of the other the td d canonical form and the lb ub canonical form abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and b v v which satisfy both i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we call such a pair a leonard pair on v we introduce two canonical forms for leonard pairs we call these the td d canonical form and the lb ub canonical form in the td d canonical form the leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix subject to a certain normalization in the lb ub canonical form the leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix subject to a certain normalization we describe the two canonical forms in detail as an application we obtain the following results given square matrices a b over k with a tridiagonal and b diagonal we display a necessary and sufficient condition for a b to represent a leonard pair given square matrices a b over k with a lower bidiagonal and b upper bidiagonal we display a necessary and sufficient condition for a b to represent a leonard pair we briefly discuss how leonard pairs correspond to the q racah polynomials and some related polynomials in the askey scheme we present some open problems concerning leonard pairs

tridiagonal pairs and the μ conjecture let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by t ito and the second author we present a result which supports the conjecture given scalars i i 0d i i 0d in f that satisfy the known constraints on the eigenvalues of a tridiagonal pair we define an f algebra t by generators and relations we consider the f algebra e0 te0 for a certain idempotent e0 t let f x1 xd denote the polynomial algebra over f involving d mutually commuting indeterminates we display a surjective f algebra homomorphism f x1 xd e0 te0 we conjecture that is an isomorphism we show that this conjecture implies the classification conjecture and that the conjecture holds for d 5

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

nonparametric resampling of random walks for spectral network clustering parametric resampling schemes have been recently introduced in complex network analysis with the aim of assessing the statistical significance of graph clustering and the robustness of community partitions we propose here a method to replicate structural features of complex networks based on the non parametric resampling of the transition matrix associated with an unbiased random walk on the graph we test this bootstrapping technique on synthetic and real world modular networks and we show that the ensemble of replicates obtained through resampling can be used to improve the performance of standard spectral algorithms for community detection

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials