three dimensional ego motion estimation from motion fields observed with multiple cameras abstract in this paper we present a robust method to estimate the three dimensional ego motion of an observer moving in a static environment this method combines the optical flow fields observed with multiple cameras to avoid the ambiguity of 3 d motion recovery due to small field of view and small depth variation in the field of view two residual functions are proposed to estimate the ego motion for different situations in the non degenerate case both the direction and the scale of the three dimensional rotation and translation can be obtained in the degenerate case rotation can still be obtained but translation can only be obtained up to a scale factor both the number of cameras and the camera placement affect the accuracy of the estimated ego motion we compare different camera configurations through simulation some results of real world experiments are also given to demonstrate the benefits of our method

cross validation optimization for large scale structured classification kernel methods we propose a highly efficient framework for penalized likelihood kernel methods applied to multi class models with a large structured set of classes as opposed to many previous approaches which try to decompose the fitting problem into many smaller ones we focus on a newton optimization of the complete model making use of model structure and linear conjugate gradients in order to approximate newton search directions crucially our learning method is based entirely on matrix vector multiplication primitives with the kernel matrices and their derivatives allowing straightforward specialization to new kernels and focusing code optimization efforts to these primitives only kernel parameters are learned automatically by maximizing the cross validation log likelihood in a gradient based way and predictive probabilities are estimated we demonstrate our approach on large scale text classification tasks with hierarchical structure on thousands of classes achieving state of the art results in an order of magnitude less time than previous work parts of this work appeared in the conference paper seeger 2007

fast exact multiplication by the hessian just storing the hessian h the matrix of second derivatives 2e wi wj of the error e with respect to each pair of weights of a large neural network is difficult since a common use of a large matrix like h is to compute its product with various vectors we derive a technique that directly calculates hv where v is an arbitrary vector to calculate hv we first define a differential operator rv f w r f w rv r 0 note that rv w hv and rv w v and then apply rv to the equations used to compute w the result is an exact and numerically stable procedure for computing hv which takes about as much computation and is about as local as a gradient evaluation we then apply the technique to a one pass gradient calculation algorithm backpropagation a relaxation gradient calculation algorithm recurrent backpropagation and two stochastic gradient calculation algorithms boltzmann machines and weight perturbation finally we show that this technique can be used at the heart of many iterative techniques for computing various properties of h obviating any need to calculate the full hessian

a branch and cut approach to the cardinality constrained circuit problem the cardinality constrained circuit problem cccp is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges the cccp is np hard we present classes of facet inducing inequalities for the convex hull of feasible circuits and a branch and cut solution approach using these inequalities

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

the merchant subtour problem we consider the problem of a travelling merchant who makes money by buying commodities where they are cheap and selling them in other places where he can make a profit the merchant ships commodities of his own choice in a van of fixed capacity given the prices of all the commodities in all of the places and the cost of driving from one place to another the problem the merchant faces each day is to select a subset of the cities that he can visit in a day and to determine the order in which the cities are visited such that the total profit is maximised we call this problem the merchant subtour problem the msp models the pricing problem of a rather complex pickup and delivery problem that was given to us by the dutch logistics company van gend loos we show that a special case of the msp has a totally unimodular constraint matrix this knowledge enables us to develop a tabu search algorithm for finding good feasible solutions to the msp and a branch and price and cut algorithm for solving the msp to optimality the relaxations solved in each node of the branch and bound tree are strengthened by lifted knapsack inequalities lifted cycle inequalities and mod k cuts we present computational results on data sets derived from our main instance of the van gend loos pickup and delivery problem

optimal visual motion estimation a note the problem of estimating 3d motion in an optimal manner using correspondences of features in two views is analyzed the importance of having an optimal estimator is twofold first for the estimation itself and second for the bound it offers on how much sensitivity one can expect from a two frame point based motion algorithm the optimal estimator turns out to be nonlinear and for that reason techniques that provide very good initial guesses for the iterative computation of the optimal estimator are developed

motion parameter estimation from global flow field data presented are two methods for the determination of the parameters of motion of a sensor given the vector flow field induced by an imaging system governed by a perspective transformation of a rigid scene both algorithms integrate global data to determine motion parameters the first the flow circulation algorithm determines the rotational parameters the second the foe search algorithm determines the translational parameters of the motion independently of the first algorithm several methods for determining when the function has the appropriate form are suggested one method involves filtering the function by a collection of circular surround zero mean receptive fields the other methods project the function onto a linear space of quadratic polynomials and measures the distance between the two functions the error function for the first two methods is a quadratic polynomial of the candidate position yielding a very rapid search strategy

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

number of solutions for motion and structure from multiple framecorrespondence much of the dynamic computer vision literature deals with the determination of motion and structure by observing two frames captured at two instants of time motion prediction and understanding can be improved significantly particularly in the presence of noise by analyzing an image sequence containing more than two frames in this paper we assume knowledge of correspondence of points on the surface of an object which is moving with constant motion i e constant translation and constant rotation around an unknown center we give a new formulation of the problem and prove that the following results hold in general for the number of solutions to motion and structure values i e values of translation rotation and depth a for three point correspondences over three views there are at most two solutions only one of which has all positive depth values b for two point correspondences over four views there is a unique solution c for one point correspondence over five views there can be up to ten solutions d for one point correspondence over six views there is a unique solution the method of solution for each of the above formulations requires the solving of a system of multivariate polynomials whose coefficients are functions of the observed data in order to determine the number of solutions to these systems we use theorems from algebraic geometry which imply that under a few mild conditions the number of solutions at one set of data points provides an upper bound on the number of solutions for almost all sets of data points thus a bound on the number of solutions is obtained when a single system is solved by a method such as homotopy continuation which we use here

on the separation of maximally violated mod k cuts separation is of fundamental importance in cutting plane based techniques for integer linear programming ilp in recent decades a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts in this paper we address the separation of chvatal rank 1 inequalities in the context of general ilp s of the form min c t x ax b x integer where a is an m n integer matrix and b an m dimensional integer vector in particular for any given integer k we study mod k cuts of the form t ax t b for any 0 1 k k 1 k m such that t a is integer following the line of research recently proposed for mod 2 cuts by applegate bixby chvatal and cook 1 and fleischer and tardos 19 we restrict to maximally violated cuts i e to inequalities which are violated by k 1 k by the given fractional point we show that for any given k such a separation requires o mn min m n time applications to both the symmetric and asymmetric tsp are discussed in particular for any given k we propose an o v 2 e time exact separation algorithm for mod k cuts which are maximally violated by a given fractional symmetric or asymmetric tsp solution with support graph g v e this implies that we can identify a maximally violated cut for the symmetric tsp whenever a maximally violated extended comb inequality exists finally facet defining mod k cuts for the symmetric and asymmetric tsp are studied

polynomial time separation of a superclass of simple comb inequalities the comb inequalities are a well known class of facet inducing inequalities for the traveling salesman problem defined in terms of certain vertex sets called the handle and the teeth we say that a comb inequality is simple if the following holds for each tooth either the intersection of the tooth with the handle has cardinality one or the part of the tooth outside the handle has cardinality one or both the simple comb inequalities generalize the classical 2 matching inequalities of edmonds edmonds j 1965 maximum matching and a polyhedron with 0 1 vertices j res nat bur standards69b 125 130 and also the so called chvatal comb inequalities in 1982 padberg and rao padberg m w m r rao 1982 odd minimum cut sets and b matchings math oper res 7 67 80 gave a polynomial time separation algorithm for the 2 matching inequalities i e an algorithm for testing if a given fractional solution to an lp relaxation violates a 2 matching inequality we extend this significantly by giving a polynomial time separation algorithm for a class of valid inequalities which includes all simple comb inequalities

new results on the windy postman problem in this paper we study the windy postman problem wpp this is a well known arc routing problem that contains the mixed chinese postman problem mcpp as a special case we extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the windy postman problem over graphs with up to four vertices and ten edges we introduce two new families of facet inducing inequalities and prove that these inequalities along with the already known odd zigzag inequalities are chvatal gomory inequalities of rank at most 2 moreover a branch and cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented finally the performance of a branch and cut algorithm over several sets of large wpp and mcpp instances with up to 3 000 nodes and 9 000 edges and arcs in the mcpp case shows that to our knowledge this is the best algorithm to date for the exact resolution of the wpp and the mcpp

directions of motion fields are hardly ever ambiguous if instead of the full motion field we consider only the direction of the motion field due to a rigid motion what can we say about the three dimensional motion information contained in it this paper provides a geometric analysis of this question based solely on the constraint that the depth of the surfaces in view is positive the motivation behind this analysis is to provide a theoretical foundation for image constraints employing only the sign of flow in various directions and justify their utilization for addressing 3d dynamic vision problems it is shown that considering as the imaging surface the whole sphere independently of the scene in view two different rigid motions cannot give rise to the same directional motion field if we restrict the image to half of a sphere or an infinitely large image plane two different rigid motions with instantaneous translational and rotational velocities t 1 1 and t 2 2 cannot give rise to the same directional motion field unless the plane through t 1 and t 2 is perpendicular to the plane through 1 and 2 i e t 1 t 2 1 2 e 0 in addition in order to give practical significance to these uniqueness results for the case of a limited field of view we also characterize the locations on the image where the motion vectors due to the different motions must have different directions if 1 2 t 1 t 2 e 0 and certain additional constraints are met then the two rigid motions could produce motion fields with the same direction for this to happen the depth of each corresponding surface has to be within a certain range defined by a second and a third order surface similar more restrictive constraints are obtained for the case of multiple motions consequently directions of motion fields are hardly ever ambiguous a byproduct of the analysis is that full motion fields are never ambiguous with a half sphere as the imaging surface

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

on the domino parity inequalities for the stsp one method which has been used very successfully for finding optimal and provably good solutions for large instances of the symmetric travelling salesman problem stsp is the branch and cut method this method requires knowledge of classes of useful valid inequalities for the polytope associated with the stsp as well as efficient separation routines for these classes of inequalities recently a new class of valid inequalities called the domino parity inequalites were introduced for the stsp an efficient separation routine is known for these constraints if certain conditions are satisfied by the point to be separated this separation routine has never been implemented or tested we present several performance enhancements for this separation routine and discuss our implementation of this improved algorithm we test our implementation and provide results which we believe demonstrate the practical usefulness of these constraints and the separation routine for the stsp within a branch and cut framework

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

optimal motion and structure estimation the causes of existing linear algorithms exhibiting various high sensitivities to noise are analyzed it is shown that even a small pixel level perturbation may override the epipolar information that is essential for the linear algorithms to distinguish different motions this analysis indicates the need for optimal estimation in the presence of noise methods are introduced for optimal motion and structure estimation under two situations of noise distribution known and unknown computationally the optimal estimation amounts to minimizing a nonlinear function for the correct convergence of this nonlinear minimization a two step approach is used the first step is using a linear algorithm to give a preliminary estimate for the parameters the second step is minimizing the optimal objective function starting from that preliminary estimate as an initial guess a remarkable accuracy improvement has been achieved by this two step approach over using the linear algorithm alone

separating clique trees and bipartition inequalities having a fixed number of handles and teeth in polynomial time many important cutting planes have been discovered for the traveling salesman problem among them are the clique tree inequalities and the more general bipartition inequalities little is known in the way of exact algorithms for separating these inequalities in polynomial time in the size of the fractional point x which is being separated an algorithm is presented here that separates bipartition inequalities of a fixed number of handles and teeth which includes clique tree inequalities of a fixed number of handles and teeth in polynomial time

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

cutting planes and the elementary closure in fixed dimension the elementary closurep of a polyhedronp is the intersection ofp with all its gomory chvatal cutting planes p is a rational polyhedron provided thatp is rational the known bounds for the number of inequalities definingp are exponential even in fixed dimension we show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension ifp is a simplicial cone we construct a polytopeq whose integral elements correspond to cutting planes ofp the vertices of the integer hullq iinclude the facets ofp a polynomial upper bound on their number can be obtained by applying a result of cook et al 1992 finally we present a polynomial algorithm in varying dimension which computes cutting planes for a simplicial cone that correspond to vertices ofq i

the circuit polytope facets given an undirected graph g v e and a cost vector c e the weighted girth problem is to find a circuit in g having minimum total cost this problem is in general x1d4a9 x1d4ab hard a promising approach to the solution of hard combinatorial optimization problems is given by the so called cutting plane methods these involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions we consider the weighted girth problem in the case where g is the complete graph kn and study the facial structure of the circuit polytope pcn and some related polyhedra in the appendix we give complete characterizations of pcn for n up to 6 we were motivated to study the weighted girth problem by the fact that this problem and variations of it appear as a subproblem namely the pricing problem in a column generation approach to the vehicle routing problem

separation algorithms for classes of stsp inequalities arising from a new stsp relaxation the problem ofseparating a class of inequalities that are valid for the symmetric traveling salesman problem stsp consists of devising an efficient method that given a fractional pointx for a relaxation of the stsp either finds an inequality in the given class of stsp inequalities that is violated byx or determines that there are no such violated inequalities we can define important classes of stsp inequalities by performing naddef s and rinaldi s 1993 node lifting operation on stsp inequalities defined on small graphs we present efficient methods forexactly separating large classes of stsp inequalities that arise from an stsp relaxation related to node lifting in particular we show how to find efficiently the most violated inequality in the class of stsp inequalities having abackbone set with a constant numberk of vertices and thus separate in polynomial time the class of all stsp inequalities whose backbone set has at mostk vertices

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

vehicle type motion estimation from multi frame images a model for vehicle type motion which assumes that the motion is a rotation around an axis through the vehicle center followed by a forward translation along the main axis of the vehicle is proposed when the rotation and the amplitude of translation are constant this type of motion is shown to be equivalent to a constant camera centered motion this indicates that a constant motion in the conventional camera centered model which is commonly considered to be artificial can in fact be a reasonable model in real life it is shown that a constant vehicle type motion can be interpreted as a constant screw motion a linear algorithm for estimating constant vehicle type motion is presented and experiments using real scene images are included as an extension vehicle type motion with constant rotation and constantly accelerated translation is discussed

estimation three dimensional motion of rigid objects from noisy observations an estimate is made of the motion of a rigid body from two noisy 2 d perspective projections using the least squares method and the algebra of r y tsai and t s huang 1984 the accuracy of the estimated motion parameters is influenced by the position of the features of the object used in the calculation four test variables are derived that indicate how the accuracy is affected and they are used for discarding inaccurate estimates monte carlo tests demonstrate the obtained accuracy

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

max flow min cut matroids polynomial testing and polynomial algorithms for maximum flow and shortest routes a polynomial test for deciding whether or not a given binary matroid has the max flow min cut property is described in addition polynomial algorithms for finding shortest routes and extreme integral max flows in max flow min cut matroids are developed the algorithms make use of a decomposition result for the max flow min cut matroids that was previously established in joint work with f t tseng the max flow algorithm also implies a strengthened version of p d seymour s original characterization of the max flow min cut matroids

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

3 d translational motion and structure from binocular image flows image flow fields from parallel stereo cameras are analyzed to determine the relative 3 d translational motion of the camera platform with respect to objects in view and to establish stereo correspondence of features in the left and right images a two step procedure is suggested in the first step translational motion parameters are determined from linear equations the coefficients of which consist of the sums of measured quantities in the two images separate equations are developed for cases when measurements of either the full optical flow or the normal flow are available this computation does not require feature to feature correspondence in addition no assumption is made about the surfaces being viewed in the second step of the calculation with the knowledge of the estimated translational motion parameters the binocular flow information is used to find features in one image that correspond to given features in the other image experimental results with synthetic and laboratory images indicate that the method provides accurate results even in the presence of noise

gaussian processes for classification mean field algorithms we derive a mean field algorithm for binary classification with gaussian processes that is based on the tap approach originally proposed in statistical physics of disordered systems the theory also yields an approximate leave one out estimator for the generalization error which is computed with no extra computational cost we show that from the tap approach it is possible to derive both a simpler naive mean field theory and support vector machines svms as limiting cases for both mean field algorithms and support vector machines simulation results for three small benchmark data sets are presented they show that one may get state of the art performance by using the leave one out estimator for model selection and the built in leave one out estimators are extremely precise when compared to the exact leave one out estimate the second result is taken as strong support for the internal consistency of the mean field approach

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

fast cross validation for kernel based algorithms cross validation cv is a widely adopted approach for selecting the optimal model however the computation of empirical cross validation error cve has high complexity due to multiple times of learner training in this paper we develop a novel approximation theory of cve and present an approximate approach to cv based on the bouligand influence function bif for kernel based algorithms we first represent the bif and higher order bifs in taylor expansions and approximate cv via the taylor expansions we then derive an upper bound of the discrepancy between the original and approximate cv furthermore we provide a novel computing method to calculate the bif for general distribution and evaluate bif criterion for sample distribution to approximate cv the proposed approximate cv requires training on the full data set only once and is suitable for a wide variety of kernel based algorithms experimental results demonstrate that the proposed approximate cv is sound and effective

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

lifted cycle inequalities for the asymmetric traveling salesman problem we investigate the family of facet defining inequalities for the asymmetric traveling salesman ats polytope obtainable by lifting the cycle inequalities we establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ats polytope known to date in particular we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities this characterization is used to investigate three new large classes of lifted cycle inequalities that we call shell fork and curtain inequalities these latter inequalities have unbounded chvatal rank furthermore the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines finally each member of the family is shown to have a counterpart for the symmetric ts sts polytope that is often new and is obtainable by mapping the inequality for the ats polytope into a certain face of the sts polytope and then lifting the resulting inequality into one for the sts polytope itself

subset row inequalities applied to the vehicle routing problem with time windows this paper presents a branch and cut and price algorithm for the vehicle routing problem with time windows the standard dantzig wolfe decomposition of the arc flow formulation leads to a set partitioning problem as the master problem and an elementary shortest path problem with resource constraints as the pricing problem we introduce the subset row inequalities which are chvatal gomory rank 1 cuts based on a subset of the constraints in the master problem applying a subset row inequality in the master problem increases the complexity of the label setting algorithm used to solve the pricing problem because an additional resource is added for each inequality we propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step like structure of the objective function of the pricing problem computational experiments have been performed on the solomon benchmarks where we were able to close several instances the results show that applying subset row inequalities in the master problem significantly improves the lower bound and in many cases makes it possible to prove optimality in the root node

separating maximally violated comb inequalities in planar graphs the traveling salesman problem tsp is a benchmark problem in combinatorial optimization it was one of the very first problems used for developing and testing approaches to solving large integer programs including cutting plane algorithms and branch and cut algorithms much of the research in this area has been focused on finding new classes of facets for the tsp polytope and much less attention has been paid to algorithms for separating from these classes of facets in this paper we consider the problem of finding violated comb inequalities if there are no violated subtour constraints in a fractional solution of the tsp a comb inequality may not be violated by more than 1 given a fractional solution in the subtour elimination polytope whose graph is planar we either find a violated comb inequality or determine that there are no comb inequalities violated by 1 our algorithm runs in on mcn time where mcn is the time to compute a cactus representation of all minimum cuts of a weighted planar graph on n vertices

recursive 3 d visual motion estimation using subspace constraints the 3 d motion of a camera within a static environment produces a sequence of time varying images that can be used for reconstructing the relative motion between the scene and the viewer the problem of reconstructing rigid motion from a sequence of perspective images may be characterized as the estimation of the state of a nonlinear dynamical system which is defined by the rigidity constraint and the perspective measurement map the time derivative of the measured output of such a system which is called the 2 d motion field and is approximated by the optical flow is bilinear in the motion parameters and may be used to specify a subspace constraint on the direction of heading independent of rotation and depth and a pseudo measurement for the rotational velocity as a function of the estimated heading the subspace constraint may be viewed as an implicit dynamical model with parameters on a differentiable manifold and the visual motion estimation problem may be cast in a system theoretic framework as the identification of such an implicit model we use techniques which pertain to nonlinear estimation and identification theory to recursively estimate 3 d rigid motion from a sequence of images independent of the structure of the scene such independence from scene structure allows us to deal with a variable number of visible feature points and occlusions in a principled way the further decoupling of the direction of heading from the rotational velocity generates a filter with a state that belongs to a two dimensional and highly constrained state space as a result the filter exhibits robustness properties which are highlighted in a series of experiments on real and noisy synthetic image sequences while the position of feature points is not part of the state of the model the innovation process of the filter describes how each feature is compatible with a rigid motion interpretation which allows us to test for outliers and makes the filter robust with respect to errors in the feature tracking optical flow reflections t junctions once motion has been estimated the 3 d structure of the scene follows easily by releasing the constraint that the visible points lie in front of the viewer one may explain some psychophysical effects on the nonrigid percept of rigidly moving objects

nonparametric least squares estimation in derivative families cost function estimation often involves data on a function and a family of its derivatives such data can substantially improve convergence rates of nonparametric estimators we propose series type estimators which incorporate the various derivative data into a single nonparametric least squares procedure convergence rates are obtained and it is shown that for low dimensional cases much of the beneficial impact is realized even if only data on ordinary first order partials are available in instances where root n consistency is attained smoothing parameters can often be chosen very easily without resort to cross validation simulations and an illustration of cost function estimation are included

binary group and chinese postman polyhedra a new proof of the characterization of the chinese postman polyhedra is given in developing this proof a theorem of gomory about homomorphic lifting of facets for group polyhedra is generalized to subproblems some results for the chinese postman problem are generalized to binary group problems in addition a connection is made between fulkerson s blocking polyhedra and a blocking pair of binary group problems a connection is also developed between minors and lifting of facets for group problems

motion fields are hardly ever ambiguous there has been much concern with ambiguity in the recovery of motion and structure from time varying images i show here that the class of surfaces leading to ambiguous motion fields is extremely restricted only certain hyperboloids of one sheet and some degenerate forms qualify furthermore the viewer must be on the surface for it to lead to a potentially ambiguous motion field thus the motion field over an appreciable image region almost always uniquely defines the instantaneous translational and rotational velocities as well as the shape of the surface up to a scale factor

direct passive navigation in this correspondence we show how to recover the motion of an observer relative to a planar surface from image brightness derivatives we do not compute the optical flow as an intermediate step only the spatial and temporal brightness gradients at a minimum of eight points we first present two iterative schemes for solving nine nonlinear equations in terms of the motion and surface parameters that are derived from a least squares fomulation an initial pass over the relevant image region is used to accumulate a number of moments of the image brightness derivatives all of the quantities used in the iteration are efficiently computed from these totals without the need to refer back to the image we then show that either of two possible solutions can be obtained in closed form we first solve a linear matrix equation for the elements of a 3 3 matrix the eigenvalue decomposition of the symmetric part of the matrix is then used to compute the motion parameters and the plane orientation a new compact notation allows us to show easily that there are at most two planar solutions

separating a superclass of comb inequalities in planar graphs many classes of valid and facet inducing inequalities are known for the family of polytopes associated with thesymmetric travelling salesman problem stsp includingsubtour elimination 2 matching andcomb inequalities for a given class of inequalities anexact separation algorithm is a procedure which given an lp relaxation vectorx finds one or more inequalities in the class which are violated byx or proves that none exist such algorithms are at the core of the highly successfulbranch and cut algorithms for the stsp however whereas polynomial time exact separation algorithms are known for subtour elimination and 2 matching inequalities the complexity of comb separation is unknown apartial answer to the comb problem is provided in this paper we define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx e 0 is planar the separation algorithm runs in o n3 time wheren is the number of vertices in the graph

a tabu search heuristic for the vehicle routing problem with stochastic demands and customers this paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands a tabu search heuristic is developed for this problem comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89 45 of cases with an average deviation of 0 38 from optimality

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

bayesian classification with gaussian processes we consider the problem of assigning an input vector to one of m classes by predicting p c x for c 1 m for a two class problem the probability of class one given x is estimated by spl sigma y x where spl sigma y 1 1 e sup y a gaussian process prior is placed on y x and is combined with the training data to obtain predictions for new x points we provide a bayesian treatment integrating over uncertainty in y and in the parameters that control the gaussian process prior the necessary integration over y is carried out using laplace s approximation the method is generalized to multiclass problems m 2 using the softmax function we demonstrate the effectiveness of the method on a number of datasets

exact algorithm for minimising the number of setups in the one dimensional cutting stock problem the cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste because changing over from one cutting pattern to another involves significant setups an auxiliary problem is to minimise the number of different patterns that are used the pattern minimisation problem is significantly more complex but it is of great practical importance in this paper we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns each of which corresponds to selecting a fixed number of copies of a specific cutting pattern the integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems at each node of the branch and bound tree the linear programming relaxation of our formulation is made tighter by adding super additive inequalities branching rules are presented that yield a balanced tree incumbent solutions are obtained using a rounding heuristic the resulting branch and price and cut procedure is used to produce optimal or approximately optimal solutions for a set of real life problems