deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

graph dual regularization non negative matrix factorization for co clustering low rank matrix factorization is one of the most useful tools in scientific computing data mining and computer vision among of its techniques non negative matrix factorization nmf has received considerable attention due to producing a parts based representation of the data recent research has shown that not only the observed data are found to lie on a nonlinear low dimensional manifold namely data manifold but also the features lie on a manifold namely feature manifold in this paper we propose a novel algorithm called graph dual regularization non negative matrix factorization dnmf which simultaneously considers the geometric structures of both the data manifold and the feature manifold we also present a graph dual regularization non negative matrix tri factorization algorithm dnmtf as an extension of dnmf moreover we develop two iterative updating optimization schemes for dnmf and dnmtf respectively and provide the convergence proofs of our two optimization schemes experimental results on uci benchmark data sets several image data sets and a radar hrrp data set demonstrate the effectiveness of both dnmf and dnmtf

an efficient matrix factorization based low rank representation for subspace clustering in recent years robust subspace clustering is an important unsupervised clustering problem in machine learning and computer vision communities the recently proposed spectral clustering based approach called low rank representation lrr yields an optimal solution for the case of independent subspaces and partially corrupted data however it has to be solved iteratively and involves singular value decomposition svd at each iteration and then suffers from high computation cost of multiple svds in this paper we propose an efficient matrix tri factorization mtf approach with a positive semidefinite psd constraint to approximate the original nuclear norm minimization nnm problem and mitigate the computation cost of performing svds specially we introduce a matrix tri factorization idea into the original low rank representation framework and then convert it into a small scale matrix nuclear norm minimization problem finally we establish an alternating direction method adm based algorithm to efficiently solve the proposed problem experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of the proposed mtf approach comparing with the state of the art algorithms

on the minimum violations ranking of a tournament this paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament the criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament such a situation is referred to as a violation the objective therefore is to determine the minimum violations ranking mvr while there are situations where this ranking would be allowed to contain ties among subsets of objects we will concern ourselves herein with linear ordering no ties a series of examples are given where this requirement would seem to be appropriate in order to put the mvr problem into proper perspective we introduce the concept of a distance on the set of tournaments a set of natural axioms is presented which any such distance measure should obey and it is proven that in the presence of these axioms a unique such measure exists it is then shown that the mvr problem is equivalent to the minimum distance problem which can be represented in several forms in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem this opens up a wide scope of possible solution procedures for the mvr problem an optimal algorithm is presented along with computational results in addition various heuristics are discussed including an improved heuristic referred to as the iterated kendall method

a parametric gp model dealing with incomplete information for group decision making abstract we consider a group decision making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices in many cases experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives limited expertise related to some problem domain unavailable data etc resulting in incomplete pairwise comparison matrices our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information for that purpose we have established an optimization problem in which a similarity function and a parametric compromise function are defined associated to this problem a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution moreover the parameters involved in the method have a clear meaning in the context of group problems

eigenvector ranking method as a measuring tool formulas for errors abstract the eigenvector method em is well known to derive information from pairwise comparison matrices in decision making processes however this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as right left asymmetry rank reversal and reversal of order of intensity of preference in this paper we associate em with some standard measuring procedure analyze this procedure from the viewpoint of measurement theory and find the actual em error we show that the above phenomena have the same cause and are eliminated when the em errors are taken into account the full decision support tool which has all components of a standard measuring tool is composed of pairwise comparisons as an initial measuring procedure em as a data processor and the obtained formulas for em errors as an error indicator we consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix both versions are equally suitable to measure and rank any comparable elements with positive numerical values and have the same mean relative errors equal to the square root of the double saaty s consistency index using the mean relative error we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking this estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives these restrictions are much stronger than previously thought

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

the dixmier property and tracial states for c algebras a t was partially supported by an nserc postdoctoral fellowship and through the epsrc grant ep n00874x 1 acknowledgements we are grateful to luis santiago for helpful discussions at an early stage of this investigation we would also like to thank the referee for providing helpful comments which have led to a number of improvements

a note on deriving weights from pairwise comparison ratio matrices abstract this paper addresses the cook and kress method regarding the use of mathematical programming to derive weights from pairwise comparison ratio matrices we note that the cook and kress formulation has the shortcoming that alternative optimal solutions may occur which would lead to an infinite set of possible weights this paper proposes to resolve the problem by formulating a phase ii optimization procedure to be solved using quadratic programming this mathematical formulation is based on the determination of a weight vector that is statistically most likely to cause the associated pairwise comparison matrix it is concluded that the cook and kress method accompanied with the proposed phase ii approach can always derive a unique set of weights from a pairwise comparison matrix

multiframe motion segmentation with missing data using powerfactorization and gpca we consider the problem of segmenting multiple rigid body motions from point correspondences in multiple affine views we cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two three or four our algorithm involves projecting all point trajectories onto a 5 dimensional subspace using the svd the powerfactorization method or ransac and fitting multiple linear subspaces representing different rigid body motions to the points in 5 using gpca unlike previous work our approach does not restrict the motion subspaces to be four dimensional and independent instead it deals gracefully with all the spectrum of possible affine motions from two dimensional and partially dependent to four dimensional and fully independent our algorithm can handle the case of missing data meaning that point tracks do not have to be visible in all images by using the powerfactorization method to project the data in addition our method can handle outlying trajectories by using ransac to perform the projection we compare our approach to other methods on a database of 167 motion sequences with full motions independent motions degenerate motions partially dependent motions missing data outliers etc on motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

a characterization of the logarithmic least squares method abstract we provide an axiomatic characterization of the logarithmic least squares method sometimes called row geometric mean used for deriving a preference vector from a pairwise comparison matrix this procedure is shown to be the only one satisfying two properties correctness in the consistent case which requires the reproduction of the inducing vector for any consistent matrix and invariance to a specific transformation on a triad that is the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3 cycle by a positive scalar

computing interval weights for incomplete pairwise comparison matrices of large dimension a weak consistency based approach multiple criteria decision making and evaluation problems dealing with a large number of objects are very demanding particularly when the use of pairwise comparison pc techniques is required a major drawback arises when it is not possible to obtain all the pcs due to time or cost limitations or to split the given problem into smaller subproblems in such cases two tools are needed to find acceptable weights of objects an efficient method for partially filling a pairwise comparison matrix pcm and a suitable method for deriving weights from this incomplete pcm this paper presents a novel interactive algorithm for large dimensional problems guided by two main ideas the sequential optimal choice of the pcs to be performed and the concept of weak consistency the proposed solution significantly reduces the number of needed pcs by adding information implied by the weak consistency after the input of each pc providing sets of feasible values for all missing pcs interval weights of objects are computed from the resulting incomplete weakly consistent pcm adapting the methodology for calculating fuzzy weights from fuzzy pcms the computed weight intervals thus cover all possible weakly consistent completions of the incomplete pcm the algorithm works both with saaty s pcms and fuzzy preference relations the performance of the algorithm is illustrated by a numerical example and a real life case study the performed simulation demonstrates that the proposed algorithm is capable of reducing the number of pcs required in pcms of dimension 15 and greater by more than 60 on average

c simplicity of free products with amalgamation and radical classes of groups abstract we give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group c algebra and provide a concrete example of an amalgam with trivial kernel such that its reduced group c algebra has a unique tracial state but is not simple moreover we show that there is a radical class of groups for which the reduced group c algebra of any group is simple precisely when the group has a trivial radical corresponding to this class

a multiple criteria decision model with ordinal preference data abstract this paper develops a model for aggregating ordinal preference ranking across multiple criteria criteria weights are developed as part of the optimization process the model utilizes three levels of ordinal priorities 1 preferences among alternatives on the basis of each criterion 2 preferences among the criteria to reflect the importance of those criteria and 3 preferences among the criteria which express the degree to which the respondent can clearly distinguish among the alternatives the model is based upon a data envelopment analysis structure which permits a flexible weighting mechanism for each alternative being ranked

efficient recovery of low rank matrix via double nonconvex nonsmooth rank minimization recently there is a rapidly increasing attraction for the efficient recovery of low rank matrix in computer vision and machine learning the popular convex solution of rank minimization is nuclear norm based minimization nnm which usually leads to a biased solution since nnm tends to overshrink the rank components and treats each rank component equally to address this issue some nonconvex nonsmooth rank nnr relaxations have been exploited widely different from these convex and nonconvex rank substitutes this paper first introduces a general and flexible rank relaxation function named weighted nnr relaxation function which is actually derived from the initial double nnr dnnr relaxations i e dnnr relaxation function acts on the nonconvex singular values function svf an iteratively reweighted svf optimization algorithm with continuation technology through computing the supergradient values to define the weighting vector is devised to solve the dnnr minimization problem and the closed form solution of the subproblem can be efficiently obtained by a general proximal operator in which each element of the desired weighting vector usually satisfies the nondecreasing order we next prove that the objective function values decrease monotonically and any limit point of the generated subsequence is a critical point combining the kurdyka ojasiewicz property with some milder assumptions we further give its global convergence guarantee as an application in the matrix completion problem experimental results on both synthetic data and real world data can show that our methods are competitive with several state of the art convex and nonconvex matrix completion methods

robust face recognition based on dynamic rank representation abstract robust face recognition is an active topic in computer vision while face occlusion is one of the most challenging problems for robust face recognition algorithm the latest research on low rank representation demonstrated its high efficiency to subspace segmentation and feature extraction motivated by previous work in this paper we consider the problem of human face recognition from frontal views with varying illumination as well as occlusion and disguise we present a novel approach for face recognition by extracting dynamic subspace of images and obtaining the discriminative parts in each individual we use these parts to represent the characteristic of discriminative components give a recognition protocol to classify face images the experiments carried on publicly available databases i e ar extended yale b and orl vilidate its accuray robustness and speed the proposed method needs lower dimensions training samples but gains a higher recognition rate than other popular approaches

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

bbtopsis a bag based technique for order preference by similarity to ideal solution abstract this contribution explores the way of accomplishing a topsis ranking of various submitted variants taking into account a our newly developed notions of variant superiority inferiority and noninferiority multiattribute canonical fuzzy bags defined on attributes and b making use of bag metrics to compute the relative closeness of the different variants to an ideal solution in particular we prove the existence and the uniqueness of a bag metric fulfilling a number of interesting properties

representing the strengths and directions of pairwise comparisons we consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects as in the ahp and other methods our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons but also their directions for example if the comparison directions satisfy transitivity then the weights produced by our method also satisfy transitivity this is not always true for other methods we also present a set of reasonable axioms for which our method is the essentially unique solution our method and axioms are closely related to those of cook and kress eur j oper res 37 1988 335 our method like theirs reduces to solving a linear program hence it is different from the approach used in the ahp for the special case that the comparison directions satisfy transitivity our method is quite simple and reduces to performing a forward pass as in the critical path method

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods