quac quick unsupervised anisotropic clustering abstract we present a novel unsupervised algorithm for quickly finding clusters in multi dimensional data it does not make the assumption of isotropy instead taking full advantage of the anisotropic gaussian kernel to adapt to local data shape and scale we employ some little used properties of the multivariate gaussian distribution to represent the data and also give as a corollary of the theory we formulate a simple yet principled means of preventing singularities in gaussian models the efficacy and robustness of the proposed method are demonstrated on both real and artificial data providing qualitative and quantitative results and comparing against the well known mean shift and k means algorithms

existence of efficient and properly efficient solutions to problems of constrained vector optimization the paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set the main attention is paid to the two major notions of optimality in vector problems pareto efficiency and proper efficiency in the sense of geoffrion employing adequate tools of variational analysis and generalized differentiation we first establish relationships between the notions of properness m tameness and the palais smale conditions formulated for the restriction of the vector cost mapping on the constraint set these results are instrumental to derive verifiable necessary and sufficient conditions for the existence of pareto efficient solutions in vector optimization furthermore the developed approach allows us to obtain new sufficient conditions for the existence of geoffrion properly efficient solutions to such constrained vector problems

multiple objective programming with polynomial objectives and constraints abstract an approach to solving multiple objective programming problems with polynomial objectives and polynomial constraints is developed the notions of pre efficient solution and local efficient solution are investigated relative to the single objective problem of benson illustrative examples are presented in which the solutions are obtained by means of homotopy continuation

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

the lagrange multiplier rule for multifunctions in banach spaces we study general constrained multiobjective optimization problems with objectives being closed multifunctions in banach spaces in terms of the coderivatives and normal cones we provide generalized lagrange multiplier rules as necessary optimality conditions of the above problems in an asplund space setting sharper results are presented

a general cone decomposition theory based on efficiency letk1 andk2 be two convex cones in some common vector space this paper is concerned with the question of finding a good decomposition with respect tok1 andk2 of a given element of the minkowski sumk1 k2 we propose the criterion of efficiency as a measure for the quality of a decomposition this criterion allows us to set up a framework from which a general cone decomposition theory is then derived

existence of solutions for vector optimization problems we define the generalized efficient solution which is more general than the weakly efficient solution for vector optimization problems and prove the existence of the generalized efficient solution for nondifferentiable vector optimization problems by using vector variational like inequalities for set valued maps

on the negative spectrum of the hierarchical schrodinger operator abstract this paper is devoted to the spectral theory of the schrodinger operator on the simplest fractal dyson s hierarchical lattice an explicit description of the spectrum eigenfunctions resolvent and parabolic kernel are provided for the unperturbed operator i e for the dyson hierarchical laplacian positive spectrum is studied for the perturbations of the hierarchical laplacian since the spectral dimension of the operator under consideration can be an arbitrary positive number the model allows a continuous phase transition from recurrent to transient underlying markov process this transition is also studied in the paper

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

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

spectral methods for graph clustering a survey graph clustering is an area in cluster analysis that looks for groups of related vertices in a graph due to its large applicability several graph clustering algorithms have been proposed in the last years a particular class of graph clustering algorithms is known as spectral clustering algorithms these algorithms are mostly based on the eigen decomposition of laplacian matrices of either weighted or unweighted graphs this survey presents different graph clustering formulations most of which based on graph cut and partitioning problems and describes the main spectral clustering algorithms found in literature that solve these problems

the domination property in multicriteria optimization abstract the domination property applies in a multicriteria problem if each feasible alternative is either efficient or else is dominated by an efficient alternative sufficient conditions for the property to hold are presented we show that if certain common conditions are met then proper efficiency implies the domination property and that under convexity conditions efficiency proper efficiency and the domination property are equivalent

interactive image segmentation using probabilistic hypergraphs this paper introduces a novel interactive framework for segmenting images using probabilistic hypergraphs which model the spatial and appearance relations among image pixels the probabilistic hypergraph provides us a means to pose image segmentation as a machine learning problem in particular we assume that a small set of pixels which are referred to as seed pixels are labeled as the object and background the seed pixels are used to estimate the labels of the unlabeled pixels by learning on a hypergraph via minimizing a quadratic smoothness term formed by a hypergraph laplacian matrix subject to the known label constraints we derive a natural probabilistic interpretation of this smoothness term and provide a detailed discussion on the relation of our method to other hypergraph and graph based learning methods we also present a front to end image segmentation system based on the proposed method which is shown to achieve promising quantitative and qualitative results on the commonly used grabcut dataset

on the existence of pareto solutions for polynomial vector optimization problems we are interested in the existence of pareto solutions to the vector optimization problem begin aligned mathrm min mathbb r m f x x in mathbb r n end aligned where f mathbb r n rightarrow mathbb r m is a polynomial map by using the tangency variety of f we first construct a semi algebraic set of dimension at most m 1 containing the set of pareto values of the problem then we establish connections between the palais smale conditions m tameness and properness for the map f based on these results we provide some sufficient conditions for the existence of pareto solutions of the problem we also introduce a generic class of polynomial vector optimization problems having at least one pareto solution

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

existence and characterization of efficient decisions with respect to cones existence and characterization theorems for the efficient nondominated set of decisions inr n are presented the existence is proved when the set of decisions satisfies some compactness conditions the efficient set is characterized in terms of the exposed efficient decisions when certain convexity and compactness conditions are satisfied

efficient graph based image segmentation this paper addresses the problem of segmenting an image into regions we define a predicate for measuring the evidence for a boundary between two regions using a graph based representation of the image we then develop an efficient segmentation algorithm based on this predicate and show that although this algorithm makes greedy decisions it produces segmentations that satisfy global properties we apply the algorithm to image segmentation using two different kinds of local neighborhoods in constructing the graph and illustrate the results with both real and synthetic images the algorithm runs in time nearly linear in the number of graph edges and is also fast in practice an important characteristic of the method is its ability to preserve detail in low variability image regions while ignoring detail in high variability regions

a survey of kernel and spectral methods for clustering clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines the focus of this paper is the partitioning clustering problem with a special interest in two recent approaches kernel and spectral methods the aim of this paper is to present a survey of kernel and spectral clustering methods two approaches able to produce nonlinear separating hypersurfaces between clusters the presented kernel clustering methods are the kernel version of many classical clustering algorithms e g k means som and neural gas spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized an explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation besides fuzzy kernel clustering methods are presented as extensions of kernel k means clustering algorithm

the linear order complementarity problem the classical complementarity problem in euclidean space can be viewed alternatively as a variational inequality or as a lattice orthogonality problem generalizations of the former have been extensively studied but infinite dimensional analogues of the latter have been largely ignored moreover as we show many well known results about the classical complementarity problem are more appropriately viewed order theoretically this is particularly true of least element solutions which are central to the present study of order complementarity in vector lattices we emphasize that the lattice theoretic descriptions we employ are very useful even in the standard finite dimensional setting

weak clarke epiderivative in set valued optimization a new notion of weak clarke epiderivative for a set valued map is introduced using the concept of clarke tangent cone the existence characterization and properties of weak clarke epiderivative are then studied finally optimality criteria are established for a constrained set valued optimization problem in terms of weak clarke epiderivative

epidifferentiability and hypodifferentiability of pseudoconvex maps in set optimization problems abstract in this paper we study set valued optimization problems with set optimization criteria with this aim we introduce two classes of pseudoconvex maps whose elements are epidifferentiable or hypodifferentiable we establish optimality conditions in terms of the epiderivatives and hypoderivatives of associated maps of infima or suprema under pseudoconvexity conditions finally by connecting set optimization criteria with vector criterion we apply these results in order to solve a vector optimization problem with constraints given by a set valued map

recent advances in graph based pattern recognition with applications in document analysis graphs are a powerful and popular representation formalism in pattern recognition particularly in the field of document analysis they have found widespread application from the formal point of view however graphs are quite limited in the sense that the majority of mathematical operations needed to build common algorithms such as classifiers or clustering schemes are not defined consequently we observe a severe lack of algorithmic procedures that can directly be applied to graphs there exists recent work however aimed at overcoming these limitations the present paper first provides a review of the use of graph representations in document analysis then we discuss a number of novel approaches suitable for making tools from statistical pattern recognition available to graphs these novel approaches include graph kernels and graph embedding with several experiments using different data sets from the field of document analysis we show that the new methods have great potential to outperform traditional procedures applied to graph representations

about contingent epiderivatives abstract in this paper we analyze some questions concerning the contingent epiderivative when the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives we also investigate the relationship between the contingent derivative and the generalized epiderivative furthermore we give existence theorems with respect to daniell cones as a particular case we study set valued maps that take values in r n

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

linear bilevel programs with multiple objectives at the upper level bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure they are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making when all objective functions are linear and constraints at both levels define polyhedra it is proved that the set of efficient solutions is non empty taking into account the properties of the feasible region of the bilevel problem some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques all the methods result in solving linear bilevel problems with a single objective function at each level

hypergraph imaging an overview hypergraph theory as originally developed by berge hypergraphe dunod paris 1987 is a theory of finite combinatorial sets modeling lot of problems of operational research and combinatorial optimization this framework turns out to be very interesting for many other applications in particular for computer vision in this paper we are going to survey the relationship between combinatorial sets and image processing more precisely we propose an overview of different applications from image hypergraph models to image analysis it mainly focuses on the combinatorial representation of an image and shows the effectiveness of this approach to low level image processing in particular to segmentation edge detection and noise cancellation

color texture segmentation using unsupervised graph cuts this paper proposes a novel approach to color texture segmentation based on graph cut techniques which finds an optimal color texture segmentation of a color textured image by regarding it as a minimum cut problem in a weighted graph a new texture descriptor based on the texton theory is introduced to efficiently represent texture attributes of the given image then the segmentation is formulated in terms of energy minimization with graph cuts where color and texton features are modelled with a multivariate finite mixture model with an unknown number of components contrary to previous supervised graph cut approaches our method finds minimum cuts using split moves in an unsupervised way the segmentation result including the number of segments is determined during the split moves without user interaction thus our method is called unsupervised graph cuts experimental results of color texture segmentation using various images including the mit vistex datasets and the berkeley datasets are presented and analyzed in terms of precision and recall to verify its effectiveness

some more density results for proper efficiencies in this paper we give sufficient conditions for a set and or a cone ensuring the density of henig s proper efficient points and positive proper efficient points in the set of efficient points in hausdorff locally convex spaces these conditions are based on the concept of the point of continuity an d generalize many existing results

the domination property for efficiency and bishop phelps theorem in locally convex spaces abstract we introduce the notion of closedness for any bounded convex set and investigate the relationship between closedness and local closedness moreover we study some properties of convex sets by using closedness local closedness and convexity we give two main theorems on the domination property and the existence of efficient points in locally convex spaces from these two main theorems we deduce a number of corollaries which improve the related known results as an application of the main results we obtain several support point theorems in locally convex spaces which generalize the famous bishop phelps theorem

a reductive approach to hypergraph clustering an application to image segmentation in the last few years hypergraph based methods have gained considerable attention in the resolution of real world clustering problems since such a mode of representation can handle higher order relationships between elements compared to the standard graph theory the most popular and promising approach to hypergraph clustering arises from concepts in spectral hypergraph theory 53 and clustering is configured as a hypergraph cut problem where an appropriate objective function has to be optimized the spectral relaxation of this optimization problem allows to get a clustering that is close to the optimum but this approach generally suffers from its high computational demands especially in real world problems where the size of the data involved in their resolution becomes too large a natural way to overcome this limitation is to operate a reduction of the hypergraph where spectral clustering should be applied over a hypergraph of smaller size in this paper we introduce two novel hypergraph reduction algorithms that are able to maintain the hypergraph structure as accurate as possible these algorithms allowed us to design a new approach devoted to hypergraph clustering based on the multilevel paradigm that operates in three steps i hypergraph reduction ii initial spectral clustering of the reduced hypergraph and iii clustering refinement the accuracy of our hypergraph clustering framework has been demonstrated by extensive experiments with comparison to other hypergraph clustering algorithms and have been successfully applied to image segmentation for which an appropriate hypergraph based model have been designed the low running times displayed by our algorithm also demonstrates that the latter unlike the standard spectral clustering approach can handle datasets of considerable size

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

graph cuts and efficient n d image segmentation combinatorial graph cut algorithms have been successfully applied to a wide range of problems in vision and graphics this paper focusses on possibly the simplest application of graph cuts segmentation of objects in image data despite its simplicity this application epitomizes the best features of combinatorial graph cuts methods in vision global optima practical efficiency numerical robustness ability to fuse a wide range of visual cues and constraints unrestricted topological properties of segments and applicability to n d problems graph cuts based approaches to object extraction have also been shown to have interesting connections with earlier segmentation methods such as snakes geodesic active contours and level sets the segmentation energies optimized by graph cuts combine boundary regularization with region based properties in the same fashion as mumford shah style functionals we present motivation and detailed technical description of the basic combinatorial optimization framework for image segmentation via s t graph cuts after the general concept of using binary graph cut algorithms for object segmentation was first proposed and tested in boykov and jolly 2001 this idea was widely studied in computer vision and graphics communities we provide links to a large number of known extensions based on iterative parameter re estimation and learning multi scale or hierarchical approaches narrow bands and other techniques for demanding photo video and medical applications

uniform convexity and the splitting problem for selections abstract we continue to investigate cases when the repovs semenov splitting problem for selections has an affirmative solution for continuous set valued mappings we consider the situation in infinite dimensional uniformly convex banach spaces we use the notion of polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets not necessary balls we study general geometric properties of uniformly convex sets we also obtain an affirmative solution of the splitting problem for selections of certain set valued mappings with uniformly convex images

a new fuzzy clustering validity index with a median factor for centroid based clustering determining the number of clusters which is usually approved by domain experts or evaluated by clustering validity indexes is an important issue in clustering analysis this study discusses the effectiveness of clustering validity indexes for centroid based partitional clustering algorithms most general purpose clustering validity indexes take the minimum maximum distance between a pair of data objects a pair of cluster centroids or an object and a centroid as an important evaluation factor however they may present unstable results especially when two centroids are allocated closely to alleviate this problem a new clustering validity index which is termed the wu and li index wli is proposed in this paper our proposed wli partially allows to some extent the existence of closely allocated centroids in the clustering results by considering not only the minimum but the median distances between a pair of centroids as well therefore possessing better stability the performances of wli and some existing clustering validity indexes are evaluated and compared by running the fuzzy c means algorithm for clustering various types of datasets including artificial datasets uci datasets and images experimental results have shown that wli has the more accurate and satisfactory performance than other indexes

a unified separation theorem for closed sets in a banach space and optimality conditions for vector optimization using the technique of variational analysis and in terms of normal cones we establish unified separation results for finitely many closed not necessarily convex sets in banach spaces which not only cover the existing nonconvex separation results and a classical convex separation theorem but also recapture the approximate projection theorem with help of the separation result for closed sets we provide necessary and sufficient conditions for approximate pareto solutions of constrained vector optimization problems in particular we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting

exact recovery in the hypergraph stochastic block model a spectral algorithm abstract we consider the exact recovery problem in the hypergraph stochastic block model hsbm with k blocks of equal size more precisely we consider a random d uniform hypergraph h with n vertices partitioned into k clusters of size s n k hyperedges e are added independently with probability p if e is contained within a single cluster and q otherwise where 0 q p 1 we present a spectral algorithm which recovers the clusters exactly with high probability given mild conditions on n k p q and d our algorithm is based on the adjacency matrix of h which is a symmetric n n matrix whose u v th entry is the number of hyperedges containing both u and v to the best of our knowledge our algorithm is the first to guarantee exact recovery when the number of clusters k n

pure point spectrum of the laplacians on fractal graphs abstract we establish the pure point spectrum of the laplacians on two point self similar fractal graphs all eigenvalues have infinite multiplicity and a countable system of orthonormal eigenfunctions with compact support is complete in the corresponding hilbert space

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

generalized benders decomposition for one class of minlps with vector conic constraint in this paper we mainly study one class of mixed integer nonlinear programming problems minlps with vector conic constraint in banach spaces duality theory of convex vector optimization problems applied to this class of minlps is deeply investigated with the help of duality we use the generalized benders decomposition method to establish an algorithm for solving this minlp several convergence theorems on the algorithm are also presented the convergence theorems generalize and extend the existing results on minlps in finite dimensional spaces