the structure of max plus hyperplanes a max plus hyperplane briefly a hyperplane is the set of all points x x1 xn rmaxn satisfying an equation of the form a1x1 anxn an 1 b1x1 bnxn bn 1 that is max a1 x1 an xn an 1 max b1 x1 bn xn bn 1 with ai bi rmax i 1 n 1 where each side contains at least one term and where ai bi for at least one index i we show that the complements of max plus semispaces at finite points z rn are building blocks for the hyperplanes in rmaxn recall that a semispace at z is a maximal with respect to inclusion max plus convex subset of rmaxn z namely observing that up to a permutation of indices we may write the equation of any hyperplane h in one of the following two forms a1x1 apxp ap 1xp 1 aqxq a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m n 1 are finite or a1x1 apxp ap 1xp 1 aqxq an 1 a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m are finite and an 1 is either finite or we give a formula that expresses a nondegenerate strictly affine hyperplane i e with m n and an 1 as a union of complements of semispaces at a point z rn called the center of h with the boundary of a union of complements of other semispaces at z using this formula we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior we give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center and we characterize the cases in which the boundary bd h of a nondegenerate strictly affine hyperplane h is also a hyperplane next we give the relations between nondegenerate strictly affine hyperplanes h their centers z and their coefficients ai in the converse direction we show that any union of complements of semispaces at a point z rn with the boundary of any union of complements of some other semispaces at that point z is a nondegenerate strictly affine hyperplane we obtain a formula for the total number of strictly affine hyperplanes we give complete lists of all strictly affine hyperplanes for the cases n 1 and n 2 we show that each linear hyperplane h in rmaxn i e with an 1 can be decomposed as the union of four parts where each part is easy to describe in terms of complements of semispaces some of them in a lower dimensional space

simultaneous structure and texture image inpainting an algorithm for the simultaneous filling in of texture and structure in regions of missing image information is presented in this paper the basic idea is to first decompose the image into the sum of two functions with different basic characteristics and then reconstruct each one of these functions separately with structure and texture filling in algorithms the first function used in the decomposition is of bounded variation representing the underlying image structure while the second function captures the texture and possible noise the region of missing information in the bounded variation image is reconstructed using image inpainting algorithms while the same region in the texture image is filled in with texture synthesis techniques the original image is then reconstructed adding back these two sub images the novel contribution of this paper is then in the combination of these three previously developed components image decomposition with inpainting and texture synthesis which permits the simultaneous use of filling in algorithms that are suited for different image characteristics examples on real images show the advantages of this proposed approach

recognizing faces prone to occlusions and common variations using optimal face subgraphs an intuitive graph optimization face recognition approach called harmony search oriented ebgm hso ebgm inspired by the classical elastic bunch graph matching ebgm graphical model is proposed in this contribution in the proposed hso ebgm a recent evolutionary approach called harmony search optimization is tailored to automatically determine optimal facial landmarks a novel notion of face subgraphs have been formulated with the aid of these automated landmarks that maximizes the similarity entailed by the subgraphs for experimental evaluation two sets of de facto databases i e ar and face recognition grand challenge frgc ver2 0 are used to validate and analyze the behavior of the proposed hso ebgm in terms of number of subgraphs varying occlusion sizes face images under controlled ideal conditions realistic partial occlusions expression variations and varying illumination conditions for a number of experiments results justify that the hso ebgm shows improved recognition performance when compared to recent state of the art face recognition approaches

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

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

on tropical fractional linear programming very recently tropical counterparts of fractional linear programs have been studied some algorithms were proposed for solving them with techniques ranging from bisection methods to homeomorphisms to formal power series in this paper some algorithms are also proposed they mainly rely in the ability of finding the greatest and smallest solutions of tropical equations a subject that was discussed in a previous work of the authors 13

on the approximability of minimizing nonzero variables or unsatisfied relations in linear systems we investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning operations research and pattern recognition in the first class min ulr one wishes given a possibly infeasible system of linear relations to find a solution that violates as few relations as possible while satisfying all the others in the second class min rvls the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible for both min ulr and min rvls the four basic types of relational operators and are considered while min rvls with equations was mentioned to be np hard in garey and johnson 1979 we established in amaldi 1992 amaldi and kann 1995 that min ulr with equalities and inequalities are np hard even when restricted to homogeneous systems with bipolar coefficients the latter problems have been shown hard to approximate in arora et al 1993 in this paper we determine strong bounds on the approximability of various variants of min rvls and min ulr including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional the various np hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints but none of them can be approximated within any constant factor unless p np particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning in particular we disprove a conjecture of van horn and martinez 1992 regarding the existence of a polynomial time algorithm to design linear classifiers or perceptrons that involve a close to minimum number of features

densities of idempotent measures and large deviations considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by maslov then idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure in this paper we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied these conditions depend on the lattice structure of the semiring and on the boolean algebra in which the measure is defined as an application we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by varadhan

low rank latent pattern approximation with applications to robust image classification this paper develops a novel method to address the structural noise in samples for image classification recently regression related classification methods have shown promising results when facing the pixelwise noise however they become weak in coping with the structural noise due to ignoring of relationships between pixels of noise image meanwhile most of them need to implement the iterative process for computing representation coefficients which leads to the high time consumption to overcome these problems we exploit a latent pattern model called low rank latent pattern approximation llpa to reconstruct the test image having structural noise the rank function is applied to characterize the structure of the reconstruction residual between test image and the corresponding latent pattern simultaneously the error between the latent pattern and the reference image is constrained by frobenius norm to prevent overfitting llpa involves a closed form solution by the virtue of a singular value thresholding operator the provided theoretic analysis demonstrates that llpa indeed removes the structural noise during classification task additionally llpa is further extended to the form of matrix regression by connecting multiple training samples and alternating direction of multipliers method with gaussian back substitution algorithm is used to solve the extended llpa experimental results on several popular data sets validate that the proposed methods are more robust to image classification with occlusion and illumination changes as compared to some existing state of the art reconstruction based methods and one deep neural network based method

multidimensional morphable models a framework for representing and matching object classes we describe a flexible model for representing images of objects of a certain class known a priori such as faces and introduce a new algorithm for matching it to a novel image and thereby perform image analysis the flexible model known as a multidimensional morphable model is learned from example images of objects of a class in this paper we introduce an effective stochastic gradient descent algorithm that automatically matches a model to a novel image several experiments demonstrate the robustness and the broad range of applicability of morphable models our approach can provide novel solutions to several vision tasks including the computation of image correspondence object verification and image compression

reconstruction of partially damaged face images based on a morphable face model this paper proposes a method for reconstructing partially damaged faces based on a morphable face model faces are modeled by linear combinations of prototypes of shape and texture with the shape and texture information from an undamaged region only we can estimate optimal coefficients for linear combinations of prototypes of shape and texture by simple projection for least square minimization lsm our experimental results show that reconstructed faces are very natural and plausible like real photos

low rank structure learning via nonconvex heuristic recovery in this paper we propose a nonconvex framework to learn the essential low rank structure from corrupted data different from traditional approaches which directly utilizes convex norms to measure the sparseness our method introduces more reasonable nonconvex measurements to enhance the sparsity in both the intrinsic low rank structure and the sparse corruptions we will respectively introduce how to combine the widely used lp norm 0 p 1 and log sum term into the framework of low rank structure learning although the proposed optimization is no longer convex it still can be effectively solved by a majorization minimization mm type algorithm with which the nonconvex objective function is iteratively replaced by its convex surrogate and the nonconvex problem finally falls into the general framework of reweighed approaches we prove that the mm type algorithm can converge to a stationary point after successive iterations the proposed model is applied to solve two typical problems robust principal component analysis and low rank representation experimental results on low rank structure learning demonstrate that our nonconvex heuristic methods especially the log sum heuristic recovery algorithm generally perform much better than the convex norm based method 0 p 1 for both data with higher rank and with denser corruptions

clustering and embedding using commute times this paper exploits the properties of the commute time between nodes of a graph for the purposes of clustering and embedding and explores its applications to image segmentation and multibody motion tracking our starting point is the lazy random walk on the graph which is determined by the heat kernel of the graph and can be computed from the spectrum of the graph laplacian we characterize the random walk using the commute time that is the expected time taken for a random walk to travel between two nodes and return and show how this quantity may be computed from the laplacian spectrum using the discrete green s function our motivation is that the commute time can be anticipated to be a more robust measure of the proximity of data than the raw proximity matrix in this paper we explore two applications of the commute time the first is to develop a method for image segmentation using the eigenvector corresponding to the smallest eigenvalue of the commute time matrix we show that our commute time segmentation method has the property of enhancing the intragroup coherence while weakening intergroup coherence and is superior to the normalized cut the second application is to develop a robust multibody motion tracking method using an embedding based on the commute time our embedding procedure preserves commute time and is closely akin to kernel pca the laplacian eigenmap and the diffusion map we illustrate the results on both synthetic image sequences and real world video sequences and compare our results with several alternative methods

robust image regression based on the extended matrix variate power exponential distribution of dependent noise dealing with partial occlusion or illumination is one of the most challenging problems in image representation and classification in this problem the characterization of the representation error plays a crucial role in most current approaches the error matrix needs to be stretched into a vector and each element is assumed to be independently corrupted this ignores the dependence between the elements of error in this paper it is assumed that the error image caused by partial occlusion or illumination changes is a random matrix variate and follows the extended matrix variate power exponential distribution this has the heavy tailed regions and can be used to describe a matrix pattern of l times m dimensional observations that are not independent this paper reveals the essence of the proposed distribution it actually alleviates the correlations between pixels in an error matrix e and makes e approximately gaussian on the basis of this distribution we derive a schatten p norm based matrix regression model with l q regularization alternating direction method of multipliers is applied to solve this model to get a closed form solution in each step of the algorithm two singular value function thresholding operators are introduced in addition the extended schatten p norm is utilized to characterize the distance between the test samples and classes in the design of the classifier extensive experimental results for image reconstruction and classification with structural noise demonstrate that the proposed algorithm works much more robustly than some existing regression based methods

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

max plus a b invariant spaces and control of timed discrete event systems the concept of a b invariant subspace or controlled invariant of a linear dynamical system is extended to linear systems over the max plus semiring although this extension presents several difficulties which are similar to those encountered in the same kind of extension to linear dynamical systems over rings it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings sufficient conditions are given for computing the maximal a b invariant subspace contained in a given space and the existence of linear state feedbacks is discussed an application to the study of transportation networks which evolve according to a timetable is considered

fuzzy systems and controllers lyapunov tools for a regionwise approach this paper reviews a class of tools which can help to check the stability of fuzzy linguistic controllers which is a difficult task due to their regionwise structure the application of conventional methods when considering a fuzzy controller as a global nonlinear controller often leads to conservative stability results the use of appropriate continuous or piecewise continuous lyapunov functions allows to address the stability of nonlinear systems under fuzzy control while taking into account the local structure of the fuzzy controllers some methods are presented which enforce the global lyapunov function to decrease when leaving one subspace for another closer to the equilibrium using concentric domains and providing less conservative conditions

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

graph laplace for occluded face completion and recognition this paper proposes a spectral graph based algorithm for face image repairing which can improve the recognition performance on occluded faces the face completion algorithm proposed in this paper includes three main procedures 1 sparse representation for partially occluded face classification 2 image based data mining and 3 graph laplace gl for face image completion the novel part of the proposed framework is gl as named from graphical models and the laplace equation and can achieve a high quality repairing of damaged or occluded faces the relationship between the gl and the traditional poisson equation is proven we apply our face repairing algorithm to produce completed faces and use face recognition to evaluate the performance of the algorithm experimental results verify the effectiveness of the gl method for occluded face completion

nuclear l1 norm joint regression for face reconstruction and recognition with mixed noise abstract occlusion real disguise and illumination are still the common difficulties encountered in face recognition the sparse representation based classifier src has shown a great potential in handling pixel level sparse noise while the nuclear norm based matrix regression nmr model has been demonstrated to be powerful for dealing with the image wise structural noise both methods however might be not very effective for handling the mixed noise the structural noise plus the sparse noise in this paper we present two nuclear l 1 norm joint matrix regression nl 1 r models for face recognition with mixed noise which are derived by using map maximum a posteriori probability estimation the first model considers the mixed noise as a whole while the second model assumes the mixed noise is an additive combination of two independent componenral nts sparse noise and structuoise the proposed models can be solved by the alternating direction method of multipliers admm we validate the effectiveness of the proposed models through a series of experiments on face reconstruction and recognition

stability analysis and systematic design of fuzzy control systems abstract a method of the stability analysis and systematic design of fuzzy control systems is presented in this paper the fuzzy control system is formulated by a set of local takagi sugeno fuzzy models t s fuzzy model with aggregation the controller is designed by state feedback for each local model separately it is shown that under some conditions the controller designed can guarantee the global stability of the closed loop fuzzy system the stability of the whole closed loop fuzzy system is also proved via the piecewise lyapunov function the simulation results for the fuzzy control of a nonlinear system the ball and beam system bb system show that this method is effective