compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

improved derivative bounds of the rational quadratic bezier curves a new derivative bound of the rational quadratic bezier curves is obtained it is an improvement over the bound which was regarded as a sharp bound in a recent paper also the new bound is superior to the existing bounds given by some other authors

learning a mahalanobis metric from equivalence constraints many learning algorithms use a metric defined over the input space as a principal tool and their performance critically depends on the quality of this metric we address the problem of learning metrics using side information in the form of equivalence constraints unlike labels we demonstrate that this type of side information can sometimes be automatically obtained without the need of human intervention we show how such side information can be used to modify the representation of the data leading to improved clustering and classification specifically we present the relevant component analysis rca algorithm which is a simple and efficient algorithm for learning a mahalanobis metric we show that rca is the solution of an interesting optimization problem founded on an information theoretic basis if dimensionality reduction is allowed within rca we show that it is optimally accomplished by a version of fisher s linear discriminant that uses constraints moreover under certain gaussian assumptions rca can be viewed as a maximum likelihood estimation of the within class covariance matrix we conclude with extensive empirical evaluations of rca showing its advantage over alternative methods

the asymptotic behavior of the composition of two resolvents abstract the asymptotic behavior of the composition of two resolvents in a hilbert space is investigated connections are made between the solutions of associated monotone inclusion problems and their dual versions the applications provided include a study of an alternating minimization procedure and a new proof of von neumann s classical result on the method of alternating projections

products of finitely many resolvents of maximal monotone mappings in reflexive banach spaces we propose two algorithms for finding common zeros of finitely many maximal monotone mappings in reflexive banach spaces these algorithms are based on the bregman distance related to a well chosen convex function and improve previous results finally we mention two applications of our algorithms for solving equilibrium problems and convex feasibility problems

on the method of alternating resolvents abstract the work of hundal h hundal an alternating projection that does not converge in norm nonlinear anal 57 1 2004 35 61 has revealed that the sequence generated by the method of alternating projections converges weakly but not strongly in general in this paper we present several algorithms based on alternating resolvents of two maximal monotone operators a and b that can be used to approximate common zeros of a and b in particular we prove that the sequences generated by our algorithms converge strongly a particular case of such algorithms enables one to approximate minimum values of certain convex functionals

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

self dual regularization of monotone operators via the resolvent average we give two self dual regularizations of maximal monotone operators on hilbert spaces these regularizations and their set valued inverses are strongly monotone single valued and lipschitz with full domain moreover these regularizations graphically converge to the original monotone operator if a maximal monotone operator has nonempty zeros these self dual regularizations can be used to find its least norm solution when the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with nonempty minimizers this translates to finding the least norm minimizer

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

simultaneous semi coupled dictionary learning for matching in canonical space cross modal recognition and matching with privileged information are important challenging problems in the field of computer vision the cross modal scenario deals with matching across different modalities and needs to take care of the large variations present across and within each modality the privileged information scenario deals with the situation that all the information available during training may not be available during the testing stage and hence algorithms need to leverage the extra information from the training stage itself we show that for multi modal data either one of the above situations may arise if one modality is absent during testing here we propose a novel framework which can handle both these scenarios seamlessly with applications to matching multi modal data the proposed approach jointly uses data from the two modalities to build a canonical representation which encompasses information from both the modalities we explore four different types of canonical representations for different types of data the algorithm computes dictionaries and canonical representation for data from both the modalities such that the transformed sparse coefficients of both the modalities are equal to that of the canonical representation the sparse coefficients are finally matched using mahalanobis metric extensive experiments on different data sets involving rgbd text image and audio image data show the effectiveness of the proposed framework

on the asymptotic behavior of nonlinear semigroups and the range of accretive operators ii abstract it is known that certain problems in partial differential equations may be interpreted as initial value problems for ordinary differential equations in banach spaces when such an evolution equation is governed by an accretive operator then its solutions give rise to a nonlinear contraction semigroup in this paper we study certain aspects of the asymptotic behavior of nonlinear semigroups and or resolvents of accretive operators we also derive new results on their behavior at the origin it turns out that the behavior of a nonlinear semigroup resembles that of the resolvent of its generator both at infinity and at the origin author

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

a strongly convergent hybrid proximal method in banach spaces this paper is devoted to the study of strong convergence in inexact proximal like methods for finding zeroes of maximal monotone operators in banach spaces convergence properties of proximal point methods in banach spaces can be summarized as follows if the operator have zeroes then the sequence of iterates is bounded and all its weak accumulation points are solutions whether or not the whole sequence converges weakly to a solution and which is the relation of the weak limit with the initial iterate are key questions we present a hybrid proximal bregman projection method allowing for inexact solutions of the proximal subproblems that guarantees strong convergence of the sequence to the closest solution in the sense of the bregman distance to the initial iterate

a distributed approach toward discriminative distance metric learning distance metric learning dml is successful in discovering intrinsic relations in data however most algorithms are computationally demanding when the problem size becomes large in this paper we propose a discriminative metric learning algorithm develop a distributed scheme learning metrics on moderate sized subsets of data and aggregate the results into a global solution the technique leverages the power of parallel computation the algorithm of the aggregated dml adml scales well with the data size and can be controlled by the partition we theoretically analyze and provide bounds for the error induced by the distributed treatment we have conducted experimental evaluation of the adml both on specially designed tests and on practical image annotation tasks those tests have shown that the adml achieves the state of the art performance at only a fraction of the cost incurred by most existing methods

improved bounds on the magnitude of the derivative of rational bezier curves abstract in this paper we derive some new derivative bounds of rational bezier curves according to some existing identities and inequalities the comparison of the new bounds with some existing ones is also presented

a von neumann alternating method for finding common solutions to variational inequalities abstract by modifying von neumann s alternating projections algorithm we obtain an alternating method for solving the recently introduced common solutions to variational inequalities problem csvip for simplicity we mainly confine our attention to the two set csvip which entails finding common solutions to two unrelated variational inequalities in hilbert space

systems of structured monotone inclusions duality algorithms and applications a general primal dual splitting algorithm for solving systems of structured coupled monotone inclusions in hilbert spaces is introduced and its asymptotic behavior is analyzed each inclusion in the primal system features compositions with linear operators parallel sums and lipschitzian operators all the operators involved in this structured model are used separately in the proposed algorithm most steps of which can be executed in parallel this provides a flexible solution method applicable to a variety of problems beyond the reach of the state of the art several applications are discussed to illustrate this point

adaptive relevance matrices in learning vector quantization we propose a new matrix learning scheme to extend relevance learning vector quantization rlvq an efficient prototype based classification algorithm toward a general adaptive metric by introducing a full matrix of relevance factors in the distance measure correlations between different features and their importance for the classification scheme can be taken into account and automated and general metric adaptation takes place during training in comparison to the weighted euclidean metric used in rlvq and its variations a full matrix is more powerful to represent the internal structure of the data appropriately large margin generalization bounds can be transferred to this case leading to bounds that are independent of the input dimensionality this also holds for local metrics attached to each prototype which corresponds to piecewise quadratic decision boundaries the algorithm is tested in comparison to alternative learning vector quantization schemes using an artificial data set a benchmark multiclass problem from the vci repository and a problem from bioinformatics the recognition of splice sites for c elegans

a proximal projection method for finding zeros of set valued operators in this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set valued operators in a reflexive banach space this algorithm which we call the proximal projection method is essentially a fixed point procedure and our convergence results are based on new generalizations of the browder s demiclosedness principle we show how the proximal projection method can be applied for solving ill posed variational inequalities and convex optimization problems with data given or computable by approximations only the convergence properties of the proximal projection method we establish also allow us to prove that the proximal point method with bregman distances whose convergence was known to occur for maximal monotone operators still converges when the operator involved in it is monotone with sequentially weakly closed graph

incorporating privileged information through metric learning in some pattern analysis problems there exists expert knowledge in addition to the original data involved in the classification process the vast majority of existing approaches simply ignore such auxiliary privileged knowledge recently a new paradigm learning using privileged information was introduced in the framework of svm this approach is formulated for binary classification and as typical for many kernel based methods can scale unfavorably with the number of training examples while speeding up training methods and extensions of svm to multiclass problems are possible in this paper we present a more direct novel methodology for incorporating valuable privileged knowledge in the model construction phase primarily formulated in the framework of generalized matrix learning vector quantization this is done by changing the global metric in the input space based on distance relations revealed by the privileged information hence unlike in svm any convenient classifier can be used after such metric modification bringing more flexibility to the problem of incorporating privileged information during the training experiments demonstrate that the manipulation of an input space metric based on privileged data improves classification accuracy moreover our methods can achieve competitive performance against the svm formulations

the extragradient method for finding a common solution of a finite family of variational inequalities and a finite family of fixed point problems in the presence of computational errors abstract in a hilbert space we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors most results known in the literature establish the convergence of algorithms when computational errors are summable in the present paper the convergence of the subgradient method is established for nonsummable computational errors we show that the subgradient method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

general projective splitting methods for sums of maximal monotone operators we describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real hilbert space unlike prior methods for this problem we neither assume n 2 nor first reduce the problem to the case n 2 our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator at the cost of a single computationally simple projection step this framework gives rise to a family of splitting methods of unprecedented flexibility numerous parameters including the proximal stepsize may vary by iteration and by operator the order of operator evaluation may vary by iteration and may be either serial or parallel the analysis essentially generalizes our prior results for the case n 2 we also include a relative error criterion for approximately evaluating resolvents which was not present in our earlier work

multiplicative iterative algorithms for convex programming abstract we study multiplicative iterative algorithms for the minimization of a differentiable convex function defined on the positive orthant of r n if the objective function has compact level sets and has a locally lipschitz continuous gradient then these algorithms converge to a solution of the minimization problem moreover the convergence is nearly monotone in the sense of kullback leibler divergence special cases of these algorithms have been applied in position emission tomography and are formally related to the em algorithm for position emission tomography

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

max min distance analysis by using sequential sdp relaxation for dimension reduction we propose a new criterion for discriminative dimension reduction max min distance analysis mmda given a data set with c classes represented by homoscedastic gaussians mmda maximizes the minimum pairwise distance of these c classes in the selected low dimensional subspace thus unlike fisher s linear discriminant analysis flda and other popular discriminative dimension reduction criteria mmda duly considers the separation of all class pairs to deal with general case of data distribution we also extend mmda to kernel mmda kmmda dimension reduction via mmda kmmda leads to a nonsmooth max min optimization problem with orthonormal constraints we develop a sequential convex relaxation algorithm to solve it approximately to evaluate the effectiveness of the proposed criterion and the corresponding algorithm we conduct classification and data visualization experiments on both synthetic data and real data sets experimental results demonstrate the effectiveness of mmda kmmda associated with the proposed optimization algorithm

some remarks on weighted lupas q bezier curves abstract recently han et al 2016 introduced weighted lupas q bezier curves as a novel generalization of the classical rational bezier curves in this note we make some remarks on their article in particular we revisit the rational quadratic case for an illustrative exposition

solving coupled composite monotone inclusions by successive fejer approximations of their kuhn tucker set we propose a new class of primal dual fejer monotone algorithms for solving systems of composite monotone inclusions our construction is inspired by a framework used by eckstein and svaiter for the basic problem of finding a zero of the sum of two monotone operators at each iteration points in the graph of the monotone operators present in the model are used to construct a half space containing the kuhn tucker set associated with the system the primal dual update is then obtained via a relaxed projection of the current iterate onto this half space an important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators

a proximal point method for the variational inequality problem in banach spaces in this paper we prove well definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive banach spaces the proximal version we consider makes use of bregman functions whose original definition for finite dimensional spaces has here been properly extended to our more general framework

strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces abstract in this paper we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a banach space then we study the strong and weak convergence of the sequences

visco penalization of the sum of two monotone operators a new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a hilbert space is investigated this curve consists of the zeros of perturbed problems in which one operator is replaced with its yosida approximation and a viscosity term is added as the perturbation vanishes the curve is shown to converge to the zero of the sum that solves a particular strictly monotone variational inequality as an off spring of this result we obtain an approximating curve for finding a particular zero of the sum of several maximal monotone operators applications to convex optimization are discussed

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

entropy in linear programs this paper treats entropy constrained linear programs from modelling as well as computational aspects the optimal solutions to linear programs with one additional entropy constraint are expressed in terms of lagrange multipliers conditions for uniqueness are given sensitivity and duality are studied the newton kantorovich method is used to obtain a locally convergent iterative procedure related problems based on maximum entropy or minimum information are discussed

a douglas rachford splitting method for solving equilibrium problems abstract we propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions we prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition our algorithm is a consequence of the douglas rachford splitting applied to this auxiliary monotone inclusion connections between monotone inclusions and equilibrium problems are studied

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

backward backward splitting in hadamard spaces abstract the backward backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in hilbert spaces we generalize this setting to hadamard spaces and prove the convergence of an error tolerant version of the backward backward method

strong convergence theorems for bregman totally quasi asymptotically nonexpansive mappings in reflexive banach spaces the purpose of this article is by using the shrinking projection method introduced by takahashi kubota and takeuchi to propose an iteration algorithm for bregman total quasi f asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive banach spaces as applications we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive banach spaces the results presented in the paper improve and extend the corresponding results of reich and sabach 2010 12 suantai et al 2012 13 nilsrakoo and saejung 2011 11 qin et al 2009 5 wang et al 2011 6 su et al 2010 7 martinez yanes and xu 2006 3 and others

an alternating projection that does not converge in norm this paper proves that the method of alternating projections between two closed convex intersecting sets does not always converge in norm weak convergence was established by bregman soviet math dokl 6 1965 688 but the status of norm convergence was undetermined an explicit counterexample is provided

distance metric learning for large margin nearest neighbor classification the accuracy of k nearest neighbor knn classification depends significantly on the metric used to compute distances between different examples in this paper we show how to learn a mahalanobis distance metric for knn classification from labeled examples the mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes knn classification using euclidean distances in our approach the metric is trained with the goal that the k nearest neighbors always belong to the same class while examples from different classes are separated by a large margin as in support vector machines svms the margin criterion leads to a convex optimization based on the hinge loss unlike learning in svms however our approach requires no modification or extension for problems in multiway as opposed to binary classification in our framework the mahalanobis distance metric is obtained as the solution to a semidefinite program on several data sets of varying size and difficulty we find that metrics trained in this way lead to significant improvements in knn classification sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster we show how to learn and combine these local metrics in a globally integrated manner

bayesian neighborhood component analysis learning a distance metric in feature space potentially improves the performance of the k nearest neighbor classifier and is useful in many real world applications many metric learning ml algorithms are however based on the point estimation of a quadratic optimization problem which is time consuming susceptible to overfitting and lacks a natural mechanism to reason with parameter uncertainty a property useful especially when the training set is small and or noisy to deal with these issues we present a novel bayesian ml bml method called bayesian neighborhood component analysis nca based on the well known nca method in which the metric posterior is characterized by the local label consistency constraints of observations encoded with a similarity graph instead of independent pairwise constraints for efficient bayesian inference we explore the variational lower bound over the log likelihood of the original nca objective experiments on several publicly available data sets demonstrate that the proposed method is able to learn robust metric measures from small size data set and or from challenging training set with labels contaminated by errors the proposed method is also shown to outperform a previous pairwise constrained bml method

most maximally monotone operators have a unique zero and a super regular resolvent abstract maximally monotone operators play important roles in optimization variational analysis and differential equations finding zeros of maximally monotone operators has been a central topic in a hilbert space we show that most resolvents are super regular that most maximally monotone operators have a unique zero and that the set of strongly monotone mapping is of the first category although each strongly monotone operator has a unique zero the results are established by applying the baire category theorem to the space of nonexpansive mappings

strong convergence of block iterative outer approximation methods for convex optimization the strong convergence of a broad class of outer approximation methods for minimizing a convex function over the intersection of an arbitrary number of convex sets in a reflexive banach space is studied in a unified framework the generic outer approximation algorithm under investigation proceeds by successive minimizations over the intersection of convex supersets of the feasibility set determined in terms of the current iterate and variable blocks of constraints the convergence analysis involves flexible constraint approximation and aggregation techniques as well as relatively mild assumptions on the constituents of the problem various well known schemes are recovered as special realizations of the generic algorithm and parallel block iterative extensions of these schemes are devised within the proposed framework the case of inconsistent constraints is also considered

on the bounds of the derivative of rational bezier curves according to some identities and inequalities about the intermediate weights and control points of the de casteljau algorithm for rational bezier curve we derive a new bound of the derivative of rational bezier curves the comparison of the new bounds with some existing ones is also presented based on some previous bounds the bound presented in this paper and many numerical examples we propose a conjecture of the bound for the derivative of rational bezier curves

a monotone skew splitting model for composite monotone inclusions in duality the principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew adjoint operator an algorithmic framework is developed for solving this generic problem in a hilbert space setting new primal dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators and convergence results are established these algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration comparisons with existing methods are made and applications to composite variational problems are demonstrated

constrained empirical risk minimization framework for distance metric learning distance metric learning dml has received increasing attention in recent years in this paper we propose a constrained empirical risk minimization framework for dml this framework enriches the state of the art studies on both theoretic and algorithmic aspects theoretically we comprehensively analyze the generalization by bounding the sample and the approximation errors with respect to the best model algorithmically we carefully derive an optimal gradient descent by using nesterov s method and provide two example algorithms that utilize the logarithmic loss and the smoothed hinge loss respectively we evaluate the new framework on data classification and image retrieval experiments results show that the new framework has competitive performance compared with the representative dml algorithms including xing s method large margin nearest neighbor classifier neighborhood component analysis and regularized metric learning

fast similarity search for learned metrics we introduce a method that enables scalable similarity search for learned metrics given pairwise similarity and dissimilarity constraints between some examples we learn a mahalanobis distance function that captures the examples underlying relationships well to allow sublinear time similarity search under the learned metric we show how to encode the learned metric parameterization into randomized locality sensitive hash functions we further formulate an indirect solution that enables metric learning and hashing for vector spaces whose high dimensionality makes it infeasible to learn an explicit transformation over the feature dimensions we demonstrate the approach applied to a variety of image data sets as well as a systems data set the learned metrics improve accuracy relative to commonly used metric baselines while our hashing construction enables efficient indexing with learned distances and very large databases

on the derivative formula of a rational bezier curve at a corner in this paper we develop the exact arbitrary order derivative formula of a rational bezier curve at its corner some examples are presented to demonstrate the computation of the exact derivatives

on certain conditions for the existence of solutions of equilibrium problems the main purpose of this paper is the study of sufficient and or necessary conditions for existence of solutions of equilibrium problems we discuss some of the assumptions of the problem under which the introduced conditions are sufficient and or necessary and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem

approximation common zero of two accretive operators in banach spaces the purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in banach spaces and we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems the results of this paper are improvements and extensions of the corresponding ones announced by many others

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

a parallel splitting method for coupled monotone inclusions a parallel splitting method is proposed for solving systems of coupled monotone inclusions in hilbert spaces and its convergence is established under the assumption that solutions exist unlike existing alternating algorithms which are limited to two variables and linear coupling our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes the breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions variational problems best approximation and network flows

derivative bound estimations on rational conic bezier curves abstract in this paper we derive a bound on derivatives of rational conic bezier curves both theoretical analysis and numerical examples show that our bound is sharper than some existed ones discussions on optimal bounds for the derivatives of rational conic bezier curves are also provided

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

new minimal bounds for the derivatives of rational bezier paths and rational rectangular bezier surfaces new minimal bounds are derived for the magnitudes of the derivatives of the rational bezier paths and the rational rectangular bezier surface patches of arbitrary degree which improve previous work of this type in many cases moreover our new bounds are explicitly given by simple and closed form expressions an important advantage of the closed form expressions is that they allow us to prove that our bounds are sharp under certain well defined conditions some numerical examples highlighting the potential of the new bounds in providing improved estimates are given in an appendix

projection and proximal point methods convergence results and counterexamples recently hundal has constructed a hyperplane h a cone k and a starting point y0 in 2 such that the sequence of alternating projections pk ph n y0 n n converges weakly to some point in h k but not in norm we show how this construction results in a counterexample to norm convergence for iterates of averaged projections hence we give an a rmative answer to a question raised by reich two decades ago furthermore new counterexamples to norm convergence for iterates of rmly nonexpansive maps a la genel and lindenstrauss and for the proximal point algorithm al a g uler are provided we also present a counterexample along with some weak and norm convergence results for the new framework of string averaging projection methods introduced by censor et al extensions to banach spaces and the situation for the hilbert ball are discussed as well 2003 elsevier ltd all rights reserved msc 47h09 47j25 90c25

asymptotic behavior of resolvents of coaccretive operators in the hilbert ball abstract we study the resolvents of coaccretive operators in the hilbert ball with special emphasis on the asymptotic behavior of their compositions and metric convex combinations we consider the case where the given coaccretive operators share a common fixed point inside the ball as well as the case where they share a common sink point on its boundary we establish weak convergence in the former case and strong convergence in the latter we also present two related convergence results for a continuous implicit scheme

a family of projective splitting methods for the sum of two maximal monotone operators a splitting method for two monotone operators a and b is an algorithm that attempts to converge to a zero of the sum a b by solving a sequence of subproblems each of which involves only the operator a or only the operator b prior algorithms of this type can all in essence be categorized into three main classes the douglas peaceman rachford class the forward backward class and the little used double backward class through a certain extended solution set in a product space we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in hilbert space our algorithms are essentially standard projection methods using splitting decomposition to construct separators we prove convergence through fejer monotonicity techniques but showing fejer convergence of a different sequence to a different set than in earlier splitting methods our projective algorithms converge under more general conditions than prior splitting methods allowing the proximal parameter to vary from iteration to iteration and even from operator to operator while retaining convergence for essentially arbitrary pairs of operators the new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases

a contraction proximal point algorithm with two monotone operators abstract it is a known fact that the method of alternating projections introduced long ago by von neumann fails to converge strongly for two arbitrary nonempty closed and convex subsets of a real hilbert space in this paper a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved if the two operators are subdifferentials of indicator functions this new algorithm coincides with the old method of alternating projections several other important algorithms such as the contraction proximal point algorithm occur as special cases of our algorithm hence our main results generalize and unify many results that occur in the literature