incremental support vector learning for ordinal regression support vector ordinal regression svor is a popular method to tackle ordinal regression problems however until now there were no effective algorithms proposed to address incremental svor learning due to the complicated formulations of svor recently an interesting accurate on line algorithm was proposed for training nu support vector classification nu svc which can handle a quadratic formulation with a pair of equality constraints in this paper we first present a modified svor formulation based on a sum of margins strategy the formulation has multiple constraints and each constraint includes a mixture of an equality and an inequality then we extend the accurate on line nu svc algorithm to the modified formulation and propose an effective incremental svor algorithm the algorithm can handle a quadratic formulation with multiple constraints where each constraint is constituted of an equality and an inequality more importantly it tackles the conflicts between the equality and inequality constraints we also provide the finite convergence analysis for the algorithm numerical experiments on the several benchmark and real world data sets show that the incremental algorithm can converge to the optimal solution in a finite number of steps and is faster than the existing batch and incremental svor algorithms meanwhile the modified formulation has better accuracy than the existing incremental svor algorithm and is as accurate as the sum of margins based formulation of shashua and levin

modularity of relatively rational vertex algebras and fusion rules of principal affine w algebras we study modularity of the characters of a vertex super algebra equipped with a family of conformal structures along the way we introduce the notions of rationality and cofiniteness relative to such a family we apply the results to determine modular transformations of trace functions on admissible modules over affine kac moody algebras and via brst reduction trace functions on minimal series representations of principal affine w algebras

a new system of generalized nonlinear quasi variational like inclusions with a monotone operators in banach spaces in this paper we study some properties of a monotone operator in banach space give the notion of proximal mapping associated with the a monotone operator and show its lipschitz continuity we also consider a new system of generalized nonlinear quasi variational like inclusions with a monotone operators in banach spaces and constructed a new iterative algorithm for solving the system of generalized nonlinear quasi variational like inclusions in banach spaces under suitable assumptions we prove the convergence of the iterative sequences generated by the algorithm

generalized strongly nonlinear quasivariational inequalities abstract in this paper we prove an existence theorem of solutions and develop the algorithms of approximating solutions for a generalized strongly nonlinear variational inequality and a generalized strongly nonlinear complementarity problem an algorithm for finding the approximate solution of generalized strongly nonlinear quasivariational inequalities and generalized strongly nonlinear quasicomplementarity problems is also given as special cases these results include the corresponding results of fang and peterson noor siddiqi and ansari and ding

modularity in orbifold theory for vertex operator superalgebras this paper is about the orbifold theory for vertex operator superalgebras given a vertex operator superalgebra v and a finite automorphism group g of v we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in g under the c2 cofinite condition we also establish that these functions afford a representation of the full modular group if v is c2 cofinite and g rational for any g g

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

semi supervised dimension reduction using trace ratio criterion in this brief we address the trace ratio tr problem for semi supervised dimension reduction we first reformulate the objective function of the recent work semi supervised discriminant analysis sda in a tr form we also observe that in sda the low dimensional data representation f is constrained to be in the linear subspace spanned by the training data matrix x i e f xt w in order to relax this hard constraint we introduce a flexible regularizer f xt w 2 which models the regression residual into the reformulated objective function with such relaxation our method referred to as tr based flexible sda tr fsda can better cope with data sampled from a certain type of nonlinear manifold that is somewhat close to a linear subspace in order to address the non trivial optimization problem in tr fsda we further develop an iterative algorithm to simultaneously solve for the low dimensional data representation f and the projection matrix w moreover we theoretically prove that our iterative algorithm converges to the optimum based on the newton raphson method the experiments on two face databases one shape image database and one webpage database demonstrate that tr fsda outperforms the existing semi supervised dimension reduction methods

discriminative semi supervised feature selection via manifold regularization feature selection has attracted a huge amount of interest in both research and application communities of data mining we consider the problem of semi supervised feature selection where we are given a small amount of labeled examples and a large amount of unlabeled examples since a small number of labeled samples are usually insufficient for identifying the relevant features the critical problem arising from semi supervised feature selection is how to take advantage of the information underneath the unlabeled data to address this problem we propose a novel discriminative semi supervised feature selection method based on the idea of manifold regularization the proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data in comparison with previous semi supervised feature selection algorithms our proposed semi supervised feature selection method is an embedded feature selection method and is able to find more discriminative features we formulate the proposed feature selection method into a convex concave optimization problem where the saddle point corresponds to the optimal solution to find the optimal solution the level method a fairly recent optimization method is employed we also present a theoretic proof of the convergence rate for the application of the level method to our problem empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi supervised feature selection method

multiclass posterior probability support vector machines tao et al have recently proposed the posterior probability support vector machine ppsvm which uses soft labels derived from estimated posterior probabilities to be more robust to noise and outliers tao et al s model uses a window based density estimator to calculate the posterior probabilities and is a binary classifier we propose a neighbor based density estimator and also extend the model to the multiclass case our bias variance analysis shows that the decrease in error by ppsvm is due to a decrease in bias on 20 benchmark data sets we observe that ppsvm obtains accuracy results that are higher or comparable to those of canonical svm using significantly fewer support vectors

auxiliary principle for generalized mixed variational like inequalities abstract in this paper we use the auxiliary principle technique to prove the existence of a solution of some new classes of variational inequalities which are called the generalized mixed variational like inequalities this technique is used to analyze an iterative algorithm several special cases which can be obtained from our main result are also discussed

strongly nonlinear quasivariational inequalities abstract in this paper we develop the algorithms for finding the approximate solution of a strongly nonlinear quasivariational inequality and a strongly nonlinear quasicomplementarity problem and we include as special cases known results in this area we also observe that these two problems are equivalent if the convex set involved in the formulation of these problem is a convex cone

modular linear differential equations of fourth order and minimal w algebras abstract a characterization of the minimal w algebras associated with the deligne exceptional series at level h 6 is obtained by using one parameter family of modular linear differential equations of order 4 in particular the characters of the ramond twisted modules of minimal w algebras related to the deligne exceptional series satisfy one of these differential equations in order to obtain the characterization the differential equations in the one parameter family which have solutions of cft type are classified whose solutions are explicitly described

torus n point functions for mathbb r graded vertex operator superalgebras and continuous fermion orbifolds we consider genus one n point functions for a vertex operator superalgebra with a real grading we compute all n point functions for rank one and rank two fermion vertex operator superalgebras in the rank two fermion case we obtain all orbifold n point functions for a twisted module associated with a continuous automorphism generated by a heisenberg bosonic state the modular properties of these orbifold n point functions are given and we describe a generalization of fay s trisecant identity for elliptic functions

modular invariance for twisted modules over a vertex operator superalgebra the purpose of this paper is to generalize zhu s theorem about characters of modules over a vertex operator algebra graded by integer conformal weights to the setting of a vertex operator superalgebra graded by rational conformal weights to recover sl 2 mathbb z invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones it also turns out to be necessary in describing the space of conformal blocks in the supersymmetric case to include certain odd traces on modules alongside traces and supertraces we prove that the set of supertrace functions thus supplemented spans a finite dimensional sl 2 mathbb z invariant space we close the paper with several examples

learning to classify ordinal data the data replication method classification of ordinal data is one of the most important tasks of relation learning this paper introduces a new machine learning paradigm specifically intended for classification problems where the classes have a natural order the technique reduces the problem of classifying ordered classes to the standard two class problem the introduced method is then mapped into support vector machines and neural networks generalization bounds of the proposed ordinal classifier are also provided an experimental study with artificial and real data sets including an application to gene expression analysis verifies the usefulness of the proposed approach

block quantized support vector ordinal regression support vector ordinal regression svor is a recently proposed ordinal regression or algorithm despite its theoretical and empirical success the method has one major bottleneck which is the high computational complexity in this brief we propose a both practical and theoretical guaranteed algorithm block quantized support vector ordinal regression bqsvor where we approximate the kernel matrix k with k tilde that is composed of k 2 constant blocks we provide detailed theoretical justification on the approximation accuracy of bqsvor moreover we prove theoretically that the or problem with the block quantized kernel matrix k tilde could be solved by first separating the data samples in the training set into k clusters with kernel k means and then performing svor on the k cluster representatives hence the algorithm leads to an optimization problem that scales only with the number of clusters instead of the data set size finally experiments on several real world data sets support the previous analysis and demonstrate that bqsvor improves the speed of svor significantly with guaranteed accuracy

the benefits of modeling slack variables in svms in this letter we explore the idea of modeling slack variables in support vector machine svm approaches the study is motivated by svm which models the slacks through a smooth correcting function that is determined by additional privileged information about the training examples not available in the test phase we take a closer look at the meaning and consequences of smooth modeling of slacks as opposed to determining them in an unconstrained manner through the svm optimization program to better understand this difference we only allow the determination and modeling of slack values on the same information that is using the same training input in the original input space we also explore whether it is possible to improve classification performance by combining in a convex combination the original svm slacks with the modeled ones we show experimentally that this approach not only leads to improved generalization performance but also yields more compact lower complexity models finally we extend this idea to the context of ordinal regression where a natural order among the classes exists the experimental results confirm principal findings from the binary case

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

multiple ordinal regression by maximizing the sum of margins human preferences are usually measured using ordinal variables a system whose goal is to estimate the preferences of humans and their underlying decision mechanisms requires to learn the ordering of any given sample set we consider the solution of this ordinal regression problem using a support vector machine algorithm specifically the goal is to learn a set of classifiers with common direction vectors and different biases correctly separating the ordered classes current algorithms are either required to solve a quadratic optimization problem which is computationally expensive or based on maximizing the minimum margin i e a fixed margin strategy between a set of hyperplanes which biases the solution to the closest margin another drawback of these strategies is that they are limited to order the classes using a single ranking variable e g perceived length in this paper we define a multiple ordinal regression algorithm based on maximizing the sum of the margins between every consecutive class with respect to one or more rankings e g perceived length and weight we provide derivations of an efficient easy to implement iterative solution using a sequential minimal optimization procedure we demonstrate the accuracy of our solutions in several data sets in addition we provide a key application of our algorithms in estimating human subjects ordinal classification of attribute associations to object categories we show that these ordinal associations perform better than the binary one typically employed in the literature

perturbed ishikawa type iterative algorithm for generalized quasivariational inclusions in this paper a perturbed ishikawa type iterative algorithm of solving a new class of generalized nonlinear implicit quasivariational inclusions are suggested and analysed first an existence theorem of solutions for generalized nonlinear implicit quasivariational inclusion is proved under suitable assumptions then the convergence of the perturbed ishikawa type iterative sequences generated by the new algorithm is proved as special cases some known results are also discussed

maximum margin clustering made practical motivated by the success of large margin methods in supervised learning maximum margin clustering mmc is a recent approach that aims at extending large margin methods to unsupervised learning however its optimization problem is nonconvex and existing mmc methods all rely on reformulating and relaxing the nonconvex optimization problem as semidefinite programs sdp though sdp is convex and standard solvers are available they are computationally very expensive and only small data sets can be handled to make mmc more practical we avoid sdp relaxations and propose in this paper an efficient approach that performs alternating optimization directly on the original nonconvex problem a key step to avoid premature convergence in the resultant iterative procedure is to change the loss function from the hinge loss to the laplacian square loss so that overconfident predictions are penalized experiments on a number of synthetic and real world data sets demonstrate that the proposed approach is more accurate much faster hundreds to tens of thousands of times faster and can handle data sets that are hundreds of times larger than the largest data set reported in the mmc literature

a maximum margin approach for semisupervised ordinal regression clustering ordinal regression or is generally defined as the task where the input samples are ranked on an ordinal scale or has found a wide variety of applications and a great deal of work has been done on it however most of the existing work focuses on supervised semisupervised or classification and the semisupervised or clustering problems have not been explicitly addressed in real world or applications labeling a large number of training samples is usually time consuming and costly and instead a set of unlabeled samples can be utilized to set up the or model moreover although the sample labels are unavailable we can sometimes get the relative ranking information of the unlabeled samples this sample ranking information can be utilized to refine the or model hence how to build an or model on the unlabeled samples and incorporate the sample ranking information into the process of improving the clustering accuracy remains a key challenge for or applications in this paper we consider the semisupervised or clustering problems with sample ranking constraints which give the relative ranking information of the unlabeled samples and put forward a maximum margin approach for semisupervised or clustering text m 2 sorc on one hand text m 2 sorc seeks a set of parallel hyperplanes to partition the unlabeled samples into clusters on the other hand a loss function is put forward to incorporate the sample ranking information into the clustering process as a result the optimization function of text m 2 sorc is formulated to maximize the margins of the closest neighboring clusters and meanwhile minimize the loss associated with the sample ranking constraints extensive experiments on or data sets show that the proposed text m 2 sorc method outperforms the traditional semisupervised clustering methods considered

ordinal neural networks without iterative tuning ordinal regression or is an important branch of supervised learning in between the multiclass classification and regression in this paper the traditional classification scheme of neural network is adapted to learn ordinal ranks the model proposed imposes monotonicity constraints on the weights connecting the hidden layer with the output layer to do so the weights are transcribed using padding variables this reformulation leads to the so called inequality constrained least squares icls problem its numerical solution can be obtained by several iterative methods for example trust region or line search algorithms in this proposal the optimum is determined analytically according to the closed form solution of the icls problem estimated from the karush kuhn tucker conditions furthermore following the guidelines of the extreme learning machine framework the weights connecting the input and the hidden layers are randomly generated so the final model estimates all its parameters without iterative tuning the model proposed achieves competitive performance compared with the state of the art neural networks methods for or

transductive ordinal regression ordinal regression is commonly formulated as a multiclass problem with ordinal constraints the challenge of designing accurate classifiers for ordinal regression generally increases with the number of classes involved due to the large number of labeled patterns that are needed the availability of ordinal class labels however is often costly to calibrate or difficult to obtain unlabeled patterns on the other hand often exist in much greater abundance and are freely available to take benefits from the abundance of unlabeled patterns we present a novel transductive learning paradigm for ordinal regression in this paper namely transductive ordinal regression tor the key challenge of this paper lies in the precise estimation of both the ordinal class label of the unlabeled data and the decision functions of the ordinal classes simultaneously the core elements of the proposed tor include an objective function that caters to several commonly used loss functions casted in transductive settings for general ordinal regression a label swapping scheme that facilitates a strictly monotonic decrease in the objective function value is also introduced extensive numerical studies on commonly used benchmark datasets including the real world sentiment prediction problem are then presented to showcase the characteristics and efficacies of the proposed tor further comparisons to recent state of the art ordinal regression methods demonstrate the introduced transductive learning paradigm for ordinal regression led to the robust and improved performance

negative correlation ensemble learning for ordinal regression in this paper two neural network threshold ensemble models are proposed for ordinal regression problems for the first ensemble method the thresholds are fixed a priori and are not modified during training the second one considers the thresholds of each member of the ensemble as free parameters allowing their modification during the training process this is achieved through a reformulation of these tunable thresholds which avoids the constraints they must fulfill for the ordinal regression problem during training diversity exists in different projections generated by each member is taken into account for the parameter updating this diversity is promoted in an explicit way using a diversity encouraging error function extending the well known negative correlation learning framework to the area of ordinal regression and inheriting many of its good properties experimental results demonstrate that the proposed algorithms can achieve competitive generalization performance when considering four ordinal regression metrics

parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings a new class of parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings is introduced by applying resolvent operator technique of h maximal monotone mapping and the property of fixed point set of set valued contractive mappings the behavior and sensitivity analysis of the solution set of the parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings are studied the continuity and lipschitz continuity of the solution set with respect to the parameter are proved under suitable assumptions our approach and results are new and improve unify and extend previous many known results in this field

iterative algorithms for a new system of nonlinear variational inclusions with a η accretive mappings in banach spaces in this paper we introduce and study a new system of nonlinear variational inclusions with a h accretive mappings in banach spaces by using the resolvent operator associated with a h accretive mappings we construct some new iterative algorithms for approximating the solution of this system of variational inclusions we also prove the existence of solutions and the convergence of the sequences generated by the algorithm in banach spaces the results presented in this paper extend and improve some known results in the literature

modularity of n point functions and theta functions for vertex operator superalgebras abstract the modularity of the n point trace functions and theta functions associated with vertex operator superalgebras are established

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

support vector ordinal regression in this letter we propose two new support vector approaches for ordinal regression which optimize multiple thresholds to define parallel discriminant hyperplanes for the ordinal scales both approaches guarantee that the thresholds are properly ordered at the optimal solution the size of these optimization problems is linear in the number of training samples the sequential minimal optimization algorithm is adapted for the resulting optimization problems it is extremely easy to implement and scales efficiently as a quadratic function of the number of examples the results of numerical experiments on some benchmark and real world data sets including applications of ordinal regression to information retrieval verify the usefulness of these approaches

reduction from cost sensitive ordinal ranking to weighted binary classification we present a reduction framework from ordinal ranking to binary classification the framework consists of three steps extracting extended examples from the original examples learning a binary classifier on the extended examples with any binary classification algorithm and constructing a ranker from the binary classifier based on the framework we show that a weighted 0 1 loss of the binary classifier upper bounds the mislabeling cost of the ranker both error wise and regret wise our framework allows not only the design of good ordinal ranking algorithms based on well tuned binary classification approaches but also the derivation of new generalization bounds for ordinal ranking from known bounds for binary classification in addition our framework unifies many existing ordinal ranking algorithms such as perceptron ranking and support vector ordinal regression when compared empirically on benchmark data sets some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms in addition the newly designed algorithms lead to better cost sensitive ordinal ranking performance as well as improved listwise ranking performance

general algorithm of solutions for nonlinear variational inequalities in banach space abstract in this paper we shall study a class of general nonlinear variational inequalities in reflexive banach spaces by applying a minimax inequality obtained by the author some existence uniqueness theorems of solutions for the general nonlinear variational inequalities are proved next by applying the auxiliary problem technique we suggest an innovative iterative algorithm to compute the approximate solutions of the general nonlinear variational inequality finally the convergence criteria is also discussed

a study on smo type decomposition methods for support vector machines decomposition methods are currently one of the major methods for training support vector machines they vary mainly according to different working set selections existing implementations and analysis usually consider some specific selection rules this paper studies sequential minimal optimization type decomposition methods under a general and flexible way of choosing the two element working set the main results include 1 a simple asymptotic convergence proof 2 a general explanation of the shrinking and caching techniques and 3 the linear convergence of the methods extensions to some support vector machine variants are also discussed

support vector machinery for infinite ensemble learning ensemble learning algorithms such as boosting can achieve better performance by averaging over the predictions of some base hypotheses nevertheless most existing algorithms are limited to combining only a finite number of hypotheses and the generated ensemble is usually sparse thus it is not clear whether we should construct an ensemble classifier with a larger or even an infinite number of hypotheses in addition constructing an infinite ensemble itself is a challenging task in this paper we formulate an infinite ensemble learning framework based on the support vector machine svm the framework can output an infinite and nonsparse ensemble through embedding infinitely many hypotheses into an svm kernel we use the framework to derive two novel kernels the stump kernel and the perceptron kernel the stump kernel embodies infinitely many decision stumps and the perceptron kernel embodies infinitely many perceptrons we also show that the laplacian radial basis function kernel embodies infinitely many decision trees and can thus be explained through infinite ensemble learning experimental results show that svm with these kernels is superior to boosting with the same base hypothesis set in addition svm with the stump kernel or the perceptron kernel performs similarly to svm with the gaussian radial basis function kernel but enjoys the benefit of faster parameter selection these properties make the novel kernels favorable choices in practice

existence and algorithm of solutions for generalized mixed implicit quasi variational inequalities1this project was supported by the national science foundation of people s republic china 1 in this paper by applying the auxiliary variational principle technique some existence theorems of solutions for a class of generalized mixed implicit quasi variational inequalities are proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria are also given as special cases of these results the open problem put forward by noor is answered positively and some results of existence and algorithm of solutions for generalized implicit quasi complementarity problems are also obtained our results are new and generalize a number of known results of mixed variational inequalities mixed quasi variational inequalities and quasi complementary problems involving single valued and set valued mappings in recent literatures