pairwise sparsity preserving embedding for unsupervised subspace learning and classification two novel unsupervised dimensionality reduction techniques termed sparse distance preserving embedding sdpe and sparse proximity preserving embedding sppe are proposed for feature extraction and classification sdpe and sppe perform in the clean data space recovered by sparse representation and enhanced euclidean distances over noise removed data are employed to measure pairwise similarities of points in extracting informative features sdpe and sppe aim at preserving pairwise similarities between data points in addition to preserving the sparse characteristics this paper calculates the sparsest representation of all vectors jointly by a convex optimization the sparsest codes enable certain local information of data to be preserved and can endow sdpe and sppe a natural discriminating power adaptive neighborhood and robust characteristic against noise and errors in delivering low dimensional embeddings we also mathematically show sdpe and sppe can be effectively extended for discriminant learning in a supervised manner the validity of sdpe and sppe is examined by extensive simulations comparison with other related state of the art unsupervised algorithms show that promising results are delivered by our techniques

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

class label versus sample label based cca when correlating the samples with the corresponding class labels canonical correlation analysis cca can be used for supervised feature extraction and subsequent classification intuitively different encoding modes for class label can result in different classification performances however actually when the samples in each class share a common class label as in usual cases a unified formulation of cca is not only derived naturally but also more importantly from it we can get some insight into the shortcoming of the existing feature extraction using cca for sequent classification the existing encodings for class label fail to reflect the difference among the samples such as in central region of class and those in mixture overlapping region among classes consequently resulting in its equivalence to the traditional linear discriminant analysis lda for some commonly used class label encodings to reflect such a difference between the samples we elaborately design an independent soft label for each sample of each class rather than a common label for all the samples of the same class a purpose of doing so is to try to promote cca classification performance the experiments show that this soft label based cca is better than or comparable to the original cca lda in terms of the recognition performance

constrained large margin local projection algorithms and extensions for multimodal dimensionality reduction a constrained large margin local projection cmlp technique for multimodal dimensionality reduction is proposed we elaborate the criterion of cmlp from a pairwise constrained marginal perspective four effective cmlp solution schemes are presented and the corresponding comparative analyses are given an equivalent weighted least squares formulation for cmlp is also detailed cmlp is originated from the criterion of locality preserving projections lpp but cmlp offers a number of attractive advantages over lpp to keep the intrinsic proximity relations of inter class and intra class similarity pairs the localized pairwise cannot link and must link constraints are applied to specify the types of those neighboring pairs by utilizing the cmlp criterion margins between inter and intra class clusters are significantly enlarged as a result multimodal distributions are effectively preserved to further optimize the cmlp criterion one feasible improvement strategy is described with kernel methods we present the kernelized extensions of our approaches mathematical comparisons and analyses between this work and the related works are also detailed extensive simulations including multivariate manifold visualization and classification on the benchmark ucl orl yale umist mit cbcl and usps datasets are conducted to verify the efficiency of our techniques the presented results reveal that our methods are highly competitive with and even outperform some widely used state of the art algorithms

precedence theorems and dynamic programming for the single machine weighted tardiness problem abstract we tackle precedence constrained sequencing on a single machine in order to minimize total weighted tardiness classic dynamic programming dp methods for this problem are limited in performance due to excessive memory requirements particularly when the precedence network is not sufficiently dense over the last decades a number of precedence theorems have been proposed which distinguish dominant precedence constraints for a job pool that is initially without precedence relation in this paper we connect and extend the findings of the foregoing two strands of literature we develop a framework for applying the precedence theorems to the precedence constrained problem to tighten the search space and we propose an exact dp algorithm that utilizes a new efficient memory management technique our procedure outperforms the state of the art algorithm for instances with medium to high network density we also empirically verify the computational gain of using different sets of precedence theorems

cellular control of manufacturing systems abstract the scope of this paper is twofold firstly we propose augmented lagrangian based decomposition techniques for solving scheduling problems in manufacturing systems and then we show that the resulting decomposition of the mathematical problem lends itself to control systems with a topological cellular structure and that it can be espoused very naturally with an object oriented programming approach

orthogonal neighborhood preserving discriminant analysis for face recognition in this paper we propose a new linear subspace analysis algorithm called orthogonal neighborhood preserving discriminant analysis onpda given a set of data points in the ambient space a weight matrix is firstly built which describes the relationship between the data points then optimal between class scatter matrix and within class scatter matrix are defined such that the neighborhood structure can be preserved in order to improve the discriminating power a new method is presented for orthogonalizing the basis eigenvectors we evaluate the performance of the proposed algorithm for face recognition with the use of different databases consistent and promising results demonstrate the effectiveness of our algorithm

sparsity preserving projections with applications to face recognition dimensionality reduction methods drs have commonly been used as a principled way to understand the high dimensional data such as face images in this paper we propose a new unsupervised dr method called sparsity preserving projections spp unlike many existing techniques such as local preserving projection lpp and neighborhood preserving embedding npe where local neighborhood information is preserved during the dr procedure spp aims to preserve the sparse reconstructive relationship of the data which is achieved by minimizing a l1 regularization related objective function the obtained projections are invariant to rotations rescalings and translations of the data and more importantly they contain natural discriminating information even if no class labels are provided moreover spp chooses its neighborhood automatically and hence can be more conveniently used in practice compared to lpp and npe the feasibility and effectiveness of the proposed method is verified on three popular face databases yale ar and extended yale b with promising results

stronger lagrangian bounds by use of slack variables applications to machine scheduling problems lagrangian relaxation is a powerful bounding technique that has been applied successfully to manynp hard combinatorial optimization problems the basic idea is to see annp hard problem as an easy to solve problem complicated by a number of nasty side constraints we show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds the trick is widely applicable but we focus on a broad class of machine scheduling problems for which it is particularly useful we provide promising computational results for three problems belonging to this class for which lagrangian bounds have appeared in the literature the single machine problem of minimizing total weighted completion time subject to precedence constraints the two machine flow shop problem of minimizing total completion time and the single machine problem of minimizing total weighted tardiness

graph optimization for dimensionality reduction with sparsity constraints graph based dimensionality reduction dr methods play an increasingly important role in many machine learning and pattern recognition applications in this paper we propose a novel graph based learning scheme to conduct graph optimization for dimensionality reduction with sparsity constraints godrsc different from most of graph based dr methods where graphs are generally constructed in advance godrsc aims to simultaneously seek a graph and a projection matrix preserving such a graph in one unified framework resulting in an automatically updated graph moreover by applying an l 1 regularizer a sparse graph is achieved which models the locality structure of data and contains natural discriminating information finally extensive experiments on several publicly available uci and face databases verify the feasibility and effectiveness of the proposed method

an exact algorithm for the precedence constrained single machine scheduling problem abstract this study proposes an efficient exact algorithm for the precedence constrained single machine scheduling problem to minimize total job completion cost where machine idle time is forbidden the proposed algorithm is based on the ssdp successive sublimation dynamic programming method and is an extension of the authors previous algorithms for the problem without precedence constraints in this method a lower bound is computed by solving a lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence constrained total weighted tardiness and total weighted earliness tardiness problems and most instances with 100 jobs of the former problem

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

a restricted lagrangean approach to the traveling salesman problem we describe an algorithm for the asymmetric traveling salesman problem tsp using a new restricted lagrangean relaxation based on the assignment problem ap the lagrange multipliers are constrained so as to guarantee the continued optimality of the initial ap solution thus eliminating the need for repeatedly solving ap in the process of computing multipliers we give several polynomially bounded procedures for generating valid inequalities and taking them into the lagrangean function with a positive multiplier without violating the constraints so as to strengthen the current lower bound upper bounds are generated by a fast tour building heuristic when the bound strengthening techniques are exhausted without matching the upper with the lower bound we branch by using two different rules according to the situation the usual subtour breaking disjunction and a new disjunction based on conditional bounds we discuss computational experience on 120 randomly generated asymmetric tsp s with up to 325 cities the maximum time used for any single problem being 82 seconds this is a considerable improvement upon earlier methods though the algorithm discussed here is for the asymmetric tsp the approach can be adapted to the symmetric tsp by using the 2 matching problem instead of ap

weighted norm inequalities for general operators on monotone functions in this paper we characterize the weights w v for which s f p w c f q v for f nonincreasing where s f 0 x y f y dy

face recognition using discriminant locality preserving projections based on maximum margin criterion in this paper we propose a new discriminant locality preserving projections based on maximum margin criterion dlpp mmc dlpp mmc seeks to maximize the difference rather than the ratio between the locality preserving between class scatter and locality preserving within class scatter dlpp mmc is theoretically elegant and can derive its discriminant vectors from both the range of the locality preserving between class scatter and the range space of locality preserving within class scatter dlpp mmc can also derive its discriminant vectors from the null space of locality preserving within class scatter when the parameter of dlpp mmc approaches experiments on the orl yale feret and pie face databases show the effectiveness of the proposed dlpp mmc

trace ratio problem revisited dimensionality reduction is an important issue in many machine learning and pattern recognition applications and the trace ratio tr problem is an optimization problem involved in many dimensionality reduction algorithms conventionally the solution is approximated via generalized eigenvalue decomposition due to the difficulty of the original problem however prior works have indicated that it is more reasonable to solve it directly than via the conventional way in this brief we propose a theoretical overview of the global optimum solution to the tr problem via the equivalent trace difference problem eigenvalue perturbation theory is introduced to derive an efficient algorithm based on the newton raphson method theoretical issues on the convergence and efficiency of our algorithm compared with prior literature are proposed and are further supported by extensive empirical results

a lagrangean based branch and bound algorithm for single machine sequencing with precedence constraints to minimize total weighted completion time the single machine sequencing problem is considered in which there are precedence constraints on the jobs the objective is to minimize the sum of weighted completion times a lower bound is obtained by successively performing a lagrangean relaxation of appropriate constraints each lagrange multiplier is chosen to provide the maximum increment to the lower bound subject to retaining the nonnegativity of the coefficients of the variables when no further suitable constraints can be introduced into the lagrangean function the variables having zero cost coefficient are used to obtain a feasible sequence which provides an upper bound the gap between the lower and upper bound is reduced by removing some constraints from the lagrangean function and replacing them with others this lower bounding procedure is used in a branch and bound algorithm computational results indicate that the algorithm can satisfactorily solve problems with up to 100 jobs

constraint score a new filter method for feature selection with pairwise constraints feature selection is an important preprocessing step in mining high dimensional data generally supervised feature selection methods with supervision information are superior to unsupervised ones without supervision information in the literature nearly all existing supervised feature selection methods use class labels as supervision information in this paper we propose to use another form of supervision information for feature selection i e pairwise constraints which specifies whether a pair of data samples belong to the same class must link constraints or different classes cannot link constraints pairwise constraints arise naturally in many tasks and are more practical and inexpensive than class labels this topic has not yet been addressed in feature selection research we call our pairwise constraints guided feature selection algorithm as constraint score and compare it with the well known fisher score and laplacian score algorithms experiments are carried out on several high dimensional uci and face data sets experimental results show that with very few pairwise constraints constraint score achieves similar or even higher performance than fisher score with full class labels on the whole training data and significantly outperforms laplacian score

twin kernel embedding in most existing dimensionality reduction algorithms the main objective is to preserve relational structure among objects of the input space in a low dimensional embedding space this is achieved by minimizing the inconsistency between two similarity dissimilarity measures one for the input data and the other for the embedded data via a separate matching objective function based on this idea a new dimensionality reduction method called twin kernel embedding tke is proposed tke addresses the problem of visualizing non vectorial data that is difficult for conventional methods in practice due to the lack of efficient vectorial representation tke solves this problem by minimizing the inconsistency between the similarity measures captured respectively by their kernel gram matrices in the two spaces in the implementation by optimizing a nonlinear objective function using the gradient descent algorithm a local minimum can be reached the results obtained include both the optimal similarity preserving embedding and the appropriate values for the hyperparameters of the kernel experimental evaluation on real non vectorial datasets confirmed the effectiveness of tke tke can be applied to other types of data beyond those mentioned in this paper whenever suitable measures of similarity dissimilarity can be defined on the input data

non linear metric learning using pairwise similarity and dissimilarity constraints and the geometrical structure of data the problem of clustering with side information has received much recent attention and metric learning has been considered as a powerful approach to this problem until now various metric learning methods have been proposed for semi supervised clustering although some of the existing methods can use both positive must link and negative cannot link constraints they are usually limited to learning a linear transformation i e finding a global mahalanobis metric in this paper we propose a framework for learning linear and non linear transformations efficiently we use both positive and negative constraints and also the intrinsic topological structure of data we formulate our metric learning method as an appropriate optimization problem and find the global optimum of this problem the proposed non linear method can be considered as an efficient kernel learning method that yields an explicit non linear transformation and thus shows out of sample generalization ability experimental results on synthetic and real world data sets show the effectiveness of our metric learning method for semi supervised clustering tasks

joint low rank and sparse principal feature coding for enhanced robust representation and visual classification recovering low rank and sparse subspaces jointly for enhanced robust representation and classification is discussed technically we first propose a transductive low rank and sparse principal feature coding lspfc formulation that decomposes given data into a component part that encodes low rank sparse principal features and a noise fitting error part to well handle the outside data we then present an inductive lspfc i lspfc i lspfc incorporates embedded low rank and sparse principal features by a projection into one problem for direct minimization so that the projection can effectively map both inside and outside data into the underlying subspaces to learn more powerful and informative features for representation to ensure that the learned features by i lspfc are optimal for classification we further combine the classification error with the feature coding error to form a unified model discriminative lspfc d lspfc to boost performance the model of d lspfc seamlessly integrates feature coding and discriminative classification so the representation and classification powers can be enhanced the proposed approaches are more general and several recent existing low rank or sparse coding algorithms can be embedded into our problems as special cases visual and numerical results demonstrate the effectiveness of our methods for representation and classification

a general maximal operator and the _ condition 1 let p v be two measures on r and let there be given for each cube q er afunction q supported in q we consider the maximal operator mf x sup ff q dv where the sup is extended over all cubes centered at x and obtain theorem 1 therearrangement inequality m y4 0 3 f dt here g denotes the non increasing rearrangement of g with respect to the measure a and is a nonincreas ing function given in terms of p v q from this one easily sees that m and tnus tne finiteness of this integral i e g l p i

kpca plus lda a complete kernel fisher discriminant framework for feature extraction and recognition this paper examines the theory of kernel fisher discriminant analysis kfd in a hilbert space and develops a two phase kfd framework i e kernel principal component analysis kpca plus fisher linear discriminant analysis lda this framework provides novel insights into the nature of kfd based on this framework the authors propose a complete kernel fisher discriminant analysis ckfd algorithm ckfd can be used to carry out discriminant analysis in double discriminant subspaces the fact that it can make full use of two kinds of discriminant information regular and irregular makes ckfd a more powerful discriminator the proposed algorithm was tested and evaluated using the feret face database and the cenparmi handwritten numeral database the experimental results show that ckfd outperforms other kfd algorithms

a branch and bound algorithm for the acyclic subgraph problem abstract a branch and bound algorithm for the acyclic subgraph problem feedback are set problem is described the branching scheme lexicographically enumerates all permutations skipping initial segments known by some easy tests not to have any optimal completion a lower bound for the number of feedback arcs is given by the size of any collection of disjoint cycles we propose a heuristic algorithm to find a large collection the size of the problems our branch and bound algorithm can solve varies from 25 to 34 nodes depending on the nature of the problem

supervised optimal locality preserving projection in the past few years the computer vision and pattern recognition community has witnessed a rapid growth of a new kind of feature extraction method the manifold learning methods which attempt to project the original data into a lower dimensional feature space by preserving the local neighborhood structure among these methods locality preserving projection lpp is one of the most promising feature extraction techniques unlike the unsupervised learning scheme of lpp this paper follows the supervised learning scheme i e it uses both local information and class information to model the similarity of the data based on novel similarity we propose two feature extraction algorithms supervised optimal locality preserving projection solpp and normalized laplacian based supervised optimal locality preserving projection nl solpp optimal here means that the extracted features via solpp or nl solpp are statistically uncorrelated and orthogonal we compare the proposed solpp and nl solpp with lpp orthogonal locality preserving projection olpp and uncorrelated locality preserving projection ulpp on publicly available data sets experimental results show that the proposed solpp and nl solpp achieve much higher recognition accuracy

bagging constraint score for feature selection with pairwise constraints constraint score is a recently proposed method for feature selection by using pairwise constraints which specify whether a pair of instances belongs to the same class or not it has been shown that the constraint score with only a small amount of pairwise constraints achieves comparable performance to those fully supervised feature selection methods such as fisher score however one major disadvantage of the constraint score is that its performance is dependent on a good selection on the composition and cardinality of constraint set which is very challenging in practice in this work we address the problem by importing bagging into constraint score and a new method called bagging constraint score bcs is proposed instead of seeking one appropriate constraint set for single constraint score in bcs we perform multiple constraint score each of which uses a bootstrapped subset of original given constraint set diversity analysis on individuals of ensemble shows that resampling pairwise constraints is helpful for simultaneously improving accuracy and diversity of individuals we conduct extensive experiments on a series of high dimensional datasets from uci repository and gene databases and the experimental results validate the effectiveness of the proposed method

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

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

maximal functions on classical lorentz spaces and hardy s inequality with weights for nonincreasing functions a characterization is given of a class of classical lorentz spaces on which the hardy littlewood maximal operator is bounded this is done by determining the weights for which hardy s inequality holds for nonincreasing functions an alternate characterization valid for nondecreasing weights is also derived

graph optimized locality preserving projections locality preserving projections lpp is a typical graph based dimensionality reduction dr method and has been successfully applied in many practical problems such as face recognition however lpp depends mainly on its underlying neighborhood graph whose construction suffers from the following issues 1 such neighborhood graph is artificially defined in advance and thus does not necessary benefit subsequent dr task 2 such graph is constructed using the nearest neighbor criterion which tends to work poorly due to the high dimensionality of original space 3 it is generally uneasy to assign appropriate values for the neighborhood size and heat kernel parameter involved in graph construction to address these problems we develop a novel dr algorithm called graph optimized locality preserving projections golpp the idea is to integrate graph construction with specific dr process into a unified framework which results in an optimized graph rather than predefined one moreover an entropy regularization term is incorporated into the objective function for controlling the uniformity level of the edge weights in graph so that a principled graph updating formula naturally corresponding to conventional heat kernel weights can be obtained finally the experiments on several publicly available uci and face data sets show the feasibility and effectiveness of the proposed method with encouraging results

dual decomposition of a single machine scheduling problem we design a fast ascent direction algorithm for the lagrangian dual problem of the single machine scheduling problem of minimizing total weighted completion time subject to precedence constraints we show that designing such an algorithm is relatively simple if a scheduling problem is formulated in terms of the job completion times rather than as an 0 1 linear program also we show that upon termination of such an ascent direction algorithm we get a dual decomposition of the original problem which can be exploited to develop approximative and enumerative approaches for it computational results exhibit that in our application the ascent direction leads to good lagrangian lower and upper bounds

computation of heterogeneous object co embeddings from relational measurements dimensionality reduction and data embedding methods generate low dimensional representations of a single type of homogeneous data objects in this work we examine the problem of generating co embeddings or pattern representations from two different types of objects within a joint common space of controlled dimensionality where the only available information is assumed to be a set of pairwise relations or similarities between instances of the two groups we propose a new method that models the embedding of each object type symmetrically to the other type subject to flexible scale constraints and weighting parameters the embedding generation relies on an efficient optimization dispatched using matrix decomposition that is also extended to support multidimensional co embeddings we also propose a scheme of heuristically reducing the parameters of the model and a simple way of measuring the conformity between the original object relations and the ones re estimated from the co embeddings in order to achieve model selection by identifying the optimal model parameters with a simple search procedure the capabilities of the proposed method are demonstrated with multiple synthetic and real world datasets from the text mining domain the experimental results and comparative analyses indicate that the proposed algorithm outperforms existing methods for co embedding generation analysis of arbitrary relational information between heterogeneous objects explicit generation of co embeddings directly from relational measurements high efficiency due to simple matrix decomposition and a small set of parameters simple scheme to implement the model identification of the parameters multiple comparisons with synthetic and real world datasets and existing algorithms

efficient and robust feature extraction by maximum margin criterion in pattern recognition feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information principal component analysis pca and linear discriminant analysis lda are the two most popular linear dimensionality reduction methods however pca is not very effective for the extraction of the most discriminant features and lda is not stable due to the small sample size problem in this paper we propose some new linear and nonlinear feature extractors based on maximum margin criterion mmc geometrically feature extractors based on mmc maximize the average margin between classes after dimensionality reduction it is shown that mmc can represent class separability better than pca as a connection to lda we may also derive lda from mmc by incorporating some constraints by using some other constraints we establish a new linear feature extractor that does not suffer from the small sample size problem which is known to cause serious stability problems for lda the kernelized nonlinear counterpart of this linear feature extractor is also established in the paper our extensive experiments demonstrate that the new feature extractors are effective stable and efficient

trace ratio criterion based generalized discriminative learning for semi supervised dimensionality reduction dealing with high dimensional data has always been a major problem in many pattern recognition and machine learning applications trace ratio criterion is a criterion that can be applicable to many dimensionality reduction methods as it directly reflects euclidean distance between data points of within or between classes in this paper we analyze the trace ratio problem and propose a new efficient algorithm to find the optimal solution based on the proposed algorithm we are able to derive an orthogonal constrained semi supervised learning framework the new algorithm incorporates unlabeled data into training procedure so that it is able to preserve the discriminative structure as well as geometrical structure embedded in the original dataset under such a framework many existing semi supervised dimensionality reduction methods such as sda lap lda ssdr ssmmc can be improved using our proposed framework which can also be used to formulate a corresponding kernel framework for handling nonlinear problems theoretical analysis indicates that there are certain relationships between linear and nonlinear methods finally extensive simulations on synthetic dataset and real world dataset are presented to show the effectiveness of our algorithms the results demonstrate that our proposed algorithm can achieve great superiority to other state of art algorithms

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

semi supervised double sparse graphs based discriminant analysis for dimensionality reduction abstract discriminant analysis da is a well known dimensionality reduction tool in pattern classification with enough efficient labeled samples the optimal projections could be found by maximizing the between class scatter variance meanwhile minimizing the within class scatter variance however the acquisition of label information is difficult in practice so semi supervised discriminant analysis has attracted much attention in recent years where both few labeled samples and many unlabeled samples are utilized during learning process sparse graph learned by sparse representation contains local structure information about data and is widely employed in dimensionality reduction in this paper semi supervised double sparse graphs sdsg based dimensionality reduction is proposed which considers both the positive and negative structure relationship of data points by using double sparse graphs aiming to explore the discriminant information among unlabeled samples joint k nearest neighbor selection strategy is proposed to select pseudo labeled samples which contain some precise discriminant information in the following procedures the data subset consisting of labeled samples and pseudo labeled samples are used instead of the original data based on two different criterions two sdsg based discriminant analysis methods are designed and denoted by sdsg dda distance based da and sdsg rda reconstruction based da which also use different strategies to reduce the effect of pseudo labels inaccuracy finally the experimental results both on uci datasets and hyperspectral images validate the effectiveness and advantage of the proposed methods compared with some classical dimensionality reduction methods

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs

orthogonal neighborhood preserving projections a projection based dimensionality reduction technique this paper considers the problem of dimensionality reduction by orthogonal projection techniques the main feature of the proposed techniques is that they attempt to preserve both the intrinsic neighborhood geometry of the data samples and the global geometry in particular we propose a method named orthogonal neighborhood preserving projections which works by first building an affinity graph for the data in a way that is similar to the method of locally linear embedding lle however in contrast with the standard lle where the mapping between the input and the reduced spaces is implicit onpp employs an explicit linear mapping between the two as a result handling new data samples becomes straightforward as this amounts to a simple linear transformation we show how we can define kernel variants of onpp as well as how we can apply the method in a supervised setting numerical experiments are reported to illustrate the performance of onpp and to compare it with a few competing methods

embeddings of lorentz type spaces involving weighted integral means we solve the problem of characterizing weights on 0 infinity for which the inequality involving two possibly different general inner weighted means integral infinity 0 integral t 0 f s m2