kolmogorov s hydrodynamic attractors this paper discusses briefly the content and the lasting influence of kolmogorov s 1958 seminar on dynamical systems and hydrodynamic instability

least squares recursive projection twin support vector machine for classification in this paper we formulate a least squares version of the recently proposed projection twin support vector machine ptsvm for binary classification this formulation leads to extremely simple and fast algorithm called least squares projection twin support vector machine lsptsvm for generating binary classifiers different from ptsvm we add a regularization term ensuring the optimization problems in our lsptsvm are positive definite and resulting better generalization ability instead of usually solving two dual problems we solve two modified primal problems by solving two systems of linear equations whereas ptsvm need to solve two quadratic programming problems along with two systems of linear equations our experiments on publicly available datasets indicate that our lsptsvm has comparable classification accuracy to that of ptsvm but with remarkably less computational time

number of degrees of freedom of two dimensional turbulence we derive upper bounds for the number of degrees of freedom of two dimensional navier stokes turbulence freely decaying from a smooth initial vorticity field omega x y 0 omega0 this number denoted by n is defined as the minimum dimension such that for n or n arbitrary n dimensional balls in phase space centered on the solution trajectory omega x y t for t 0 contract under the dynamics of the system linearized about omega x y t in other words n is the minimum number of greatest lyapunov exponents whose sum becomes negative it is found that n or c1re when the phase space is endowed with the energy norm and n or c2re 1 ln re 1 3 when the phase space is endowed with the enstrophy norm here c1 and c2 are constant and re is the reynolds number defined in terms of omega0 the system length scale and the viscosity nu the linear or nearly linear dependence of n on re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number this result is in a sharp contrast to the forced case for which well known estimates for the hausdorff dimension dh of the global attractor scale highly superlinearly with nu 1 we argue that the extra dependence of dh on nu 1 is not an intrinsic property of the turbulent dynamics rather it is a removable artifact brought about by the use of a time independent forcing as a model for energy and enstrophy injection that drives the turbulence

locality regularized linear regression discriminant analysis for feature extraction abstract locality regularized linear regression classification llrc is an effective classifier that shows great potential for face recognition however the original feature space cannot guarantee the classification efficiency of llrc to alleviate this problem we propose a novel dimensionality reduction method called locality regularized linear regression discriminant analysis llrda for feature extraction the proposed llrda is developed according to the decision rule of llrc and seeks to generate a subspace that is discriminant for llrc specifically the intra class and inter class local reconstruction scatters are first defined to characterize the compactness and separability of samples respectively then the objective function for llrda is derived by maximizing the inter class local reconstruction scatter and simultaneously minimizing the intra class local reconstruction scatter extensive experimental results on cmu pie orl feret and yale b face databases validate the effectiveness of our proposed method

generalizing discriminant analysis using the generalized singular value decomposition discriminant analysis has been used for decades to extract features that preserve class separability it is commonly defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters the requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions we examine a number of optimization criteria and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement the result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data we use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives and conclude with a discussion of their relative merits

recursive projection twin support vector machine via within class variance minimization in this paper a novel binary classifier coined projection twin support vector machine ptsvm is proposed the idea is to seek two projection directions one for each class such that the projected samples of one class are well separated from those of the other class in its respective subspace in order to further boost performance a recursive algorithm for ptsvm is proposed to generate more than one projection axis for each class to overcome the singularity problem principal component analysis pca is utilized to transform the data in the original space into a low dimensional subspace wherein the optimization problem of ptsvm is convex and can be solved efficiently the experimental results on several uci benchmark data sets and usps digit database show the feasibility and effectiveness of the proposed method

incremental linear discriminant analysis a fast algorithm and comparisons it has always been a challenging task to develop a fast and an efficient incremental linear discriminant analysis ilda algorithm for this purpose we conduct a new study for linear discriminant analysis lda in this paper and develop a new ilda algorithm we propose a new batch lda algorithm called lda qr lda qr is a simple and fast lda algorithm which is obtained by computing the economic qr factorization of the data matrix followed by solving a lower triangular linear system the relationship between lda qr and uncorrelated lda ulda is also revealed based on lda qr we develop a new incremental lda algorithm called ilda qr the main features of our ilda qr include that 1 it can easily handle the update from one new sample or a chunk of new samples 2 it has efficient computational complexity and space complexity and 3 it is very fast and always achieves competitive classification accuracy compared with ulda algorithm and existing ilda algorithms numerical experiments based on some real world data sets demonstrate that our ilda qr is very efficient and competitive with the state of the art ilda algorithms in terms of classification accuracy computational complexity and space complexity

graph embedded nonparametric mutual information for supervised dimensionality reduction in this paper we propose a novel algorithm for dimensionality reduction that uses as a criterion the mutual information mi between the transformed data and their corresponding class labels the mi is a powerful criterion that can be used as a proxy to the bayes error rate furthermore recent quadratic nonparametric implementations of mi are computationally efficient and do not require any prior assumptions about the class densities we show that the quadratic nonparametric mi can be formulated as a kernel objective in the graph embedding framework moreover we propose its linear equivalent as a novel linear dimensionality reduction algorithm the derived methods are compared against the state of the art dimensionality reduction algorithms with various classifiers and on various benchmark and real life datasets the experimental results show that nonparametric mi as an optimization objective for dimensionality reduction gives comparable and in most of the cases better results compared with other dimensionality reduction methods

locally linear discriminant analysis for multimodally distributed classes for face recognition with a single model image we present a novel method of nonlinear discriminant analysis involving a set of locally linear transformations called locally linear discriminant analysis llda the underlying idea is that global nonlinear data structures are locally linear and local structures can be linearly aligned input vectors are projected into each local feature space by linear transformations found to yield locally linearly transformed classes that maximize the between class covariance while minimizing the within class covariance in face recognition linear discriminant analysis lida has been widely adopted owing to its efficiency but it does not capture nonlinear manifolds of faces which exhibit pose variations conventional nonlinear classification methods based on kernels such as generalized discriminant analysis gda and support vector machine svm have been developed to overcome the shortcomings of the linear method but they have the drawback of high computational cost of classification and overfitting our method is for multiclass nonlinear discrimination and it is computationally highly efficient as compared to gda the method does not suffer from overfitting by virtue of the linear base structure of the solution a novel gradient based learning algorithm is proposed for finding the optimal set of local linear bases the optimization does not exhibit a local maxima problem the transformation functions facilitate robust face recognition in a low dimensional subspace under pose variations using a single model image the classification results are given for both synthetic and real face data

1d correlation filter based class dependence feature analysis for face recognition in this paper a novel one dimensional correlation filter based class dependence feature analysis 1d cfa method is presented for robust face recognition compared with original cfa that works in the two dimensional 2d image space 1d cfa encodes the image data as vectors in 1d cfa a new correlation filter called optimal extra class origin output tradeoff filter oeotf which is designed in the low dimensional principal component analysis pca subspace is proposed for effective feature extraction experimental results on benchmark face databases such as feret ar and frgc show that oeotf based 1d cfa consistently outperforms other state of the art face recognition methods this demonstrates the effectiveness and robustness of the novel method

uncorrelated linear discriminant analysis based on weighted pairwise fisher criterion in this paper we propose a novel uncorrelated weighted linear discriminant analysis uwlda method for feature extraction and recognition the uwlda first introduces a weighting function to restrain the dominant role of the classes with larger distance and then searches the optimal discriminant vectors under the conjugative orthogonal constrains in the null space of the within class scatter matrix and its conjugative orthogonal complement space respectively as a result the proposed technique not only derive the optimal and lossless discriminative information but also guarantee that all extracted features are statistically uncorrelated experiments on feret face database and ar face database are performed to test and evaluate the proposed algorithm the results demonstrate the effectiveness of uwlda

discriminative histograms of local dominant orientation d hldo for biometric image feature extraction this paper presents a simple and robust method namely discriminative histograms of local dominant orientation d hldo for biometric image feature extraction in d hldo the local dominant orientation map and the corresponding relative energy map are obtained by applying the singular value decomposition svd to the collected gradient vectors over a local patch the dominant orientation map and the relative energy map are then used to construct the concatenated histogram features local mean based nearest neighbor discriminant analysis lm nnda is finally employed to reduce the redundancy information and get the low dimensional and discriminative features the proposed method is applied to face finger knuckle print and palm biometrics and is examined using the ar cmu pie and frgcv2 0 face image databases the polyu palmprint database and the polyu finger knuckle print database experimental results demonstrate the effectiveness of the proposed d hldo method

instability for the navier stokes equations on the 2 dimensional torus and a lower bound for the hausdorff dimension of their global attractors we prove instability of stationary solutions of the navier stokes equations on the domain 0 2 0 2 with periodic boundary condition for a class of external forces for large reynolds number moreover we give a lower bound for the hausdorff dimension of the global attractors

an efficient approach for face recognition based on common eigenvalues in this paper a simple technique is proposed for face recognition among many human faces it is based on the polynomial coefficients covariance matrix and algorithm on common eigenvalues the main advantage of the proposed approach is that the identification of similarity between human faces is carried out without computing actual eigenvalues and eigenvectors a symmetric matrix is calculated using the polynomial coefficients based companion matrices of two compared images the nullity of a calculated symmetric matrix is used as similarity measure for face recognition the value of nullity is very small for dissimilar images and distinctly large for similar face images the feasibility of the propose approach is demonstrated on three face databases i e the orl database the yale database b and the feret database experimental results have shown the effectiveness of the proposed approach for feature extraction and classification of the face images having large variation in pose and illumination

face recognition using kernel direct discriminant analysis algorithms techniques that can introduce low dimensional feature representation with enhanced discriminatory power is of paramount importance in face recognition fr systems it is well known that the distribution of face images under a perceivable variation in viewpoint illumination or facial expression is highly nonlinear and complex it is therefore not surprising that linear techniques such as those based on principle component analysis pca or linear discriminant analysis lda cannot provide reliable and robust solutions to those fr problems with complex face variations in this paper we propose a kernel machine based discriminant analysis method which deals with the nonlinearity of the face patterns distribution the proposed method also effectively solves the so called small sample size sss problem which exists in most fr tasks the new algorithm has been tested in terms of classification error rate performance on the multiview umist face database results indicate that the proposed methodology is able to achieve excellent performance with only a very small set of features being used and its error rate is approximately 34 and 48 of those of two other commonly used kernel fr approaches the kernel pca kpca and the generalized discriminant analysis gda respectively

hyperplane navigation a method to set individual scores in fmri group datasets abstract recent studies have demonstrated that spatial patterns of fmri bold activity distribution over the brain may be used to classify different groups or mental states these studies are based on the application of advanced pattern recognition approaches and multivariate statistical classifiers most published articles in this field are focused on improving the accuracy rates and many approaches have been proposed to accomplish this task nevertheless a point inherent to most machine learning methods and still relatively unexplored in neuroimaging is how the discriminative information can be used to characterize groups and their differences in this work we introduce the maximum uncertainty linear discrimination analysis mlda and show how it can be applied to infer groups patterns by discriminant hyperplane navigation in addition we show that it naturally defines a behavioral score i e an index quantifying the distance between the states of a subject from predefined groups we validate and illustrate this approach using a motor block design fmri experiment data with 35 subjects

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

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

multi class pairwise linear dimensionality reduction using heteroscedastic schemes linear dimensionality reduction ldr techniques have been increasingly important in pattern recognition pr due to the fact that they permit a relatively simple mapping of the problem onto a lower dimensional subspace leading to simple and computationally efficient classification strategies although the field has been well developed for the two class problem the corresponding issues encountered when dealing with multiple classes are far from trivial in this paper we argue that as opposed to the traditional ldr multi class schemes if we are dealing with multiple classes it is not expedient to treat it as a multi class problem per se rather we shall show that it is better to treat it as an ensemble of chernoff based two class reductions onto different subspaces whence the overall solution is achieved by resorting to either voting weighting or to a decision tree strategy the experimental results obtained on benchmark datasets demonstrate that the proposed methods are not only efficient but that they also yield accuracies comparable to that obtained by the optimal bayes classifier

a discrete theory of irregular sampling abstract we derive algorithms for the iterative reconstruction of discrete band limited signals from their irregular samples they converge at a geometric rate and all constants are computable explicitly we also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite toeplitz matrices

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

generalized discriminant analysis using a kernel approach we present a new method that we call generalized discriminant analysis gda to deal with nonlinear discriminant analysis using kernel function operator the underlying theory is close to the support vector machines svm insofar as the gda method provides a mapping of the input vectors into high dimensional feature space in the transformed space linear properties make it easy to extend and generalize the classical linear discriminant analysis lda to nonlinear discriminant analysis the formulation is expressed as an eigenvalue problem resolution using a different kernel one can cover a wide class of nonlinearities for both simulated data and alternate kernels we give classification results as well as the shape of the decision function the results are confirmed using real data to perform seed classification

nuclear norm based 2 dpca for extracting features from images the 2 d principal component analysis 2 dpca is a widely used method for image feature extraction however it can be equivalently implemented via image row based principal component analysis this paper presents a structured 2 d method called nuclear norm based 2 dpca n 2 dpca which uses a nuclear norm based reconstruction error criterion the nuclear norm is a matrix norm which can provide a structured 2 d characterization for the reconstruction error image the reconstruction error criterion is minimized by converting the nuclear norm based optimization problem into a series of f norm based optimization problems in addition n 2 dpca is extended to a bilateral projection based n 2 dpca n b2 dpca the virtue of n b2 dpca over n 2 dpca is that an image can be represented with fewer coefficients n 2 dpca and n b2 dpca are applied to face recognition and reconstruction and evaluated using the extended yale b cmu pie frgc and ar databases experimental results demonstrate the effectiveness of the proposed methods

semi supervised linear discriminant analysis for dimension reduction and classification when facing high dimensional data dimension reduction is necessary before classification among dimension reduction methods linear discriminant analysis lda is a popular one that has been widely used lda aims to maximize the ratio of the between class scatter and total data scatter in projected space and the label of each data is necessary however in real applications the labeled data are scarce and unlabeled data are in large quantity so lda is hard to be used under such case in this paper we propose a novel method named semi supervised linear discriminant analysis slda which can use limited number of labeled data and a quantity of the unlabeled ones for training so that lda can accommodate to the situation of a few labeled data available assuming that f represents the calculated class indicator matrix of the training data and y denotes the true label of the labeled data the objective function contains two parts one is the criterion of lda which is a function of projection w and a class indicator matrix f the other is the difference between the true data label and calculated label of these labeled data as far as we know there is no closed form solution to the objective function to solve such problem we develop an iterative algorithm which calculates the class indicator matrix and the projection alternatively the convergence of the proposed iterative algorithm is proved and confirmed by experiments the experimental results on eight datasets show that the performance of slda is superior to that of traditional lda and some state of the art semi supervised algorithms highlightsa semi supervised variant of lda named semi supervised lda is proposed our method can use limited number of labeled data and a quantity of the unlabeled ones for training we develop an iterative algorithm which calculates the class indicator matrix and the projection alternatively

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

on minimum class locality preserving variance support vector machine in this paper a so called minimum class locality preserving variance support machine mclpv svm algorithm is presented by introducing the basic idea of the locality preserving projections lpp which can be seen as a modified class of support machine svm and or minimum class variance support machine mcvsvm mclpv svm in contrast to svm and mcvsvm takes the intrinsic manifold structure of the data space into full consideration and inherits the characteristics of svm and mcvsvm we discuss in the paper the linear case the small sample size case and the nonlinear case of the mclpv svm similar to mcvsvm the mclpv svm optimization problem in the small sample size case is solved by using dimensionality reduction through principal component analysis pca and one in the nonlinear case is transformed into an equivalent linear mclpv svm problem under kernel pca kpca experimental results on real datasets indicate the effectiveness of the mclpv svm by comparing it with svm and mcvsvm

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

extensivity of two dimensional turbulence this study is concerned with how the attractor dimension of the two dimensional navier stokes equations depends on characteristic length scales including the system integral length scale the forcing length scale and the dissipation length scale upper bounds on the attractor dimension derived by constantin foias and temam are analysed it is shown that the optimal attractor dimension estimate grows linearly with the domain area suggestive of extensive chaos for a sufficiently large domain if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed for sufficiently small domain area a slightly super extensive estimate becomes optimal in the extensive regime the attractor dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined on physical grounds in terms of the average rate of shear this dissipation length scale which is not necessarily the scale at which the energy or enstrophy dissipation takes place can be identified with the dimension correlation length scale the square of which is interpreted according to the concept of extensive chaos as the area of a subsystem with one degree of freedom furthermore these length scales can be identified with a minimum length scale of the flow which is rigorously deduced from the concept of determining nodes

a multi manifold discriminant analysis method for image feature extraction in this paper we propose a multi manifold discriminant analysis mmda method for an image feature extraction and pattern recognition based on graph embedded learning and under the fisher discriminant analysis framework in an mmda the within class graph and between class graph are respectively designed to characterize the within class compactness and the between class separability seeking for the discriminant matrix to simultaneously maximize the between class scatter and minimize the within class scatter in addition in an mmda the within class graph can represent the sub manifold information while the between class graph can represent the multi manifold information the proposed mmda is extensively examined by using the feret ar and orl face databases and the polyu finger knuckle print databases the experimental results demonstrate that an mmda is effective in feature extraction leading to promising image recognition performance

perturbation lda learning the difference between the class empirical mean and its expectation fisher s linear discriminant analysis lda is popular for dimension reduction and extraction of discriminant features in many pattern recognition applications especially biometric learning in deriving the fisher s lda formulation there is an assumption that the class empirical mean is equal to its expectation however this assumption may not be valid in practice in this paper from the perturbation perspective we develop a new algorithm called perturbation lda p lda in which perturbation random vectors are introduced to learn the effect of the difference between the class empirical mean and its expectation in fisher criterion this perturbation learning in fisher criterion would yield new forms of within class and between class covariance matrices integrated with some perturbation factors moreover a method is proposed for estimation of the covariance matrices of perturbation random vectors for practical implementation the proposed p lda is evaluated on both synthetic data sets and real face image data sets experimental results show that p lda outperforms the popular fisher s lda based algorithms in the undersampled case

fractional order embedding canonical correlation analysis and its applications to multi view dimensionality reduction and recognition due to the noise disturbance and limited number of training samples within set and between set sample covariance matrices in canonical correlation analysis cca usually deviate from the true ones in this paper we re estimate within set and between set covariance matrices to reduce the negative effect of this deviation specifically we use the idea of fractional order to respectively correct the eigenvalues and singular values in the corresponding sample covariance matrices and then construct fractional order within set and between set scatter matrices which can obviously alleviate the problem of the deviation on this basis a new approach is proposed to reduce the dimensionality of multi view data for classification tasks called fractional order embedding canonical correlation analysis fecca the proposed method is evaluated on various handwritten numeral face and object recognition problems extensive experimental results on the cenparmi uci at t ar and coil 20 databases show that fecca is very effective and obviously outperforms the existing joint dimensionality reduction or feature extraction methods in terms of classification accuracy moreover its improvements for recognition rates are statistically significant on most cases below the significance level 0 05 a new method namely fecca is presented for multi view dimensionality reduction fractional order within set and between set scatter matrices are constructed by sample spectrum modeling the extracted features by fecca have strong discriminant power for recognition experimental results show fecca has better recognition rates than the existing joint feature extraction methods

discriminative common vectors for face recognition in face recognition tasks the dimension of the sample space is typically larger than the number of the samples in the training set as a consequence the within class scatter matrix is singular and the linear discriminant analysis lda method cannot be applied directly this problem is known as the small sample size problem in this paper we propose a new face recognition method called the discriminative common vector method based on a variation of fisher s linear discriminant analysis for the small sample size case two different algorithms are given to extract the discriminative common vectors representing each person in the training set of the face database one algorithm uses the within class scatter matrix of the samples in the training set while the other uses the subspace methods and the gram schmidt orthogonalization procedure to obtain the discriminative common vectors then the discriminative common vectors are used for classification of new faces the proposed method yields an optimal solution for maximizing the modified fisher s linear discriminant criterion given in the paper our test results show that the discriminative common vector method is superior to other methods in terms of recognition accuracy efficiency and numerical stability

locally adaptive metric nearest neighbor classification nearest neighbor classification assumes locally constant class conditional probabilities this assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality severe bias can be introduced under these conditions when using the nearest neighbor rule we propose a locally adaptive nearest neighbor classification method to try to minimize bias we use a chi squared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones as a result the class conditional probabilities are smoother in the modified neighborhoods whereby better classification performance can be achieved the efficacy of our method is validated and compared against other techniques using both simulated and real world data

1d lda vs 2d lda when is vector based linear discriminant analysis better than matrix based recent advances have shown that algorithms with 2d matrix based representation perform better than the traditional 1d vector based ones in particular 2d lda has been widely reported to outperform 1d lda however would the matrix based linear discriminant analysis be always superior and when would 1d lda be better in this paper we investigate into these questions and have a comprehensive comparison between 1d lda and 2d lda in theory and in experiments we analyze the heteroscedastic problem in 2d lda and formulate mathematical equalities to explore the relationship between 1d lda and 2d lda then we point out potential problems in 2d lda it is shown that 2d lda has eliminated the information contained in the covariance information between different local geometric structures such as the rows or the columns which is useful for discriminant feature extraction whereas 1d lda could preserve such information interestingly this new finding indicates that 1d lda is able to gain higher fisher score than 2d lda in some extreme case furthermore sufficient conditions on which 2d lda would be bayes optimal for two class classification problem are derived and comparison with 1d lda in this aspect is also analyzed this could help understand how 2d lda is expected to achieve at its best further discover its relationship with 1d lda and well support other findings after the theoretical analysis comprehensive experimental results are reported by fairly and extensively comparing 1d lda with 2d lda in contrast to the existing view that some 2d lda based algorithms would perform better than 1d lda when the number of training samples for each class is small or when the number of discriminant features used is small we show that it is not always true and show that some standard 1d lda based algorithms could perform better in those cases on some challenging data sets

face recognition using a kernel fractional step discriminant analysis algorithm feature extraction is among the most important problems in face recognition systems in this paper we propose an enhanced kernel discriminant analysis kda algorithm called kernel fractional step discriminant analysis kfda for nonlinear feature extraction and dimensionality reduction not only can this new algorithm like other kernel methods deal with nonlinearity required for many face recognition tasks it can also outperform traditional kda algorithms in resisting the adverse effects due to outlier classes moreover to further strengthen the overall performance of kda algorithms for face recognition we propose two new kernel functions cosine fractional power polynomial kernel and non normal gaussian rbf kernel we perform extensive comparative studies based on the yaleb and feret face databases experimental results show that our kfda algorithm outperforms traditional kernel principal component analysis kpca and kda algorithms moreover further improvement can be obtained when the two new kernel functions are used

feature extraction based on laplacian bidirectional maximum margin criterion maximum margin criterion mmc based feature extraction is more efficient than linear discriminant analysis lda for calculating the discriminant vectors since it does not need to calculate the inverse within class scatter matrix however mmc ignores the discriminative information within the local structures of samples and the structural information embedding in the images in this paper we develop a novel criterion namely laplacian bidirectional maximum margin criterion lbmmc to address the issue we formulate the image total laplacian matrix image within class laplacian matrix and image between class laplacian matrix using the sample similar weight that is widely used in machine learning the proposed lbmmc based feature extraction computes the discriminant vectors by maximizing the difference between image between class laplacian matrix and image within class laplacian matrix in both row and column directions experiments on the feret and yale face databases show the effectiveness of the proposed lbmmc based feature extraction method

attractor dimension estimates for two dimensional shear flows abstract we study the large time behavior of boundary and pressure gradient driven incompressible fluid flows in elongated two dimensional channels with emphasis on estimates for their degrees of freedom i e the dimension of the attractor for the solutions of the navier stokes equations for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form c re 3 2 where c is a universal constant denotes the aspect ratio of the channel length width and re is the reynolds number based on the channel width and the imposed outer velocity scale for fixed pressure gradient driven channel flows we obtain an upper bound of form c re2 where c is another universal positive constant and the reynolds number is based on a velocity defined by the infimum over all possible trajectories of the time averaged mass flux per unit channel width we discuss these results in terms of physical arguments based on small length scales in turbulent flows

an optimization criterion for generalized discriminant analysis on undersampled problems an optimization criterion is presented for discriminant analysis the criterion extends the optimization criteria of the classical linear discriminant analysis lda through the use of the pseudoinverse when the scatter matrices are singular it is applicable regardless of the relative sizes of the data dimension and sample size overcoming a limitation of classical lda the optimization problem can be solved analytically by applying the generalized singular value decomposition gsvd technique the pseudoinverse has been suggested and used for undersampled problems in the past where the data dimension exceeds the number of data points the criterion proposed in this paper provides a theoretical justification for this procedure an approximation algorithm for the gsvd based approach is also presented it reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster this reduced problem yields much smaller matrices to which the gsvd can be applied efficiently experiments on text data with up to 7 000 dimensions show that the approximation algorithm produces results that are close to those produced by the exact algorithm

k local hyperplane distance nearest neighbor classifier oriented local discriminant analysis k local hyperplane distance nearest neighbor hknn classifier is an improved k nearest neighbor knn algorithm that has been successfully applied to pattern classification this paper embeds the decision rule of hknn classifier into the discriminant analysis model to develop a new feature extractor the obtained feature extractor is called k local hyperplane distance nearest neighbor classifier oriented local discriminant analysis holda in which a regularization item is imposed on the original hknn algorithm to obtain a more reliable distance metric based on this distance metric the homo class and hetero class local scatters are characterized in holda by maximizing the ratio of the hetero class local scatter to the homo class local scatter we obtain a subspace which is suitable for feature extraction and classification in general this paper provides a framework for building a feature extractor from the decision rule of a classifier by this means the feature extractor and classifier can be seamlessly integrated experimental results on four databases demonstrate that the integrated pattern recognition system is effective

on the relation between discriminant analysis and mutual information for supervised linear feature extraction this paper provides a unifying view of three discriminant linear feature extraction methods linear discriminant analysis heteroscedastic discriminant analysis and maximization of mutual information we propose a model independent reformulation of the criteria related to these three methods that stresses their similarities and elucidates their differences based on assumptions for the probability distribution of the classification data we obtain sufficient conditions under which two or more of the above criteria coincide it is shown that these conditions also suffice for bayes optimality of the criteria our approach results in an information theoretic derivation of linear discriminant analysis and heteroscedastic discriminant analysis finally regarding linear discriminant analysis we discuss its relation to multidimensional independent component analysis and derive suboptimality bounds based on information theory

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

on image analysis by moments research has been performed investigating the use of moments for pattern recognition in recent years the basic problem of the influence of discretization and noise on moment accuracy as object descriptors has been barely investigated in this paper the detailed error analysis involved in the moment method is discussed several new techniques to increase the accuracy and efficiency of moment descriptor are proposed we utilize these results for the problem of image reconstruction from the orthogonal legendre moments computed from discrete and noisy data the automatic selection of an optimal number of moments is also discussed

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

some specific mathematical constraints on 2d turbulence abstract we derive upper and lower bounds for ensemble averages of energy enstrophy and palinstrophy for the 2d periodic navier stokes equations this is carried out both in the general case and in the case where the energy power law for fully developed turbulence holds in the turbulent case the bounds are sharp up to a logarithm and provide a new lower bound on the landau lifschitz degrees of freedom we also prove two properties of the inertial term under the turbulence assumption one is that as the grashof number is increased the ensemble average of this term approaches the force the other is that an estimate of it via the ladyzhenskaya inequality is sharp on a considerable portion of the global attractor

minimum class variance support vector machines in this paper a modified class of support vector machines svms inspired from the optimization of fisher s discriminant ratio is presented the so called minimum class variance svms mcvsvms the mcvsvms optimization problem is solved in cases in which the training set contains less samples that the dimensionality of the training vectors using dimensionality reduction through principal component analysis pca afterward the mcvsvms are extended in order to find nonlinear decision surfaces by solving the optimization problem in arbitrary hilbert spaces defined by mercer s kernels in that case it is shown that under kernel pca the nonlinear optimization problem is transformed into an equivalent linear mcvsvms problem the effectiveness of the proposed approach is demonstrated by comparing it with the standard svms and other classifiers like kernel fisher discriminant analysis in facial image characterization problems like gender determination eyeglass and neutral facial expression detection

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

discrete irregular sampling with larger gaps abstract discrete irregular sampling is the problem of recovering a band limited discrete signal from its nonuniformly sampled values we introduce a new method for discrete irregular sampling it features a new sampling geometry where the sampling points are chosen to come in pairs a frequency domain formulation reduces the problem to a system of linear equations for which the matrix is of toeplitz type and is positive definite as soon as there are enough samples an explicit estimate of the condition number of that matrix is given under some condition on the maximum sampling gap this condition allows gaps larger than the nyquist gap we recombine the new method and the conjugate method to deduce an efficient iterative reconstruction scheme for band limited discrete signals numerical results are provided to illustrate the efficiency of the method

improved discriminate analysis for high dimensional data and its application to face recognition many pattern recognition applications involve the treatment of high dimensional data and the small sample size problem principal component analysis pca is a common used dimension reduction technique linear discriminate analysis lda is often employed for classification pca plus lda is a famous framework for discriminant analysis in high dimensional space and singular cases in this paper we examine the theory of this framework and find out that even if there is no small sample size problem the pca dimension reduction cannot guarantee the subsequent successful application of lda we thus develop an improved discriminate analysis method by introducing an inverse fisher criterion and adding a constrain in pca procedure so that the singularity phenomenon will not occur experiment results on face recognition suggest that this new approach works well and can be applied even when the number of training samples is one per class

face recognition using scale adaptive directional and textural features a novel approach to face recognition problem using directional and texture information from face images is proposed in this paper in order to capture the directionality specially designed using local polynomial approximation technique scale adaptive digital filters are used for texture features extraction a low dimensional and computationally effective local descriptor is utilized textural and directional features are captured at the holistic and part based levels resulting in a robust face descriptor the proposed method is tested on a number of standard test face datasets orl xm2vts extended yale cmu pie ar and feret for different scenarios and its performance is compared with several state of the art techniques

why can lda be performed in pca transformed space abstract pca plus lda is a popular framework for linear discriminant analysis lda in high dimensional and singular case in this paper we focus on building a theoretical foundation for this framework moreover we point out the weakness of the previous lda based methods and suggest a complete pca plus lda algorithm experimental results on orl face image database indicate that the proposed method is more effective than the previous ones

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

linear dimensionality reduction by maximizing the chernoff distance in the transformed space linear dimensionality reduction ldr techniques are quite important in pattern recognition due to their linear time complexity and simplicity in this paper we present a novel ldr technique which though linear aims to maximize the chernoff distance in the transformed space thus augmenting the class separability in such a space we present the corresponding criterion which is maximized via a gradient based algorithm and provide convergence and initialization proofs we have performed a comprehensive performance analysis of our method combined with two well known classifiers linear and quadratic on synthetic and real life data and compared it with other ldr techniques the results on synthetic and standard real life data sets show that the proposed criterion outperforms the latter when combined with both linear and quadratic classifiers

bifurcations of a steady state solution to the two dimensional navier stokes equations this paper studies the spatially periodic navier stokes flows in r2 driven by a unidirectional external force this dynamical system admits a steady state solution u0 for all reynolds numbers u0 is the basic flow in our consideration and primary bifurcations of u0 are investigated in particular it is found that there exists a flow invariant subspace containing cos mx ny or sin mx ny and the occurrence of stability and bifurcations of u0 in such a subspace essentially depends on the choice of the integers m and n our findings are obtained by analysis together with numerical computation

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

rectified nearest feature line segment for pattern classification this paper points out and analyzes the advantages and drawbacks of the nearest feature line nfl classifier to overcome the shortcomings a new feature subspace with two simple and effective improvements is built to represent each class the proposed method termed rectified nearest feature line segment rnfls is shown to possess a novel property of concentration as a result of the added line segments features which significantly enhances the classification ability another remarkable merit is that rnfls is applicable to complex tasks such as the two spiral distribution which the original nfl cannot deal with properly finally experimental comparisons with nfl nn nearest neighbor k nn and nnl nearest neighbor line using both artificial and real world data sets demonstrate that rnfls offers the best performance

recursive concave convex fisher linear discriminant with applications to face handwritten digit and terrain recognition in classification previous studies have shown that an eigenvalue based technique can be cast as an related svm type problem and that by solving this svm type problem the performance can be improved significantly in this paper we develop a recursive concave convex fisher linear discriminant dr rpfld for dimension reduction technique of high dimensional data to extract as many meaningful features as possible which incorporates the fundamental idea behind fisher linear discriminant and casts the fisher linear discriminant as a concave convex programming problem based on the hinge loss the solution of our method follows from solving the related svm type optimization problems iteratively which means the proposed method can be viewed as the combination of multiple related svm type problems the special formulation of our method provides convenience for constructing sparse multi class fisher linear discriminant directly due to use of a recursive procedure the number of features available from rpfld is independent of the number of classes meaning that in contrast to the original fisher linear discriminant the number of features available from our method has no upper bound we evaluate our algorithm on the yale and orl face image databases handwritten digit database and terrain image dataset experimental results show that rpfld outperforms other fisher linear discriminant algorithms

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 collaborative representation based projections method for feature extraction in graph embedding based methods we usually need to manually choose the nearest neighbors and then compute the edge weights using the nearest neighbors via l2 norm e g lle it is difficult and unstable to manually choose the nearest neighbors in high dimensional space so how to automatically construct a graph is very important in this paper first we give a l2 graph like l1 graph l2 graph calculates the edge weights using the total samples avoiding manually choosing the nearest neighbors second a l2 graph based feature extraction method is presented called collaborative representation based projections crp like spp crp aims to preserve the collaborative representation based reconstruction relationship of data crp utilizes a l2 norm graph to characterize the local compactness information crp maximizes the ratio between the total separability information and the local compactness information to seek the optimal projection matrix crp is much faster than spp since crp calculates the objective function with l2 norm while spp calculate the objective function with l1 norm experimental results on feret ar yale face databases and the polyu finger knuckle print database demonstrate that crp works well in feature extraction and leads to a good recognition performance we give a l2 norm graph based on collaborative representation we propose a collaborative representation based projections crp for feature extraction crp is a rayleigh quotient form and can be calculated via generalized eigenvalue decomposition

bayes optimality in linear discriminant analysis we present an algorithm that provides the one dimensional subspace where the bayes error is minimized for the c class problem with homoscedastic gaussian distributions our main result shows that the set of possible one dimensional spaces v for which the order of the projected class means is identical defines a convex region with associated convex bayes error function g v this allows for the minimization of the error function using standard convex optimization algorithms our algorithm is then extended to the minimization of the bayes error in the more general case of heteroscedastic distributions this is done by means of an appropriate kernel mapping function this result is further extended to obtain the d dimensional solution for any given d by iteratively applying our algorithm to the null space of the d l dimensional solution we also show how this result can be used to improve upon the outcomes provided by existing algorithms and derive a low computational cost linear approximation extensive experimental validations are provided to demonstrate the use of these algorithms in classification data analysis and visualization

bagging null space locality preserving discriminant classifiers for face recognition in this paper we propose a novel bagging null space locality preserving discriminant analysis bagnlpda method for facial feature extraction and recognition the bagnlpda method first projects all the training samples into the range space of a so called locality preserving total scatter matrix without losing any discriminative information the projected training samples are then randomly sampled using bagging to generate a set of bootstrap replicates null space discriminant analysis is performed in each replicate and the results of them are combined using majority voting as a result the proposed method aggregates a set of complementary null space locality preserving discriminant classifiers experiments on feret and pie subsets demonstrate the effectiveness of bagnlpda

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

complete large margin linear discriminant analysis using mathematical programming approach in this paper we develop a novel dimensionality reduction dr framework coined complete large margin linear discriminant analysis clmlda inspired by several recently proposed dr methods clmlda constructs two mathematical programming models by maximizing the minimum distance between each class center and the total class center respectively in the null space of within class scatter matrix and its orthogonal complementary space in this way clmlda not only makes full use of the discriminative information contained in the whole feature space but also overcome the weakness of linear discriminant analysis lda in dealing with the class separation problem the solutions of clmlda follow from solving two nonconvex optimization problems each of which is transformed to a series of convex quadratic programming problems by using the constrained concave convex procedure first and then solved by off the shelf optimization toolbox experiments on both toy and several publicly available databases demonstrate its feasibility and effectiveness

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

comments on globally maximizing locally minimizing unsupervised discriminant projection with application to face and palm biometrics this paper develops an unsupervised discriminant projection udp technique for dimensionality reduction of high dimensional data in small sample size cases udp can be seen as a linear approximation of a multimanifolds based learning framework which takes into account both the local and nonlocal quantities udp characterizes the local scatter as well as the nonlocal scatter seeking to find a projection that simultaneously maximizes the nonlocal scatter and minimizes the local scatter this characteristic makes udp more intuitive and more powerful than the most up to date method locality preserving projection lpp which considers only the local scatter for clustering or classification tasks the proposed method is applied to face and palm biometrics and is examined using the yale feret and ar face image databases and the polyu palmprint database the experimental results show that udp consistently outperforms lpp and pca and outperforms lda when the training sample size per class is small this demonstrates that udp is a good choice for real world biometrics applications

linear reconstruction measure steered nearest neighbor classification framework the linear reconstruction measure lrm which determines the nearest neighbors of the query sample in all known training samples by sorting the minimum l 2 norm error linear reconstruction coefficients is introduced in this paper the intuitive interpretation and mathematical proofs are presented to reveal the efficient working mechanism of lrm through analyzing the physical meaning of coefficients and regularization items we find that lrm provides more useful information and advantages than the conventional similarity measure model which calculates the distance between two entities i e conventional point to point c ptp inspired by the advantages of lrm the linear reconstruction measure steered nearest neighbor classification framework lrm nncf is designed with eight classifiers according to different decision rules and models of lrm evaluation on several face databases and the experimental results demonstrate that these proposed classifiers can achieve greater performance than the c ptp based 1 nns and competitive recognition accuracy and robustness compared with the state of the art classifiers

multiclass classifiers based on dimension reduction with generalized lda linear discriminant analysis lda has been widely used for dimension reduction of data sets with multiple classes the lda has been recently extended to various generalized lda methods that are applicable regardless of the relative sizes between the data dimension and the number of data items in this paper we propose several multiclass classifiers based on generalized lda glda algorithms taking advantage of the dimension reducing transformation matrix without requiring additional training or parameter optimization a marginal linear discriminant classifier mldc a bayesian linear discriminant classifier bldc and a one dimensional bldc are introduced for multiclass classification our experimental results illustrate that these classifiers produce higher ten fold cross validation accuracy than knn and centroid based classifiers in the reduced dimensional space obtained from glda

optimal bounds on the dimension of the attractor of the navier stokes equations abstract in this article we derive optimal upper bounds on the dimension of the attractor for the navier stokes equations in two dimensional domains these bounds fully agree with the lower bounds obtained by babin and vishik 1983 see also ghidaglia and temam and liu 1993 as in babin and vishik 1983 we consider here elongated domains and leaving the density of volume forces and the viscosity fixed we let the shape ratio of the domain become large so that the grashof number is large the estimates derived here are based on the general methods for estimating attractors dimensions as in constantin et al 1988 on a new version of the lieb thirring inequalities for elongated domains and on techniques developed for such domains in rs 1993 and temam and ziane 1996 at the end of the article we also give some partial results in the three dimensional case for which we need a physical assumption on the reynolds number introduced in ghidaglia and temam

median null sw based method for face feature recognition with the progress of science and technology artificial intelligence is being paid more and more attention people want to use computers to deal with complex practical problems so linear discriminant analysis lda is widely used as a dimensionality reduction technique in image and text recognition classification tasks however a weakness of lda model is that the class average vector in the formula completely depends on class sample average under special circumstances such as noise bright light some outliers will appear in the practical input databases therefore by employing several given practical samples the class sample average is not enough to estimate the class average accurately so the recognition performance of lda model will decline compared to human intelligence computers are far short of necessary fundamental knowledge of judgment which people normally acquire during the formative years of their lives in order to solve the problem and also to render lda model more robust we propose a within class scatter matrix null space median method m n s w which first transforms the original space by employing a basis of within class scatter matrix null space and then in the transformed space the maximum of between class scatter matrix is pursued in the second stage within class median vector is used in the traditional lda model experiments on orl feret and yale face data sets are performed to test and evaluate the effectiveness of the proposed method

scatter balance an angle based supervised dimensionality reduction subspace selection is widely applied in data classification clustering and visualization the samples projected into subspace can be processed efficiently in this paper we research the linear discriminant analysis lda and maximum margin criterion mmc algorithms intensively and analyze the effects of scatters to subspace selection meanwhile we point out the boundaries of scatters in lda and mmc algorithms to illustrate the differences and similarities of subspace selection in different circumstances besides the effects of outlier classes on subspace selection are also analyzed according to the above analysis we propose a new subspace selection method called angle linear discriminant embedding alde on the basis of angle measurement alde utilizes the cosine of the angle to get new within class and between class scatter matrices and avoids the small sample size problem simultaneously to deal with high dimensional data we extend alde to a two stage alde ts alde the synthetic data experiments indicate that alde can balance the within class and between class scatters and be robust to outlier classes the experimental results based on uci machine learning repository and image databases show that ts alde has a lower time complexity than alde while processing high dimensional data

from classifiers to discriminators a nearest neighbor rule induced discriminant analysis the current discriminant analysis method design is generally independent of classifiers thus the connection between discriminant analysis methods and classifiers is loose this paper provides a way to design discriminant analysis methods that are bound with classifiers we begin with a local mean based nearest neighbor lm nn classifier and use its decision rule to supervise the design of a discriminator therefore the derived discriminator called local mean based nearest neighbor discriminant analysis lm nnda matches the lm nn classifier optimally in theory in contrast to that lm nnda is a nn classifier induced discriminant analysis method we further show that the classical fisher linear discriminant analysis flda is a minimum distance classifier i e nearest class mean classifier induced discriminant analysis method the proposed lm nnda method is evaluated using the cenparmi handwritten numeral database the nust603 handwritten chinese character database the eth80 object category database and the feret face image database the experimental results demonstrate the performance advantage of lm nnda over other feature extraction methods with respect to the lm nn or nn classifier

sharp lower bounds for the dimension of the global attractor of the sabra shell model of turbulence in this work we derive lower bounds for the hausdorff and fractal dimensions of the global attractor of the sabra shell model of turbulence in different regimes of parameters we show that for a particular choice of the forcing term and for sufficiently small viscosity term the sabra shell model has a global attractor of large hausdorff and fractal dimensions proportional to log 1 for all values of the governing parameter e except for e 1 the obtained lower bounds are sharp matching the upper bounds for the dimension of the global attractor obtained in our previous work moreover the complexity of the dynamics of the shell model increases as the viscosity tends to zero and we describe a precise scenario of successive bifurcations for different parameters regimes in the three dimensional regime of parameters this scenario changes when the parameter e becomes sufficiently close to 0 or to 1 we also show that in the two dimensional regime of parameters for a certain non zero forcing term the long term dynamics of the model becomes trivial for every value of the viscosity

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

a comparison of generalized linear discriminant analysis algorithms linear discriminant analysis lda is a dimension reduction method which finds an optimal linear transformation that maximizes the class separability however in undersampled problems where the number of data samples is smaller than the dimension of data space it is difficult to apply lda due to the singularity of scatter matrices caused by high dimensionality in order to make lda applicable several generalizations of lda have been proposed recently in this paper we present theoretical and algorithmic relationships among several generalized lda algorithms and compare their computational complexities and performances in text classification and face recognition towards a practical dimension reduction method for high dimensional data an efficient algorithm is proposed which reduces the computational complexity greatly while achieving competitive prediction accuracies we also present nonlinear extensions of these lda algorithms based on kernel methods it is shown that a generalized eigenvalue problem can be formulated in the kernel based feature space and generalized lda algorithms are applied to solve the generalized eigenvalue problem resulting in nonlinear discriminant analysis performances of these linear and nonlinear discriminant analysis algorithms are compared extensively

discriminant waveletfaces and nearest feature classifiers for face recognition feature extraction discriminant analysis and classification rules are three crucial issues for face recognition we present hybrid approaches to handle three issues together for feature extraction we apply the multiresolution wavelet transform to extract the waveletface we also perform the linear discriminant analysis on waveletfaces to reinforce discriminant power during classification the nearest feature plane nfp and nearest feature space nfs classifiers are explored for robust decisions in presence of wide facial variations their relationships to conventional nearest neighbor and nearest feature line classifiers are demonstrated in the experiments the discriminant waveletface incorporated with the nfs classifier achieves the best face recognition performance

rapid and brief communication generalizing relevance weighted lda in this paper a new variant on linear discriminant analysis lda that we refer to as generalizing relevance weighted lda or grw lda is proposed grw lda extends the applicability to cases that lda cannot handle by combining the advantages of two recent lda enhancements namely generalized singular value decomposition based lda and relevance weighted lda experimental results on feret face database demonstrate the effectiveness of the proposed method

instability of two phase flows a lower bound on the dimension of the global attractor of the cahn hilliard navier stokes system abstract we consider a model for the flow of a mixture of two viscous and incompressible fluids in a two or three dimensional channel like domain the model consists of the navier stokes equations governing the fluid velocity coupled with a convective cahn hilliard equation for the relative density of atoms of one of the fluids we prove the instability of certain stationary solutions for such a system endowed with periodic boundary conditions on elongated domains 0 2 0 0 2 or 0 2 0 0 2 0 2 0 for a special class of periodic body forces provided that 0 and 0 are small enough as a consequence we deduce a lower bound for the hausdorff dimension of the global attractor