discrete hessian eigenmaps method for dimensionality reduction for a given set of data points lying on a low dimensional manifold embedded in a high dimensional space the dimensionality reduction is to recover a low dimensional parametrization from the data set the recently developed hessian eigenmaps method is a mathematically rigorous method that also sets a theoretical framework for the nonlinear dimensionality reduction problem in this paper we develop a discrete version of the hessian eigenmaps method and present an analysis giving conditions under which the method works as intended as an application a procedure to modify the standard constructions of k nearest neighborhoods is presented to ensure that hessian lle can recover the original coordinates up to an affine transformation

face recognition using a fuzzy fisherface classifier in this study we are concerned with face recognition using fuzzy fisherface approach and its fuzzy set based augmentation the well known fisherface method is relatively insensitive to substantial variations in light direction face pose and facial expression this is accomplished by using both principal component analysis and fisher s linear discriminant analysis what makes most of the methods of face recognition including the fisherface approach similar is an assumption about the same level of typicality relevance of each face to the corresponding class category we propose to incorporate a gradual level of assignment to class being regarded as a membership grade with anticipation that such discrimination helps improve classification results more specifically when operating on feature vectors resulting from the pca transformation we complete a fuzzy k nearest neighbor class assignment that produces the corresponding degrees of class membership the comprehensive experiments completed on orl yale and cnu chungbuk national university face databases show improved classification rates and reduced sensitivity to variations between face images caused by changes in illumination and viewing directions the performance is compared vis a vis other commonly used methods such as eigenface and fisherface

robust nonnegative patch alignment for dimensionality reduction dimensionality reduction is an important method to analyze high dimensional data and has many applications in pattern recognition and computer vision in this paper we propose a robust nonnegative patch alignment for dimensionality reduction which includes a reconstruction error term and a whole alignment term we use correntropy induced metric to measure the reconstruction error in which the weight is learned adaptively for each entry for the whole alignment we propose locality preserving robust nonnegative patch alignment lp rna and sparsity preserviing robust nonnegative patch alignment sp rna which are unsupervised and supervised respectively in the lp rna we propose a locally sparse graph to encode the local geometric structure of the manifold embedded in high dimensional space in particular we select large p nearest neighbors for each sample then obtain the sparse representation with respect to these neighbors the sparse representation is used to build a graph which simultaneously enjoys locality sparseness and robustness in the sp rna we simultaneously use local geometric structure and discriminative information in which the sparse reconstruction coefficient is used to characterize the local geometric structure and weighted distance is used to measure the separability of different classes for the induced nonconvex objective function we formulate it into a weighted nonnegative matrix factorization based on half quadratic optimization we propose a multiplicative update rule to solve this function and show that the objective function converges to a local optimum several experimental results on synthetic and real data sets demonstrate that the learned representation is more discriminative and robust than most existing dimensionality reduction methods

spectral clustering on multiple manifolds spectral clustering sc is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data though sc methods have been successfully applied to a large number of challenging clustering scenarios it is noteworthy that they will fail when there are significant intersections among different clusters in this paper based on the analysis that sc methods are able to work well when the affinity values of the points belonging to different clusters are relatively low we propose a new method called spectral multi manifold clustering smmc which is able to handle intersections in our model the data are assumed to lie on or close to multiple smooth low dimensional manifolds where some data manifolds are separated but some are intersecting then local geometric information of the sampled data is incorporated to construct a suitable affinity matrix finally spectral method is applied to this affinity matrix to group the data extensive experiments on synthetic as well as real datasets demonstrate the promising performance of smmc

a regularized approach for geodesic based semisupervised multimanifold learning geodesic distance as an essential measurement for data dissimilarity has been successfully used in manifold learning however most geodesic distance based manifold learning algorithms have two limitations when applied to classification 1 class information is rarely used in computing the geodesic distances between data points on manifolds and 2 little attention has been paid to building an explicit dimension reduction mapping for extracting the discriminative information hidden in the geodesic distances in this paper we regard geodesic distance as a kind of kernel which maps data from linearly inseparable space to linear separable distance space in doing this a new semisupervised manifold learning algorithm namely regularized geodesic feature learning algorithm is proposed the method consists of three techniques a semisupervised graph construction method replacement of original data points with feature vectors which are built by geodesic distances and a new semisupervised dimension reduction method for feature vectors experiments on the mnist usps handwritten digit data sets mit cbcl face versus nonface data set and an intelligent traffic data set show the effectiveness of the proposed algorithm

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

direct discriminant locality preserving projection with hammerstein polynomial expansion discriminant locality preserving projection dlpp is a linear approach that encodes discriminant information into the objective of locality preserving projection and improves its classification ability to enhance the nonlinear description ability of dlpp we can optimize the objective function of dlpp in reproducing kernel hilbert space to form a kernel based discriminant locality preserving projection kdlpp however kdlpp suffers the following problems 1 larger computational burden 2 no explicit mapping functions in kdlpp which results in more computational burden when projecting a new sample into the low dimensional subspace and 3 kdlpp cannot obtain optimal discriminant vectors which exceedingly optimize the objective of dlpp to overcome the weaknesses of kdlpp in this paper a direct discriminant locality preserving projection with hammerstein polynomial expansion hpddlpp is proposed the proposed hpddlpp directly implements the objective of dlpp in high dimensional second order hammerstein polynomial space without matrix inverse which extracts the optimal discriminant vectors for dlpp without larger computational burden compared with some other related classical methods experimental results for face and palmprint recognition problems indicate the effectiveness of the proposed hpddlpp

mesogenic lattice models with partly antinematic interactions producing uniaxial nematic phases the present paper considers nematogenic lattice models involving particles of d 2h symmetry whose centers of mass are associated with a three dimensional simple cubic lattice the pair potential is isotropic in orientation space and restricted to nearest neighbors let two orthonormal triads define orientations of a pair of interacting particles the simplest potential models proposed in the literature can be reduced to a linear combination involving the squares of the scalar products between corresponding unit vectors only and depending on three parameters by now various sets of potential parameters have been proposed and studied in the literature some of which capable of producing biaxial orientational order at sufficiently low temperature on the other hand in experimental terms mesogenic biaxial molecules mostly produce uniaxial mesophases thus we address here two very simple cases involving a nematic calamitic term as well as one model p0m or two model ppm antinematic ones whose coefficients are set equal in magnitude when only one antinematic coefficient is used the third one is set to zero the calamitic term favors the alignment of two corresponding molecular axes whereas antinematic terms or geometric constraints tend to keep two other pairs of axes mutually orthogonal the models were investigated by molecular field treatments and monte carlo simulation and found to predict a first or second order transitions between uniaxial nematic and isotropic phases the molecular field treatments yielded results in reasonable agreement with simulation

semantic preserving distance metric learning and applications how do we accurately browse a large set of images or efficiently annotate the images from an image library image clustering methods are invaluable tools for applications such as content based image retrieval and image annotation to perform these tasks it is critical to have proper features to describe the visual and semantic content of images and to define an accurate distance metric to measure the dissimilarity between any two images however existing methods which adopt the features of color histograms edge direction histograms and shape context lack the ability to describe semantic content to solve this problem we propose a new approach that utilizes user provided pairwise constraints to describe the semantic relationship between two images a semantic preserving distance metric learning sp dml algorithm is developed to explore the complementary characteristics of the visual features and pairwise constraints in a unified feature space in this space the learned distance metric can be used to measure the dissimilarity between two images specifically the manifold structure adopted in sp dml is revealed by the image s visual features to integrate semantic contents in distance metric learning sp dml utilizes pairwise constraints to build semantic patches and align these patches to obtain the optimal distance metric for the new feature space experimental results in image clustering demonstrate that the performance of sp dml is appealing

nonlinear embedding preserving multiple local linearities locally linear embedding lle is one of the effective and efficient algorithms for nonlinear dimensionality reduction this paper discusses the stability of lle focusing on the optimal weights for extracting local linearity behind the considered manifold it is proven that there are multiple sets of weights that are approximately optimal and can be used to improve the stability of lle a new algorithm using multiple weights is then proposed together with techniques for constructing multiple weights this algorithm is called as nonlinear embedding preserving multiple local linearities neml neml improves the preservation of local linearity and is more stable than lle a short analysis for neml is also given for isometric manifolds neml is compared with the local tangent space alignment ltsa in methodology since both of them adopt multiple local constraints numerical examples are given to show the improvement and efficiency of neml

fidelity under isospectral perturbations a random matrix study the set of hamiltonians generated by all unitary transformations from a single hamiltonian is the largest set of isospectral hamiltonians we can form taking advantage of the fact that the unitary group can be generated from hermitian matrices we can take the ones generated by the gaussian unitary ensemble with a small parameter as small perturbations similarly the transformations generated by hermitian antisymmetric matrices from orthogonal matrices form isospectral transformations among symmetric matrices based on this concept we can obtain the fidelity decay of a system that decays under a random isospectral perturbation with well defined properties regarding time reversal invariance if we choose the hamiltonian itself also from a classical random matrix ensemble then we obtain solutions in terms of form factors in the limit of large matrices

ensemble manifold regularized sparse low rank approximation for multiview feature embedding in computer vision and pattern recognition researches the studied objects are often characterized by multiple feature representations with high dimensionality thus it is essential to encode that multiview feature into a unified and discriminative embedding that is optimal for a given task to address this challenge this paper proposes an ensemble manifold regularized sparse low rank approximation emr slra algorithm for multiview feature embedding the emr slra algorithm is based on the framework of least squares component analysis in particular the low dimensional feature representation and the projection matrix are obtained by the low rank approximation of the concatenated multiview feature matrix by considering the complementary property among multiple features emr slra simultaneously enforces the ensemble manifold regularization on the output feature embedding in order to further enhance its robustness against the noise the group sparsity is introduced into the objective formulation to impose direct noise reduction on the input multiview feature matrix since there is no closed form solution for emr slra this paper provides an efficient emr slra optimization procedure to obtain the output feature embedding experiments on the pattern recognition applications confirm the effectiveness of the emr slra algorithm compare with some other multiview feature dimensionality reduction approaches highlightsthis paper proposes a novel emr slra algorithm for multiview feature embedding the least squares component analysis is generalized to multiview version the ensemble manifold regularization is enforced to explore the complementarity the group sparsity is introduced to promote the robustness against the noise an efficient iterative procedure is developed to solve emr slra

quantitative analysis of nonlinear embedding a lot of nonlinear embedding techniques have been developed to recover the intrinsic low dimensional manifolds embedded in the high dimensional space however the quantitative evaluation criteria are less studied in literature the embedding quality is usually evaluated by visualization which is subjective and qualitative the few existing evaluation methods to estimate the embedding quality neighboring preservation rate for example are not widely applicable in this paper we propose several novel criteria for quantitative evaluation by considering the global smoothness and co directional consistence of the nonlinear embedding algorithms the proposed criteria are geometrically intuitive simple and easy to implement with a low computational cost experiments show that our criteria capture some new geometrical properties of the nonlinear embedding algorithms and can be used as a guidance to deal with the embedding of the out of samples

low rank matrix factorization with multiple hypergraph regularizer this paper presents a novel low rank matrix factorization method named multihmmf which incorporates multiple hypergraph manifold regularization to the low rank matrix factorization in order to effectively exploit high order information among the data samples the hypergraph is introduced to model the local structure of the intrinsic manifold specifically multiple hypergraph regularization terms are separately constructed to consider the local invariance the optimal intrinsic manifold is constructed by linearly combining multiple hypergraph manifolds then the regularization term is incorporated into a truncated singular value decomposition framework resulting in a unified objective function so that matrix factorization is changed into an optimization problem alternating optimization is used to solve the optimization problem with the result that the low dimensional representation of data space is obtained the experimental results of image clustering demonstrate that the proposed method outperforms state of the art data representation methods we propose a novel low rank matrix factorization method we incorporate multiple hypergraph manifold regularization to the matrix factorization we adopt alternating optimization to solve the optimization problem

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

biaxial and uniaxial phases produced by partly repulsive mesogenic models involving d2h molecular symmetries the present paper considers biaxial nematogenic lattice models involving particles of d 2h symmetry whose centers of mass are associated with a three dimensional simple cubic lattice the pair potential is isotropic in orientation space and restricted to nearest neighbors let two orthonormal triads define orientations of a pair of interacting particles the investigated potential models are quadratic with respect to the nine scalar products between the two sets of unit vectors actually based on available geometric identities these expressions can be reduced to diagonal form containing only the scalar products between corresponding unit vectors and depending on three parameters over the years this comparatively simple functional form has also proven to be rather versatile by now various sets of potential parameters capable of producing mesogenic behavior of some kind have been proposed and studied in the literature a new and simplified form was recently proposed and investigated by sonnet virga durand and de matteis a m sonnet e g virga and g e durand phys rev e 67 061701 2003 g de matteis and e g virga phys rev e 71 061703 2005 and is known to support a biaxial phase at sufficiently low temperature following the idea of the above authors we have studied a more extended range of parameters including cases where biaxiality cannot be sustained in the pair ground state in cases where a biaxial phase survives an appropriate mean field analysis may predict the existence of a direct second order transition to the isotropic phase as well as a second order sequence isotropic to uniaxial to biaxial a second order phase transition is also predicted which involves isotropic and uniaxial phases only monte carlo simulations have been carried out as well for a few points in the parameter space and found to produce results which partly confirm mean field predictions

feature extraction by learning lorentzian metric tensor and its extensions we develop a supervised dimensionality reduction method called lorentzian discriminant projection ldp for feature extraction and classification our method represents the structures of sample data by a manifold which is furnished with a lorentzian metric tensor different from classic discriminant analysis techniques ldp uses distances from points to their within class neighbors and global geometric centroid to model a new manifold to detect the intrinsic local and global geometric structures of data set in this way both the geometry of a group of classes and global data structures can be learnt from the lorentzian metric tensor thus discriminant analysis in the original sample space reduces to metric learning on a lorentzian manifold we also establish the kernel tensor and regularization extensions of ldp in this paper the experimental results on benchmark databases demonstrate the effectiveness of our proposed method and the corresponding extensions

matrix perturbation analysis of local tangent space alignment abstract we consider the performance of local tangent space alignment zhang zha 1 one of several manifold learning algorithms which have been proposed as a dimension reduction method matrix perturbation theory is applied to obtain a worst case upper bound on the angle between the computed linear invariant subspace and the linear invariant subspace that is associated with the embedded intrinsic parametrization our result is the first performance bound that has been derived

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

level repulsion in constrained gaussian random matrix ensembles introducing sets of constraints we define new classes of random matrix ensembles the constrained gaussian unitary cgue and the deformed gaussian unitary dgue ensembles the latter interpolate between the gue and the cgue we derive a sufficient condition for gue type level repulsion to persist in the presence of constraints for special classes of constraints we extend this approach to the orthogonal and to the symplectic ensembles a generalized fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones we find that in the dgues level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints

efficient simplicial reconstructions of manifolds from their samples an algorithm for manifold learning is presented given only samples of a finite dimensional differentiable manifold and no a priori knowledge of the manifold s geometry or topology except for its dimension the goal is to find a description of the manifold the learned manifold must approximate the true manifold well both geometrically and topologically when the sampling density is sufficiently high the proposed algorithm constructs a simplicial complex based on approximations to the tangent bundle of the manifold an important property of the algorithm is that its complexity depends on the dimension of the manifold rather than that of the embedding space successful examples are presented in the cases of learning curves in the plane curves in space and surfaces in space in addition a case when the algorithm fails is analyzed

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

gabor feature based classification using the enhanced fisher linear discriminant model for face recognition this paper introduces a novel gabor fisher 1936 classifier gfc for face recognition the gfc method which is robust to changes in illumination and facial expression applies the enhanced fisher linear discriminant model efm to an augmented gabor feature vector derived from the gabor wavelet representation of face images the novelty of this paper comes from 1 the derivation of an augmented gabor feature vector whose dimensionality is further reduced using the efm by considering both data compression and recognition generalization performance 2 the development of a gabor fisher classifier for multi class problems and 3 extensive performance evaluation studies in particular we performed comparative studies of different similarity measures applied to various classifiers we also performed comparative experimental studies of various face recognition schemes including our novel gfc method the gabor wavelet method the eigenfaces method the fisherfaces method the efm method the combination of gabor and the eigenfaces method and the combination of gabor and the fisherfaces method the feasibility of the new gfc method has been successfully tested on face recognition using 600 feret frontal face images corresponding to 200 subjects which were acquired under variable illumination and facial expressions the novel gfc method achieves 100 accuracy on face recognition using only 62 features

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

silhouette analysis for human action recognition based on supervised temporal t sne and incremental learning this paper develops a human action recognition method for human silhouette sequences based on supervised temporal t stochastic neighbor embedding st tsne and incremental learning inspired by the sne and its variants st tsne is proposed to learn the underlying relationship between action frames in a manifold where the class label information and temporal information are introduced to well represent those frames from the same action class as to the incremental learning an important step for action recognition we introduce three methods to perform the low dimensional embedding of new data two of them are motivated by local methods locally linear embedding and locality preserving projection those two techniques are proposed to learn explicit linear representations following the local neighbor relationship and their effectiveness is investigated for preserving the intrinsic action structure the rest one is based on manifold oriented stochastic neighbor projection to find a linear projection from high dimensional to low dimensional space capturing the underlying pattern manifold extensive experimental results and comparisons with the state of the art methods demonstrate the effectiveness and robustness of the proposed st tsne and incremental learning methods in the human action silhouette analysis

universal manifold embedding for geometrically deformed functions assume we have a set of observations for example images of different objects each undergoing a different geometric deformation yet all the deformations belong to the same family as a result of the action of these deformations the set of different observations on each object is generally a manifold in the ambient space of observations in this paper we show that in those cases where the set of deformations admits a finite dimensional representation there is a mapping from the space of observations to a low dimensional linear space the manifold corresponding to each object is mapped to a distinct linear subspace of euclidean space the dimension of the subspace is the same as that of the manifold this mapping which we call universal manifold embedding enables the estimation of geometric deformations using the classical linear theory the universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework the embedding of the space of observations depends on the deformation model and is independent of the specific observed object hence it is universal we study two cases of this embedding that of elastic deformations of 1 d signals and the case of affine deformations of n dimensional signals

spherical and hyperbolic embeddings of data many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects for many types of data these dissimilarities are not euclidean i e they do not represent the distances between points in a euclidean space and therefore cannot be isometrically embedded in a euclidean space examples include shape dissimilarities graph distances and mesh geodesic distances in this paper we provide a means of embedding such non euclidean data onto surfaces of constant curvature we aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data the space can be either of positive curvature spherical or of negative curvature hyperbolic we give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation for objects which do not reside exactly on the manifold we develop a optimisation based procedure for approximate embedding on a hyperspherical manifold we use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space this process is efficient enough to allow us to embed data sets of several thousand objects we apply our method to a variety of data including time warping functions shape similarities graph similarity and gesture similarity data in each case the embedding maintains the local structure of the data while placing the points in a metric space

supervised locally linear embedding with probability based distance for classification we present a novel dimension reduction method for classification based on probability based distance and the technique of locally linear embedding lle logistic discrimination ld is adopted for estimating the probability distribution as well as for classification on the reduced data different from the supervised locally linear embedding slle that is only used for the dimension reduction of training data our probability based locally linear embedding plle can be applied on both training and testing data five microarray data sets in high dimensional spaces the iris data and a real set of handwritten digits are experimented the numerical results show the proposed methodology performs better compared with the ld classifiers applied on the lower dimensional embedding coordinates computed by lle or slle

structure of wave functions in 1 2 body random matrix ensembles random matrix ensembles defined by a mean field one body plus a chaos generating random two body interaction called embedded ensembles of 1 2 body interactions predict for wave functions in the chaotic domain an essentially one parameter gaussian forms for the energy dependence of the number of principal components npc and the localization length l h defined by information entropy which are two important measures of chaos in finite interacting many particle systems numerical embedded ensemble calculations and nuclear shell model results for npc and l h are compared with the theory these analyses clearly point out that for realistic finite interacting many particle systems in the chaotic domain wave function structure is given by 1 2 body embedded random matrix ensembles

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

proximity based frameworks for generating embeddings from multi output data this paper is about supervised and semi supervised dimensionality reduction dr by generating spectral embeddings from multi output data based on the pairwise proximity information two flexible and generic frameworks are proposed to achieve supervised dr sdr for multilabel classification one is able to extend any existing single label sdr to multilabel via sample duplication referred to as mesd the other is a multilabel design framework that tackles the sdr problem by computing weight proximity matrices based on simultaneous feature and label information referred to as mope as a generalization of many current techniques a diverse set of different schemes for label based proximity calculation as well as a mechanism for combining label based and feature based weight information by considering information importance and prioritization are proposed for mope additionally we summarize many current spectral methods for unsupervised dr udr single multilabel sdr and semi supervised dr ssdr and express them under a common template representation as a general guide to researchers in the field we also propose a general framework for achieving ssdr by combining existing sdr and udr models and also a procedure of reducing the computational cost via learning with a target set of relation features the effectiveness of our proposed methodologies is demonstrated with experiments with document collections for multilabel text categorization from the natural language processing domain

similarity preserving principal curve an optimal 1 d feature extractor for data representation this paper discusses the problem of what kind of learning model is suitable for the tasks of feature extraction for data representation and suggests two evaluation criteria for nonlinear feature extractors reconstruction error minimization and similarity preservation based on the suggested evaluation criteria a new type of principal curve similarity preserving principal curve sppc is proposed sppcs minimize the reconstruction error under the condition that the similarity between similar samples are preserved in the extracted features thus giving researchers effective and reliable cognition of the inner structure of data sets the existence and properties of sppcs are analyzed a practical learning algorithm is proposed and high dimensional extensions of sppcs are also discussed experimental results show the virtues of sppcs in preserving inner structures of data sets and discovering manifolds with high nonlinearity

feature extraction using constrained maximum variance mapping in this paper an efficient feature extraction method named as constrained maximum variance mapping cmvm is developed the proposed algorithm can be viewed as a linear approximation of multi manifolds learning based approach which takes the local geometry and manifold labels into account the cmvm and the original manifold learning based approaches have a point in common that the locality is preserved moreover the cmvm is globally maximizing the distances between different manifolds after the local scatters have been characterized the proposed method focuses on developing a linear transformation that can maximize the dissimilarities between all the manifolds under the constraint of locality preserving compared to most of the up to date manifold learning based methods this trick makes contribution to pattern classification from two aspects on the one hand the local structure in each manifold is still kept on the other hand the discriminant information between manifolds can be explored finally feret face database cmu pie face database and usps handwriting data are all taken to examine the effectiveness and efficiency of the proposed method experimental results validate that the proposed approach is superior to other feature extraction methods such as linear discriminant analysis lda locality preserving projection lpp unsupervised discriminant projection udp and maximum variance projection mvp

nonlinear dimensionality reduction by locally linear inlaying high dimensional data is involved in many fields of information processing however sometimes the intrinsic structures of these data can be described by a few degrees of freedom to discover these degrees of freedom or the low dimensional nonlinear manifold underlying a high dimensional space many manifold learning algorithms have been proposed here we describe a novel algorithm locally linear inlaying lli which combines simple geometric intuitions and rigorously established optimality to compute the global embedding of a nonlinear manifold using a divide and conquer strategy lli gains some advantages in itself first its time complexity is linear in the number of data points and hence lli can be implemented efficiently second lli overcomes problems caused by the nonuniform sample distribution third unlike existing algorithms such as isometric feature mapping isomap local tangent space alignment ltsa and locally linear coordination llc lli is robust to noise in addition to evaluate the embedding results quantitatively two criteria based on information theory and kolmogorov complexity theory respectively are proposed furthermore we demonstrated the efficiency and effectiveness of our proposal by synthetic and real world data sets

evolutionary cross domain discriminative hessian eigenmaps is it possible to train a learning model to separate tigers from elks when we have 1 labeled samples of leopard and zebra and 2 unlabelled samples of tiger and elk at hand cross domain learning algorithms can be used to solve the above problem however existing cross domain algorithms cannot be applied for dimension reduction which plays a key role in computer vision tasks e g face recognition and web image annotation this paper envisions the cross domain discriminative dimension reduction to provide an effective solution for cross domain dimension reduction in particular we propose the cross domain discriminative hessian eigenmaps or cdhe for short cdhe connects training and test samples by minimizing the quadratic distance between the distribution of the training set and that of the test set therefore a common subspace for data representation can be well preserved furthermore we basically expect the discriminative information used to separate leopards and zebra can be shared to separate tigers and elks and thus we have a chance to duly address the above question margin maximization principle is adopted in cdhe so the discriminative information for separating different classes e g leopard and zebra here can be well preserved finally cdhe encodes the local geometry of each training class e g leopard and zebra here in the local tangent space which is locally isometric to the data manifold and thus cdhe preserves the intraclass local geometry the objective function of cdhe is not convex so the gradient descent strategy can only find a local optimal solution in this paper we carefully design an evolutionary search strategy to find a better solution of cdhe experimental evidence on both synthetic and real word image datasets demonstrates the effectiveness of cdhe for cross domain web image annotation and face recognition

geometry and radiometry invariant matched manifold detection consider a set of deformable objects undergoing geometric and radiometric transformations as a result of the action of these transformations the set of different realizations of each object is generally a manifold in the space of observations assuming the geometric deformations an object undergoes belong to some finite dimensional family it has been shown that the universal manifold embedding ume provides a set of nonlinear operators that universally maps each of the different manifolds where each manifold is generated by the set all of possible appearances of a single object into a distinct linear subspace of an euclidean space in this paper we generalize this framework to the case where the observed object undergoes both an affine geometric transformation and a monotonic radiometric transformation and present a novel framework for the detection and recognition of the deformable objects applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations but is geometry covariant with the affine transformation the set of all possible observations on that object is mapped by the ume into a single linear subspace invariant with respect to both the geometric and radiometric transformations the embedding of the space of observations is independent of the specific observed object hence it is universal the invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations the observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities since each tile is mapped by the radiometry invariant ume to a distinct linear subspace the detection and tracking problems are solved by evaluating distances between linear subspaces classification in this context becomes a problem of determining which labeled subspace in a grassmannian is closest to a subspace in the same grassmannian where the latter has been generated by radiometry invariant ume from an unlabeled observation

parallel vector field embedding we propose a novel local isometry based dimensionality reduction method from the perspective of vector fields which is called parallel vector field embedding pfe we first give a discussion on local isometry and global isometry to show the intrinsic connection between parallel vector fields and isometry the problem of finding an isometry turns out to be equivalent to finding orthonormal parallel vector fields on the data manifold therefore we first find orthonormal parallel vector fields by solving a variational problem on the manifold then each embedding function can be obtained by requiring its gradient field to be as close to the corresponding parallel vector field as possible theoretical results show that our method can precisely recover the manifold if it is isometric to a connected open subset of euclidean space both synthetic and real data examples demonstrate the effectiveness of our method even if there is heavy noise and high curvature

molecular field theory approach to the landau theory of liquid crystals uniaxial and biaxial nematics nematic liquid crystal phase diagrams in temperature biaxiality space are usually complex we construct a landau theory based on the analogous molecular field theory for orthorhombic biaxial nematic fluids a formal procedure yields coefficients some of which unusually can be tensorial in this landau expansion correctly predicts the complete set of invariants formed from the ordering tensors and avoids ad hoc parametrization of the molecular biaxiality by regularizing the landau expansion to avoid unwanted order parameter divergences at low temperatures we predict phase behavior over the whole range of biaxiality the resulting phase diagrams have the same topology as those of molecular field theory

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

a fuzzy supervised learning method with dynamical parameter estimation for nonlinear discriminant analysis in this paper we develop a novel fuzzy supervised learning algorithm based on the dynamical parameter estimation first a reformative supervised fuzzy lda algorithm rf lda for the training samples is proposed compared with the conventional fuzzy lda algorithm the presented algorithm computes the discriminant vectors associated with the membership grade from each training sample which is theoretically effective in overcoming the classification limitation originating from the imprecise samples second when a novel fuzzy lda model is required in order to take some decision about the feature extraction and classification the dynamical parameter estimation method of the fuzzy lda model should recursively process the measured data as they become available in the line of previous arguments we approach the problem of control parameter estimation of rf lda by considering the formulation of a hopfield neural network hnn which is named hrf lda third considering the fact that the kernel fisher discriminant kfd is effective in extracting the nonlinear discriminative information of the feature space by using kernel trick a kernel version of hrf lda is presented subsequently which has the potential to outperform the traditional fuzzy learning algorithms especially in the cases of nonlinear small sample sizes the advantage of this learning algorithm is that it successfully utilizes the improved kernel fuzzy lda algorithm as a supervised feature extraction tool meanwhile by means of the control parameter estimation we address the problem that the particular value of offset in the calculation of the grade of membership is dynamically assigned extensive experimental studies conducted on the orl nust603 feret yale and xm2vts face images show the effectiveness of the proposed fuzzy integrated algorithm

local coordinates alignment with global preservation for dimensionality reduction dimensionality reduction is vital in many fields and alignment based methods for nonlinear dimensionality reduction have become popular recently because they can map the high dimensional data into a low dimensional subspace with the property of local isometry however the relationships between patches in original high dimensional space cannot be ensured to be fully preserved during the alignment process in this paper we propose a novel method for nonlinear dimensionality reduction called local coordinates alignment with global preservation we first introduce a reasonable definition of topology preserving landmarks tpls which not only contribute to preserving the global structure of datasets and constructing a collection of overlapping linear patches but they also ensure that the right landmark is allocated to the new test point then an existing method for dimensionality reduction that has good performance in preserving the global structure is used to derive the low dimensional coordinates of tpls local coordinates of each patch are derived using tangent space of the manifold at the corresponding landmark and then these local coordinates are aligned into a global coordinate space with the set of landmarks in low dimensional space as reference points the proposed alignment method called landmarks based alignment can produce a closed form solution without any constraints while most previous alignment based methods impose the unit covariance constraint which will result in the deficiency of global metrics and undesired rescaling of the manifold experiments on both synthetic and real world datasets demonstrate the effectiveness of the proposed algorithm

inductive manifold learning using structured support vector machine most manifold learning techniques are used to transform high dimensional data sets into low dimensional space in the use of such techniques after unseen data samples are added to the data set retraining is usually necessary however retraining is a time consuming process and no guarantee of the transformation into the exactly same coordinates thus presenting a barrier to the application of manifold learning as a preprocessing step in predictive modeling to solve this problem learning a mapping from high dimensional representations to low dimensional coordinates is proposed via structured support vector machine after training a mapping low dimensional representations of unobserved data samples can be easily predicted experiments on several datasets show that the proposed method outperforms the existing out of sample extension methods highlightswe propose a general framework for out of sample extensions for any manifold learning methods we learn a mapping from original space to its manifolds by using structured svm experiments on several datasets show that the proposed method outperforms the existing ones

minimum principle for indefinite mean field free energies the extension of bogoliubov jr s minimax principle to biaxial nematic liquid crystals has recently made it possible to derive the universal mean field phase diagram for all quadrupolar interactions between nematogenic biaxial molecules including those rendering the mean field free energy fancyscript f 0 indefinite to justify this extension so far largely based on heuristic arguments we prove here a minimum principle that also applies when fancyscript f 0 is indefinite bogoliubov jr s minimax principle then emerges as a characterization of this minimum principle the theory presented here is more general than its application to biaxial nematic liquid crystals could reveal it essentially builds upon a concavity property of fancyscript f 0 in the order tensor associated with the repulsive component of the molecular interaction

spectral statistics of the two body random ensemble revisited using longer spectra we reanalyze spectral properties of the two body random ensemble studied 30 years ago at the center of the spectra the old results are largely confirmed and we show that the nonergodicity is essentially due to the variance of the lowest moments of the spectra the longer spectra allow us to test and reach the limits of validity of french s correction for the number variance at the edge of the spectra we discuss the problems of unfolding in more detail with a gaussian unfolding of each spectrum the nearest neighbor spacing distribution between ground state and first exited state is shown to be stable using such an unfolding the distribution tends toward a semi poisson distribution for longer spectra for comparison with the nuclear table ensemble we could use such unfolding obtaining similar results as in the early papers but an ensemble with realistic splitting gives reasonable results if we just normalize the spacings in accordance with the procedure used for the data

image clustering based on sparse patch alignment framework image clustering methods are efficient tools for applications such as content based image retrieval and image annotation recently graph based manifold learning methods have shown promising performance in extracting features for image clustering typical manifold learning methods adopt appropriate neighborhood size to construct the neighborhood graph which captures local geometry of data distribution because the density of data points distribution may be different in different regions of the manifold a fixed neighborhood size may be inappropriate in building the manifold in this paper we propose a novel algorithm named sparse patch alignment framework for the embedding of data lying in multiple manifolds specifically we assume that for each data point there exists a small neighborhood in which only the points that come from the same manifold lie approximately in a low dimensional affine subspace based on the patch alignment framework we propose an optimization strategy for constructing local patches which adopt sparse representation to select a few neighbors of each data point that span a low dimensional affine subspace passing near that point after that the whole alignment strategy is utilized to build the manifold experiments are conducted on four real world datasets and the results demonstrate the effectiveness of the proposed method

nonlinear topological component analysis application to age invariant face recognition we introduce a novel formalism that performs dimensionality reduction and captures topological features such as the shape of the observed data to conduct pattern classification this mission is achieved by 1 reducing the dimension of the observed variables through a kernelized radial basis function technique and expressing the latent variables probability distribution in terms of the observed variables 2 disclosing the data manifold as a 3 d polyhedron via the alpha shape constructor and extracting topological features and 3 classifying a data set using a mixture of multinomial distributions we have applied our methodology to the problem of age invariant face recognition experimental results obtained demonstrate the efficiency of the proposed methodology named nonlinear topological component analysis when compared with some state of the art approaches

minimal coupling model of the biaxial nematic phase a minimal coupling model exhibiting isotropic uniaxial and biaxial nematic phases is analyzed in detail and its relation to existing models known in the literature is clarified its intrinsic symmetry properties are exploited to restrict the relevant ranges of coupling constants further on properties of the model are thoroughly investigated by means of bifurcation theory as proposed by kayser and raveche phys rev a 17 2067 1978 and mulder phys rev a 39 360 1989 as a first step toward this goal the bifurcation theory is applied to a general formulation of density functional theory in terms of direct correlation functions on a general formal level the theory is then analyzed to show that the bifurcation points from the reference high symmetry equilibrium phase to a low symmetry structure depend only on the properties of the one particle distribution function and the direct pair correlation function of the reference phase the character of the bifurcation whether spinodal critical tricritical isolated landau point etc depends in addition on a few higher order direct correlation functions explicit analytical results are derived for the case when only the leading l 2 terms of the potential mean field analysis or of the direct pair correlation function expansion in the symmetry adapted basis are retained formulas are compared with the numerical calculations for the mean field momentum l 2 potential model in which case they are exact in particular bifurcations from the isotropic and uniaxial nematic to the biaxial nematic phases are discussed the possibility of the recently reported nematic uniaxial nematic biaxial tricritical point a m sonnet e g virga and g e durand phys rev e 67 061701 2003 is analyzed as well

learning the dynamical system behind sensory data this letter looks at the physics behind sensory data by identifying the parameters that govern the physical system and estimating them from sensory observations we extend takens s delay embedding theorem to the dynamical systems controlled by parameters an embedding of the product space of the phase and the parameter spaces of the dynamical system can be obtained by the delay embedding map provided that the parameter of the dynamical system changes slowly the reconstruction error is bounded for slowly varying parameters a manifold learning technique is applied to the embedding obtained to extract a low dimensional global coordinate system representing the product space the phase space of the deterministic dynamics can be contracted by using the adjacency relationship in time which enables recovery of only the parameter space as examples the manifolds of synthetic and real world vowels with time varying fundamental frequency f0 are analyzed and the f0 contours are extracted by an unsupervised algorithm experimental results show that the proposed method leads to robust performance under various noise conditions and rapid changes of f0 compared with the current state of the art f0 estimation algorithms

semi supervised classification via local spline regression this paper presents local spline regression for semi supervised classification the core idea in our approach is to introduce splines developed in sobolev space to map the data points directly to be class labels the spline is composed of polynomials and green s functions it is smooth nonlinear and able to interpolate the scattered data points with high accuracy specifically in each neighborhood an optimal spline is estimated via regularized least squares regression with this spline each of the neighboring data points is mapped to be a class label then the regularized loss is evaluated and further formulated in terms of class label vector finally all of the losses evaluated in local neighborhoods are accumulated together to measure the global consistency on the labeled and unlabeled data to achieve the goal of semi supervised classification an objective function is constructed by combining together the global loss of the local spline regressions and the squared errors of the class labels of the labeled data in this way a transductive classification algorithm is developed in which a globally optimal classification can be finally obtained in the semi supervised learning setting the proposed algorithm is analyzed and addressed into the laplacian regularization framework comparative classification experiments on many public data sets and applications to interactive image segmentation and image matting illustrate the validity of our method

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

uncorrelated fuzzy neighborhood preserving analysis based feature projection for driver drowsiness recognition driver drowsiness is reported as one of the main causal factors in many traffic accidents as it progressively impairs the driver s awareness about external events drowsiness detection can be approached through monitoring physiological signals while driving to correlate drowsiness with the change in the corresponding patterns of the electroencephalogram eeg electrooculogram eog and electrocardiogram ecg signals the main challenge in such an approach is to extract a set of features that can highly discriminate between the different drowsiness levels this paper proposes a new fuzzy neighborhood preserving analysis fnpa feature projection method that is used to extract the discriminant information relevant to the loss of attention caused by drowsiness unlike existing methods fnpa considers the fuzzy memberships of the input measurements into the different classes while constructing the graph laplacian thus it is able to identify both the discriminant and the geometrical structure of the input data while accounting for the overlapping nature of the drowsiness patterns furthermore in order to address the singularity problem that occurs in many real world problems the singular value decomposition svd and later the qr decomposition are utilized to extract a set of statistically uncorrelated features presenting the uncorrelated fnpa ufnpa in the current preliminary study with datasets collected from 31 subjects only while performing a driving simulation task the proposed method is capable of accurately classifying the drowsiness levels using a small number of features with an average accuracy of 93 on the other hand the possibility of developing a subject independent drowsiness recognition system is also investigated when the problem is converted into a binary classification task as imposed by the number of drowsiness levels exhibited by the drivers with accuracies ranging from 82 to 84

dimension reduction for aerodynamic design optimization the search for an optimal design in a high dimensional design space of a multivariate problem requires a sample size proportional or even exponential to the number of variables of the problem this curse of dimensionality places a computational burden on the cost of optimization especially when the problem uses expensive high fidelity simulations and may force one to try to reduce the dimensions of a problem traditional variable screening techniques reduce the dimensionality of the problem by removing variables that seem irrelevant to the design problem this practice fails when all the variables are equally relevant in the problem or when some variables are relevant only in some parts of the design space the present work describes a dimension reduction method called generative topographic mapping based on non linear latent models which transform a high dimensional data set into a low dimensional latent space without removing any variables it is first illustrated on a two dimensional branin function and then applied to a thirty dimensional airfoil problem the method is then compared with a global optimizer a genetic algorithm other dimension reduction methods principle component analysis and gaussian process latent variable models and with kriging surrogate models the method improves when the initial sample used for dimension reduction is filtered to contain only good designs

face recognition a convolutional neural network approach we present a hybrid neural network for human face recognition which compares favourably with other methods the system combines local image sampling a self organizing map som neural network and a convolutional neural network the som provides a quantization of the image samples into a topological space where inputs that are nearby in the original space are also nearby in the output space thereby providing dimensionality reduction and invariance to minor changes in the image sample and the convolutional neural network provides partial invariance to translation rotation scale and deformation the convolutional network extracts successively larger features in a hierarchical set of layers we present results using the karhunen loeve transform in place of the som and a multilayer perceptron mlp in place of the convolutional network for comparison we use a database of 400 images of 40 individuals which contains quite a high degree of variability in expression pose and facial details we analyze the computational complexity and discuss how new classes could be added to the trained recognizer

local graph embedding based on maximum margin criterion via fuzzy set abstract recently many algorithms based on locally graph embedding are proposed for dimensional reduction in nonlinear data however these algorithms are not effective when dealing with face images affected by variations in illumination conditions poses or perspectives and different facial expressions so distant data points are not deemphasized efficiently by locally graph embedding algorithms and it may degrade the performance of classification in order to solve the aforementioned problem this paper proposes a new efficient dimension reduction method local graph embedding method based on maximum margin criterion via fuzzy set for face recognition firstly the goal of this algorithm is preserved under nearest neighbor premise by constructing the fuzzy intrinsic graph and the fuzzy penalty graph secondly two novel fuzzy laplacian scatter matrices are calculated using fuzzy k nearest neighbor fknn in the proposed method finally maximum margin criterion mmc is used to avoid the small size sample problem the results of face recognition experiments on the orl yale and ar face databases demonstrate the effectiveness of our proposed method

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

shape classification by manifold learning in multiple observation spaces manifold learning is a non linear method with the aim of finding a constructive way to embed the data from a high dimensional space into a low dimensional one the improvement of shape classification is the major approach in this paper which is based on the continuity in feature space in accordance with the continuity in semantic one in this regard a non linear approach is employed to map the shape feature vectors to a new space while their semantics become similar with human opinion the euclidean distance between two feature vectors would be closed shapes are described by four contour based and region based techniques by the proposed method in other word they are described in four observation spaces furthermore a parameter space is learnt from multiple observation spaces based on fusion of dissimilarities in a supervised manner experimental results show the validity and efficiency of the proposed approach for shape classification over a variety of standard shape datasets

spectral fluctuation properties of constrained unitary ensembles of gaussian distributed random matrices we investigate the spectral fluctuation properties of constrained ensembles of random matrices defined by the condition that a number nq of matrix elements vanish identically that condition is imposed in unitarily invariant form in the limit of large matrix dimension we show that as long as nq is smaller than a critical value at which the quadratic level repulsion of the gaussian unitary ensemble of random matrices may be destroyed all spectral fluctuation measures have the same form as for the gaussian unitary ensemble

review of the k body embedded ensembles of gaussian random matrices the embedded ensembles were introduced by mon and french 1975 ann phys ny 95 90 as physically more plausible stochastic models of many body systems governed by one and two body interactions than provided by standard random matrix theory we review several approaches aimed at determining the spectral density the spectral fluctuation properties and the ergodic properties of these ensembles moments methods numerical simulations the replica trick the eigenvector decomposition of the matrix of second moments and supersymmetry the binary correlation approximation and the study of correlations between matrix elements

su n wigner racah algebra for the matrix of second moments of embedded gaussian unitary ensemble of random matrices recently pluhar and weidenmuller ann phys n y 297 344 2002 showed that the eigenvectors of the matrix of second moments of embedded gaussian unitary ensemble of random matrices generated by k body interactions egue k for m fermions in n single particle states are su n wigner coefficients and derived also an expression for the eigenvalues going beyond this work we will show that the eigenvalues of this matrix are square of a su n racah coefficient and thus the matrix of second moments of egue k is solved completely by su n wigner racah algebra

local multidimensional scaling for nonlinear dimension reduction graph drawing and proximity analysis in the past decade there has been a resurgence of interest in nonlinear dimension reduction among new proposals are local linear embedding isomap and kernel principal components analysis which all construct global low dimensional embeddings from local affine or metric information we introduce a competing method called local multidimensional scaling lmds like lle isomap and kpca lmds constructs its global embedding from local information but it uses instead a combination of mds and force directed graph drawing we apply the force paradigm to create localized versions of mds stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces we solve the problem of tuning parameter selection with a meta criterion that measures how well the sets of k nearest neighbors agree between the data and the embedding tuned lmds seems to be able to outperform mds pca lle isomap and kpca as illustrated with two well known image datasets the meta criterion can also be

quadratic projection based feature extraction with its application to biometric recognition this paper presents a novel quadratic projection based feature extraction framework where a set of quadratic matrices is learned to distinguish each class from all other classes we formulate quadratic matrix learning qml as a standard semidefinite programming sdp problem however the conventional interior point sdp solvers do not scale well to the problem of qml for high dimensional data to solve the scalability of qml we develop an efficient algorithm termed dualqml based on the lagrange duality theory to extract nonlinear features to evaluate the feasibility and effectiveness of the proposed framework we conduct extensive experiments on biometric recognition experimental results on three representative biometric recognition tasks including face palmprint and ear recognition demonstrate the superiority of the dualqml based feature extraction algorithm compared to the current state of the art algorithms highlightsa novel quadratic projection based feature extraction framework is developed a set of quadratic matrices is learnt to distinguish each class from other classes quadratic matrix learning qml is formulated as an sdp problem an efficient algorithm is developed to solve qml based on lagrange duality theory experiments on biometric recognition show the effectiveness of our algorithm

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach

data driven facial animation via semi supervised local patch alignment abstract this paper reports a novel data driven facial animation technique which drives a neutral source face to get the expressive target face using a semi supervised local patch alignment framework we define the local patch and assume that there exists a linear transformation between a patch of the target face and the intrinsic embedding of the corresponding patch of the source face based on this assumption we compute the intrinsic embeddings of source patches and align these embeddings to form the result during the course of alignment we use a set of motion data as shape regularizer to impel the result to approach the unknown target face the intrinsic embedding can be computed through both locally linear embedding and local tangent space alignment experimental results indicate that the proposed framework can obtain decent face driving results quantitative and qualitative evaluations of the proposed framework demonstrate its superiority to existing methods

discriminative zernike and pseudo zernike moments for face recognition usually magnitude coefficients of some selected orders of zms and pzms have been used as invariant image features the careful selection of the set of features with higher discrimination competence may increase the recognition performance in this paper the authors have used a statistical method to estimate the discrimination strength of all the extracted coefficients of zms and pzms whereas for classification only the coefficients with estimated higher discrimination strength are used in the feature vector the performance of these selected discriminative zms dzms and discriminative pzms dpzms features have been compared to that of their corresponding conventional approaches on yale orl and feret databases against illumination expression scale and pose variations an extension to these dzms and dpzms have been proposed by combining them with pca and fld it has been observed from the exhaustive experimentation that the recognition rate is improved by 2 6 at reduced dimensions and with less computational complexity than that of using the successive zms and pzms features

stable local dimensionality reduction approaches dimensionality reduction is a big challenge in many areas a large number of local approaches stemming from statistics or geometry have been developed however in practice these local approaches are often in lack of robustness since in contrast to maximum variance unfolding mvu which explicitly unfolds the manifold they merely characterize local geometry structure moreover the eigenproblems that they encounter are hard to solve we propose a unified framework that explicitly unfolds the manifold and reformulate local approaches as the semi definite programs instead of the above mentioned eigenproblems three well known algorithms locally linear embedding lle laplacian eigenmaps le and local tangent space alignment ltsa are reinterpreted and improved within this framework several experiments are presented to demonstrate the potential of our framework and the improvements of these local algorithms

eigenvalue bounds for an alignment matrix in manifold learning abstract this paper presents a spectral analysis for an alignment matrix that arises in reconstruction of a global coordinate system from local coordinate systems through alignment in manifold learning some characterizations of its eigenvalues and its null space as well as a lower bound for the smallest positive eigenvalue are given which generalize earlier results of li et al 2007 4 to include a more general situation that arises in alignments of local sections of different dimensions our results provide a theoretical understanding of the local tangent space alignment ltsa method zhang and zha 2004 12 for nonlinear manifold learning and address some computational issues related to the method

generalized entanglement as a framework for complex quantum systems purity versus delocalization measures we establish contact between the delocalization properties of pure quantum states as quantified by their number of principal components and the average generalized entanglement properties as quantified by purity measures relative to different observable sets we find that correlations between products of state vector components with respect to hamming distance play an important role in the structure of subsystem based purity measures in particular we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged over a maximal set of mutually unbiased product bases furthermore we provide a method for computing the expected amount of generalized entanglement with respect to an arbitrary observable set for random pure states specific examples and an explicit application to a disordered quantum spin chain are discussed

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

local linear regression on manifolds and its geometric interpretation high dimensional data analysis has been an active area and the main focus areas have been variable selection and dimension reduction in practice it occurs often that the variables are located on an unknown lower dimensional nonlinear manifold under this manifold assumption one purpose of this article is regression and gradient estimation on the manifold and another is developing a new tool for manifold learning as regards the first aim we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold and performing the popular local linear regression llr on a tangent plane estimate an immediate consequence is a dramatic reduction in the computational time when the ambient space dimension p d we provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature boundary and nonuniform sampling effects we propose a bandwidth selector that can handle heteroscedastic errors with re

face recognition based on the uncorrelated discriminant transformation abstract the extraction of discriminant features is the most fundamental and important problem in face recognition this paper presents a method to extract optimal discriminant features for face images by using the uncorrelated discriminant transformation and kl expansion experiments on the orl database and the nust603 database have been performed experimental results show that the uncorrelated discriminant transformation is superior to the foley sammon discriminant transformation and the new method to extract uncorrelated discriminant features for face images is very effective an error rate of 2 5 is obtained with the experiments on the orl database an average error rate of 1 2 is obtained with the experiments on the nust603 database experiments show that by extracting uncorrelated discriminant features face recognition could be performed with higher accuracy on lower than 16 16 resolution mosaic images it is suggested that for the uncorrelated discriminant transformation the optimal face image resolution can be regarded as the resolution m n which makes the dimensionality n mn of the original image vector space be larger and closer to the number of known face classes

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

extension and generalization of the gay berne potential in this paper we report a generalized form for the range parameter governing the pair interaction between soft ellipsoidal particles for nonequivalent uniaxial particles we extend the berne pechukas gaussian overlap formalism to obtain an explicit expression for this range parameter we confirm that this result is identical to that given by an approach that is not widely recognized based on an approximation to the perram wertheim hard ellipsoid contact function we further illustrate the power of the latter route by using it to write down the range parameter for the interaction between two nonequivalent biaxial particles an explicit interaction potential for nonequivalent uniaxial particles is obtained by importing the uniaxial range parameter result into the standard gay berne form a parametrization of this potential is investigated for a rod disk interaction

classification of symmetric positive definite matrices based on bilinear isometric riemannian embedding abstract because covariance features with the form of symmetric positive definite matrices lie on riemannian manifold classification on riemannian manifold could possess high performance in many applications unfortunately the applicability of classification methods developed on riemannian manifold is limited by their huge computational complexities particularly with the feature data on high dimensional riemannian manifold to alleviate the problem of computational cost in this paper a simple yet efficient dimensionality reduction algorithm bilinear isometric riemannian embedding is derived to construct a low dimensional embedding from high dimensional riemannian manifold to this end we model the bilinear isometric mapping to identify a low dimensional embedding that maximizes the preservation of riemannian geodesic distance a supervised classification method embedding discriminant analysis is then proposed based on the low dimensional embedding experimental results on image and electroencephalogram reveal that the proposed algorithms can efficiently extract the distance preserving embedding and obtain higher classification performance

dimensionality reduction an interpretation from manifold regularization perspective in this paper we propose to unify various dimensionality reduction algorithms by interpreting the manifold regularization mr framework in a new way although the mr framework was originally proposed for learning we utilize it to give a unified treatment for many dimensionality reduction algorithms from linear to nonlinear supervised to unsupervised and single class to multi class approaches in addition the framework can provide a general platform to design new dimensionality reduction algorithms the framework is expressed in the form of a regularized fitting problem in a reproducing kernel hilbert space it consists of one error part and two regularization terms the complexity term and the smoothness term the error part measures the difference between the estimated low dimensional data distribution and the true high dimensional data distribution or the difference between the estimated and targeted low dimensional representations of data the complexity term is a measurement of the complexity of the feature mapping for dimensionality reduction and the smoothness term reflects the intrinsic structure of data based on the framework we propose a manifold regularized kernel least squares mr kls method which can efficiently learn an explicit feature mapping in the semi supervised sense experiments show that our approach is effective for out of sample extrapolation c 2014 elsevier inc all rights reserved

quantum chaos delocalization and entanglement in disordered heisenberg models we investigate disordered one and two dimensional heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many body and quantum information perspectives special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics delocalization and entanglement in the presence of nontrivial symmetries the implication of the basis dependence of state delocalization indicators such as the number of principal components is addressed and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin and fermion purity measures their dependence on delocalization and on energy spectrum statistics being examined a distinctive correlation between entanglement delocalization and integrability is uncovered which may be generic to systems described by the two body random ensemble and may point to a new diagnostic tool for quantum chaos analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full hilbert space are also established and compared with random matrix theory predictions

finding representative landmarks of data on manifolds data driven non parametric models such as manifold learning algorithms are promising data analysis tools however to fit an off training set data point in a learned model one must first locate the point in the training set this query has a time cost proportional to the problem size which limits the model s scalability in this paper we address the problem of selecting a subset of data points as the landmarks helping locate the novel points on the data manifolds we propose a new category of landmarks defined with the following property the way the landmarks represent the data in the ambient euclidean space should resemble the way they represent the data on the manifold given the data points and the subset of landmarks we provide procedures to test whether the proposed property presents for the choice of landmarks if the data points are organized with a neighbourhood graph as it is often conducted in practice we interpret the proposed property in terms of the graph topology we also discuss the extent to which the topology is preserved for landmark set passing our test procedure another contribution of this work is to develop an optimization based scheme to adjust an existing landmark set which can improve the reliability for representing the manifold data experiments on the synthetic data and the natural data have been done the results support the proposed properties and algorithms

trustworthy dimension reduction for visualization different data sets abstract a new nonlinear dimension reduction dr method which is called trustworthy stochastic proximity embedding tspe is introduced in this paper to visualize different types of data sets tspe overcome the main shortcomings of the dr by sending the false neighbour points to the correct locations and preserving the neighbourhood relation to the true neighbours which are inside the local neighbourhood the visualization of our proposed method displays the trustworthy useful and meaningful colours where the objects of the image can be easily distinguished the performances of tspe and 20 dimension reduction methods are compared and the efficiency of the proposed method in both visualization accuracy and computational cost is shown the results showed the ability of our method in preserving neighbourhood relation where they revealed more interested information in real data set the efficiency of the visualization of tensor images data sets by tspe might help the specialist to make a good decision about a patient s treatment the comparison with experimental data set as three dimensions of curved cylinder showed the ability of tspe to unfold this complex data set efficiently whilst preserving most information of the original data set

evolutionary pursuit and its application to face recognition introduces evolutionary pursuit ep as an adaptive representation method for image encoding and classification in analogy to projection pursuit ep seeks to learn an optimal basis for the dual purpose of data compression and pattern classification it should increase the generalization ability of the learning machine as a result of seeking the trade off between minimizing the empirical risk encountered during training and narrowing the confidence interval for reducing the guaranteed risk during testing it therefore implements strategies characteristic of ga for searching the space of possible solutions to determine the optimal basis it projects the original data into a lower dimensional whitened principal component analysis pca space directed random rotations of the basis vectors in this space are searched by ga where evolution is driven by a fitness function defined by performance accuracy empirical risk and class separation confidence interval accuracy indicates the extent to which learning has been successful while separation gives an indication of expected fitness the method has been tested on face recognition using a greedy search algorithm to assess both accuracy and generalization capability the data includes for each subject images acquired at different times or under different illumination conditions ep has better recognition performance than pca eigenfaces and better generalization abilities than the fisher linear discriminant fisherfaces