application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

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

constructing descriptive and discriminative nonlinear features rayleigh coefficients in kernel feature spaces we incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination employing a unified framework in terms of a nonlinearized variant of the rayleigh coefficient we propose nonlinear generalizations of fisher s discriminant and oriented pca using support vector kernel functions extensive simulations show the utility of our approach

reconstruction of planar conductivities in subdomains from incomplete data we consider the problem of recovering a sufficiently smooth isotropic conductivity from interior knowledge of the magnitude of the current density field j generated by an imposed voltage potential f on the boundary in any dimension n geq2 we show that equipotential sets are global area minimizers in the conformal metric determined by j in two dimensions assuming the boundary voltage is almost two to one we prove uniqueness of the minimization problem this yields two results on reconstruction from incomplete data in the first case j is known in all of omega but the almost two to one f is know only on subintervals of the boundary the second case assumes that j is known only in an appropriate subdomain tilde omega our method works provided that tilde omega contains entire equipotential curves joining boundary points based on solving two point boundary value problems for the geodesic system we give a procedure to determine whether tilde omega satisfies this pro

the small sample size problem of ica a comparative study and analysis on the small sample size problems such as appearance based recognition empirical studies have shown that ica projections have trivial effect on improving the recognition performance over whitened pca however what causes the ineffectiveness of ica is still an open question in this study we find out that this small sample size problem of ica is caused by a special distributional phenomenon of the high dimensional whitened data all data points are similarly distant and nearly perpendicular to each other in this situation ica algorithms tend to extract the independent features simply by the projections that isolate single or very few samples apart and congregate all other samples around the origin without any concern on the clustering structure our comparative study further shows that the ica projections usually produce misleading features whose generalization ability is generally worse than those derived by random projections thus further selection of the ica features is possibly meaningless to address the difficulty in pursuing low dimensional features we introduce a locality pursuit approach which applies the locality preserving projections in the high dimensional whitened space experimental results show that the locality pursuit performs better than ica and other conventional approaches such as eigenfaces laplacianfaces and fisherfaces

multilinear spatial discriminant analysis for dimensionality reduction in the last few years great efforts have been made to extend the linear projection technique lpt for multidimensional data i e tensor generally referred to as the multilinear projection technique mpt the vectorized nature of lpt requires high dimensional data to be converted into vector and hence may lose spatial neighborhood information of raw data mpt well addresses this problem by encoding multidimensional data as general tensors of a second or even higher order in this paper we propose a novel multilinear projection technique called multilinear spatial discriminant analysis msda to identify the underlying manifold of high order tensor data msda considers both the nonlocal structure and the local structure of data in the transform domain seeking to learn the projection matrices from all directions of tensor data that simultaneously maximize the nonlocal structure and minimize the local structure different from multilinear principal component analysis mpca that aims to preserve the global structure and tensor locality preserving projection tlpp that is in favor of preserving the local structure msda seeks a tradeoff between the nonlocal global and local structures so as to drive its discriminant information from the range of the non local structure and the range of the local structure this spatial discriminant characteristic makes msda have more powerful manifold preserving ability than tlpp and mpca theoretical analysis shows that traditional mpts such as multilinear linear discriminant analysis tlpp mpca and tensor maximum margin criterion could be derived from the msda model by setting different graphs and constraints extensive experiments on face databases orl cmu pie and the extended yale b and the weizmann action database demonstrate the effectiveness of the proposed msda method

least gradient problems with neumann boundary condition abstract we study existence of minimizers of the least gradient problem inf v b v g x d v where b v g v b v g v 1 x p r n r is a convex continuous and homogeneous function of degree 1 with respect to the p variable and g satisfies the compatibility condition g d s 0 we prove that for every 0 g l there are infinitely many minimizers in b v moreover there exists a divergence free vector field t l n that determines the structure of level sets of all minimizers i e t determines d u d u d u a e in for every minimizer u we also prove some existence results for general 1 laplacian type equations with neumann boundary condition a numerical algorithm is presented that simultaneously finds t and a minimizer of the above least gradient problem applications of the results in conductivity imaging are discussed

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

multi class fukunaga koontz discriminant analysis for enhanced face recognition linear subspace learning methods such as fisher s linear discriminant analysis lda unsupervised discriminant projection udp and locality preserving projections lpp have been widely used in face recognition applications as a tool to capture low dimensional discriminant information however when these methods are applied in the context of face recognition they often encounter the small sample size problem in order to overcome this problem a separate principal component analysis pca step is usually adopted to reduce the dimensionality of the data however such a step may discard dimensions that contain important discriminative information that can aid classification performance in this work we propose a new idea which we named multi class fukunaga koontz discriminant analysis fkda by incorporating the fukunaga koontz transform within the optimization for maximizing class separation criteria in lda udp and lpp in contrast to traditional lda udp and lpp our approach can work with very high dimensional data as input without requiring a separate dimensionality reduction step to make the scatter matrices full rank in addition the fkda formulation seeks optimal projection direction vectors that are orthogonal which the existing methods cannot guarantee and it has the capability of finding the exact solutions to the trace ratio objective in discriminant analysis problems while traditional methods can only deal with a relaxed and inexact ratio trace objective we have shown using six face database in the context of large scale unconstrained face recognition face recognition with occlusions and illumination invariant face recognition under closed set semi open set and open set recognition scenarios that our proposed fkda significantly outperforms traditional linear discriminant subspace learning methods as well as five other competing algorithms highlightssolve small sample size problem in lda udp lpp using fkt formulation can work with high dimensional data without inverting any scatter matrices finds optimal projection direction vectors that are orthogonal finds exact solutions to the objective in the form of trace ratio improvement in unconstrained face recognition scenarios

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

on the solutions to 1 laplacian equation with l1 data in the present paper we study the behaviour as p goes to 1 of the renormalized solutions to the problems 0 1 div up p 2 up fin up 0on where p 1 is a bounded open set of rn n 2 with lipschitz boundary and f belongs to l1 we prove that these renormalized solutions pointwise converge up to subsequences to a function u with a suitable definition of solution we also prove that u is a solution to a limit problem moreover we analyze the situation occurring when more regular data f are considered

cardinality of phase transition of ising models on closed cayley trees abstract we provide a new approach to study phase transition for ising models on closed cayley trees it is shown that there may exist more than one limiting measure for an ising model on an infinite cayley tree in both ferromagnetic and antiferromagnetic cases the exact critical values of the parameters corresponding to the pair interactions are obtained for both cases

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

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

optimizing the data dependent kernel under a unified kernel optimization framework the kernel functions play a central role in kernel methods accordingly over the years the optimization of kernel functions has been a promising research area ideally fisher discriminant criteria can be used as an objective function to optimize the kernel function to augment the margin between different classes unfortunately fisher criteria are optimal only in the case that all the classes are generated from underlying multivariate normal distributions of common covariance matrix but different means and each class is expressed by a single cluster due to the assumptions fisher criteria obviously are not a suitable choice as a kernel optimization rule in some applications such as the multimodally distributed data in order to solve this problem recently many improved discriminant criteria dc have been also developed therefore to apply these discriminant criteria to kernel optimization in this paper based on a data dependent kernel function we propose a unified kernel optimization framework which can use any discriminant criteria formulated in a pairwise manner as the objective functions under the kernel optimization framework to employ different discriminant criteria one has to only change the corresponding affinity matrices without having to resort to any complex derivations in feature space experimental results based on some benchmark data demonstrate the efficiency of our method

global and local structure preservation for feature selection the recent literature indicates that preserving global pairwise sample similarity is of great importance for feature selection and that many existing selection criteria essentially work in this way in this paper we argue that besides global pairwise sample similarity the local geometric structure of data is also critical and that these two factors play different roles in different learning scenarios in order to show this we propose a global and local structure preservation framework for feature selection glspfs which integrates both global pairwise sample similarity and local geometric data structure to conduct feature selection to demonstrate the generality of our framework we employ methods that are well known in the literature to model the local geometric data structure and develop three specific glspfs based feature selection algorithms also we develop an efficient optimization algorithm with proven global convergence to solve the resulting feature selection problem a comprehensive experimental study is then conducted in order to compare our feature selection algorithms with many state of the art ones in supervised unsupervised and semisupervised learning scenarios the result indicates that 1 our framework consistently achieves statistically significant improvement in selection performance when compared with the currently used algorithms 2 in supervised and semisupervised learning scenarios preserving global pairwise similarity is more important than preserving local geometric data structure 3 in the unsupervised scenario preserving local geometric data structure becomes clearly more important and 4 the best feature selection performance is always obtained when the two factors are appropriately integrated in summary this paper not only validates the advantages of the proposed glspfs framework but also gains more insight into the information to be preserved in different feature selection tasks

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

supervised dimension reduction of intrinsically low dimensional data high dimensional data generated by a system with limited degrees of freedom are often constrained in low dimensional manifolds in the original space in this article we investigate dimension reduction methods for such intrinsically low dimensional data through linear projections that preserve the manifold structure of the data for intrinsically one dimensional data this implies projecting to a curve on the plane with as few intersections as possible we are proposing a supervised projection pursuit method that can be regarded as an extension of the single index model for nonparametric regression we show results from a toy and two robotic applications

on weak convergence of the douglas rachford method we prove that the sequences generated by the douglas rachford method converge weakly to zero of the sum of two maximal monotone operators using new tools introduced in recent works of eckstein and the author the assumption of maximal monotonicity of the sum is also removed using a recent result of bauschke

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

fusing local patterns of gabor magnitude and phase for face recognition gabor features have been known to be effective for face recognition however only a few approaches utilize phase feature and they usually perform worse than those using magnitude feature to investigate the potential of gabor phase and its fusion with magnitude for face recognition in this paper we first propose local gabor xor patterns lgxp which encodes the gabor phase by using the local xor pattern lxp operator then we introduce block based fisher s linear discriminant bfld to reduce the dimensionality of the proposed descriptor and at the same time enhance its discriminative power finally by using bfld we fuse local patterns of gabor magnitude and phase for face recognition we evaluate our approach on feret and frgc 2 0 databases in particular we perform comparative experimental studies of different local gabor patterns we also make a detailed comparison of their combinations with bfld as well as the fusion of different descriptors by using bfld extensive experimental results verify the effectiveness of our lgxp descriptor and also show that our fusion approach outperforms most of the state of the art approaches

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

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

learning discriminative singular value decomposition representation for face recognition abstract face representation is a critical step in face recognition recently singular value decomposition svd based representation methods have attracted researchers attentions for their power of alleviating the facial variations the svd representation reveals that the svd basis set is important for the recognition purpose and the corresponding singular values svs are regulated to form a more effective representation image however there exists a common problem in the existing svd based representation methods they all empirically make a rule to regulate the svs which is obviously not optimal in theory to address this problem in this paper we propose a novel method named learning discriminative singular value decomposition representation ldsvdr for face recognition we build an individual svd basis set for each image and then learn a common set of svs by taking account of the information in the basis sets according to a discriminant criterion across the training images the proposed model is solved by sequential quadratic programming sqp method extensive experiments are conducted on three popular face databases and the results demonstrate the effectiveness of our method when dealing with variations of illumination occlusion disguise and face sketch recognition task

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

can the virtual labels obtained by traditional lp approaches be well encoded in wlr semisupervised dimension reduction via virtual label regression first derives the virtual labels of unlabeled data by employing a newly designed label propagation lp approach called special random walk srw and then encodes them in a weighted linear regression model nie et al 2011 highlighted two important characteristics of srw nonexistent in the previous lp approaches outlier detection and probability value output which guarantee the elegant encoding of the resultant virtual labels in the weighted label regression however in this brief we show that the relationship between the srw and the previous work on lp is very close naturally a problem deserving investigation is whether traditional lp approaches are indeed unable to share the above two characteristics of srw we aim to address this problem

continuity of minimizers to weighted least gradient problems abstract we revisit the question of existence and regularity of minimizers to the weighted least gradient problem with dirichlet boundary condition inf a x d u u b v u g where g c and a c 2 is a weight function that is bounded away from zero under suitable geometric conditions on the domain r n we construct continuous solutions of the above problem for any dimension n 2 by extending the sternberg williams ziemer technique sternberg et al 1992 to this setting of inhomogeneous variations we show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a 2 n 1 i n this result complements the approach in jerrard et al 2018 since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities the proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by leon simon in simon 1987

maximal linear embedding for dimensionality reduction over the past few decades dimensionality reduction has been widely exploited in computer vision and pattern analysis this paper proposes a simple but effective nonlinear dimensionality reduction algorithm named maximal linear embedding mle mle learns a parametric mapping to recover a single global low dimensional coordinate space and yields an isometric embedding for the manifold inspired by geometric intuition we introduce a reasonable definition of locally linear patch maximal linear patch mlp which seeks to maximize the local neighborhood in which linearity holds the input data are first decomposed into a collection of local linear models each depicting an mlp these local models are then aligned into a global coordinate space which is achieved by applying mds to some randomly selected landmarks the proposed alignment method called landmarks based global alignment lga can efficiently produce a closed form solution with no risk of local optima it just involves some small scale eigenvalue problems while most previous aligning techniques employ time consuming iterative optimization compared with traditional methods such as isomap and lle our mle yields an explicit modeling of the intrinsic variation modes of the observation data extensive experiments on both synthetic and real data indicate the effectivity and efficiency of the proposed algorithm

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

a two stage linear discriminant analysis via qr decomposition linear discriminant analysis lda is a well known method for feature extraction and dimension reduction it has been used widely in many applications involving high dimensional data such as image and text classification an intrinsic limitation of classical lda is the so called singularity problems that is it fails when all scatter matrices are singular many lda extensions were proposed in the past to overcome the singularity problems among these extensions pca lda a two stage method received relatively more attention in pca lda the lda stage is preceded by an intermediate dimension reduction stage using principal component analysis pca most previous lda extensions are computationally expensive and not scalable due to the use of singular value decomposition or generalized singular value decomposition in this paper we propose a two stage lda method namely lda qr which aims to overcome the singularity problems of classical lda while achieving efficiency and scalability simultaneously the key difference between lda qr and pca lda lies in the first stage where lda qr applies qr decomposition to a small matrix involving the class centroids while pca lda applies pca to the total scatter matrix involving all training data points we further justify the proposed algorithm by showing the relationship among lda qr and previous lda methods extensive experiments on face images and text documents are presented to show the effectiveness of the proposed algorithm

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

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

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

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

category specific object segmentation via unsupervised discriminant shape category specific object segmentation has been a long standing research topic in pattern recognition this paper presents an unsupervised discriminant shape uds to address category specific object segmentation by incorporating the proposed shape prior into an intuitive energy minimization framework recently based on the region proposal methods deep convolutional neural networks cnns provide access to candidate segments in categories of interest from images however the segments obtained from bottom up proposals tend to undershoot or overshoot objects and are easily classified into one specific class to address this problem we propose an unsupervised discriminant projection based clustering algorithm udc to obtain more precise shape prior to guide the segmentation and the class specific proposals are clustered based on their projections onto the discriminant projection direction based on the set of proposals we then obtain the prior information of foreground uds with an easy voting scheme the derived uds prior is finally utilized in the subsequent energy minimizing formulation based figure ground segmentation we conduct extensive and comprehensive evaluations on the msrc object discovery fashionista and pascal s datasets demonstrating the effectiveness and robustness of the uds based segmentation highlightsa novel unsupervised discriminant shape is proposed for segmenting images an unsupervised discriminant clustering method is built for precise shape prior our model can produce reliable segmentation for category specific objects evaluations on 4 datasets demonstrate the effectiveness of our method

on a degenerate problem in the calculus of variations we establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations the proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets we exploit the uniqueness and the regularity results to explore some of their qualitative properties in particular we emphasize the link between the supports of the solutions and the cheeger 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

a weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging we consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes the required interior data can be obtained by means of mri measurements on the boundary we only require knowledge of the electrodes their impedances and the corresponding average input currents from the mathematical point of view this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the complete electrode model cem of somersalo cheney and isaacson siam j appl math 52 1992 pp 1023 1040 we show that this variational problem has nonunique solutions the surprising discovery is that the physical data is still sufficient to determine the geometry of the connected components of the level sets of the minimizers we thus obtain an interesting phase retrieval result knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude we characterize locally the nonuniqueness in the variational problem in two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity the proofs involve a maximum principle and a new regularity up to the boundary result for the cem boundary conditions a nonlinear algorithm is proposed and implemented to illustrate the theoretical results

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

a novel supervised dimensionality reduction algorithm graph based fisher analysis in this paper a novel supervised dimensionality reduction dr algorithm called graph based fisher analysis gbfa is proposed more specifically we redefine the intrinsic and penalty graph and trade off the importance degrees of the same class points to the intrinsic graph and the importance degrees of the not same class points to the penalty graph by a strictly monotone decreasing function then the novel feature extraction criterion based on the intrinsic and penalty graph is applied for the non linearly separable problems we study the kernel extensions of gbfa with respect to positive definite kernels and indefinite kernels respectively in addition experiments are provided for analyzing and illustrating our results

on boundary detachment phenomena for the total variation flow with dynamic boundary conditions abstract we combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions we analyze the behavior of facets at the parts of the boundary where these conditions are imposed we devote particular attention to the radially symmetric data we observe that the boundary layer detachment actually can happen at concave parts of the boundary

a shape and texture based enhanced fisher classifier for face recognition this paper introduces a new face coding and recognition method the enhanced fisher classifier efc which employs the enhanced fisher linear discriminant model efm on integrated shape and texture features shape encodes the feature geometry of a face while texture provides a normalized shape free image the dimensionalities of the shape and the texture spaces are first reduced using principal component analysis constrained by the efm for enhanced generalization the corresponding reduced shape and texture features are then combined through a normalization procedure to form the integrated features that are processed by the efm for face recognition experimental results using 600 face images corresponding to 200 subjects of varying illumination and facial expressions show that 1 the integrated shape and texture features carry the most discriminating information followed in order by textures masked images and shape images and 2 the new coding and face recognition method efc performs the best among the eigenfaces method using l sub 1 or l sub 2 distance measure and the mahalanobis distance classifiers using a common covariance matrix for all classes or a pooled within class covariance matrix in particular efc achieves 98 5 recognition accuracy using only 25 features

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

global plus local linear discriminant analysis lda is one of the most popular supervised feature extraction techniques used in machine learning and pattern classification however lda only captures global geometrical structure information of the data and ignores the geometrical structure information of local data points though many articles have been published to address this issue most of them are incomplete in the sense that only part of the local information is used we show here that there are total three kinds of local information namely local similarity information local intra class pattern variation and local inter class pattern variation we first propose a new method called enhanced within class lda ewlda algorithm to incorporate the local similarity information and then propose a complete framework called complete global local lda cglda algorithm to incorporate all these three kinds of local information experimental results on two image databases demonstrate the effectiveness of our algorithms there are three kinds of local information that exist for linear discriminant analysis for incorporating the local similarity information we propose the ewlda algorithm for incorporating the three kinds of local information we propose the cglda algorithm

recursion relations for the q state potts model on a closed cayley tree abstract recursion relations which determine the free energy of a q state potts model on a closed cayley tree are derived the fixed points of these relations and their relative stabilities are discussed in detail

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

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

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

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

local linear discriminant analysis framework using sample neighbors the linear discriminant analysis lda is a very popular linear feature extraction approach the algorithms of lda usually perform well under the following two assumptions the first assumption is that the global data structure is consistent with the local data structure the second assumption is that the input data classes are gaussian distributions however in real world applications these assumptions are not always satisfied in this paper we propose an improved lda framework the local lda llda which can perform well without needing to satisfy the above two assumptions our llda framework can effectively capture the local structure of samples according to different types of local data structure our llda framework incorporates several different forms of linear feature extraction approaches such as the classical lda and principal component analysis the proposed framework includes two llda algorithms a vector based llda algorithm and a matrix based llda mllda algorithm mllda is directly applicable to image recognition such as face recognition our algorithms need to train only a small portion of the whole training set before testing a sample they are suitable for learning large scale databases especially when the input data dimensions are very high and can achieve high classification accuracy extensive experiments show that the proposed algorithms can obtain good classification results

phase transitions for the ising model on the closed cayley tree abstract the closed cayley tree with the ising model which was defined by jelitto for a branching ratio of two is here generalized to an arbitrary branching ratio the occurrence of a discontinuity in the spin spin correlation functions is demonstrated and the free energy is obtained the topology of the network with its interesting correlation results makes the model a candidate for further investigation

subspace based discrete transform encoded local binary patterns representations for robust periocular matching on nist s face recognition grand challenge in this paper we employ several subspace representations principal component analysis unsupervised discriminant projection kernel class dependence feature analysis and kernel discriminant analysis on our proposd discrete transform encoded local binary patterns dt lbp to match periocular region on a large data set such as nist s face recognition grand challenge frgc ver2 database we strictly follow frgc experiment 4 protocol which involves 1 to 1 matching of 8014 uncontrolled probe periocular images to 16 028 controlled target periocular images 128 million pairwise face match comparisons the performance of the periocular region is compared with that of full face with different illumination preprocessing schemes the verification results on periocular region show that subspace representation on dt lbp outperforms lbp significantly and gains a giant leap from traditional subspace representation on raw pixel intensity additionally our proposed approach using only the periocular region is almost as good as full face with only 2 5 reduction in verification rate at 0 1 false accept rate yet we gain tolerance to expression occlusion and capability of matching partial faces in crowds in addition we have compared the best standalone dt lbp descriptor with eight other state of the art descriptors for facial recognition and achieved the best performance the two general frameworks are our major contribution 1 a general framework that employs various generative and discriminative subspace modeling techniques for dt lbp representation and 2 a general framework that encodes discrete transforms with local binary patterns for the creation of robust descriptors

face recognition using nearest feature space embedding face recognition algorithms often have to solve problems such as facial pose illumination and expression pie to reduce the impacts many researchers have been trying to find the best discriminant transformation in eigenspaces either linear or nonlinear to obtain better recognition results various researchers have also designed novel matching algorithms to reduce the pie effects in this study a nearest feature space embedding called nfs embedding algorithm is proposed for face recognition the distance between a point and the nearest feature line nfl or the nfs is embedded in the transformation through the discriminant analysis three factors including class separability neighborhood structure preservation and nfs measurement were considered to find the most effective and discriminating transformation in eigenspaces the proposed method was evaluated by several benchmark databases and compared with several state of the art algorithms according to the compared results the proposed method outperformed the other algorithms

graph regularized multiset canonical correlations with applications to joint feature extraction abstract multiset canonical correlation analysis mcca is a powerful technique for analyzing linear correlations among multiple representation data however it usually fails to discover the intrinsic geometrical and discriminating structure of multiple data spaces in real world applications in this paper we thus propose a novel algorithm called graph regularized multiset canonical correlations grmccs which explicitly considers both discriminative and intrinsic geometrical structure in multiple representation data grmcc not only maximizes between set cumulative correlations but also minimizes local intraclass scatter and simultaneously maximizes local interclass separability by using the nearest neighbor graphs on within set data thus it can leverage the power of both mcca and discriminative graph laplacian regularization extensive experimental results on the ar cmu pie yale b at t and eth 80 datasets show that grmcc has more discriminating power and can provide encouraging recognition results in contrast with the state of the art algorithms

the constrained least gradient problem in rn we consider the constrained least gradient problem inf u dx u c 0 1 u 1 a e u g on which arises as the relaxation of a nonconvex problem in optimal design we establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem we also establish the uniqueness of solutions and discuss their structure

low rank preserving embedding abstract in this paper we consider the problem of linear dimensionality reduction with the novel technique of low rank representation which is a promising tool of discovering subspace structures of given data existing approaches based on graph embedding usually capture structure of data via stacking the local structure of each datum such as neighborhood graph l1 graph and l2 graph yet they lack explicit discrimination between those local structures and suffer from corrupted samples to this end we propose a new linear dimensionality reduction method by virtue of the lowest rank representation lrr of data which is dubbed low rank preserving embedding lrpe different from the traditional routes lrpe achieves all data self representations jointly and can thus extract the global structure of a data set as a whole the global low rank constraint explicitly enforces the lrr matrix to be block diagonal form so that the samples with a similar intrinsic structure which are more likely to be from the same class are described by a similar set of bases hence lrpe is discriminative even if no class labels are provided benefiting from the robust lrr lrpe is also robust to various noises and errors in data besides we rewritten all related methods into a unified formulation followed by a detailed solution and clear comparisons finally we conduct extensive experiments on publicly available data sets for data visualization and classification the inspiring experimental results show the effectiveness the cheap computation and the robustness of the proposed method

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

tree structured nuclear norm approximation with applications to robust face recognition structured sparsity as an extension of standard sparsity has shown the outstanding performance when dealing with some highly correlated variables in computer vision and pattern recognition however the traditional mixed l1 l2 or l1 text l infty group norm becomes weak in characterizing the internal structure of each group since they cannot alleviate the correlations between variables recently nuclear norm has been validated to be useful for depicting a spatially structured matrix variable it considers the global structure of the matrix variable but overlooks the local structure to combine the advantages of structured sparsity and nuclear norm this paper presents a tree structured nuclear norm approximation tsna model assuming that the representation residual with tree structured prior is a random matrix variable and follows a dependent matrix distribution the extended alternating direction method of multipliers is utilized to solve the proposed model an efficient bound condition based on the extended restricted isometry constants is provided to show the exact recovery of the proposed model under the given noisy case in addition tsna is connected with some newest methods such as sparse representation based classifier nuclear l1 norm joint regression and nuclear norm based matrix regression which can be regarded as the special cases of tsna experiments with face reconstruction and recognition demonstrate the benefits of tsna over other approaches

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

the ising model on a closed cayley tree abstract the free energy for the ising model on a closed tree studied by jellito is transformed to a form which is explicitly nondifferentiable at the critical temperature

nonlinear dimensionality reduction using a temporal coherence principle temporal coherence principle is an attractive biologically inspired learning rule to extract slowly varying features from quickly varying input data in this paper we develop a new nonlinear neighborhood preserving nnp technique by utilizing the temporal coherence principle to find an optimal low dimensional representation from the original high dimensional data nnp is based on a nonlinear expansion of the original input data such as polynomials of a given degree it can be solved by the eigenvalue problem without using gradient descent and is guaranteed to find the global optimum nnp can be viewed as a nonlinear dimensionality reduction framework which takes into consideration both time series and data sets without an obvious temporal structure according to different situations we introduce three algorithms of nnp named nnp 1 nnp 2 and nnp 3 the objective function of nnp 1 is equal to slow feature analysis sfa and it works well for time series such as image sequences nnp 2 artificially constructs time series consisting of neighboring points for data sets without a clear temporal structure such as image data nnp 3 is proposed for classification tasks which can minimize the distances of neighboring points in the embedding space and ensure that the remaining points are as far apart as possible simultaneously furthermore the kernel extension of nnp is also discussed in this paper the proposed algorithms work very well on some image sequences and image data sets compared to other methods meanwhile we perform the classification task on the mnist handwritten digit database using the supervised nnp algorithms the experimental results demonstrate that nnp is an effective technique for nonlinear dimensionality reduction tasks

learning locality constrained collaborative representation for robust face recognition abstract the models of low dimensional manifold and sparse representation are two well known concise models that suggest that each data can be described by a few characteristics manifold learning is usually investigated for dimension reduction by preserving some expected local geometric structures from the original space into a low dimensional one the structures are generally determined by using pairwise distance e g euclidean distance alternatively sparse representation denotes a data point as a linear combination of the points from the same subspace in practical applications however the nearby points in terms of pairwise distance may not belong to the same subspace and vice versa consequently it is interesting and important to explore how to get a better representation by integrating these two models together to this end this paper proposes a novel coding algorithm called locality constrained collaborative representation lccr which introduce a kind of local consistency into coding scheme to improve the discrimination of the representation the locality term derives from a biologic observation that the similar inputs have similar codes the objective function of lccr has an analytical solution and it does not involve local minima the empirical studies based on several popular facial databases show that lccr is promising in recognizing human faces with varying pose expression and illumination as well as various corruptions and occlusions