l1 norm and maximum margin criterion based discriminant locality preserving projections via trace lasso discriminant locality preserving projections based on maximum margin criterion dlpp mmc is a useful feature extraction method since it has shown good performances in pattern recognition the conventional dlpp mmc however is not robust to noises and outliers since its objective function is based on l2 norm in this paper we propose a novel l1 norm and maximum margin criterion based discriminant locality preserving projections via trace lasso dlpp mmc l1tl l1 norm rather than l2 norm is used in the formulation of dlpp mmc l1tl which makes it be robust to noises and outliers besides in order to improve the performance of dlpp mmc l1tl further we use trace lasso to regularize the basis vectors trace lasso which can balance l1 norm and l2 norm and consider sparsity and correlation of data simultaneously is a recently proposed norm an iterative procedure for solving dlpp mmc l1tl is also proposed in this paper the experiment results on some data sets demonstrate the effectiveness of dlpp mmc l1tl we propose a l1 norm and mmc based dlpp method via trace lasso our proposed algorithm can simultaneously consider the sparsity and correlation we also propose an efficient procedure for solving the proposed method

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

principal component analysis based on l1 norm maximization a method of principal component analysis pca based on a new l1 norm optimization technique is proposed unlike conventional pca which is based on l2 norm the proposed method is robust to outliers because it utilizes l1 norm which is less sensitive to outliers it is invariant to rotations as well the proposed l1 norm optimization technique is intuitive simple and easy to implement it is also proven to find a locally maximal solution the proposed method is applied to several datasets and the performances are compared with those of other conventional methods

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

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

discriminant locality preserving projections based on l1 norm maximization conventional discriminant locality preserving projection dlpp is a dimensionality reduction technique based on manifold learning which has demonstrated good performance in pattern recognition however because its objective function is based on the distance criterion using l2 norm conventional dlpp is not robust to outliers which are present in many applications this paper proposes an effective and robust dlpp version based on l1 norm maximization which learns a set of local optimal projection vectors by maximizing the ratio of the l1 norm based locality preserving between class dispersion and the l1 norm based locality preserving within class dispersion the proposed method is proven to be feasible and also robust to outliers while overcoming the small sample size problem the experimental results on artificial datasets binary alphadigits dataset feret face dataset and polyu palmprint dataset have demonstrated the effectiveness of the proposed method

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

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

linear discriminant analysis based on l1 norm maximization linear discriminant analysis lda is a well known dimensionality reduction technique which is widely used for many purposes however conventional lda is sensitive to outliers because its objective function is based on the distance criterion using l2 norm this paper proposes a simple but effective robust lda version based on l1 norm maximization which learns a set of local optimal projection vectors by maximizing the ratio of the l1 norm based between class dispersion and the l1 norm based within class dispersion the proposed method is theoretically proved to be feasible and robust to outliers while overcoming the singular problem of the within class scatter matrix for conventional lda experiments on artificial datasets standard classification datasets and three popular image databases demonstrate the efficacy of the proposed method

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

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

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

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

l1 norm kernel discriminant analysis via bayes error bound optimization for robust feature extraction a novel discriminant analysis criterion is derived in this paper under the theoretical framework of bayes optimality in contrast to the conventional fisher s discriminant criterion the major novelty of the proposed one is the use of l1 norm rather than l2 norm which makes it less sensitive to the outliers with the l1 norm discriminant criterion we propose a new linear discriminant analysis l1 lda method for linear feature extraction problem to solve the l1 lda optimization problem we propose an efficient iterative algorithm in which a novel surrogate convex function is introduced such that the optimization problem in each iteration is to simply solve a convex programming problem and a close form solution is guaranteed to this problem moreover we also generalize the l1 lda method to deal with the nonlinear robust feature extraction problems via the use of kernel trick and hereafter proposed the l1 norm kernel discriminant analysis l1 kda method extensive experiments on simulated and real data sets are conducted to evaluate the effectiveness of the proposed method in comparing with the state of the art methods

an uncertainty principle for quaternion fourier transform we review the quaternionic fourier transform qft using the properties of the qft we establish an uncertainty principle for the right sided qft this uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion valued signals in the spatial and frequency domains it is shown that only a gaussian quaternion signal minimizes the uncertainty

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

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

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

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

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

efficient implementation of quaternion fourier transform convolution and correlation by 2 d complex fft the concepts of quaternion fourier transform qft quaternion convolution qcv and quaternion correlation which are based on quaternion algebra have been found to be useful for color image processing however the necessary computational algorithms and their complexity still need some attention we develop efficient algorithms for qft qcv and quaternion correlation the conventional complex two dimensional 2 d fourier transform ft is used to implement these quaternion operations very efficiently with these algorithms we only need two complex 2 d fts to implement a qft six complex 2 d fts to implement a one side qcv or a quaternion correlation and 12 complex 2 d fts to implement a two side qcv and the efficiency of these quaternion operations is much improved meanwhile we also discuss two additional topics the first one is about how to use qft and qcv for quaternion linear time invariant qlti system analysis this topic is important for quaternion filter design and color image processing besides we also develop the spectrum product qcv it is an improvement of the conventional form of qcv for any arbitrary input functions it always corresponds to the product operation in the frequency domain it is very useful for quaternion filter design

windowed fourier transform of two dimensional quaternionic signals in this paper we generalize the classical windowed fourier transform wft to quaternion valued signals called the quaternionic windowed fourier transform qwft using the spectral representation of the quaternionic fourier transform qft we derive several important properties such as reconstruction formula reproducing kernel isometry and orthogonality relation taking the gaussian function as window function we obtain quaternionic gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic gabor basis we apply the qwft properties and the right sided qft to establish a heisenberg type uncertainty principle for the qwft finally we briefly introduce an application of the qwft to a linear time varying system

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

wigner distributions and ambiguity functions of 2 d quaternionic and monogenic signals the paper presents new notions of wigner distributions and corresponding ambiguity functions defined by quaternionic fourier transforms of correlation products of recently defined quaternionic and monogenic two dimensional 2 d signals the properties of new defined wigner distributions are compared with wigner distributions of 2 d analytic signals with single quadrant spectra it is well known that wigner distributions of complex signals are real functions differently the wigner distributions of quaternionic and monogenic signals may be quaternionic valued functions however it may happen that some 2 d slices of 4 d wigner distributions are real functions

robust and sparse linear discriminant analysis via an alternating direction method of multipliers in this paper we propose a robust linear discriminant analysis rlda through bhattacharyya error bound optimization rlda considers a nonconvex problem with the l 1 norm operation that makes it less sensitive to outliers and noise than the l 2 norm linear discriminant analysis lda in addition we extend our rlda to a sparse model rslda both rlda and rslda can extract unbounded numbers of features and avoid the small sample size sss problem and an alternating direction method of multipliers admm is used to cope with the nonconvexity in the proposed formulations compared with the traditional lda our rlda and rslda are more robust to outliers and noise and rslda can obtain sparse discriminant directions these findings are supported by experiments on artificial data sets as well as human face databases

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

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

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

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

optimal mean two dimensional principal component analysis with f norm minimization our method simultaneously optimizes the projection matrix and mean in the criterion objective our method directly considers the reconstruction errors of data while most existing l1 norm 2dpca methods do not our method is not only robust but also retains 2dpcas desirable properties such as rotational invariance we solve the solution by a non greedy algorithm which has closed form solution and good convergence two dimensional principal component analysis 2dpca employs the squared f norm as distance metric for feature extraction and is widely used in the field of pattern analysis and recognition especially face image analysis but it is sensitive to the presence of outliers due to the fact that squared f norm remarkably enlarges the role of outliers in the criterion function to handle this problem we propose a robust formulation for 2dpca namely optimal mean 2dpca with f norm minimization omf 2dpca in omf 2dpca distance in spatial dimensions attribute dimensions is measured in f norm while the summation over different data points uses 1 norm moreover we center the data using the optimized mean rather than the fixed mean this helps further improve robustness of our method to solve omf 2dpca we propose a fast iterative algorithm which has a closed form solution in each iteration experimental results on face image databases illustrate its effectiveness and advantages

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

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

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

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

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

solutions of integral and integro differential equation systems by using differential transform method in this study differential transform method dtm is applied to both integro differential and integral equation systems the method is further expanded with a formulation to treat fredholm integrals if the system considered has a solution in terms of the series expansion of known functions this powerful method catches the exact solution so as to show this capability and robustness some systems of integral and integro differential equations are solved as numerical examples

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

window functions represented by b spline functions characteristics of window functions provided by b spline functions are analyzed in comparison to other known windows for the same highest sidelobe level those windows have a mainlobe bandwidth between those of gaussian and cos sup a x windows are as good as those of cos sup a x ones in the sidelobe falloff

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

two dimensional quaternion wavelet transform abstract in this paper we introduce the continuous quaternion wavelet transform cqwt we express the admissibility condition in terms of the right sided quaternion fourier transform we show that its fundamental properties such as inner product norm relation and inversion formula can be established whenever the quaternion wavelets satisfy a particular admissibility condition we present several examples of the cqwt as an application we derive a heisenberg type uncertainty principle for these extended wavelets