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

on special bessel periods and the gross prasad conjecture for mathrm so 2n 1 times mathrm so 2 in this paper we investigate one instance of the global gross prasad conjecture our main result proves that if an irreducible cuspidal automorphic representation pi of an odd dimensional special orthogonal group whose local component pi w at some finite place w is generic admits the special bessel model corresponding to a quadratic extension e over a base field f then the central l value l 1 2 pi l 1 2 pi times chi e does not vanish here chi e denotes the quadratic character of mathbb a f times corresponding to e as an application we obtain the equivalence between the non vanishing of the special bessel period and that of the corresponding central l value when pi is associated to a full modular holomorphic siegel cusp form of degree two which is a hecke eigenform and e is an imaginary quadratic extension of mathbb q

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

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

locally conformal kahler manifolds with potential a locally conformally kahler lck manifold m is one which is covered by a kahler manifold widetilde m with the deck transformation group acting conformally on widetilde m if m admits a holomorphic flow acting on widetilde m conformally it is called a vaisman manifold neither the class of lck manifolds nor that of vaisman manifolds is stable under small deformations we define a new class of lck manifolds called lck manifolds with potential which is closed under small deformations all vaisman manifolds are lck with potential we show that an lck manifold with potential admits a covering which can be compactified to a stein variety by adding one point this is used to show that any lck manifold m with potential dim m 3 can be embedded into a hopf manifold thus improving similar results for vaisman manifolds ornea and verbitsky math ann 332 121 143 2005

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

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

generalized bergman kernels on symplectic manifolds we study the near diagonal asymptotic expansion of the generalized bergman kernel of the renormalized bochner laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold we show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them as a consequence we calculate the density of states function of the bochner laplacian and establish a symplectic version of the convergence of the induced fubini study metric we also discuss generalizations of the asymptotic expansion for non compact or singular manifolds as well as their applications our approach is inspired by the analytic localization techniques of bismut and lebeau

solving the small sample size problem in face recognition using generalized discriminant analysis the goal of face recognition is to distinguish persons via their facial images each person s images form a cluster and a new image is recognized by assigning it to the correct cluster since the images are very high dimensional it is necessary to reduce their dimension linear discriminant analysis lda has been shown to be effective at dimension reduction while preserving the cluster structure of the data it is classically defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters the requirement that one of these matrices be nonsingular restricts its application to datasets in which the dimension of the data does not exceed the sample size for face recognition however the dimension typically exceeds the number of images in the database resulting in what is referred to as the small sample size problem recently the applicability of lda has been extended by using the generalized singular value decomposition gsvd to circumvent the nonsingularity requirement thus making lda directly applicable to face recognition data our experiments confirm that lda gsvd solves the small sample size problem very effectively as compared with other current methods

averages of fourier coefficients of siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables let d be a a negative discriminant and let t vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant d we prove an asymptotic formula for d rightarrow infty for the average over t of the number of representations of t by an integral positive definite quaternary quadratic form and obtain bounds for averages of fourier coefficients of linear combinations of siegel theta series we also find an asymptotic estimate from below on the number of binary forms of fixed discriminant d which are represented by a given quaternary form in particular we can show that for growing d a positive proportion of the binary quadratic forms of discriminant d is represented by the given quaternary quadratic form

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

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

riesz transforms and spectral multipliers of the hodge laguerre operator abstract on r d endowed with the laguerre probability measure we define a hodge laguerre operator l acting on differential forms here is the laguerre exterior differentiation operator defined as the classical exterior differential except that the partial derivatives x i are replaced by the laguerre derivatives x i x i and is the adjoint of with respect to inner product on forms defined by the euclidean structure and the laguerre measure we prove dimension free bounds on l p 1 p for the riesz transforms l 1 2 and l 1 2 as applications we prove the strong hodge de rahm kodaira decomposition for forms in l p and deduce existence and regularity results for the solutions of the hodge and de rham equations in l p we also prove that for suitable functions m the operator m l is bounded on l p 1 p

learning a mahalanobis distance metric for data clustering and classification distance metric is a key issue in many machine learning algorithms this paper considers a general problem of learning from pairwise constraints in the form of must links and cannot links as one kind of side information a must link indicates the pair of the two data points must be in a same class while a cannot link indicates that the two data points must be in two different classes given must link and cannot link information our goal is to learn a mahalanobis distance metric under this metric we hope the distances of point pairs in must links are as small as possible and those of point pairs in cannot links are as large as possible this task is formulated as a constrained optimization problem in which the global optimum can be obtained effectively and efficiently finally some applications in data clustering interactive natural image segmentation and face pose estimation are given in this paper experimental results illustrate the effectiveness of our algorithm

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

spectral flow and dixmier traces abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi finite von neumann algebra our results have several applications we deduce a formula for the chern character of an odd l 1 summable breuer fredholm module in terms of a hochschild 1 cycle we explain how to derive a wodzicki residue for pseudo differential operators along the orbits of an ergodic r n action on a compact space x finally we give a short proof of an index theorem of lesch for generalised toeplitz operators

a unified framework for semi supervised dimensionality reduction in practice many applications require a dimensionality reduction method to deal with the partially labeled problem in this paper we propose a semi supervised dimensionality reduction framework which can efficiently handle the unlabeled data under the framework several classical methods such as principal component analysis pca linear discriminant analysis lda maximum margin criterion mmc locality preserving projections lpp and their corresponding kernel versions can be seen as special cases for high dimensional data we can give a low dimensional embedding result for both discriminating multi class sub manifolds and preserving local manifold structure experiments show that our algorithms can significantly improve the accuracy rates of the corresponding supervised and unsupervised approaches

on the compactification of hyperconcave ends and the theorems of siu yau and nadel we show that the pseudoconcave holes of some naturally arising class of manifolds called hyperconcave ends can be filled in including the case of complex dimension two as a consequence we obtain a stronger version of the compactification theorem of siu yau and extend nadel s theorems to dimension two

compactification of complete kahler manifolds of negative ricci curvature soit m une variete de kahler complete de dimension m et soit la forme de kahler on suppose 1 ric 0 2 m est tres fortement m 2 pseudoconcave 3 le recouvrement universel de m est de stein alors m est biholomorphe a une variete quasiprojective

semi supervised orthogonal discriminant analysis via label propagation trace ratio is a natural criterion in discriminant analysis as it directly connects to the euclidean distances between training data points this criterion is re analyzed in this paper and a fast algorithm is developed to find the global optimum for the orthogonal constrained trace ratio problem based on this problem we propose a novel semi supervised orthogonal discriminant analysis via label propagation differing from the existing semi supervised dimensionality reduction algorithms our algorithm propagates the label information from the labeled data to the unlabeled data through a specially designed label propagation and thus the distribution of the unlabeled data can be explored more effectively to learn a better subspace extensive experiments on toy examples and real world applications verify the effectiveness of our algorithm and demonstrate much improvement over the state of the art algorithms

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

growth of maps distortion in groups and symplectic geometry in the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism the uniform norm of the differential of its n th iteration the word length of its n th iteration where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms we find lower bounds for the growth rates of these sequences in a number of situations these bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism they are obtained by using recent results of schwarz on floer homology as an application we prove non existence of certain non linear symplectic representations for finitely generated groups

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

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

the method weak lefschetz theorems and the topology of kahler manifolds in no nori studied the fundamental group of complements of nodal curves with ample normal bundle in smooth projective surfaces the main tool was the following weak lefschetz theorem theorem nori suppose u x is a local biholomorphism from a connected complex manifold u into a connected smooth projective variety x of dimension at least 2 and u contains a connected effective divisor y with compact support and ample normal bundle then for every zariski open subset z of x the image of 1 1 z in 1 z is of finite index

characteristic classes and distortion of diffeomorphisms let m and n be manifolds of bounded geometry that is the absolute value of their curvatures as well as their covariant derivatives are bounded from above and their injectivity radii are bounded from below a map f m n has bounded d stortion if there is an upper bound on the norm of the derivative d f a bounded distortion diffeomorphism cb d d is assumed to have an inverse which also has bounded distortion similarly homotopies are assumed to have bounded distortion note that this implies that homotopic maps are a finite distance apart we prove the following theorem that describes when characteristic classes of a compact manifold obstruct the existence of bounded distortion diffeomorphisms of universal covers theorem there are compact manifolds a and b with fundamental group r and a map f a b with jcpcb i pca for some rational pontrjagin class p whose universal cover is b d homotopic to a b d diffeomorphism if and only if r is nonamenable

positive factorizations of pseudoperiodic homeomorphisms we generalize a classical result concerning smooth germs of surfaces by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization as a consequence of this and a topological characterization of these monodromies by anne pichon we conclude that a pseudoperiodic homeomorphism on an oriented surface with boundary with positive fractional dehn twist coefficients and screw numbers admits a positive factorization we use the main theorem to give a sufficiency criterion for certain pseudoperiodic homeomorphisms with negative screw numbers to admit a positive factorization

supervised dimensionality reduction via sequential semidefinite programming many dimensionality reduction problems end up with a trace quotient formulation since it is difficult to directly solve the trace quotient problem traditionally the trace quotient cost function is replaced by an approximation such that the generalized eigenvalue decomposition can be applied in contrast we directly optimize the trace quotient in this work it is reformulated as a quasi linear semidefinite optimization problem which can be solved globally and efficiently using standard off the shelf semidefinite programming solvers also this optimization strategy allows one to enforce additional constraints for example sparseness constraints on the projection matrix we apply this optimization framework to a novel dimensionality reduction algorithm the performance of the proposed algorithm is demonstrated in experiments by several uci machine learning benchmark examples usps handwritten digits as well as orl and yale face data

embeddability of some strongly pseudoconvex cr manifolds we obtain an embedding theorem for compact strongly pseudoconvex cr manifolds which are boundaries of some complete hermitian manifolds we use this to compactify some negatively curved kahler manifolds with compact strongly pseudoconvex boundary an embedding theorem for sasakian manifolds is also derived

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

normal cr structures on compact 3 manifolds we study normal cr compact manifolds in dimension 3 for a choice of a cr reeb vector field we associate a sasakian metric on them and we classify those metrics as a consequence the underlying manifolds are topologically finite quotients of s 3 or of a non flat circle bundle over a riemann surface of positive genus in the latter case we prove that their cr automorphisms group is a finite extension of s 1 and we classify the normal cr structures on these manifolds

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

the dirichlet series of koecher and maas and modular forms of weight 3 2 the basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series in order that this question makes sense one has of course to specify which type of theta series should be taken into account we want here to follow in a special case the approach taken e g in b61 w a l l which in its general form asks which cuspidal modular forms f of degree n and weight k for the group foc n can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m 2 k and the same level n moreover one wants to derive an explicit expression for such a linear combination in terms of the petersson inner products of f with the theta series involved the representation theoretic version of this question is to ask which cuspidat automorphic representations of the adelic symplectic or metaplectic group

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

orthogonal laplacianfaces for face recognition following the intuition that the naturally occurring face data may be generated by sampling a probability distribution that has support on or near a submanifold of ambient space we propose an appearance based face recognition method called orthogonal laplacianface our algorithm is based on the locality preserving projection lpp algorithm which aims at finding a linear approximation to the eigenfunctions of the laplace beltrami operator on the face manifold however lpp is nonorthogonal and this makes it difficult to reconstruct the data the orthogonal locality preserving projection olpp method produces orthogonal basis functions and can have more locality preserving power than lpp since the locality preserving power is potentially related to the discriminating power the olpp is expected to have more discriminating power than lpp experimental results on three face databases demonstrate the effectiveness of our proposed algorithm

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

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

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

torus fibrations and localization of index ii local index for acyclic compatible system we give a framework of localization for the index of a dirac type operator on an open manifold suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets each of which has a structure of the total space of a torus bundle under an acyclic condition we define the index of the dirac type operator by using the witten type deformation and show that the index has several properties such as excision property and a product formula in particular we show that the index is localized on the compact set

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

semi supervised local multi manifold isomap by linear embedding for feature extraction we explore the discriminative feature extraction problem a semisupervised local multimanifold isomap by linear embedding is proposed our model can use labeled and unlabeled data to deliver manifold features our model aims to minimize pairwise intraclass distances in the same manifold our model aims to maximize the distances between different manifolds in this paper we mainly propose a semi supervised local multi manifold isomap learning framework by linear embedding termed ssmm isomap that can apply the labeled and unlabeled training samples to perform the joint learning of neighborhood preserving local nonlinear manifold features and a linear feature extractor the formulation of ssmm isomap aims at minimizing pairwise distances of intra class points in the same manifold and maximizing the distances over different manifolds to enhance the performance of nonlinear manifold feature learning we also incorporate the neighborhood reconstruction error to preserve local topology structures between both labeled and unlabeled samples to enable our ssmm isomap to extract local manifold features from the outside new data we also add a feature approximation error that correlates manifold features with embedded features by the jointly learnt feature extractor thus the learnt linear extractor can extract the local manifold features from the new data efficiently by direct embedding to optimize the proposed objective function two effective schemes are presented i e scaling by majorizing a complicated function and eigen decomposition notice that the comparison of the proposed two solvers is also described we mainly evaluate ssmm isomap for manifold feature learning data clustering and classification extensive simulation results verify the effectiveness of our ssmm isomap algorithm compared with other related feature learning techniques

learning from normalized local and global discriminative information for semi supervised regression and dimensionality reduction semi supervised dimensionality reduction is one of the important topics in pattern recognition and machine learning during the past decade laplacian regularized least square laprls and semi supervised discriminant analysis sda are the two widely used semi supervised dimensionality reduction methods in this paper we show that sda and laprls can be unified into a constrained manifold regularized least square framework the manifold term however cannot fully utilize the underlying discriminative information we thus introduce a new and effective semi supervised dimensionality reduction method called learning from local and global information llgdi to solve the problem the proposed llgdi method adopts a set of local classification functions to preserve both local geometrical and discriminative information of dataset it also adopts a global classification function to preserve the global discriminative information and an uncorrelated constraint to calculate the projection matrix for simultaneously solving regression and dimensionality reduction problem as a result the llgdi method is able to preserve local discriminative manifold information as well as the global discriminative information theoretical analysis and extensive simulations presented in the paper show the effectiveness of the llgdi algorithm the results also demonstrate llgdi can achieve superior performance compared with other existing methods

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

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

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n