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

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

correspondence between fuzzy measures and classical measures abstract in this paper we study the question whether given a fuzzy measure as defined in 3 and 4 there exists a classical measure such that the fuzzy measure of a measurable fuzzy set equals the classical measure of the area below the membership function of the results are that in the case of finite additivity there is a one to one correspondence between classical measures and fuzzy measures whereas in the case of countable additivity this result only holds for generated fuzzy algebras finally some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy algebra to the generated fuzzy algebra are discussed

pseudofundamental triangular norms and g t tribes abstract we generalize the theory of t tribes of butnariu and klement this generalization may serve as a basis of a common theory of t measures and decomposable measures of fuzzy events

tensor linear laplacian discrimination tlld for feature extraction discriminant feature extraction plays a central role in pattern recognition and classification in this paper we propose the tensor linear laplacian discrimination tlld algorithm for extracting discriminant features from tensor data tlld is an extension of linear discriminant analysis lda and linear laplacian discrimination lld in directions of both nonlinear subspace learning and tensor representation based on the contextual distance the weights for the within class scatters and the between class scatter can be determined to capture the principal structure of data clusters this makes tlld free from the metric of the sample space which may not be known moreover unlike lld the parameter tuning of tlld is very easy experimental results on face recognition texture classification and handwritten digit recognition show that tlld is effective in extracting discriminative features

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

low rank structure learning via nonconvex heuristic recovery in this paper we propose a nonconvex framework to learn the essential low rank structure from corrupted data different from traditional approaches which directly utilizes convex norms to measure the sparseness our method introduces more reasonable nonconvex measurements to enhance the sparsity in both the intrinsic low rank structure and the sparse corruptions we will respectively introduce how to combine the widely used lp norm 0 p 1 and log sum term into the framework of low rank structure learning although the proposed optimization is no longer convex it still can be effectively solved by a majorization minimization mm type algorithm with which the nonconvex objective function is iteratively replaced by its convex surrogate and the nonconvex problem finally falls into the general framework of reweighed approaches we prove that the mm type algorithm can converge to a stationary point after successive iterations the proposed model is applied to solve two typical problems robust principal component analysis and low rank representation experimental results on low rank structure learning demonstrate that our nonconvex heuristic methods especially the log sum heuristic recovery algorithm generally perform much better than the convex norm based method 0 p 1 for both data with higher rank and with denser corruptions

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

feature learning via partial differential equation with applications to face recognition a novel partial differential equation method is proposed for feature learning the feature is discriminative and invariant under rotation translation and illumination the method is efficient in low resolution images and when the samples are few to our best knowledge this is the first work that applies pde to feature learning feature learning is a critical step in pattern recognition such as image classification however most of the existing methods cannot extract features that are discriminative and at the same time invariant under some transforms this limits the classification performance especially in the case of small training sets to address this issue in this paper we propose a novel partial differential equation pde based method for feature learning the feature learned by our pde is discriminative also translationally and rotationally invariant and robust to illumination variation to our best knowledge this is the first work that applies pde to feature learning and image recognition tasks specifically we model feature learning as an evolution process governed by a pde which is designed to be translationally and rotationally invariant and is learned via minimizing the training error hence extracts discriminative information from data after feature extraction we apply a linear classifier for classification we also propose an efficient algorithm that optimizes the whole framework our method is very effective when the training samples are few the experimental results of face recognition on the four benchmark face datasets show that the proposed method outperforms the state of the art feature learning methods in the case of low resolution images and when the training samples are limited

spectral multiplicity for gln r we study the behavior of the cuspidal spectrum of t where is associated to gln i and v is cofinite but not compact by a technique that modifies the lax phillips technique and uses ideas from wave equation techniques if r is the dimension of jp na k is the counting function for the laplacian attached to a hilbert space ha ma x is the multiplicity function and ho is the space of cusp forms we obtain the following results theorem 1 there exists a space of functions containing all cusp forms such that n x cr vol ai fc a a loga

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

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

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

a characterization of the ordering of continuous t norms abstract given two t norms t 1 and t 2 it is quite often difficult or even impossible to check directly whether t 1 t 2 in schweizer and sklar 1983 a necessary and sufficient condition is given for continuous archimedean t norms to be comparable we simplify this result proving a sufficient condition which involves only one argument of a one place function rather than two arguments this result is applied to show the monotonicity of some well known classes of t norms next we generalize the result of schweizer and sklar to the case of all continuous t norms and present also a sufficient condition which is again easier to check

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

diffusion maps and coarse graining a unified framework for dimensionality reduction graph partitioning and data set parameterization we provide evidence that nonlinear dimensionality reduction clustering and data set parameterization can be solved within one and the same framework the main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise our construction which is based on a markov random walk on the data offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry we show that clustering in embedding spaces is equivalent to compressing operators the objective of data partitioning and clustering is to coarse grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy we show that the quantization distortion in diffusion space bounds the error of compression of the operator thus giving a rigorous justification for k means clustering in diffusion space and a precise measure of the performance of general clustering algorithms

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

structure constrained low rank representation benefiting from its effectiveness in subspace segmentation low rank representation lrr and its variations have many applications in computer vision and pattern recognition such as motion segmentation image segmentation saliency detection and semisupervised learning it is known that the standard lrr can only work well under the assumption that all the subspaces are independent however this assumption cannot be guaranteed in real world problems this paper addresses this problem and provides an extension of lrr named structure constrained lrr sc lrr to analyze the structure of multiple disjoint subspaces which is more general for real vision data we prove that the relationship of multiple linear disjoint subspaces can be exactly revealed by sc lrr with a predefined weight matrix as a nontrivial byproduct we also illustrate that sc lrr can be applied for semisupervised learning the experimental results on different types of vision problems demonstrate the effectiveness of our proposed method

fundamental triangular norm based tribes and measures abstract we complete the results of butnariu and klement about t s tribes and t s measures where t s s 0 is a frank s fundamental triangular norm for a denumerable universum x we show that any t s tribe is a semigenerated one and hence any t s measure is a generated one

triangular norm based measures and their markov kernel representation abstract we approach the problem whether left continuous triangular norm based valuations called t measures or t probability measures defined on triangular normbased tribes of the unit cube can be disintegrated by markov kernels we prove that each t measure based on a fundamental triangular norm these triangular norms t together with their corresponding triangular conorms s satisfy the functional equation t x y s x y x y can be uniquely represented as a sum of a disintegrable t measure and a hard core which is either identically zero or which is monotonically irreducible i e cannot be disintegrated

an algebraic generalization of the notion of tribe tribes became a successful basis of measure theory on fuzzy sets during their investigation new requirements arose we discuss possible modifications of the original definition of a tribe we suggest a more general structure which admits to apply different triangular norms in different parts emphasis is put on point free notions

a local weyl s law the angular distribution and multiplicity of cusp forms on product spaces let h be a finite volume symmetric space with h the product of half planes let 1 be the laplacian on the ith half plane and assume that we have a cusp from o so we have i o i o for i 1 2 n

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

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1