radial basis function based adaptive fuzzy systems and their applications to system identification and prediction abstract in this paper we describe a neuro fuzzy system with adaptive capability to extract fuzzy if then rules from input and output sample data through learning the proposed system called radial basis function rbf based adaptive fuzzy system afs employs the gaussian functions to represent the membership functions of the premise part of fuzzy rules three architectural deviations of the rbf based afs are also presented according to different consequence types such as constant first order linear function and fuzzy variable these provide versatility of the network to handle arbitrary fuzzy inference schemes we present examples of system identification and time series prediction to illustrate how to solve these problems and to demonstrate its validity and effectiveness using the rbf based afs

a ga based fuzzy modeling approach for generating tsk models this paper proposes a new genetic based modeling method for building simple and well defined tsk models with scatter type input partitions our approach manages all attributes characterizing the structure of a tsk model simultaneously particularly it determines the number of rules the input partition the participating inputs in each rule and the consequent parameters the model building process is divided into two phases in phase one the structure learning task is formulated as a multi objective optimization problem which is resolved using a novel genetic based structure learning gbsl scheme apart from the mean square error mse and the number of rules three additional criteria are introduced in the fitness function for measuring the quality of the partitions optimization of these measures leads to models with representative rules small overlapping and efficient data cover in order to obtain models with accurate data fitting and good local performance the consequent parameters are determined using a local mse function while the overall model is evaluated on the basis of a global mse function the search capabilities of the suggested structure learning scheme are significantly enhanced by including a highly effective local search operator implemented by a micro genetic algorithm and four problem specific operators finally a genetic based parameter learning gbpl scheme is suggested in phase two which performs fine tuning of the initial models obtained after structure learning the performance of the proposed modeling approach is evaluated using a static example and a well known dynamic benchmark problem simulation results demonstrate that our models outperform those suggested by other methods with regard to simplicity model structure and accuracy

neuro levelset system based segmentation in dynamic susceptibility contrast enhanced and diffusion weighted magnetic resonance images in this study neuro levelset method is proposed and evaluated for segmentation and grading of brain tumors on reconstructed images of dynamic susceptibility contrast dsc and diffusion weighted dw magnetic resonance images the proposed neuro levelset method comprises of two independent phases of processing at first reconstructed images have been independently processed by three different artificial neural network systems such as multilayer perceptron mlp self organizing map som and radial basis function rbf the images used for these tasks were the cerebral blood volume cbv time to peak ttp percentage of base at peak pbp and apparent diffusion coefficient adc images this processing step ensued in formation of segmentation images of brain tumors further in the second phase these coarse segmented images of each artificial neural network system have been independently subjected as speed images to levelset method in order to optimize the segmentation performance this has resulted in construction of three distinct neuro levelset methods such as mlp levelset som levelset and rbf levelset method proposed neuro levelset methods performed better in segmenting tumor edema necrosis csf and normal tissues as compared to independent artificial neural network systems among three neuro levelset methods rbf levelset system has performed well with average sensitivity and specificity values of 91 43 2 94 and 94 43 1 90 respectively

analysis of input output clustering for determining centers of rbfn the key point in design of radial basis function networks is to specify the number and the locations of the centers several heuristic hybrid learning methods which apply a clustering algorithm for locating the centers and subsequently a linear least squares method for the linear weights have been previously suggested these hybrid methods can be put into two groups which will be called as input clustering ic and input output clustering ioc depending on whether the output vector is also involved in the clustering process the idea of concatenating the output vector to the input vector in the clustering process has independently been proposed by several papers in the literature although none of them presented a theoretical analysis on such procedures but rather demonstrated their effectiveness in several applications the main contribution of this paper is to present an approach for investigating the relationship between clustering process on input output training samples and the mean squared output error in the context of a radial basis function network rbfn we may summarize our investigations in that matter as follows 1 a weighted mean squared input output quantization error which is to be minimized by ioc yields an upper bound to the mean squared output error 2 this upper bound and consequently the output error can be made arbitrarily small zero in the limit case by decreasing the quantization error which can be accomplished through increasing the number of hidden units

two dimensional supervised local similarity and diversity projection this paper presents a novel manifold learning method namely two dimensional supervised local similarity and diversity projection 2dslsdp for feature extraction the proposed method defines two weighted adjacency graphs namely similarity graph and diversity graph the affinity matrix of similarity graph is determined by the spatial relationship between vertices of this graph while affinity matrix of diversity graph is determined by the diversity information of vertices of its graph using these two graphs the proposed method constructs local similarity scatter and diversity scatter respectively a concise feature extraction criterion is then raised via minimizing the ratio of the local similarity scatter to local diversity scatter thus 2dslsdp can well preserve not only the adjacency similarity structure but also the diversity of data points which is important for the classification experiments on the ar and umist databases show the effectiveness of the proposed method

periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

on the bezoutian matrix for chebyshev polynomials abstract in the present paper a bezoutian type matrix relative to chebyshev polynomials of the second kind is introduced by the form of generating function a barnett type factorization formula and a triangular type formula are obtained for this generalized bezout matrix which generalize the corresponding results for classical bezout matrix two numerical examples are given to illustrate the correctness of our results

semiclassical asymptotics of perturbed cat maps we derive an exact representation for tr u n where u is the quantum propagator associated with an anosov perturbed cat map this takes the form of a sum over the fixed points of the n th iterate of the classical transformation the contribution of each one being given by an n fold multiple integral we focus in particular on the case when n 1 an asymptotic evaluation of the integral in question then leads to a complete semiclassical series expansion the first term of which corresponds to the gutzwiller tabor trace formula it is demonstrated that this series diverges but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1 a simple universal approximation to the late terms is then derived this explains the divergence of the semiclassical expansion in terms of complex tunnelling periodic orbits and implies the existence of unusual relations between different orbit actions it also allows us to recover the semiclassical contributions from the complex orbits explicitly using borel resummation these exponentially subdominant terms are shown to exhibit the stokes phenomenon which causes them to depend sensitively on the size of the perturbation parameter finally we develop an alternative expansion based on the orbits of the unperturbed cat map rather than diverging this is shown to converge absolutely thus making possible an exact calculation of the quantum trace using only classical mechanics its properties are however distinctly anti semiclassical

general quantization of canonical maps on a two torus canonical maps on a two torus in phase space are quantized under most general conditions recent results by keating et al 1999 nonlinearity 12 579 are thus fully extended in two directions a the translational component of a general canonical map is included in the quantization b all values of planck s constant consistent with the toral boundary conditions bcs are considered generically these values are rational numbers whose numerator must satisfy a number theoretical condition besides the condition on planck s constant the quantization is possible only for particular allowed bcs on the torus the general equation determining these bcs is derived allowed bcs may not exist in some cases representative examples are the irrational skew translations and kronecker maps exact versions of egorov s theorem are shown to hold under some conditions composition and representation properties of the quantization scheme are studied

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

on the bezoutian for polynomial matrices abstract the bezoutian b of two polynomial matrices can be described as a solution of a linear matrix equation this fact yields a new proof of the barnett factorization of b

identifying fuzzy models utilizing genetic programming abstract fuzzy models offer a convenient way to describe complex nonlinear systems moreover they permit the user to deal with uncertainty and vagueness due to these advantages fuzzy models are employed in various fields of applications e g control forecasting and pattern recognition nevertheless it has to be emphasized that the identification of a fuzzy model is a complex optimization task with many local minima genetic programming provides a way to solve such complex optimization problems in this work the use of genetic programming to identify the input variables the rule base and the involved membership functions of a fuzzy model is proposed for this purpose several new reproduction operators are introduced

two dimensional maximum local variation based on image euclidean distance for face recognition manifold learning concerns the local manifold structure of high dimensional data and many related algorithms are developed to improve image classification performance none of them however consider both the relationships among pixels in images and the geometrical properties of various images during learning the reduced space in this paper we propose a linear approach called two dimensional maximum local variation 2dmlv for face recognition in 2dmlv we encode the relationships among pixels in images using the image euclidean distance instead of conventional euclidean distance in estimating the variation of values of images and then incorporate the local variation which characterizes the diversity of images and discriminating information into the objective function of dimensionality reduction extensive experiments demonstrate the effectiveness of our approach

the design methodology of radial basis function neural networks based on fuzzy k nearest neighbors approach various approaches to partitioning of high dimensional input space have been studied with the intent of developing homogeneous clusters formed over input and output spaces of variables encountered in system modeling in this study we propose a new design methodology of a fuzzy model where the input space is partitioned by making use of some classification algorithm especially fuzzy k nearest neighbors k nn classifier being guided by some auxiliary information granules formed in the output space this classifier being regarded in the context of this design as a supervision mechanism is used to capture the distribution of data over the output space this type of supervision is beneficial when developing the structure in the input space it is demonstrated that data involved in a partition constructed by the fuzzy k nn method realized in the input space show a high level of homogeneity with regard to the data present in the output space this enhances the performance of the fuzzy rule based model whose premises in the rules involve partitions formed by the fuzzy k nn the design is illustrated with the aid of numeric examples that also provide a detailed insight into the performance of the fuzzy models and quantify several crucial design issues

unsupervised 2d dimensionality reduction with adaptive structure learning in recent years unsupervised two dimensional 2d dimensionality reduction methods for unlabeled large scale data have made progress however performance of these degrades when the learning of similarity matrix is at the beginning of the dimensionality reduction process a similarity matrix is used to reveal the underlying geometry structure of data in unsupervised dimensionality reduction methods because of noise data it is difficult to learn the optimal similarity matrix in this letter we propose a new dimensionality reduction model for 2d image matrices unsupervised 2d dimensionality reduction with adaptive structure learning drasl instead of using a predetermined similarity matrix to characterize the underlying geometry structure of the original 2d image space our proposed approach involves the learning of a similarity matrix in the procedure of dimensionality reduction to realize a desirable neighbors assignment after dimensionality reduction we add a constraint to our model such that there are exact formula see text connected components in the final subspace to accomplish these goals we propose a unified objective function to integrate dimensionality reduction the learning of the similarity matrix and the adaptive learning of neighbors assignment into it an iterative optimization algorithm is proposed to solve the objective function we compare the proposed method with several 2d unsupervised dimensionality methods k means is used to evaluate the clustering performance we conduct extensive experiments on coil20 at t feret usps and yale data sets to verify the effectiveness of our proposed method

a novel training algorithm for rbf neural network using a hybrid fuzzy clustering approach this paper introduces a novel clustering based algorithm to train gaussian type radial basis function neural networks in contrast to existing approaches we develop a specialized learning strategy that combines the merits of fuzzy and crisp clustering crisp clustering is a fast process yet very sensitive to initialization on the other hand fuzzy clustering reduces the dependency on initialization however it constitutes a slow learning process the proposed strategy aims to search for a trade off among these two potentially different effects the produced clusters possess fuzzy and crisp areas and therefore the final result is a hybrid partition where the fuzzy and crisp conditions coexist the hybrid clusters are directly involved in the estimation process of the neural network s parameters specifically the center elements of the basis functions coincide with cluster centers while the respective widths are calculated by taking into account the topology of the hybrid clusters to this end the network s design becomes a fast and efficient procedure the proposed method is successfully applied to a number of experimental cases where the produced networks prove to be highly accurate and compact in size

a trace formula for the semiclassical limit of some hermitian operators the aim of this paper is to provide an algorithm for calculating the leading order contributions for the semiclassical series of a functional of trace class hermitian operators we base our study on the linear space structure of the space of bounded hermitian operators on quantum hilbert space we work with a basis of operators with natural leading order weyl symbols performing stationary phase calculations with no assumption on their smoothness the stationary points are tilted orbits near the periodic ones in phase space we manage to get control of the error in the technique which scales like some power a of 1 n with 1 2 a 1 the calculations are directly applicable to quantum maps and we provide some examples in finite dimension with quantum perturbed cat maps

finite time h asynchronous state estimation for discrete time fuzzy markov jump neural networks with uncertain measurements abstract this paper is concerned with the problem of the h asynchronous state estimation for fuzzy markov jump neural networks fmjnns with uncertain measurements over a finite time interval in terms of a bernoulli distributed white sequence the phenomenon of the randomly occurring uncertainties in the output equation is represented by exploiting a random variable with known occurrence probabilities the main focus of this paper is to present a state estimator such that the resulting error system is finite time bounded and satisfies an h performance requirement then by employing the stochastic analysis technique sufficient conditions are provided to ensure that the state estimator is designed by means of solving a convex optimization problem an example is finally given to explain the effectiveness and potentiality of the proposed design method

orbit bifurcations and spectral statistics systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are in the semiclassical limit a combination of poisson and random matrix with relative weightings determined by the corresponding measures of regular and chaotic orbits we here identify an additional component in long range spectral statistics associated with periodic orbit bifurcations which can be semiclassically large this is illustrated for a family of perturbed cat maps it has been conjectured that in the semiclassical limit the quantum spectral statistics of classically integrable systems are generically poissonian and that those of classically chaotic systems are generically given by the average over an appropriate random matrix ensemble 1 4 in between these two extremes lie systems whose phase space is mixed that is in which regular and irregular motion coexist such systems are said to exhibit soft chaos 5 for these it has been suggested that the quantum spectral statistics are a combination of poisson and random matrix with relative weightings determined by the corresponding measures of the regular and chaotic orbits 6 our purpose here is to identify in this case an additional component in the long range statistics that is associated with periodic orbit bifurcations and which can be semiclassically large a semiclassical theory for long range spectral statistics has been developed 1 7 9 based on gutzwiller s trace formula 10 which relates the quantum density of states d e d p n e en to classical periodic orbits

the construction of a quantum markov partition we present a method for constructing a quantum markov partition its elements are obtained by quantizing the characteristic function of the classical rectangles the result is a set of quantum operators which behave asymptotically as projectors over the classical rectangles apart from edge and corner effects we investigate their spectral properties and different methods of construction the quantum partition is shown to induce a symbolic decomposition of the quantum evolution operator in particular an exact expression for the traces of the propagator is obtained having the same structure as the gutzwiller periodic orbit sum

orthogonal tensor neighborhood preserving embedding for facial expression recognition in this paper a generalized tensor subspace model is concluded from the existing tensor dimensionality reduction algorithms with this model we investigate the orthogonality of the bases of the high order tensor subspace and propose the orthogonal tensor neighborhood preserving embedding otnpe algorithm we evaluate the algorithm by applying it to facial expression recognition where both the 2nd order gray level raw pixels and the encoded 3rd order tensor formed gabor features of facial expression images are utilized the experiments show the excellent performance of our algorithm for the dimensionality reduction of the tensor formed data especially when they lie on some smooth and compact manifold embedded in the high dimensional tensor space

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

on bezoutian reduction with the vandermonde matrix abstract the vandermonde matrix reduces by congruence the bezoutian matrix when the zeros of the highest degree associated polynomial are simple in this paper a generalization of this result to the case of multiple zeros is presented

dispersive hyperasymptotics and the anharmonic oscillator hyperasymptotic summation of steepest descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation we apply the method to energy eigenvalues of the anharmonic oscillator for which there is no known integral representation but for which there is a dispersion relation hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to borel summation of the rayleigh schrodinger perturbation series

a comparative study of feature extraction methods and their application to p rbf nns in face recognition problem this study is concerned with a design of a face recognition algorithm realized based on feature extraction with the aid of the 2 directional 2 dimensional linear discriminant analysis referred to as 2d 2lda the 2dlda algorithm yields high accuracy but comes with an unresolved problem of handling large feature matrices present in face recognition problems the proposed 2d 2lda algorithm is computationally more efficient and produces more powerful discrimination decision the proposed p rbf nns is used as a recognition module the architecture of this module consists of three functional components the coefficients of the p rbf nns model are obtained by fuzzy inference method forming the inference part of fuzzy rules the essential design parameters including learning rate momentum fuzzification coefficient and the feature selection mechanism of the networks are optimized by means of differential evolution de the experimental results reported for benchmark face datasets the yale orl database and ic ci dataset demonstrate the effectiveness and efficiency of 2d 2lda algorithm compared with other pre processing approaches such as lpp 2d pca 2d lpp sr pca 2d 2pca and fusion of pca lda the experimental results show that 2d 2lda based p rbf nns achieves higher performance than being reported by other methods

optimal design of radial basis function neural networks for fuzzy rule extraction in high dimensional data abstract the design of an optimal radial basis function neural network rbfnf is not a straightforward procedure in this paper we take advantage of the functional equivalence between rbfn and fuzzy inference systems to propose a novel efficient approach to rbfn design for fuzzy rule extraction the method is based on advanced fuzzy clustering techniques solutions to practical problems are proposed by combining these different solutions a general methodology is derived the efficiency of our method is demonstrated on challenging synthetic and real world data sets

fuzzy system modeling by fuzzy partition and ga hybrid schemes abstract this paper presents an approach to building multi input and single output fuzzy models such a model is composed of fuzzy implications and its output is inferred by simplified reasoning the implications are automatically generated by the structure and parameter identification in structure identification the optimal or near optimal number of fuzzy implications is determined in view of valid partition of data set the parameters defining the fuzzy implications are identified by a ga genetic algorithm hybrid scheme to minimize mean square errors globally numerical examples are provided to evaluate the feasibility of the proposed approach comparison shows that the suggested approach can produce a fuzzy model with higher accuracy and a smaller number of fuzzy implications than the ones achieved previously in other methods the proposed approach has also been applied to construct a fuzzy model for the navigation control of a mobile robot the validity of the resultant model is demonstrated by experimentation

resultants companion matrices and jacobson chains the multiplicative property of triangular toeplitz matrices is used to derive a polyproduct rule for companion matrices this rule is then used to obtain some new identities involving resultants and companion matrices and to investigate the block structure of jacobson chains relating companion and hypercompanion matrices

fuzzy rule base simplification using multidimensional scaling and constrained optimization this paper proposes a novel approach for the development of highly accurate and interpretable fuzzy models with gaussian fuzzy sets under the framework of multidimensional scaling and non linear constrained optimization upon the assumption that an accurate initial fuzzy model has already been designed we introduce an effective methodology to approximate the similarity measure between fuzzy sets the resulting similarity degrees guide the quantification of the dissimilarity degrees between rule antecedents we then put the multidimensional scaling in place in order to transform the rule antecedents into points in a low dimensional euclidean space the elaboration on the distribution of these points is carried out by means of an objective function based fuzzy clustering using a cluster validity index rules that correspond to points belonging to the same cluster are similar and therefore are unified through a specialized merging process with respect to each dimension the aforementioned merging process acts to create a topology of fuzzy sets that enables us to elicit interpretability constraints which are used to minimize the model s performance index in terms of non linear constrained optimization the established fuzzy model appears to possess a simple and transparent i e interpretable structure while maintaining a highly accurate behavior the overall method is rigorously tested and evaluated through a number of simulation experiments that involve low and high dimensional data sets

polynomial bezoutian matrix with respect to a general basis we introduce a class of so called polynomial bezoutian matrices the operator representation relative to a pair of dual bases and the generalized barnett factorization formula for this kind of matrix are derived an intertwining relation and generalized bezoutian reduction via polynomial vandermonde matrix are presented

on bezoutians van der monde matrices and the lienard chipart stability criterion we show how the bezoutian can be reduced by congruence to diagonal form with the congruence given by a van der monde matrix this result is applied to obtain new proofs of some classical stability criteria and in particular of the lienardchipart theorem

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

a novel radial basis function neural network for discriminant analysis a novel radial basis function neural network for discriminant analysis is presented in this paper in contrast to many other researches this work focuses on the exploitation of the weight structure of radial basis function neural networks using the bayesian method it is expected that the performance of a radial basis function neural network with a well explored weight structure can be improved as the weight structure of a radial basis function neural network is commonly unknown the bayesian method is therefore used in this paper to study this a priori structure two weight structures are investigated in this study i e a single gaussian structure and a two gaussian structure an expectation maximization learning algorithm is used to estimate the weights the simulation results showed that the proposed radial basis function neural network with a weight structure of two gaussians outperformed the other algorithms

feature extraction using two dimensional maximum embedding difference in this paper we propose a novel method combining graph embedding and difference criterion techniques for image feature extraction namely two dimensional maximum embedding difference 2dmed this method directly extracts the optimal projective vectors from 2d image matrices by simultaneously considering characteristic that is the intraclass compactness graph the margin graph and inter class separability graph respectively in this method it is not necessary to convert the image matrix into high dimensional image vector so that much computational time would be saved in addition the proposed method preserves the manifold reconstruction relationships in the low dimensional subspace experimental results on the orl yale face and usps database show the effectiveness of the proposed method

generalized single hidden layer feedforward networks for regression problems in this paper traditional single hidden layer feedforward network slfn is extended to novel generalized slfn gslfn by employing polynomial functions of inputs as output weights connecting randomly generated hidden units with corresponding output nodes the significant contributions of this paper are as follows 1 a primal gslfn p gslfn is implemented using randomly generated hidden nodes and polynomial output weights whereby the regression matrix is augmented by full or partial input variables and only polynomial coefficients are to be estimated 2 a simplified gslfn s gslfn is realized by decomposing the polynomial output weights of the p gslfn into randomly generated polynomial nodes and tunable output weights 3 both p and s gslfn are able to achieve universal approximation if the output weights are tuned by ridge regression estimators and 4 by virtue of the developed batch and online sequential ridge elm br elm and osr elm learning algorithms high performance of the proposed gslfns in terms of generalization and learning speed is guaranteed comprehensive simulation studies and comparisons with standard slfns are carried out on real world regression benchmark data sets simulation results demonstrate that the innovative gslfns using br elm and osr elm are superior to standard slfns in terms of accuracy training speed and structure compactness

random matrix theory in semiclassical quantum mechanics of chaotic systems the statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles the author explains this correspondence including the fact that long ranged spectral statistics show deviations from the results of random matrix theory the method depends on the ambiguity of quantisation of a given classical system although the energy levels depend on the particular quantisation used the spectral statistics are assumed to depend only on the classical motion the ambiguity of quantisation can be represented by a small perturbation acting on the hamiltonian this perturbation executes a random walk which causes its matrix elements to undergo a random walk if the matrix elements are completely uncorrelated the spectral statistics are those of random matrix theory in the case of classically chaotic systems the matrix elements are constrained by sum rules related to classical periodic orbits this leads to deviations of the long ranged spectral statistics from the predictions of random matrix theory

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

robust bilateral lp norm two dimensional linear discriminant analysis abstract traditional linear discriminant analysis lda may suffer from a sensitivity to outliers and the small sample size sss problem while the lp norm measure for 0

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

bezoutian representation via vandermonde matrices abstract the result on benoutian representation via vandermonde matrices is provided with a proof which is based on polynomial models

identification of fuzzy models using a successive tuning method with a variant identification ratio the paper is concerned with the identification of fuzzy inference systems fuzzy models realized with the aid of the successive tuning method using a variant identification ratio that is based on hierarchical fair competition based parallel genetic algorithms hfc pga and information granulation the hfc pga is a certain multi population version of parallel genetic algorithms pga which is suitable for a simultaneous optimization of both the structure of the fuzzy model as well as its parameters the granulation of information is realized with the aid of the c means clustering algorithm information granules formed in this way become essential at further stages of the construction of the fuzzy models by forming the centers modal values of the fuzzy sets constituting individual rules of the inference schemes further optimization of the fuzzy model deals with an adjustment of a suite of parameters such as the number of input variables to be used in the model a collection of specific subsets of the input variables and the number of membership functions being used by these variables and the order and parameters of the polynomial occurring in the conclusions of the corresponding rules an iterative development of the fuzzy model deals with its structural as well as parametric optimization via hfc pga the c means algorithm and a standard least square estimation method to evaluate the performance of the proposed model we exploit well known and commonly used data sets such as gas furnace mackey glass time series as well as medical imaging system a comparative analysis demonstrates that the proposed model leads to the superior performance when compared with other fuzzy models reported in the literature

isomorphism theorems and fuzzy k ideals of k semirings we define and study the fuzzy maximal k ideal of a k semiring also we characterize the quotient k semiring r a of a k semiring r by a fuzzy maximal k ideal a and we have some isomorphism theorems in commutative k semirings and further we have some properties of the quotient k semiring r a of all cosets of a fuzzy semiprimary semiprime primary prime k ideal a in a commutative k semiring r

high orders of the weyl expansion for quantum billiards resurgence of periodic orbits and the stokes phenomenon a formalism is developed for calculating high coefficients c r of the weyl high energy expansion for the trace of the resolvent of the laplace operator in a domain b with smooth boundary b the c r are used to test the following conjectures a the sequence of c r diverges factorially controlled by the shortest accessible real or complex periodic geodesic b if this is a 2 bounce orbit it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the mobius strip which is the space of chords of b c this orbit gives an exponential contribution to the remainder when the weyl series truncated at its least term is subtracted from the resolvent the exponential switches on smoothly according to an error function where it is smallest that is across the negative energy axis stokes line these conjectures are motivated by recent results in asymptotics they survive tests for the circle billiard and for a family of curves with 2 and 3 bulges where the dominant orbit is not always the shortest and is sometimes complex for some systems which are not smooth billiards e g a particle on a ring or in a billiard where b is a polygon the weyl series terminates and so no geodesics are accessible for a particle on a compact surface of constant negative curvature only the complex geodesics are accessible from the weyl series

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

symbolic classification clustering and fuzzy radial basis function network symbolic fuzzy classification is proposed using fuzzy radial basis function network with fuzzy c medoids clustering at the hidden layer symbolic objects include linguistic nominal boolean and interval type of features along with quantitative attributes classification and clustering in this domain involve the use of symbolic dissimilarity between the objects fuzzy memberships are used for appropriately handling uncertainty inherent in real life decisions the fuzzy radial basis function frbf network here comprises an integration of the principles of radial basis function rbf network and fuzzy c medoids clustering for handling non numeric data the optimal number of hidden nodes is determined by using clustering validity indices like normalized modified hubert s statistic and davies bouldin index in the symbolic framework the effectiveness of the symbolic fuzzy classification is demonstrated on real life benchmark data sets comparison is provided with the performance of a decision tree

a pyramid approach to subpixel registration based on intensity we present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set which can be either images two dimensional or volumes three dimensional it uses an explicit spline representation of the images in conjunction with spline processing and is based on a coarse to fine iterative strategy pyramid approach the minimization is performed according to a new variation ml of the marquardt levenberg algorithm for nonlinear least square optimization the geometric deformation model is a global three dimensional 3 d affine transformation that can be optionally restricted to rigid body motion rotation and translation combined with isometric scaling it also includes an optional adjustment of image contrast differences we obtain excellent results for the registration of intramodality positron emission tomography pet and functional magnetic resonance imaging fmri data we conclude that the multiresolution refinement strategy is more robust than a comparable single stage method being less likely to be trapped into a false local optimum in addition our improved version of the marquardt levenberg algorithm is faster

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

a semiclassical quantization of area preserving maps abstract a semiclassical path integral formalism is developed for a class of area preserving maps the quasi energy spectrum of these systems is represented as a sum over closed orbits in the manner of gutzwiller the method is illustrated with some numerical tests of the standard map

design of a new adaptive neuro fuzzy inference system based on a solution for clustering in a data potential field in this study we propose a new method for building adaptive neuro fuzzy inference systems anfis via datasets in order to improve the performance of conventional anfis to handle noisy data we focus on ameliorating the cluster data space established from a given dataset to achieve this we propose a weighted clustering process in the joint input output data space thus during the clustering process the cluster with the smallest potential distance which is a combination of the euclidean distance and the size of the clusters has priority when obtaining the surveyed sample based on this principle we formulate a new algorithm for synthesizing an anfis via the proposed data potential field called anfis pf which has the following features it establishes a data potential field that covers the entire initial data space a cluster data space is built based on the generated data potential field and the anfis is synthesized using this cluster data space finally we performed experiments using datasets with and without noise to demonstrate the effectiveness of the proposed method in several applications including dynamic response noisy datasets obtained from a magnetorheological damper

application of multi class fuzzy support vector machine classifier for fault diagnosis of wind turbine this paper presents an approach for fault diagnosis of wind turbine wt based on multi class fuzzy support vector machine fsvm classifier in this method empirical mode decomposition is adopted to extract fault feature vectors from vibration signals fsvm is used for solving classification problem with outliers or noises where kernel fuzzy c means clustering algorithm and particle swarm optimization algorithm are applied to calculate fuzzy membership and optimize the parameters of kernel function of fsvm respectively in addition to study the performance of the proposed approach another two widely used methods named back propagation neural network and standard support vector machine are studied and compared discrete wavelet transform is also used to extract fault feature vectors to validate the proposed approach a direct drive wt test rig is constructed and the experiments are carried out the experimental results show that the proposed approach is an effective fault diagnosis method for wt which has a better performance and can achieve higher diagnostic accuracy a multi class fsvm classifier constructed by one against other method is used for fault diagnosis of wind turbine kfcm algorithm is extended to calculate fuzzy membership values of training samples for a multi class fsvm classifier pso algorithm is applied to optimize the parameters of the kernel function of fsvm fault diagnosis steps are presented and the effectiveness of the proposed method is validated

semiclassical quantization of sawtooth map with inclusion of next leading order and its failure the semiclassical quantization of sawtooth maps is reconsidered from the viewpoint of the higher order terms in the asymptotic series we carry out the semiclassical quantization in the leading and next leading orders however we fail in both cases the reasons are discussed in detail

rapid and brief communication two dimensional locality preserving projections 2dlpp with its application to palmprint recognition this paper proposes a novel algorithm for image feature extraction namely the two dimensional locality preserving projections 2dlpp which directly extracts the proper features from image matrices based on locality preserving criterion experimental results on the polyu palmprint database show the effectiveness of the proposed algorithm

zero order tsk type fuzzy system learning using a two phase swarm intelligence algorithm this paper proposes zero order takagi sugeno kang tsk type fuzzy system learning using a two phase swarm intelligence algorithm tpsia the first phase of tpsia learns fuzzy system structure and parameters by on line clustering aided ant colony optimization aco phase two aims to further optimize all of the free parameters in the fuzzy system using particle swarm optimization pso in clustering aided aco caco fuzzy system structure is learned through on line clustering once a new rule is generated by clustering the consequent is selected from a discrete set of candidate values by aco in aco the path of an ant is regarded as a combination of consequent values selected from every rule caco helps to locate good initial fuzzy systems for subsequent phase learning in phase two initial particles in pso are randomly generated according to the best solution found by caco all free parameters in the designed fuzzy system are optimally tuned by pso simulations on fuzzy control of three nonlinear plants are conducted to verify tpsia performance comparisons with other learning algorithms demonstrate tpsia performance

discussion comment on two dimensional locality preserving projections 2dlpp with its application to palmprint recognition this paper proposes a novel algorithm for image feature extraction namely the two dimensional locality preserving projections 2dlpp which directly extracts the proper features from image matrices based on locality preserving criterion experimental results on the polyu palmprint database show the effectiveness of the proposed algorithm

rapid and brief communications a fourier lda approach for image recognition fourier transform and linear discrimination analysis lda are two commonly used techniques of image processing and recognition based on them we propose a fourier lda approach fla for image recognition it selects appropriate fourier frequency bands with favorable linear separability by using a two dimensional separability judgment then it extracts two dimensional linear discriminative features to perform the classification our experimental results on different image data prove that fla obtains better classification performance than other linear discrimination methods

fuzzy radial basis function neural networks with information granulation and its parallel genetic optimization fuzzy modeling of complex systems is a challenging task which involves important problems of dimensionality reduction and calls for various ways of improving the accuracy of modeling the ig frbfnn a hybrid architecture of the ig fis fuzzy inference system and frbfnn fuzzy radial basis function neural networks is proposed to address these problems the paper is concerned with the analysis and design of ig frbfnns and their optimization by means of the hierarchical fair competition based parallel genetic algorithm hfc pga in the proposed network the membership functions of the premise part of the fuzzy rules of the ig based frbfnn model directly rely on the computation of the relevant distance between data points and the use of four types of polynomials such as constant linear quadratic and modified quadratic are considered for the consequent part of fuzzy rules moreover the weighted least square wls learning is exploited to estimate the coefficients of the polynomial forming the conclusion part of the rules since the performance of the ig rbfnn model is affected by some key design parameters such as a specific subset of input variables the fuzzification coefficient of the fcm the number of rules and the order of polynomial of the consequent part of fuzzy rules it becomes beneficial to carry out both structural as well as parametric optimization of the network in this study the hfc pga is used as a comprehensive optimization vehicle the performance of the proposed model is illustrated by means of several representative numerical examples

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field abstract we introduce a so called generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field some characterization of this kind of matrix such as the barnett type factorization and the intertwining relation with generalized hypercompanion matrix are obtained the diagonal reduction formula via the generalized confluent vandermonde matrix similar to that of classical bezoutian is presented the method used is based on polynomial module and operator representation

orbit bifurcations and the scarring of wave functions we extend the semiclassical theory of scarring of quantum eigenfunctions n q by classical periodic orbits to include situations where these orbits undergo generic bifurcations it is shown that n q 2 averaged locally with respect to position q and the energy spectrum en has structure around bifurcating periodic orbits with an amplitude and length scale whose dependence is determined by the bifurcation in question specifically the amplitude scales as and the length scale as and values of the scar exponents and are computed for a variety of generic bifurcations in each case the scars are semiclassically wider than those associated with isolated and unstable periodic orbits moreover their amplitude is at least as large and in most cases larger in this sense bifurcations may be said to give rise to superscars the competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed we argue that there is a resulting universal scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase space dynamics finally a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

low rank embedding for robust image feature extraction robustness to noises outliers and corruptions is an important issue in linear dimensionality reduction since the sample specific corruptions and outliers exist the class special structure or the local geometric structure is destroyed and thus many existing methods including the popular manifold learning based linear dimensionality methods fail to achieve good performance in recognition tasks in this paper we focus on the unsupervised robust linear dimensionality reduction on corrupted data by introducing the robust low rank representation lrr thus a robust linear dimensionality reduction technique termed low rank embedding lre is proposed in this paper which provides a robust image representation to uncover the potential relationship among the images to reduce the negative influence from the occlusion and corruption so as to enhance the algorithm s robustness in image feature extraction lre searches the optimal lrr and optimal subspace simultaneously the model of lre can be solved by alternatively iterating the argument lagrangian multiplier method and the eigendecomposition the theoretical analysis including convergence analysis and computational complexity of the algorithms is presented experiments on some well known databases with different corruptions show that lre is superior to the previous methods of feature extraction and therefore it indicates the robustness of the proposed method the code of this paper can be downloaded from http www scholat com laizhihui

on training rbf neural networks using input output fuzzy clustering and particle swarm optimization this paper elaborates on the use of particle swarm optimization in training gaussian type radial basis function neural networks under the umbrella of input output fuzzy clustering the problem being investigated concerns the selection of basis function centers that contribute most in network s performance given that the clustering process in the input space is guided by the clustering in the output space to accomplish this task we quantify the effect of the input space fuzzy partition upon network s square error in terms of an objective function that describes the ability of the partition to accurately reconstruct the input training samples we then theoretically prove that the minimization of the above function acts to minimize an upper bound of the network s square error therefore the resulting solution corresponds to a minimal square error while at the same time it maintains the structure of the input data due to the peculiarity of the aforementioned objective function we treat it as the fitness function used by the particle swarm optimizer the proposed methodology encompasses three design steps the first step implements an independent fuzzy clustering in the output space to obtain a set of cluster centers in the second step unlike other approaches the above centers are directly involved in the estimation of the membership degrees in the input output space in the third step these membership degrees are used by the particle swarm optimizer in order to obtain optimal values for the centers to summarize the novelty of our contribution lies in a the way we handle the information flow from output to input space and b the way we handle the effect of the input space partition upon network s performance the experiments indicate that the fitness function decreases as the number of hidden node increases finally a comparison between the proposed method and other sophisticated approaches shows its statistically significant superiority

automatic design of hierarchical takagi sugeno type fuzzy systems using evolutionary algorithms this paper presents an automatic way of evolving hierarchical takagi sugeno fuzzy systems ts fs the hierarchical structure is evolved using probabilistic incremental program evolution pipe with specific instructions the fine tuning of the if then rule s parameters encoded in the structure is accomplished using evolutionary programming ep the proposed method interleaves both pipe and ep optimizations starting with random structures and rules parameters it first tries to improve the hierarchical structure and then as soon as an improved structure is found it further fine tunes the rules parameters it then goes back to improve the structure and the rules parameters this loop continues until a satisfactory solution hierarchical ts fs model is found or a time limit is reached the proposed hierarchical ts fs is evaluated using some well known benchmark applications namely identification of nonlinear systems prediction of the mackey glass chaotic time series and some classification problems when compared to other neural networks and fuzzy systems the developed hierarchical ts fs exhibits competing results with high accuracy and smaller size of hierarchical architecture

identification of fuzzy systems by means of an auto tuning algorithm and its application to nonlinear systems abstract the study concerns a design procedure of rule based systems the proposed rule based fuzzy modeling implements system structure and parameter identification in the efficient form of if then statements and exploits the theory of system optimization and fuzzy implication rules two types of methods for rule based fuzzy modeling are studied this classification concerns the form of the conclusion part of the rules that can be either constant or formed by some linear functions both triangular and gaussian like membership function are studied the optimization hinges on an auto tuning algorithm that covers a modified constrained optimization method known as a complex method the study introduces a weighted performance index objective function that helps achieve a sound balance between the quality of results produced for the training and testing set this methodology sheds light on the role and impact of different parameters of the model on its performance in particular the mapping and predicting capabilities of the rule based computing the study is illustrated with the aid of several representative numerical examples

generalized polynomial bases and the bezoutian abstract a foundation polynomial is used to induce polynomial bases for fn 1 x the vector space of polynomials of degree less than n over an arbitrary field f the associated bases are then used to block diagonalize the bezout matrix of two polynomials under congruence

building consensus in group decision making with an allocation of information granularity abstract consensus is defined as a cooperative process in which a group of decision makers develops and agrees to support a decision in the best interest of the whole it is a questioning process more than an affirming process in which the group members usually modify their choices until a high level of agreement within the group is achieved given the importance of forming an accepted decision by the entire group the consensus problem has attained a great attention as it is a major goal in group decision making in this study we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consensus in group decision making by using fuzzy preference relations to represent the opinions of the decision makers we develop a concept of a granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule say an interval fuzzy set rough set and alike instead of a single numeric value as being more abstract the granular format of the preference model offers the required flexibility to increase the level of agreement within the group using the fact that we select the most suitable numeric representative of the fuzzy preference relation

polynomial based radial basis function neural networks p rbf nns realized with the aid of particle swarm optimization in this study we design polynomial based radial basis function neural networks p rbf nns based on a fuzzy inference mechanism the essential design parameters including learning rate momentum coefficient and fuzzification coefficient of the underlying clustering method are optimized by means of the particle swarm optimization the proposed p rbf nns dwell upon structural findings about training data that are expressed in terms of a partition matrix resulting from fuzzy clustering in this case being the fuzzy c means fcm the network is of functional nature as the weights between the hidden layer and the output are some polynomials the use of the polynomial weights becomes essential in capturing the nonlinear nature of data encountered in regression or classification problems from the perspective of linguistic interpretation the proposed network can be expressed as a collection of if then fuzzy rules the architecture of the networks discussed here embraces three functional modules reflecting the three phases of input output mapping realized in rule based architectures namely condition formation conclusion creation and aggregation the proposed classifier is applied to some synthetic and machine learning datasets and its results are compared with those reported in the previous studies

sparse approximation to the eigensubspace for discrimination two dimensional 2 d image matrix based projection methods for feature extraction are widely used in many fields of computer vision and pattern recognition in this paper we propose a novel framework called sparse 2 d projections s2dp for image feature extraction different from the existing 2 d feature extraction methods s2dp iteratively learns the sparse projection matrix by using elastic net regression and singular value decomposition theoretical analysis shows that the optimal sparse subspace approximates the eigensubspace obtained by solving the corresponding generalized eigenequation with the s2dp framework many 2 d projection methods can be easily extended to sparse cases moreover when each row column of the image matrix is regarded as an independent high dimensional vector 1 d vector it is proven that the vector based eigensubspace is also approximated by the sparse subspace obtained by the same method used in this paper theoretical analysis shows that when compared with the vector based sparse projection learning methods s2dp greatly saves both computation and memory costs this property makes s2dp more tractable for real world applications experiments on well known face databases indicate the competitive performance of the proposed s2dp over some 2 d projection methods when facial expressions lighting conditions and time vary

feature extraction using two dimensional local graph embedding based on maximum margin criterion in this paper we propose a novel method for image feature extraction namely the two dimensional local graph embedding which is based on maximum margin criterion and thus not necessary to convert the image matrix into high dimensional image vector and directly avoid computing the inverse matrix in the discriminant criterion this method directly learns the optimal projective vectors from 2d image matrices by simultaneously considering local graph embedding and maximum margin criterion the proposed method avoids huge feature matrix problem in eigenfaces fisherfaces laplacianfaces maximum margin criterion mmc and inverse matrix in 2d fisherfaces 2d laplacianfaces and 2d local graph embedding discriminant analysis 2dlgeda so that computational time would be saved for feature extraction experimental results on the yale and the usps databases show the effectiveness of the proposed method under various experimental conditions

hyperasymptotics for multidimensional integrals exact remainder terms and the global connection problem the method of steepest descents for single dimensional laplace type integrals involving an asymptotic parameter k was extended by berry howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non local saddlepoints this led to an improved asymptotic expansion hyperasymptotics which gave exponentially accurate numerical and analytic results based on the topography of the saddle distribution in the single complex plane of the integrand in this paper we generalize these results to similar well behaved multidimensional integrands with quadratic critical points integrated over infinite complex domains as previously pointed out the extra complex dimensions give rise to interesting problems and phenomena first the conventionally defined surfaces of steepest descent are no longer unique second the stokes s phenomenon whereby contributions from subdominant saddles enter the asymptotic representation is of codimension one third we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points the new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder thus the growth of the late terms in the expansion can be identified and a hyperasymptotic scheme implemented we show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem riemann sheet structure and homology associated with the multidimensional topography of the integrand